Motion Mountain Physics Textbook Volume 5 - Motion Inside Matter – Pleasure, Technology and Stars

Authors Christoph Schiller

License CC-BY-NC-ND-3.0

Plaintext

Christoph Schiller

MOTION MOUNTAIN
the adventure of physics – vol.v
motion inside matter –
pleasure, technology and stars

www.motionmountain.net
Christoph Schiller

Motion Mountain

The Adventure of Physics
Volume V

Motion Inside Matter –
Pleasure, Technology
and Stars

Edition 31, available as free pdf
with films at www.motionmountain.net
Editio trigesima prima.

Proprietas scriptoris © Chrestophori Schiller
primo anno Olympiadis trigesimae secundae.

Omnia proprietatis iura reservantur et vindicantur.
Imitatio prohibita sine auctoris permissione.
Non licet pecuniam expetere pro aliqua, quae
partem horum verborum continet; liber
pro omnibus semper gratuitus erat et manet.

Thirty-first edition.

Copyright © 1990–2021 by Christoph Schiller,
from the third year of the 24th Olympiad
to the first year of the 32nd Olympiad.

This pdf file is licensed under the Creative Commons
Attribution-Noncommercial-No Derivative Works 3.0 Germany
Licence, whose full text can be found on the website
creativecommons.org/licenses/by-nc-nd/3.0/de,
with the additional restriction that reproduction, distribution and use,
in whole or in part, in any product or service, be it
commercial or not, is not allowed without the written consent of
the copyright owner. The pdf file was and remains free for everybody
to read, store and print for personal use, and to distribute
electronically, but only in unmodified form and only at no charge.
To Britta, Esther and Justus Aaron

τῷ ἐμοὶ δαὶμονι
Die Menschen stärken, die Sachen klären.
PR E FAC E

“ ”
Primum movere, deinde docere.*
Antiquity

T
his book series is for anybody who is curious about motion in nature. How do
hings, people, animals, images and empty space move? The answer leads

Motion Mountain – The Adventure of Physics
o many adventures, and this volume presents those about motion inside everyday
matter, inside people and animals, and inside stars and nuclei.
Motion inside bodies – dead or alive – is tiny: thus it is described by quantum theory.
Quantum theory describes all motion with the quantum of action ℏ, the smallest change
observed in nature. Building on this basic idea, the text first shows how to describe life,
death and pleasure. Then, the text explains the observations of chemistry, materials sci-
ence, astrophysics and particle physics. In the structure of physics, these topics corres-
pond to the three ‘quantum’ points in Figure 1. The story of motion found inside living
cells, inside the coldest gases and throughout the hottest stars is told here in a way that
is simple, up to date and captivating.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In order to be simple, the text focuses on concepts, while keeping mathematics to the
necessary minimum. Understanding the concepts of physics is given precedence over
using formulae in calculations. The whole text is within the reach of an undergraduate.
In order to be up to date, the text is enriched by the many gems – both theoretical and
empirical – that are scattered throughout the scientific literature.
In order to be captivating, the text tries to startle the reader as much as possible. Read-
ing a book on general physics should be like going to a magic show. We watch, we are
astonished, we do not believe our eyes, we think, and finally we understand the trick.
When we look at nature, we often have the same experience. Indeed, every page presents
at least one surprise or provocation for the reader to think about. Numerous interesting
challenges are proposed.
The motto of the text, die Menschen stärken, die Sachen klären, a famous statement
on pedagogy, translates as: ‘To fortify people, to clarify things.’ Clarifying things – and
adhering only to the truth – requires courage, as changing the habits of thought produces
fear, often hidden by anger. But by overcoming our fears we grow in strength. And we
experience intense and beautiful emotions. All great adventures in life allow this, and
exploring motion is one of them. Enjoy it.

Christoph Schiller
* ‘First move, then teach.’ In modern languages, the mentioned type of moving (the heart) is called motiv-
ating; both terms go back to the same Latin root.
8 preface

Complete, unified description of motion
Adventures: describing precisely all motion, understanding
the origin of colours, space -time and particles, enjoying
extreme thinking, calculating masses and couplings,
catching a further, tiny glimpse of bliss (vol. VI).

PHYSICS: An arrow indicates an
Describing motion with precision, increase in precision by
i.e., using the least action principle. adding a motion limit.

Quantum theory
General relativity with classical gravity Quantum field theory
Adventures: the Adventures: bouncing (the ‘standard model’)
neutrons, under- Adventures: building

Motion Mountain – The Adventure of Physics
night sky, measu-
ring curved and standing tree accelerators, under-
wobbling space, growth (vol. V). standing quarks, stars,
exploring black bombs and the basis of
holes and the life, matter & radiation
universe, space (vol. V).
and time (vol. II).

Classical gravity Special relativity Quantum theory
Adventures: Adventures: light, Adventures: biology,
climbing, skiing, magnetism, length birth, love, death,
c contraction, time

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
space travel, chemistry, evolution,
the wonders of limits dilation and enjoying colours, art,
astronomy and G fast E0 = mc2 h, e, k paradoxes, medicine
geology (vol. I). limits motion (vol. II). limit and high-tech business
uniform tiny (vol. IV and vol. V).
motion motion

Galilean physics, heat and electricity
The world of everyday motion: human scale, slow and weak.
Adventures: sport, music, sailing, cooking, describing
beauty and understanding its origin (vol. I);
using electricity, light and computers,
understanding the brain and people (vol. III).

F I G U R E 1 A complete map of physics, the science of motion, as ﬁrst proposed by Matvei Bronshtein
(b. 1907 Vinnytsia, d. 1938 Leningrad). The Bronshtein cube starts at the bottom with everyday motion,
and shows the connections to the ﬁelds of modern physics. Each connection increases the precision of
the description and is due to a limit to motion that is taken into account. The limits are given for
uniform motion by the gravitational constant G, for fast motion by the speed of light c, and for tiny
motion by the Planck constant h, the elementary charge e and the Boltzmann constant k.
preface 9

Using this b o ok
Marginal notes refer to bibliographic references, to other pages or to challenge solutions.
In the colour edition, marginal notes, pointers to footnotes and links to websites are
typeset in green. Over time, links on the internet tend to disappear. Most links can be
recovered via www.archive.org, which keeps a copy of old internet pages. In the free
pdf edition of this book, available at www.motionmountain.net, all green pointers and
links are clickable. The pdf edition also contains all films; they can be watched directly
in Adobe Reader.
Solutions and hints for challenges are given in the appendix. Challenges are classified
as easy (e), standard student level (s), difficult (d) and research level (r). Challenges for
which no solution has yet been included in the book are marked (ny).

Advice for learners
Learning allows us to discover what kind of person we can be. Learning widens know-

Motion Mountain – The Adventure of Physics
ledge, improves intelligence and provides a sense of achievement. Therefore, learning
from a book, especially one about nature, should be efficient and enjoyable. Avoid bad
learning methods like the plague! Do not use a marker, a pen or a pencil to highlight or
underline text on paper. It is a waste of time, provides false comfort and makes the text
unreadable. And do not learn from a screen. In particular, never, ever, learn from the in-
ternet, from videos, from games or from a smartphone. Most of the internet, almost all
videos and all games are poisons and drugs for the brain. Smartphones are dispensers of
drugs that make people addicted and prevent learning. Nobody putting marks on paper
or looking at a screen is learning efficiently or is enjoying doing so.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In my experience as a pupil and teacher, one learning method never failed to trans-
form unsuccessful pupils into successful ones: if you read a text for study, summarize
every section you read, in your own words and images, aloud. If you are unable to do
so, read the section again. Repeat this until you can clearly summarize what you read in
your own words and images, aloud. And enjoy the telling aloud! You can do this alone
or with friends, in a room or while walking. If you do this with everything you read, you
will reduce your learning and reading time significantly; you will enjoy learning from
good texts much more and hate bad texts much less. Masters of the method can use it
even while listening to a lecture, in a low voice, thus avoiding to ever take notes.

Advice for teachers
A teacher likes pupils and likes to lead them into exploring the field he or she chose. His
or her enthusiasm is the key to job satisfaction. If you are a teacher, before the start of a
lesson, picture, feel and tell yourself how you enjoy the topic of the lesson; then picture,
feel and tell yourself how you will lead each of your pupils into enjoying that topic as
much as you do. Do this exercise consciously, every day. You will minimize trouble in
your class and maximize your teaching success.
This book is not written with exams in mind; it is written to make teachers and stu-
dents understand and enjoy physics, the science of motion.
10 preface

Feedback
The latest pdf edition of this text is and will remain free to download from the internet.
I would be delighted to receive an email from you at fb@motionmountain.net, especially
on the following issues:
Challenge 1 s — What was unclear and should be improved?
— What story, topic, riddle, picture or film did you miss?
Also help on the specific points listed on the www.motionmountain.net/help.html web
page is welcome. All feedback will be used to improve the next edition. You are welcome
to send feedback by mail or by sending in a pdf with added yellow notes, to provide
illustrations or photographs, or to contribute to the errata wiki on the website. If you
would like to translate a chapter of the book in your language, please let me know.
On behalf of all readers, thank you in advance for your input. For a particularly useful
contribution you will be mentioned – if you want – in the acknowledgements, receive a

Motion Mountain – The Adventure of Physics
reward, or both.

Support
Your donation to the charitable, tax-exempt non-profit organisation that produces, trans-
lates and publishes this book series is welcome. For details, see the web page www.
motionmountain.net/donation.html. The German tax office checks the proper use of
your donation. If you want, your name will be included in the sponsor list. Thank you in
advance for your help, on behalf of all readers across the world.
The paper edition of this book is available, either in colour or in black and white,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
from www.amazon.com, in English and in certain other languages. And now, enjoy the
reading.
C ON T E N T S

7 Preface
Using this book 9 • Advice for learners 9 • Advice for teachers 9 • Feedback 10 •
Support 10
11 Contents
15 1 Motion for enjoying life

Motion Mountain – The Adventure of Physics
15 From quantum physics to biological machines and miniaturization
Reproduction 18 • Quantum machines 19 • How do we move? – Molecular mo-
tors 21 • Linear molecular motors 23 • A rotational molecular motor: ATP syn-
thase 25 • Rotational motors and parity breaking 27 • Curiosities and fun chal-
lenges about biology 29
37 The physics of pleasure
The nerves and the brain 41 • Living clocks 42 • When do clocks exist? 44
• The precision of clocks 45 • Why are predictions so difficult, especially of the
future? 46 • Decay and the golden rule 47 • The present in quantum theory 48
• Why can we observe motion? 49 • Rest and the quantum Zeno effect 50 • Con-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
sciousness – a result of the quantum of action 51 • Why can we observe motion?
– Again 51 • Curiosities and fun challenges about quantum experience 52 • Sum-
mary on biology and pleasure 57
58 2 Changing the world with quantum effects
58 Chemistry – from atoms to DNA
Atomic bonds 59 • Ribonucleic acid and deoxyribonucleic acid 62 • Curiosities
and fun challenges about chemistry 63
67 Materials science
Why does the floor not fall? 67 • Rocks and stones 69 • Crystal formation 72 •
Some interesting crystals 75 • How can we look through matter? 83 • What is ne-
cessary to make matter invisible? 85 • What moves inside matter? 86 • Curiosities
and fun challenges about materials science 87
100 Quantum technology
Transistors 100 • Motion without friction – superconductivity and superfluid-
ity 103 • The fractional quantum Hall effect 107 • How does matter behave at
the lowest temperatures? 108 • Lasers and other spin-one vector boson launch-
ers 108 • From lamps to lasers 113 • The three lightbulb scams 114 • Applications
of lasers 115 • Challenges, dreams and curiosities about quantum technology 116 •
Summary on changing the world with quantum effects 120
122 3 Quantum electrodynamics – the origin of virtual reality
Ships, mirrors and the Casimir effect 122 • The Lamb shift 125 • The QED Lag-
rangian and its symmetries 126 • Interactions and virtual particles 127 • Va-
12 contents

cuum energy: infinite or zero? 128 • Moving mirrors 128 • Photons hitting
photons 130 • Is the vacuum a bath? 131 • Renormalization – why is an electron
so light? 131 • Curiosities and fun challenges of quantum electrodynamics 132 •
How can one move on perfect ice? – The ultimate physics test 134 • A summary
of quantum electrodynamics 135 • Open questions in QED 137
140 4 Quantum mechanics with gravitation – first steps
Falling atoms 140 • Playing table tennis with neutrons 140 • The gravitational
phase of wave functions 142 • The gravitational Bohr atom 143 • Curiosities
about quantum theory and gravity 143 • Gravitation and limits to disorder 145
• Measuring acceleration with a thermometer: Fulling–Davies–Unruh radi-
ation 146 • Black holes aren’t black 147 • The lifetime of black holes 150 • Black
holes are all over the place 151 • Fascinating gamma-ray bursts 151 • Material
properties of black holes 154 • How do black holes evaporate? 155 • The inform-
ation paradox of black holes 155 • More paradoxes 156 • Quantum mechanics of
gravitation 157 • Do gravitons exist? 157 • Space-time foam 158 • Decoherence
of space-time 159 • Quantum theory as the enemy of science fiction 159 • No

Motion Mountain – The Adventure of Physics
vacuum means no particles 160 • Summary on quantum theory and gravity 161
162 5 The structure of the nucleus – the densest clouds
A physical wonder – magnetic resonance imaging 162 • The size of nuclei and
the discovery of radioactivity 164 • Nuclei are composed 169 • Nuclei can move
alone – cosmic rays 172 • Nuclei decay – more on radioactivity 180 • Radiometric
dating 182 • Why is hell hot? 184 • Nuclei can form composites 185 • Nuclei have
colours and shapes 186 • The four types of motion in the nuclear domain 187 •
Nuclei react 188 • Bombs and nuclear reactors 191 • Curiosities and challenges
on nuclei and radioactivity 192 • Summary on nuclei 199

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
200 6 The sun, the stars and the birth of matter
The Sun 200 • Motion in and on the Sun 203 • Why do the stars shine? 208 • Why
are fusion reactors not common yet? 211 • Where do our atoms come from? 213
• Curiosities about the Sun and the stars 216 • Summary on stars and nucleosyn-
thesis 218
219 7 The strong interaction – inside nuclei and nucleons
The feeble side of the strong interaction 219 • Bound motion, the particle zoo
and the quark model 220 • The essence of quantum chromodynamics 223 • The
Lagrangian of quantum chromodynamics 225 • Experimental consequences of the
quark model 227 • Confinement of quarks – and elephants 229 • Asymptotic free-
dom 231 • The sizes and masses of quarks 233 • The mass, shape and colour of
protons 234 • Curiosities about the strong interaction 235 • A summary of QCD
and its open issues 238
240 8 The weak nuclear interaction and the handedness of nature
Transformation of elementary particles 240 • The weakness of the weak nuclear in-
teraction 241 • Distinguishing left from right 245 • Distinguishing particles and
antiparticles, CP violation 248 • Weak charge and mixings 250 • Symmetry break-
ing – and the lack of electroweak unification 252 • The Lagrangian of the weak and
electromagnetic interactions 253 • Curiosities about the weak interaction 255 •
A summary of the weak interaction 259
261 9 The standard model of particle physics – as seen on television
Summary and open questions 265
contents 13

268 10 Dreams of unification
Grand unification 268 • Comparing predictions and data 269 • The state of
grand unification 270 • Searching for higher symmetries 271 • Supersym-
metry 271 • Other attempts 273 • Dualities – the most incredible symmetries
of nature 273 • Collective aspects of quantum field theory 274 • Curiosities about
unification 275 • A summary on unification, mathematics and higher symmet-
ries 276
278 11 Bacteria, flies and knots
Bumblebees and other miniature flying systems 278 • Swimming 282 • Rotation,
falling cats and the theory of shape change 287 • Swimming in curved space 291 •
Turning a sphere inside out 292 • Clouds 293 • Vortices and the Schrödinger
equation 294 • Fluid space-time 298 • Dislocations and solid space-time 298 •
Polymers 300 • Knots and links 302 • The hardest open problems that you can
tell your grandmother 304 • Curiosities and fun challenges on knots and wobbly
entities 305 • Summary on wobbly objects 309

Motion Mountain – The Adventure of Physics
311 12 Quantum physics in a nutshell – again
Quantum field theory in a few sentences 311 • Achievements in precision 314 •
What is unexplained by quantum theory and general relativity? 316 • The physics
cube 318 • The intense emotions due to quantum field theory and general relativ-
ity 320 • What awaits us? 323
325 a Units, measurements and constants
SI units 325 • The meaning of measurement 328 • Planck’s natural units 328 •
Other unit systems 330 • Curiosities and fun challenges about units 331 • Preci-
sion and accuracy of measurements 332 • Limits to precision 333 • Physical con-
stants 334 • Useful numbers 341

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
342 b Composite particle properties
358 c Algebras, shapes and groups
358 Algebras
Lie algebras 361 • Classification of Lie algebras 362
363 Topology – what shapes exist?
Topological spaces 364 • Manifolds 365 • Holes, homotopy and homology 367
368 Types and classification of groups
Lie groups 369 • Connectedness 370 • Compactness 370
375 Mathematical curiosities and fun challenges
377 Challenge hints and solutions
386 Bibliography
412 Credits
Acknowledgements 412 • Film credits 413 • Image credits 413
417 Subject index
Photo
missing

Motion Inside Matter –
Pleasure, Technology and Stars

In our quest to understand how things move
as a result of a smallest change value in nature, we discover
how pleasure appears,
why the floor does not fall but keeps on carrying us,
that interactions are exchanges of radiation particles,
that matter is not permanent,
how quantum effects increase human wealth and health,
why empty space pulls mirrors together,
why the stars shine,
where the atoms inside us come from,
how quantum particles make up the world,
and why swimming and flying is not so easy.
Chapter 1

MOT ION F OR E N JOY I NG L I F E

“ ”
Homo sum, humani nil a me alienum puto.**
Terence

S
ince we have explored quantum effects in the previous volume, let us now have

Motion Mountain – The Adventure of Physics
ome serious fun with applied quantum physics. The quantum of action ℏ has
ignificant consequences for medicine, biology, chemistry, materials science, engin-
eering and the light emitted by stars. Also art, the colours and materials it uses, and the
creative process in the artist, are based on the quantum of action. From a physics stand-
point, all these domains study motion inside matter.
Inside matter, we observe, above all, tiny motions of quantum particles.*** Therefore
the understanding and the precise description of matter requires quantum physics. In the
following, we will only explore a cross-section, but it will be worth it. We start our ex-
ploration of motion inside matter with three special forms that are of special importance
Vol. IV, page 15 to us: life, reproduction and death. We mentioned at the start of quantum physics that

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
none of these forms of motion can be described by classical physics. Indeed, life, repro-
duction and death are quantum effects. In addition, every perception, every sense, and
thus every kind of pleasure is a quantum effect. The same is true for all our actions. Let
us find out why.

from quantum physics to biol o gical machines and
miniaturization
We know that all of quantum theory can be resumed in one sentence:

⊳ In nature, action or change below ℏ = 1.1 ⋅ 10−34 Js is not observed.

In the following, we want to understand how this observation explains life, pleasure and
death. An important consequence of the quantum of action is well-known.

⊳ If it moves, it is made of quantons, or quantum particles.

** ‘I am a man and nothing human is alien to me.’ Terence is Publius Terentius Afer (b. c. 190 Carthago,
d. 159 bce Greece), the important roman poet. He writes this in his play Heauton Timorumenos, verse 77.
*** The photograph on page 14 shows a soap bubble, the motion of the fluid in it, and the interference
colours; it was taken and is copyright by Jason Tozer for Creative Review/Sony.
16 1 motion for enjoying life

Starting size: the dot on the letter i – final size:

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 2 Metabolic growth can lead from single cells, about 0.1 mm in diameter, to living beings of
25 m in size, such as the baobab or the blue whale (© Ferdinand Reus, NOAA).
from quantum physics to biological machines and miniaturization 17

Step by step we will discover how these statements are reflected in the behaviour of living
beings. But what are living beings?
Living beings are physical systems that show metabolism, information processing,
information exchange, reproduction and motion. All these properties can be condensed
in a single statement:

⊳ A living being is a collection of machines that is able to self-reproduce.

By self -reproduction, we mean that a system uses its own metabolism to reproduce.
There are examples of objects which reproduce and which nobody would call living. Can
Challenge 2 s you find some examples? To avoid misunderstandings, whenever we say ‘reproduction’
in the following, we always mean ‘self-reproduction’.
Before we explore the definition of living beings in more detail, we stress that self-
reproduction is simplified if the system is miniaturized. Therefore, most living beings
are extremely small machines for the tasks they perform. This is especially clear when

Motion Mountain – The Adventure of Physics
living beings are compared to human-made machines. The smallness of living beings is
often astonishing, because the design and construction of human-made machines has
considerably fewer requirements.
1. Human-made machines do not need to be able to reproduce; as a result, they can be
made of many parts and can include rotating macroscopic parts. This is in contrast
to living beings, who are all made of a single piece of matter, and cannot use wheels,
propellers, gearwheels or even screws.
2. Human made machines can make use of metals, ceramics, poisonous compounds
and many other materials that living beings cannot use.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
3. Human machines do not need to self-assemble and grow; in contrast, living beings
always need to carry a built-in chemical factory with them.
4. Human machines can be assembled and can operate at various temperatures, in
strong contrast to living beings.
Despite these extreme engineering restrictions, living beings hold many miniaturization
world records for machines:
— The brain has the highest processing power per volume of any calculating device so
far. Just look at the size of chess champion Gary Kasparov and the size of the computer
against which he played and lost. Or look at the size of any computer that attempts
to speak.
— The brain has the densest and fastest memory of any device so far. The set of com-
pact discs (CDs) or digital versatile discs (DVDs) that compare with the brain is many
Vol. III, page 268 thousand times larger in volume.
— Motors in living beings are several orders of magnitude smaller than human-built
ones. Just think about the muscles in the legs of an ant.
— The motion of living beings beats the acceleration of any human-built machine by
orders of magnitude. No machine achieves the movement changes of a grasshopper,
a fly or a tadpole.
— Living beings that fly, swim or crawl – such as fruit flies, plankton or amoebas – are
still thousands of times smaller than anything comparable that is built by humans.
18 1 motion for enjoying life

In particular, already the navigation systems built by nature are far smaller than any-
thing built by human technology.
— Living being’s sensor performance, such as that of the eye or the ear, has been sur-
passed by human machines only recently. For the nose, this feat is still far in the fu-
ture. Nevertheless, the sensor sizes developed by evolution – think also about the ears
or eyes of a common fly – are still unbeaten.
Challenge 3 s — Can you spot more examples?
The superior miniaturization of living beings – compared to human-built machines –
is due to their continuous strife for efficient construction. The efficiency has three main
aspects. First of all, in the structure of living beings, everything is connected to everything.
Each part influences many others. Indeed, the four basic processes in life, namely meta-
bolic, mechanical, hormonal and electric, are intertwined in space and time. For ex-
ample, in humans, breathing helps digestion; head movements pump liquid through the
spine; a single hormone influences many chemical processes. Secondly, all parts in living
systems have more than one function. For example, bones provide structure and produce

Motion Mountain – The Adventure of Physics
Challenge 4 e
blood; fingernails are tools and shed chemical waste. Living systems use many such op-
Challenge 5 e timizations. Last but not least, living machines are well miniaturized because they make
efficient use of quantum effects. Indeed, every single function in living beings relies on
the quantum of action. And every such function is extremely well miniaturized. We ex-
plore a few important cases.

R eproduction

“ ”
Finding a mate is life’s biggest prize.
The view of biologists.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
All the astonishing complexity of life is geared towards reproduction. Reproduction, more
precisely, self-reproduction, is the ability of an object to build other objects similar to itself.
Vol. IV, page 122 Quantum theory told us that it is only possible to build a similar object, since an exact
copy would contradict the quantum of action. But this limitation is not a disadvantage:
an imperfect copy is required for life; indeed, a similar, thus imperfect copy is essential
for biological evolution, and thus for change and specialization.
Reproduction is characterized by random changes, called mutations, that distinguish
one generation from the next. The statistics of mutations, for example Mendel’s ‘laws’ of
heredity, and the lack of intermediate states, are direct consequences of quantum theory.
In other words, reproduction and heredity are quantum effects.
Reproduction requires growth, and growth needs metabolism. Metabolism is a chem-
ical process, and thus a quantum process, to harness energy, harness materials, realize
Page 58 growth, heal injuries and realize reproduction.
Since reproduction requires an increase in mass, as shown in Figure 2, all reproducing
objects show both metabolism and growth. In order that growth can lead to an object
similar to the original, a construction plan is necessary. This plan must be similar to the
plan used by the previous generation. Organizing growth with a construction plan is only
possible if nature is made of smallest entities which can be assembled following that plan.
We thus deduce that reproduction and growth implies that matter is made of smallest
entities. If matter were not made of smallest entities, there would be no way to realize
from quantum physics to biological machines and miniaturization 19

F I G U R E 3 A quantum machine (© Elmar
Bartel).

reproduction. The observation of reproduction thus implies the existence of atoms and

Motion Mountain – The Adventure of Physics
the necessity of quantum theory! Indeed, without the quantum of action there would be
no DNA molecules and there would be no way to inherit our own properties – our own
construction plan – to children.
Passing on a plan from generation to generation requires that living beings have ways
to store information. Living beings must have some built-in memory storage. We know
Vol. I, page 404 already that a system with memory must be made of many particles: there is no other
way to store information and secure its stability over time. The large number of particles
is necessary to protect the information from the influences of the outside world.
Our own construction plan is stored in DNA molecules in the nucleus and the mi-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tochondria of each of the millions of cells inside our body. We will explore some details
Page 62 below. The plan is thus indeed stored and secured with the help of many particles. There-
fore, reproduction is first of all a transfer of parent’s DNA to the next generation. The
transfer is an example of motion. It turns out that this and all other examples of motion
in our bodies occur in the same way, namely with the help of molecular machines.

Q uantum machines
Living beings move. In order to reproduce, living beings must be able to move in self-
directed ways.

⊳ A system able to perform self-directed motion is called a machine.

All self-reproducing beings, such as the one of Figure 3, are thus machines. Even ma-
chines that do not grow still need fuel, and thus need a metabolism. All machines, living
or not, are based on quantum effects.
How do living machines work? From a fundamental physics point of view, we need
only a few sections of our walk so far to describe them: we need QED and sometimes
universal gravity. Simply stated, life is an electromagnetic process taking place in weak
gravity.* But the details of this statement are tricky and interesting.

* In fact, also the nuclear interactions play some role for life: cosmic radiation is one source for random
20 1 motion for enjoying life

TA B L E 1 Motion and molecular machines found in living beings.

Motion type Examples I n vo lv e d m o t o r s

Growth collective molecular processes in linear molecular motors, ion
cell growth, cell shape change, cell pumps
motility
gene turn-on and turn-off linear molecular motors
ageing linear molecular motors
Construction material transport muscles, linear molecular motors
(polysaccharides, lipids, proteins,
nucleic acids, others)
forces and interactions between pumps in cell membranes
biomolecules and cells
Functioning metabolism (respiration, muscles, ATP synthase, ion pumps

Motion Mountain – The Adventure of Physics
digestion)
muscle working linear molecular motors, ion
pumps
thermodynamics of whole living muscles
system and of its parts
nerve signalling ion motion, ion pumps
brain working, thinking ion motion, ion pumps
memory: long-term potentiation chemical pumps
memory: synapse growth linear molecular motors

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
hormone production chemical pumps
illnesses cell motility, chemical pumps
viral infection of a cell rotational molecular motors for
RNA transport

Defence the immune system cell motility, linear molecular
motors
blood clotting chemical pumps
bronchial cleaning cilial motors
Sensing eye chemical pumps, ion pumps
ear hair motion sensors, ion pumps,
rotary molecular motors
smell ion pumps
touch ion pumps
Reproduction information storage and retrieval linear and rotational molecular
motors inside nuclei
cell division, organelle motion linear molecular motors,
polymerase
sperm motion linear molecular motors
courting, using brain and muscles linear molecular motors, ion
pumps
from quantum physics to biological machines and miniaturization 21

We can say that living beings are systems that move against their environment faster
Ref. 2 than molecules do. Observation shows that living systems move faster the bigger they
are. Observation also shows that living beings achieve this speed by making use of a
huge number of tiny machines, often made of one or only a few molecules, that work
together. These machines realize the numerous processes that are part of life.
An overview of processes taking place in living beings is given in Table 1. Above all,
the table shows that the processes are due to molecular machines.

⊳ A living being is a collection of a huge number of specialized molecular ma-
chines.

Molecular machines are among the most fascinating devices found in nature. Table 1 also
shows that nature only needs a few such devices to realize all the motion types used by
humans and by all other living beings: molecular pumps and molecular motors. Given

Motion Mountain – The Adventure of Physics
the long time that living systems have been around, these devices are extremely efficient.
They are found in every cell, including those of Figure 5. The specialized molecular ma-
chines in living beings are ion pumps, chemical pumps and rotational and linear mo-
lecular motors. Ion and chemical pumps are found in membranes and transport matter
Ref. 3 across membranes. Rotational and linear motors move structures against membranes.
Even though there is still a lot to be learned about molecular machines, the little that is
known is already spectacular enough.

How d o we move? – Molecular motors

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
How do our muscles work? What is the underlying motor? One of the beautiful results
of modern biology is the elucidation of this issue.

⊳ Muscles work because they contain molecules which change shape when
supplied with energy.

This shape change is repeatable. A clever combination and repetition of these molecular
shape changes is then used to generate macroscopic motion.

⊳ Each shape-changing molecule is a molecular motor.

There are three basic classes of molecular motors in nature: linear motors, rotational
motors and pumps.
mutations, which are so important in evolution. Plant growers often use radioactive sources to increase
Ref. 1 mutation rates. Radioactivity can also terminate life or be of use in medicine.
The nuclear interactions are also implicitly involved in life in several other ways. The nuclear interactions
were necessary to form the atoms – carbon, oxygen, etc. – required for life. Nuclear interactions are behind
the main mechanism for the burning of the Sun, which provides the energy for plants, for humans and for
all other living beings (except a few bacteria in inaccessible places).
Summing up, the nuclear interactions occasionally play a role in the appearance and in the destruction
of life; but they usually play no role for the actions or functioning of particular living beings.
22 1 motion for enjoying life

F I G U R E 4 Left: myosin and actin are the two protein molecules that realize the most important linear
molecular motor in living beings, including the motion in muscles. Right: the resulting motion step is

Motion Mountain – The Adventure of Physics
5.5 nm long; it has been slowed down by about a factor of ten (image and QuickTime ﬁlm © San Diego
State University, Jeff Sale and Roger Sabbadini).

1. Linear molecular motors are at the basis of muscle motion; an example is given in
Figure 4. Other linear motors separate genes during cell division. Linear motors also
move organelles inside cells and displace cells through the body during embryo growth,
when wounds heal, and in all other cases of cell motility. Also assembler molecules, for
example those that replicate DNA, can be seen as linear motors.
A typical molecular motor consumes around 100 to 1000 ATP molecules per second,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
thus about 10 to 100 aW. The numbers are small; there are more astonishing if we take
Challenge 6 s into account that the power due to the white noise of the surrounding water is 10 nW. In
other words, in every molecular motor, the power of the environmental noise is eight to
nine orders of magnitude higher than the power consumed by the motor! The ratio shows
what a fantastic piece of machinery such a molecular motor is. At our scale, this would
correspond to a car that drives, all the time, through an ongoing storm and earthquake.
Vol. I, page 92 2. We encountered rotational motors already earlier on; nature uses them to rotate
Ref. 4 the cilia of many bacteria as well as sperm tails. Researchers have also discovered that
evolution produced molecular motors which turn around DNA helices like a motorized
bolt would turn around a screw. Such motors are attached at the end of some viruses and
insert the DNA into virus bodies when they are being built by infected cells, or extract
Ref. 5 the DNA from the virus after it has infected a cell. The most important rotational motor,
Page 25 and the smallest known so far – 10 nm across and 8 nm high – is ATP synthase, a protein
that synthesizes most ATP in cells.
3. Molecular pumps are equally essential to life. They pump chemicals, such as ions
or specific molecules, into every cell or out of it, using energy. They do so even if the
concentration gradient tries to do the opposite. Molecular pumps are thus essential in
ensuring that life is a process far from equilibrium. Malfunctioning molecular pumps are
responsible for many health problems, for example for the water loss in cholera infection.
In the following, we explore a few specific molecular motors found in cells. How mo-
Ref. 6 lecules produce movement in linear motors was uncovered during the 1980s. The results
from quantum physics to biological machines and miniaturization 23

Motion Mountain – The Adventure of Physics
F I G U R E 5 A sea urchin egg surrounded by sperm, or molecular motors in action: molecular motors
make sperm move, make fecundation happen, and make cell division occur (photo by Kristina Yu,
© Exploratorium www.exploratorium.edu).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
started a wave of research on all other molecular motors found in nature. The research
showed that molecular motors differ from most everyday motors: molecular motors do
not involve temperature gradients, as car engines do, they do not involve electrical cur-
rents, as electrical motors do, and they do not rely on concentration gradients, as chem-
ically induced motion, such as the rising of a cake, does.

Linear molecular motors
The central element of the most important linear molecular motor is a combination of
two protein molecules, namely myosin and actin. Myosin changes between two shapes
and literally walks along actin. It moves in regular small steps, as shown in Figure 4. The
motion step size has been measured, with the help of some beautiful experiments, to al-
Ref. 6 ways be an integer multiple of 5.5 nm. A step, usually forward, but sometimes backwards,
results whenever an ATP (adenosine triphosphate) molecule, the standard biological fuel,
hydrolyses to ADP (adenosine diphosphate) and releases the energy contained in the
chemical bond. The force generated is about 3 to 4 pN; the steps can be repeated several
times a second. Muscle motion is the result of thousand of millions of such elementary
steps taking place in concert.
Why does this molecular motor work? The molecular motor is so small that the noise
due to the Brownian motion of the molecules of the liquid around it is extremely intense.
Indeed, the transformation of disordered molecular motion into ordered macroscopic
24 1 motion for enjoying life

Fixed position

U(t 1 )

U(t 2 )
Brownian motion
can take place

U(t 3 )

Motion Mountain – The Adventure of Physics
Most probable next fixed position
if particle moved

F I G U R E 6 Two types of Brownian motors: switching potential (left) and tilting potential (right).

motion is one of the great wonders of nature.
Evolution is smart: with three tricks it takes advantage of Brownian motion and trans-
forms it into macroscopic motion. (Molecular motors are therefore also called Brownian
motors.) The first trick of evolution is the use of an asymmetric, but periodic potential, a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
so-called ratchet.* The second trick of evolution is a temporal variation of the potential
of the ratchet, together with an energy input to make it happen. The two most import-
Ref. 7 ant realizations are shown in Figure 6. Molecular motors thus work away from thermal
equilibrium. The third trick is to take a large number of these molecular motors and to
add their effects.
The periodic potential variation in a molecular motor ensures that for a short, recur-
ring time interval the free Brownian motion of the moving molecule – typically 1 μm/s
– affects its position. Subsequently, the molecule is fixed again. In most of the short time
intervals of free Brownian motion, the position will not change. But if the position does
change, the intrinsic asymmetry of the ratchet shape ensures that with high probability
the molecule advances in the preferred direction. (The animation of Figure 4 lacks this
irregularity.) Then the molecule is fixed again, waiting for the next potential change. On
average, the myosin molecule will thus move in one direction. Nowadays the motion
of single molecules can be followed in special experimental set-ups. These experiments
confirm that muscles use such a ratchet mechanism. The ATP molecule adds energy to
the system and triggers the potential variation through the shape change it induces in
the myosin molecule. Nature then takes millions of these ratchets together: that is how
our muscles work.
Engineering and evolution took different choices. A moped contains one motor. An
Challenge 7 e expensive car contains about 100 motors. A human contains at least 1016 motors.

* It was named after Ratchet Gearloose, the famous inventor from Duckburg.
from quantum physics to biological machines and miniaturization 25

F I G U R E 7 A classical ratchet, here of the piezoelectric kind,
moves like a linear molecular motor (© PiezoMotor).

Another well-studied linear molecular motor is the kinesin–microtubule system that
carries organelles from one place to the other within a cell. Like in the previous example,
also in this case chemical energy is converted into unidirectional motion. Researchers
were able to attach small silica beads to single molecules and to follow their motion.
Using laser beams, they could even apply forces to these single molecules. Kinesin was
found to move with around 800 nm/s, in steps lengths which are multiples of 8 nm, using

Motion Mountain – The Adventure of Physics
one ATP molecule at a time, and exerting a force of about 6 pN.
Quantum ratchet motors do not exist only in living systems; they also exist as human-
built systems. Examples are electrical ratchets that move single electrons and optical
ratchets that drive small particles. These applications are pursued in various experi-
mental research programmes.
Also classical ratchets exist. One example is found in every mechanical clock; also
many ballpoint pens contain one. Another example of ratchet with asymmetric of mech-
anical steps uses the Leidenfrost effect to rapidly move liquid droplets, as shown in the
Vol. I, page 377 video www.thisiscolossal.com/2014/03/water-maze. A further example is shown in Fig-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ure 7; indeed, many piezoelectric actuators work as ratchets and the internet is full of
videos that show how they work. Piezoelectric ratchets, also called ultrasound motors,
are found in precision stages for probe motion and inside certain automatic zoom ob-
jectives in expensive photographic cameras. Also many atomic force microscopes and
scanning electron microscopes use ratchet actuators.
Molecular motors are essential for the growth and the working of nerves and the
Ref. 8 brain. A nerve contains large numbers of dozens of molecular motors types from all three
main families: dynein, myosin and kinesin. These motors transport chemicals, called
‘cargos’, along axons and all have loading and unloading mechanisms at their ends. They
are necessary to realize the growth of nerves, for example from the spine to the tip of
the toes. Other motors control the growth of synapses, and thus ensure that we have
long-term memory. Malfunctioning molecular motors are responsible for Alzheimer dis-
ease, Huntington disease, multiple sclerosis, certain cancers and many other diseases due
either to genetic defects or to environmental poisons.
In short, without molecular motors, we could neither move nor think.

A rotational molecular motor: ATP synthase
In cells, the usual fuel for most chemical reactions is adenosine triphosphate, or ATP.
In plants, most ATP is produced on the membranes of cell organelles called chloroplasts,
and in animal cells, in the so-called mitochondria. These are the power plants in most
cells. ATP also powers most bacteria. It turns out that ATP is synthesized by a protein
26 1 motion for enjoying life

Motion Mountain – The Adventure of Physics
F I G U R E 8 The structure of ATP synthase (© Joachim Weber).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
located in membranes. The protein is itself powered by protons, H+ , which form the
basic fuel of the human body, whereas ATP is the high-level fuel. For example, most other
molecular motors are powered by ATP. ATP releases its energy by being changed into
to adenosine diphosphate, or ADP. The importance of ATP is simple to illustrate: every
human synthesizes, during a typical day, an amount of ATP that is roughly equal to his
or her body mass.
The protein that synthesizes ATP is simply called ATP synthase. In fact, ATP synthase
differs slightly from organism to organism; however, the differences are so small that
they can be neglected in most cases. (An important variation are those pumps where
Na+ ions replace protons.) Even though ATP synthase is a highly complex protein, its
function it easy to describe: it works like a paddle wheel that is powered by a proton
Ref. 9 gradient across the membrane. Figure 8 gives an illustration of the structure and the
process. The research that led to these discoveries was rewarded with the 1997 Nobel
Prize in Chemistry.
In fact, ATP synthase also works in the reverse: if there is a large ATP gradient, it pumps
protons out of the cell. In short, ATP synthase is a rotational motor and molecular pump at
the same time. It resembles the electric starter motor, powered by the battery, found in the
older cars; in those cars, during driving, the electric motor worked as a dynamo charging
the battery. (The internet also contains animations of the rotation of ATP synthase.) ATP
from quantum physics to biological machines and miniaturization 27

A B C

D E

F A G H
J

Motion Mountain – The Adventure of Physics
R L I

F I G U R E 9 A: The asymmetric arrangement of internal organs in the human body: Normal arrangement,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
situs solitus, as common in most humans, and the mirrored arrangement, situs inversus. Images B to E
are scanning electron micrographs of mouse embryos. B: Healthy embryos at this stage already show a
right-sided tail. C: In contrast, mutant embryos with defective cilial motors remain unturned, and the
heart loop is inverted, as shown by the arrow. D: Higher-magniﬁcation images and schematic
representations of a normal heart loop. E: Similar image showing an inverted loop in a mutant embryo.
Images F to I are scanning electron micrographs of a mouse node. F: A low-magniﬁcation view of a 7.5
day-old mouse embryo observed from the ventral side, with the black rectangle indicating the node.
The orientation is indicated with the letters A for anterior, P for posterior, L for left and R for right; the
scale bar is 100 μm. G: A higher-magniﬁcation image of the mouse node; the scale bar is 20 μm. H: A
still higher-magniﬁcation view of healthy nodal cilia, indicated by arrows, and of the nodal pit cells; the
scale bar is 5 μm. I: The nodal pit cells of mutant embryos lacking cilia. J: Illustration of the molecular
transport inside a healthy ﬂagellum (© Hirokawa Nobutaka).

synthase has been studied in great detail. For example, it is known that it produces three
ATP molecules per rotation, that it produces a torque of around 20𝑘𝑇/2π, where 𝑘𝑇 is
the kinetic energy of a molecule at temperature 𝑇. There are at least 1016 such motors in
an adult body. The ATP synthase paddle wheel is one of the central building blocks of life.

Rotational motors and parit y breaking
Why is our heart on the left side and our liver on the right side? The answer of this old
question is known only since a few years. The left-right asymmetry, or chirality, of human
bodies must be connected to the chirality of the molecules that make up life. In all living
28 1 motion for enjoying life

A 0 B C

R L

4
D

Motion Mountain – The Adventure of Physics
E
Nodal flow Microvillum

NVP

Cilium

F I G U R E 10 A: Optical microscope images of ﬂowing nodal vesicular parcels (NVPs). L and R indicate the
orientation. The NVPs, indicated by arrowheads, are transported to the left side by the nodal ﬂow. The
scale bar is 10 μm. B: A scanning electron micrograph of the ventral surface of nodal pit cells. The red
arrowheads indicate NVP precursors. The scale bar is 2 μm. C and D: Transmission electron micrographs
of nodal pit cells. The scale bar is 1 μm. E: A schematic illustration of NVP ﬂow induced by the cilia. The
NVPs are released from dynamic microvilli, are transported to the left side by the nodal ﬂow due to the
cilia, and ﬁnally are fragmented with the aid of cilia at the left periphery of the node. The green halos
indicate high calcium concentration – a sign of cell activation that subsequently starts organ formation
(© Hirokawa Nobutaka).

beings, sugars, proteins and DNS/DNA are chiral molecules, and in all living beings, only
one of the two molecular mirror types is actually used. But how does nature translate the
chirality of molecules into the chirality of a body? The answer was deduced only recently
by Hirokawa Nobutaka and his team; and surprisingly, rotational molecular motors are
Ref. 10 the key to the puzzle.
The position of the internal organs is fixed during the early development of the em-
bryo. At an early stage, a central part of the embryo, the so-called node, is covered with
rotational cilia, i.e., rotating little hairs. They are shown in Figure 9. In fact, all verteb-
from quantum physics to biological machines and miniaturization 29

rates have a node at some stage of embryo development. The nodal cilia are powered by
a molecular motor; they all rotate in the same (clockwise) direction about ten times per
second. The rotation direction is a consequence of the chirality of the molecules that are
contained in the motors. However, since the cilia are inclined with respect to the surface
– towards the tail end of the embryo – the rotating cilia effectively move the fluid above
the node towards the left body side of the embryo. The fluid of this newly discovered
nodal flow contains substances – nodal vesicular parcels – that accumulate on the left
side of the node and subsequently trigger processes that determine the position of the
heart. If the cilia do not rotate due to a genetic defect, or if the flow is reversed by external
means, the heart and other organs get misplaced. This connection also explains all the
other consequences of such genetic defects or interventions.
In other words, through the rotation of the cilia and the mentioned mechanism, the
chirality of molecules is mapped to the chirality of the whole vertebrate organism. It
might even be that similar processes occur also elsewhere in nature, for example in the
development of the brain asymmetry. This is still a field of intense research. In summary,

Motion Mountain – The Adventure of Physics
molecular motors are truly central to our well-being and life.

Curiosities and fun challenges ab ou t biolo gy

“
Una pelliccia è una pelle che ha cambiato

”
bestia.*
Girolamo Borgogelli Avveduti

With modern microscopic methods it is possible to film, in all three dimensions, the
evolution of the eye of a fruit fly until it walks away as a larva. Such a film allows to follow
every single cell that occurs during the 20 hours: the film shows how cells move around

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
during development and show every single cell division. Watch this amazing film, taken
at the EMBL in Heidelberg, at youtu.be/MefTPoeVQ3w.
∗∗
Biological evolution can be summarized in three principles:
1. All living beings are different – also in a species.
2. All living beings have a tough life – due to competition.
3. Living beings with an advantage will survive and reproduce.
The last principle is often called the ‘survival of the fittest’. As a result of these three
principles, with each generation, species and living beings can change. The result of ac-
cumulated generational change is called biological evolution. In particular, these three
principles explain the change from unicellular to multicellular life, from fish to land an-
imals, and from animals to people.
Quantum effects are fundamental in all three principles of evolution. Of course, life
and metabolism are quantum effects. The differences mentioned in the first principle
Vol. IV, page 122 are due to quantum physics: perfect copies of macroscopic systems are impossible. The
second principle mentions competition; that is a kind of measurement, which, as we saw,
Vol. IV, page 20 is only possible due to the existence of a quantum of action. The third principle mentions

* ‘A fur is a skin that has changed beast.’
30 1 motion for enjoying life

reproduction: that is again a quantum effect, based on the copying of genes, which are
Page 18 quantum structures. In short, biological evolution is a process due to the quantum of
action.
∗∗
Challenge 8 d How would you determine which of two identical twins is the father of a baby?
∗∗
Can you give at least five arguments to show that a human clone, if there will ever be one,
Challenge 9 s is a completely different person than the original?
It is well known that the first ever cloned cat, copycat, born in 2002, looked completely
different from the ‘original’ (in fact, its mother). The fur colour and its patch pattern were
completely different from that of the mother. Analogously, identical human twins have
different finger prints, iris scans, blood vessel networks and intrauterine experiences,
among others.

Motion Mountain – The Adventure of Physics
Many properties of a mammal are not determined by genes, but by the environment
of the pregnancy, in particular by the womb and the birth experience. Womb influences
include fur patches, skin fold shapes, but also character traits. The influence of the birth
experience on the character is well-known and has been studied by many psychologists.
∗∗
Discuss the following argument: If nature were classical instead of quantum, there would
not be just two sexes – nor any other discrete number of them, as in some lower animals
– but there would be a continuous range of them. In a sense, there would be an infinite

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 10 s number of sexes. True?
∗∗
Here is a well-known unanswered question on evolution: how did the first kefir grains
form? Kefir grains produce the kefir drink when covered with milk for about 8 to 12
hours. The grains consist of a balanced mixture of about 40 types of bacteria and yeasts.
All kefir grains in the world are related. But how did the first ones form, about 1000 years
Challenge 11 r ago?
∗∗
Molecular motors are quite capable. The molecular motors in the sooty shearwater
(Puffinus griseus), a 45 cm long bird, allow it to fly 74 000 km in a year, with a measured
record of 1094 km a day.
∗∗
When the ciliary motors that clear the nose are overwhelmed and cannot work any more,
they send a distress signal. When enough such signals are sent, the human body triggers
the sneezing reaction. The sneeze is a reaction to blocked molecular motors.
∗∗
The growth of human embryos is one of the wonders of the world. The website embryo.
soad.umich.edu provides extensive data, photos, animations and magnetic resonance
from quantum physics to biological machines and miniaturization 31

images on the growth process.
∗∗
Challenge 12 s Do birds have a navel?
∗∗
All animals with the possibility of regenerating themselves from a small piece, such as
Planaria, reproduce asexually, by dividing. All animals that reproduce sexually are un-
able to regenerating the whole animal from a small part.
∗∗
Challenge 13 s All animals that move with limbs are left-right symmetric. Why?
∗∗
Many molecules found in living beings, such as sugar, have mirror molecules. However,

Motion Mountain – The Adventure of Physics
in all living beings only one of the two sorts is found. Life is intrinsically asymmetric.
Challenge 14 s How can this be?
∗∗
How is it possible that the genetic difference between man and chimpanzee is regularly
given as about 1 %, whereas the difference between man and woman is one chromosome
Challenge 15 s in 46, in other words, about 2.2 %?
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
What is the longest time a single bacterium has survived? It is more than the 5000 years
of the bacteria found in Egyptian mummies. For many years, the survival time was es-
Ref. 11 timated to lie at over 25 million years, a value claimed for the bacteria spores resurrected
from the intestines in insects enclosed in amber. Then it was claimed to lie at over 250
million years, the time estimated that certain bacteria discovered in the 1960s by Heinz
Dombrowski in (low-radioactivity) salt deposits in Fulda, in Germany, have hibernated
there before being brought back to life in the laboratory. A similar result has been re-
cently claimed by the discovery of another bacterium in a North-American salt deposit
Ref. 12 in the Salado formation.
However, these values are now disputed, as DNA sequencing has shown that these
bacteria were probably due to sample contamination in the laboratory, and were not part
Ref. 13 of the original sample. So the question of the longest survival time of bacteria is still open.
∗∗
In 1967, a TV camera was deposited on the Moon. Unknown to everybody, it contained
a small patch of Streptococcus mitis. Three years later, the camera was brought back to
Earth. The bacteria were still alive. They had survived for three years without food, water
or air. Life can be resilient indeed. This widely quoted story is so unbelievable that it was
Ref. 15 checked again in 2011. The conclusion: the story is false; the bacteria were added by
mistake in the laboratory after the return of the camera.
∗∗
32 1 motion for enjoying life

TA B L E 2 Approximate numbers of living species.

Life group Described species E s t i m at e d s p e c i e s
min. max.
Viruses 4 ⋅ 103 50 ⋅ 103 1 ⋅ 106
Prokaryotes (‘bacteria’) 4 ⋅ 103 50 ⋅ 103 3 ⋅ 106
Fungi 72 ⋅ 103 200 ⋅ 103 2.7 ⋅ 106
Protozoa 40 ⋅ 103 60 ⋅ 103 200 ⋅ 103
Algae 40 ⋅ 103 150 ⋅ 103 1 ⋅ 106
Plants 270 ⋅ 103 300 ⋅ 103 500 ⋅ 103
Nematodes 25 ⋅ 103 100 ⋅ 103 1 ⋅ 106
Crustaceans 40 ⋅ 103 75 ⋅ 103 200 ⋅ 103
Arachnids 75 ⋅ 103 300 ⋅ 103 1 ⋅ 106
Insects 950 ⋅ 103 2 ⋅ 106 100 ⋅ 106
70 ⋅ 103 100 ⋅ 103 200 ⋅ 103

Motion Mountain – The Adventure of Physics
Molluscs
Vertebrates 45 ⋅ 103 50 ⋅ 103 55 ⋅ 103
Others 115 ⋅ 103 200 ⋅ 103 800 ⋅ 103
Total 1.75 ⋅ 106 3.6 ⋅ 106 112 ⋅ 106

Bacteria Archaea Eucarya
Green
non-sulfur Animals Ciliates
bacteria Green plants
Methano-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Gram-positive Methano- microbiales extreme
Fungi
bacteria bacteriales Flagellates
Halophiles
Purple bacteria Thermo-
proteus
Cyanobacteria Microsporidia
Flavobacteria Pyro-
dictum Methanococcales
and relatives Thermococcales

Thermotogales

F I G U R E 11 A modern version of the evolutionary tree.

In biology, classifications are extremely useful. (This is similar to the situation in astro-
physics, but in full contrast to the situation in physics.) Table 2 gives an overview of the
Ref. 16 magnitude of the task. This wealth of material can be summarized in one graph, shown
in Figure 11. Newer research seems to suggest some slight changes to the picture. So far
however, there still is only a single root to the tree.
∗∗
from quantum physics to biological machines and miniaturization 33

Muscles produce motion through electrical stimulation. Can technical systems do the
same? Candidate are appearing: so-called electroactive polymers change shape when they
are activated with electrical current or with chemicals. They are lightweight, quiet and
simple to manufacture. However, the first arm wrestling contest between human and
artificial muscles, held in 2005, was won by a teenage girl. The race to do better is ongoing.
∗∗
Life is not a clearly defined concept. The definition used above, the ability to self-
reproduce, has its limits. Can it be applied to old animals, to a hand cut off by mistake, to
sperm, to ovules or to the first embryonal stages of a mammal? The definition of life also
gives problems when trying to apply it to single cells. Can you find a better definition? Is
Challenge 16 e the definition of living beings as ‘what is made of cells’ useful?
∗∗
Every example of growth is a type of motion. Some examples are extremely complex. Take

Motion Mountain – The Adventure of Physics
the growth of acne. It requires a lack of zinc, a weak immune system, several bacteria, as
well as the help of Demodex brevis, a mite (a small insect) that lives in skin pores. With
a size of 0.3 mm, somewhat smaller than the full stop at the end of this sentence, this
and other animals living on the human face can be observed with the help of a strong
magnifying glass.
∗∗
Humans have many living beings on board. For example, humans need bacteria to live.
It is estimated that 90 % of the bacteria in the human mouth alone are not known yet;

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
only about 1000 species have been isolated so far.
Bacteria are essential for our life: they help us to digest and they defend us against
Ref. 17 illnesses due to dangerous bacteria. In fact, the number of bacteria in a human body is
estimated to be 3.8(2.0) ∗ 1013 , more than 99 % of which are in the gut. The number of
cells in a adult, average human body is estimated to be 3.0(0.3) ∗ 1013 – of which 70 to
85 % are red blood cells. In short, a human body contains more bacteria than own cells!
Nevertheless, the combined mass of all bacteria in a human body is estimated to be only
around 0.2 kg, because gut bacteria are much smaller than human cells.
Of the around 100 groups of bacteria in nature, the human body mainly contains
species from four of them: actinobacteria, bacteroidetes, firmicutes and proteobacteria.
They play a role in obesity, malnutrition, heart disease, diabetes, multiple sclerosis, aut-
ism and many other conditions. These connections are an important domain of present
research.
∗∗
How do trees grow? When a tree – biologically speaking, a monopodal phanerophyte –
grows and produces leaves, between 40 % and 60 % of the mass it consists of, namely the
water and the minerals, has to be lifted upwards from the ground. (The rest of the mass
comes from the CO2 in the air.) How does this happen? The materials are pulled upwards
by the water columns inside the tree; the pull is due to the negative pressure that is created
when the top of the column evaporates. This is called the transpiration-cohesion-tension
34 1 motion for enjoying life

model. (This summary is the result of many experiments.) In other words, no energy is
needed for the tree to pump its materials upwards.
Trees do not need energy to transport water. As a consequence, a tree grows purely
by adding material to its surface. This implies that when a tree grows, a branch that is
formed at a given height is also found at that same height during the rest of the life of
Challenge 17 e that tree. Just check this observation with the trees in your garden.
∗∗
Mammals have a narrow operating temperature. In contrast to machines, humans func-
Challenge 18 d tion only if the internal temperature is within a narrow range. Why? And does this re-
quirement also apply to extraterrestrials – provided they exist?
∗∗
Challenge 19 r How did the first cell arise? This important question is still open. As a possible step to-
wards the answer, researchers have found several substances that spontaneously form

Motion Mountain – The Adventure of Physics
closed membranes in water. Such substances also form foams. It might well be that life
formed in foam. Other options discussed are that life formed underwater, at the places
where magma rises into the ocean. Elucidating the origins of cells is one of the great open
riddles of biology – though the answer will not be of much use.
∗∗
Challenge 20 s Could life have arrived to Earth from outer space?
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Is there life elsewhere in the universe? The answer is clear. First of all, there might be
life elsewhere, though the probability is extremely small, due to the long times involved
and the requirements for a stable stellar system, a stable planetary system, and a stable
geological system. In addition, so far, all statements that claim to have detected an ex-
ample were lies. Not mistakes, but actual lies. The fantasy of extraterrestrial life poses an
interesting challenge to everybody: Why would an extraterrestrial being be of interest to
Challenge 21 e you? If you can answer, realize the motivation in some other way, now, without waiting.
If you cannot answer, do something else.
∗∗
What could holistic medicine mean to a scientist, i.e., avoiding nonsense and false beliefs?
Holistic medicine means treating illness with view on the whole person. That translates
to four domains:
— physical support, to aid mechanical or thermal healing processes in the body;
— chemical support, with nutrients or vitamins;
— signalling support, with electrical or chemical means, to support the signalling system
of the body;
— psychological support, to help all above processes.
When all theses aspects are taken care of, healing is as rapid and complete as possible.
However, one main rule remains: medicus curat, natura sanat.*

* ‘The physician helps, but nature heals.’
from quantum physics to biological machines and miniaturization 35

∗∗
Life is, above all, beautiful. For example, the book by Claire Nouvian, The Deep:
The Extraordinary Creatures of the Abyss, presented at www.press.uchicago.edu/books/
nouvian/index.html allows one to savour the beauty of life deep in the ocean.
∗∗
What are the effects of environmental pollution on life? Answering this question is an
intense field of modern research. Here are some famous stories.
— Herbicides and many genetically altered organisms kill bees. For this reason, bees are
dying (since 2007) in the United States; as a result, many crops – such as almonds
and oranges – are endangered there. In countries where the worst herbicides and
genetically modified crops have been banned, bees have no problems. An example is
France, where the lack of bees posed a threat to the wine industry.
— Chemical pollution leads to malformed babies. In mainland China, one out of 16
children is malformed for this reason (in 2007). In Japan, malformations have been

Motion Mountain – The Adventure of Physics
much reduced – though not completely – since strict anti-pollution laws have been
passed.
— Radioactive pollution kills. In Russia, the famous Lake Karachay had to be partly
Page 195 filled with concrete because its high radioactivity killed anybody that walked along it
for an hour.
— Smoking kills – though slowly. Countries that have lower smoking rates or that have
curbed smoking have reduced rates for cancer and several other illnesses.
— Eating tuna is dangerous for your health, because of the heavy metals it contains.
— Cork trees are disappearing. The wine industry has started large research programs

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
to cope with this problem.
— Even arctic and antarctic animals have livers full of human-produced chemical pois-
ons.
— Burning fossil fuels raises the CO2 level of the atmosphere. This leads to many effects
for the Earth’s climate, including a slow rise of average temperature and sea level.
Ecological research is uncovering many additional connections. Let us hope that the
awareness for these issues increases across the world.
∗∗
Some researchers prefer to define living beings as self-reproducing systems, others prefer
to define them as metabolic systems. Among the latter, Mike Russell and Eric Smith pro-
pose the following definition of life: ‘The purpose of life is to hydrogenate carbon dioxide.’
In other terms, the aim of life is to realize the reaction

CO2 + 4 H2 → CH4 + 2 H2 O . (1)

This beautifully dry description is worth pondering – and numerous researchers are in-
deed exploring the consequences of this view.
∗∗
36 1 motion for enjoying life

Not only is death a quantum process, also aging is one. Research in the details of this vast
field is ongoing. A beautiful example is the loss of leaves in autumn. The loss is triggered
by ethene, a simple gas. You can trigger leave loss yourself, for example by putting cut
apples – a strong source of ethene – together with a rose branch in a plastic bag: the roses
will lose their leaves.
∗∗
The reanimation of somebody whose heart and breathing stopped is an useful movement
sequence, called cardiopulmonary resuscitation. Do learn it.
∗∗
New research has shown that motion is important to staying healthy. In particular, it is
important to do sports, but it is even more important to reduce the time of being seated.
People who sit many hours per day have increased risk to get diabetes, breast cancer,
white mass reduction in the brain, dementia, and various other diseases. Research into

Motion Mountain – The Adventure of Physics
the dangerous effects of sitting is still in its infancy. For example, research has shown
that sitting in front of a tv, in front of a PC or in a car for many hours a day cannot be
compensated by doing sport.
∗∗
To stay fit and to ensure that you feel fit, enjoy life and enjoy the books by Mark Verstegen.
∗∗
Are there living beings that contain metal parts? Or is every metal object automatically

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 22 e not part of a living being? Astonishingly, there are exceptions. Enjoy the search.
∗∗
Which species of living being is the most successful, if we measure success as the spe-
cies’ biomass? This simple question has no known answer. Among animals, cattle (Bos
taurus), humans (Homo sapiens) and Antarctic krill (Euphasia superba) have similar bio-
mass values, but it is not clear whether these are the highest values. No good data seem to
exist for plants – except for crops. It is almost sure that several species of bacteria, such as
from the marine genus Prochlorococcus or some other bacteria species found in soil, and
several species of fungi achieve much higher biomass values. But no reliable overview is
available.
∗∗
Trees move in many interesting ways. For example, trees fight with their neighbours
over space and access to light and nutrients. This occurs with most vehemence if the
neighbour is of another species. Most trees do not like to be touched by other trees –
but there are exceptions, such as beeches. For example, when beeches fight with oaks,
after a few years, the oak is left with little space and light, and the beech has taken over
most of it. But trees also help neighbours, for example in case of sickness, by provid-
ing nutrients and water. Many more fascinating stories about trees – including the way
the communicate via aiborne chemical signals such as ethylene (ethene) – are told by
Peter Wohlleben, Das geheime Leben der Bäume: Was sie fühlen, wie sie kommun-
the physics of pleasure 37

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 12 Branch and leaf position of a birch at the end of the day (black) and in the early morning
(red), with magniﬁed sections shown on the right (© Eetu Puttonen et al.).

izieren – die Entdeckung einer verborgenen Welt, Ludwig Verlag, 2015.
∗∗
Trees sleep at night and get up in the morning. The observation is known since centur-
ies; a beautiful measurement of the effect, using a laser scanner, was performed by Eetu
Ref. 18 Puttonen and his group. Their results, shown in Figure 12, show that the height of a typ-
ical birch branch and its leaves in the early morning is up to 10 cm lower than during the
day. The measurements also show that the trees move most in the early morning, when
they wake up. The origin of these effects seems to be the difference of water intake during
day and night.

the physics of pleasure

“
What is mind but motion in the intellectual

”
sphere?
Oscar Wilde, The Critic as Artist.
38 1 motion for enjoying life

Pleasure is a quantum effect. The reason is simple. Pleasure comes from the senses. All
senses measure. And all measurements rely on quantum theory.
The human body, like an expensive car, is full of sensors. Evolution has built these
sensors in such a way that they trigger pleasure sensations whenever we do with our
body what we are made for. Of course, no researcher will admit that he studies pleasure.
Therefore the researcher will say that he or she studies the senses, and that he or she is
doing perception research. But pleasure and all human sensors exist to let life continue.
Pleasure is highest when life is made to continue. In the distant past, the appearance of
new sensors in living systems has always had important effects of evolution, for example
during the Cambrian explosion.
Research into pleasure and biological sensors is a fascinating field that is still evolving;
here we can only have a quick tour of the present knowledge.
The ear is so sensitive and at the same time so robust against large signals that the
experts are still studying how it works. No known sound sensor can cover an energy

Motion Mountain – The Adventure of Physics
range of 1013 ; indeed, the detected sound intensities range from 1 pW/m2 (some say
50 pW/m2 ) to 10 W/m2 , the corresponding air pressures vary from 20 𝜇Pa to 60 Pa. The
lowest intensity that can be heard is that of a 20 W sound source heard at a distance of
10 000 km, if no sound is lost in between. Audible sound wavelengths span from 17 m
(for 20 Hz) to 17 mm (for 20 kHz). In this range, the ear, with its 16 000 to 20 000 hair
cells and 30 000 cochlear neurons, is able to distinguish at least 1500 pitches. But the ear
is also able to distinguish nearby frequencies, such as 400 and 401 Hz, using a special
pitch sharpening mechanism.
The eye is a position dependent photon detector. Each eye contains around 126 million

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
separate detectors on the retina. Their spatial density is the highest possible that makes
sense, given the diameter of the lens of the eye. They give the eye a resolving power of
1 󸀠 , or 0.29 mrad, and the capacity to consciously detect down to 60 incident photons in
0.15 s, or 4 absorbed photons in the same time interval.
Each eye contains 120 million highly sensitive general light intensity detectors, the
rods. They are responsible for the mentioned high sensitivity. Rods cannot distinguish
colours. Before the late twentieth century, human built light sensors with the same sens-
itivity as rods had to be helium cooled, because technology was not able to build sensors
at room temperature that were as sensitive as the human eye.
Vol. III, page 199 The human eye contains about 6 million not so sensitive colour detectors, the cones,
whose distribution we have seen earlier on. The different chemicals in the three cone
types (red, green, blue) lead to different sensor speeds; this can be checked with the
Ref. 19 simple test shown in Figure 13. The sensitivity difference between the colour-detecting
cones and the colour-blind rods is the reason that at night all cats are grey.
The images of the eye are only sharp if the eye constantly moves in small random
motions. If this motion is stopped, for example with chemicals, the images produced by
the eye become unsharp.
The eye also contains about 1 million retinal ganglion cells. All signals from the eye
are transmitted through 1 million optical nerve fibres to a brain region, the virtual cortex,
that contains over 500 million cells.
Human touch sensors are distributed over the skin, with a surface density which varies
the physics of pleasure 39

F I G U R E 13 The
different speed of the
eye’s colour sensors, the
cones, lead to a strange
effect when this picture
(in colour version) is
shaken right to left in
weak light.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 14 The ﬁve sensors of touch in humans, from the most to the least common ones: Meissner’s
corpuscles, Merkel cells, Rufﬁni corpuscles, Pacinian corpuscles and hair receptors.

from one region to the other. The density is lowest on the back and highest in the face
and on the tongue. The hand has about 17 000 tactile receptors, most of them at the fin-
ger tips. There are separate sensors for light touch (Meissner’s corpuscles) and pressure
(Merkel cells), for deformation (Ruffini corpuscles), for vibration (Pacinian corpuscles),
and for tickling (unmyelinated fibers); there are additional separate sensors for heat, for
coldness,* and for pain. Some of the sensors, whose general appearance is shown in Fig-

* There are four sensors for heat; one is triggered above 27°C, one above 31°C, one above 42°C, and one
40 1 motion for enjoying life

ure 14, react proportionally to the stimulus intensity, some differentially, giving signals
only when the stimulus changes. Many of these sensors are also found inside the body
– for example on the tongue. The sensors are triggered when external pressure deforms
them; this leads to release of Na+ and K+ ions through their membranes, which then
leads to an electric signal that is sent via nerves to the brain.
The human body also contains orientation sensors in the ear, extension sensors in each
muscle, and pain sensors distributed with varying density over the skin and inside the
body.
The taste sensor mechanisms of tongue are only partially known. The tongue is known
to produce six taste signals* – sweet, salty, bitter, sour, proteic and fatty – and the mech-
anisms are just being unravelled. The sense for proteic, also called umami, has been dis-
Ref. 20 covered in 1907, by Ikeda Kikunae; the sense for ‘fat’ has been discovered only in 2005.
The tongue, palate and cheeks have about 10 000 taste buds, 90 % of which are on the
tongue. Each taste bud has between 50 and 150 receptors; their diameter is around 10 μm.

Motion Mountain – The Adventure of Physics
In ancient Greece, Democritus imagined that taste depends on the shape of atoms.
Today it is known that sweet taste is connected with certain shape of molecules. Modern
research is still unravelling the various taste receptors in the tongue. At least three differ-
ent sweetness receptors, dozens of bitterness receptors, and one proteic and one fattiness
receptor are known. In contrast, the sour and salty taste sensation are known to be due
to ion channels. Despite all this knowledge, no sensor with a distinguishing ability of the
same degree as the tongue has yet been built by humans. A good taste sensor would have
great commercial value for the food industry. Research is also ongoing to find substances
to block taste receptors; one aim is to reduce the bitterness of medicines or of food.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The nose has about 350 different smell receptors and a total of about 40 million re-
ceptor cells. (Dogs have 25 times more.) Through the possible combinations it is estim-
ated that the nose can detect about 10 000 different smells.** Together with the six signals
that the sense of taste can produce, the nose also produces a vast range of taste sensations.
It protects against chemical poisons, such as smoke, and against biological poisons, such
as faecal matter. In contrast, artificial gas sensors exist only for a small range of gases.
Good artificial taste and smell sensors would allow checking wine or cheese during their
Challenge 24 ny production, thus making their inventor extremely rich. At the moment, humans, with all
their technology at their disposal, are not even capable of producing sensors as good as
those of a bacterium; it is known that Escherichia coli can sense at least 30 substances in
its environment.

above 52°C. The sensor for temperatures above 42°C, TRPV1, is also triggered by capsaicin, the sharp
chemical in chilli peppers.
There seems to be only one sensor for coldness, the ion channel TRPM8, triggered between 8 and 26°C.
It is also triggered by menthol, a chemical contained in mojito and mint. Coldness neurons, i.e., neurons
with TRPM8 at their tips, can be seen with special techniques using fluorescence and are known to arrive
into the teeth; they provide the sensation you get at the dentist when he applies his compressed air test.
* Taste sensitivity is not separated on the tongue into distinct regions; this is an incorrect idea that has been
copied from book to book for over a hundred years. You can perform a falsification by yourself, using sugar
Challenge 23 s or salt grains.
** Linda Buck and Richard Axel received the 2004 Nobel Prize in Physiology or Medicine for their unrav-
elling of the working of the sense of smell.
the physics of pleasure 41

Vol. III, page 32 Other animals feature additional types of sensors. Sharks can feel electrical fields. Many
snakes have sensors for infrared light, such as the pit viper or vampire bats. These sensors
are used to locate prey or food sources. Some beetles, such as Melanophila acuminata,
can also detect infrared; they use this sense to locate the wildfires they need to make
their eggs hatch. Also other insects have such organs. Pigeons, trout and sharks can feel
magnetic fields, and use this sense for navigation. Many birds and certain insects can see
UV light. Bats and dolphins are able to hear ultrasound up to 100 kHz and more. Whales
Vol. I, page 325 and elephants can detect and localize infrasound signals.
In summary, the sensors with which nature provides us are state of the art; their sens-
itivity and ease of use is the highest possible. Since all sensors trigger pleasure or help
to avoid pain, nature obviously wants us to enjoy life with the most intense pleasure
Ref. 21 possible. Studying physics is one way to do this.

“
There are two things that make life worth living:

”
Mozart and quantum mechanics.
Victor Weisskopf*

Motion Mountain – The Adventure of Physics
The nerves and the brain

“
There is no such thing as perpetual tranquillity
of mind while we live here; because life itself is
but motion, and can never be without desire,

”
nor without fear, no more than without sense.
Thomas Hobbes, Leviathan.

The main unit processing all the signals arriving from the sensors, the brain, is essential
for all feelings of pleasure. The human brain has the highest complexity of all brains

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
known.** In addition, the processing power and speed of the human brain is still larger
than any device build by man.
We saw already earlier on how electrical signals from the sensors are transported into
Vol. I, page 315 the brain. In the brain itself, the arriving signals are classified and stored, sometimes for a
short time, sometimes for a long time. Most storage mechanisms take place in the struc-
Vol. III, page 265 ture and the connection strength between brain cells, the synapses, as we have seen. The
process remaining to understand is the classification, a process we usually call thinking.
For certain low level classifications, such as geometrical shapes for the eye or sound har-
monies for the ear, the mechanisms are known. But for high-level classifications, such
as the ones used in conceptual thinking, the aim is not yet achieved. It is not yet known
how to describe the processes of reading or understanding in terms of signal motions.

* Victor Friedrich Weisskopf (b. 1908 Vienna, d. 2002 Cambridge), acclaimed theoretical physicist who
worked with Einstein, Born, Bohr, Schrödinger and Pauli. He catalysed the development of quantum elec-
trodynamics and nuclear physics. He worked on the Manhattan project but later in life intensely cam-
paigned against the use of nuclear weapons. During the cold war he accepted the membership in the Soviet
Academy of Sciences. He was professor at MIT and for many years director of CERN, in Geneva. He wrote
several successful physics textbooks. The author heard him making the above statement in 1982, during one
of his lectures.
** This is not in contrast with the fact that a few whale species have brains with a larger mass. The larger
mass is due to the protection these brains require against the high pressures which appear when whales
dive (some dive to depths of 1 km). The number of neurons in whale brains is considerably smaller than in
human brains.
42 1 motion for enjoying life

Research is still in full swing and will probably remain so for a large part of the twenty-
first century.
In the following we look at a few abilities of our brain, of our body and of other bodies
that are important for the types of pleasure that we experience when we study motion.

Living clo cks

“ ”
L’horologe fait de la réclame pour le temps.*
Georges Perros

Vol. I, page 44 We have given an overview of living clocks already at the beginning of our adventure.
They are common in bacteria, plants and animals. And as Table 3 shows, without biolo-
gical clocks, neither life nor pleasure would exist.
When we sing a musical note that we just heard we are able to reproduce the original
frequency with high accuracy. We also know from everyday experience that humans are
Ref. 22 able to keep the beat to within a few per cent for a long time. When doing sport or when

Motion Mountain – The Adventure of Physics
dancing, we are able to keep the timing to high accuracy. (For shorter or longer times,
the internal clocks are not so precise.) All these clocks are located in the brain.
Brains process information. Also computers do this, and like computers, all brains
need a clock to work well. Every clock is made up of the same components. It needs an
oscillator determining the rhythm and a mechanism to feed the oscillator with energy.
In addition, every clock needs an oscillation counter, i.e., a mechanism that reads out
the clock signal, and a means of signal distribution throughout the system is required,
synchronizing the processes attached to it. Finally, a clock needs a reset mechanism. If the
clock has to cover many time scales, it needs several oscillators with different oscillation

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
frequencies and a way to reset their relative phases.
Even though physicists know fairly well how to build good clocks, we still do not know
many aspects of biological clocks. Most biological oscillators are chemical systems; some,
Ref. 23 like the heart muscle or the timers in the brain, are electrical systems. The general elu-
cidation of chemical oscillators is due to Ilya Prigogine; it has earned him a Nobel Prize
for chemistry in 1977. But not all the chemical oscillators in the human body are known
yet, not to speak of the counter mechanisms. For example, a 24-minute cycle inside each
human cell has been discovered only in 2003, and the oscillation mechanism is not yet
fully clear. (It is known that a cell fed with heavy water ticks with 27–minute instead of
Ref. 24 24–minute rhythm.) It might be that the daily rhythm, the circadian clock, is made up
of or reset by 60 of these 24–minute cycles, triggered by some master cells in the human
body. The clock reset mechanism for the circadian clock is also known to be triggered by
daylight; the cells in the eye who perform this resetting action have been pinpointed only
in 2002. The light signal from these cells is processed by the superchiasmatic nuclei, two
dedicated structures in the brain’s hypothalamus. The various cells in the human body
act differently depending on the phase of this clock.
The clocks with the longest cycle in the human body control ageing. One of the more
famous ageing clock limits the number of divisions that a cell can undergo. Indeed, the
number of cell divisions is finite for most cell types of the human body and typically lies
between 50 and 200. (An exception are reproductory cells – we would not exist if they
* ‘Clocks are ads for time.’
the physics of pleasure 43

TA B L E 3 Examples of biological rhythms and clocks.

Living being O s c i l l at i n g s ys t e m Period
Sand hopper (Talitrus saltator) knows in which direction to flee from circadian
the position of the Sun or Moon
Human (Homo sapiens) gamma waves in the brain 0.023 to 0.03 s
alpha waves in the brain 0.08 to 0.13 s
heart beat 0.3 to 1.5 s
delta waves in the brain 0.3 to 10 s
blood circulation 30 s
cellular circahoral rhythms 1 to 2 ks
rapid-eye-movement sleep period 5.4 ks
nasal cycle 4 to 14 ks
growth hormone cycle 11 ks
suprachiasmatic nucleus (SCN), 90 ks

Motion Mountain – The Adventure of Physics
circadian hormone concentration,
temperature, etc.; leads to jet lag
skin clock circadian
monthly period 2.4(4) Ms
built-in aging 3.2(3) Gs
Common fly (Musca domestica) wing beat 30 ms
Fruit fly (Drosophila wing beat for courting 34 ms
melanogaster)
Most insects (e.g. wasps, fruit winter approach detection (diapause) by yearly

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
flies) length of day measurement; triggers
metabolism changes
Algae (Acetabularia) Adenosinetriphosphate (ATP)
concentration
Moulds (e.g. Neurospora crassa) conidia formation circadian
Many flowering plants flower opening and closing circadian
Tobacco plant flower opening clock (photoperiodism); annual
triggered by length of days, discovered
in 1920 by Garner and Allard
Arabidopsis circumnutation circadian
growth a few hours
Telegraph plant (Desmodium side leaf rotation 200 s
gyrans)
Forsythia europaea, F. suspensa, Flower petal oscillation, discovered by 5.1 ks
F. viridissima, F. spectabilis Van Gooch in 2002

would not be able to divide endlessly.) The cell division counter has been identified; it
is embodied in the telomeres, special structures of DNA and proteins found at both ends
of each chromosome. These structures are reduced by a small amount during each cell
division. When the structures are too short, cell division stops. The purely theoretical
44 1 motion for enjoying life

prediction of this mechanism by Alexei Olovnikov in 1971 was later proven by a number
of researchers. (Only the latter received the Nobel Prize in medicine, in 2009, for this
confirmation.) Research into the mechanisms and the exceptions to this process, such as
cancer and sexual cells, is ongoing.
Not all clocks in human bodies have been identified, and not all mechanisms are
known. For example, basis of the monthly period in women is interesting, complex, and
unclear.
Other fascinating clocks are those at the basis of conscious time. Of these, the brain’s
stopwatch or interval timer has been most intensely studied. Only recently was its mech-
anism uncovered by combining data on human illnesses, human lesions, magnetic res-
Ref. 25 onance studies and effects of specific drugs. The basic interval timing mechanism takes
place in the striatum in the basal ganglia of the brain. The striatum contains thousands
of timer cells with different periods. They can be triggered by a ‘start’ signal. Due to their
large number, for small times of the order of one second, every time interval has a differ-
ent pattern across these cells. The brain can read these patterns and learn them. In this

Motion Mountain – The Adventure of Physics
way we can time music or specific tasks to be performed, for example, one second after
a signal.
Even though not all the clock mechanisms in humans are known, biological clocks
share a property with all human-built and all non-living clocks: they are limited by
quantum mechanics. Even the simple pendulum is limited by quantum theory. Let us
explore the topic.

When d o clo cks exist?

“ ”
Die Zukunft war früher auch besser.*

When we explored general relativity we found out that purely gravitational clocks do not
Vol. II, page 282 exist, because there is no unit of time that can be formed using the constants 𝑐 and 𝐺.
Clocks, like any measurement standard, need matter and non-gravitational interactions
to work. This is the domain of quantum theory. Let us see what the situation is in this
case.
Ref. 26 First of all, in quantum theory, the time is not an observable. Indeed, the time oper-
ator is not Hermitean. In other words, quantum theory states that there is no physical
observable whose value is proportional to time. On the other hand, clocks are quite com-
mon; for example, the Sun or Big Ben work to most people’s satisfaction. Observations
thus encourages us to look for an operator describing the position of the hands of a clock.
However, if we look for such an operator we find a strange result. Any quantum system
having a Hamiltonian bounded from below – having a lowest energy – lacks a Hermitean
operator whose expectation value increases monotonically with time. This result can be
Challenge 25 ny proven rigorously, as a mathematical theorem.
Take a mechanical pendulum clock. In all such clocks the weight has to stop when
the chain end is reached. More generally, all clocks have to stop when the battery or the
energy source is empty. In other words, in all real clocks the Hamiltonian is bounded

* ‘Also the future used to be better in the past.’ Karl Valentin (b. 1882 Munich, d. 1948 Planegg), playwright,
writer and comedian.
the physics of pleasure 45

from below. And the above theorem from quantum theory then states that such a clock
cannot really work.
In short, quantum theory shows that exact clocks do not exist in nature. Quantum
theory states that any clock can only be approximate. Time cannot be measured exactly;
time can only be measured approximately. Obviously, this result is of importance for high
precision clocks. What happens if we try to increase the precision of a clock as much as
possible?
High precision implies high sensitivity to fluctuations. Now, all clocks have an oscil-
lator inside, e.g., a motor, that makes them work. A high precision clock thus needs a high
precision oscillator. In all clocks, the position of this oscillator is read out and shown on
the dial. Now, the quantum of action implies that even the most precise clock oscillator
has a position indeterminacy. The precision of any clock is thus limited.
Worse, like any quantum system, any clock oscillator even has a small, but finite prob-
ability to stop or to run backwards for a while. You can check this conclusion yourself.
Just have a look at a clock when its battery is almost empty, or when the weight driving

Motion Mountain – The Adventure of Physics
the pendulum has almost reached the bottom position. The clock will start doing funny
things, like going backwards a bit or jumping back and forward. When the clock works
normally, this behaviour is strongly suppressed; however, it is still possible, though with
Challenge 26 e low probability. This is true even for a sundial.
In summary, clocks necessarily have to be macroscopic in order to work properly. A
clock must be as large as possible, in order to average out its fluctuations. Astronom-
ical systems are good examples. A good clock must also be well-isolated from the en-
vironment, such as a freely flying object whose coordinate is used as time variable. For
example, this is regularly done in atomic optical clocks.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The precision of clo cks
Given the limitations due to quantum theory, what is the ultimate accuracy 𝜏 of a clock?
To start with, the indeterminacy relation provides the limit on the mass of a clock. The
Challenge 27 ny clock mass 𝑀 must obey
ℏ
𝑀> 2 (2)
𝑐𝜏

Challenge 28 e which is obviously always fulfilled in everyday life. But we can do better. Like for a pen-
dulum, we can relate the accuracy 𝜏 of the clock to its maximum reading time 𝑇. The
Ref. 27 idea was first published by Salecker and Wigner. They argued that

ℏ 𝑇
𝑀> (3)
𝑐2 𝜏 𝜏
where 𝑇 is the time to be measured. You might check that this condition directly requires
Challenge 29 e that any clock must be macroscopic.
Let us play with the formula by Salecker and Wigner. It can be rephrased in the fol-
lowing way. For a clock that can measure a time 𝑡, the size 𝑙 is connected to the mass 𝑚
46 1 motion for enjoying life

by
ℏ𝑡
𝑙>√ . (4)
𝑚

Ref. 28 How close can this limit be achieved? It turns out that the smallest clocks known, as well
as the clocks with most closely approach this limit, are bacteria. The smallest bacteria,
the mycoplasmas, have a mass of about 8 ⋅ 10−17 kg, and reproduce every 100 min, with a
precision of about 1 min. The size predicted from expression (4) is between 0.09 μm and
0.009 μm. The observed size of the smallest mycoplasmas is 0.3 μm. The fact that bacteria
can come so close to the clock limit shows us again what a good engineer evolution has
been.
Note that the requirement by Salecker and Wigner is not in contrast with the possib-
ility to make the oscillator of the clock very small; researchers have built oscillators made
Ref. 29 of a single atom. In fact, such oscillations promise to be the most precise human built
Page 42 clocks. But the oscillator is only one part of any clock, as explained above.

Motion Mountain – The Adventure of Physics
In the real world, the clock limit can be tightened even more. The whole mass 𝑀
cannot be used in the above limit. For clocks made of atoms, only the binding energy
between atoms can be used. This leads to the so-called standard quantum limit for clocks;
it limits the accuracy of their frequency 𝜈 by

𝛿𝜈 Δ𝐸
=√ (5)
𝜈 𝐸tot

where Δ𝐸 = ℏ/𝑇 is the energy indeterminacy stemming from the finite measuring time

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑇 and 𝐸tot = 𝑁𝐸bind is the total binding energy of the atoms in the metre bar. So far,
the quantum limit has not yet been achieved for any clock, even though experiments are
getting close to it.
In summary, clocks exist only in the limit of ℏ being negligible. In practice, the errors
made by using clocks and metre bars can be made as small as required; it suffices to make
the clocks large enough. Clock built into human brains comply with this requirement.
We can thus continue our investigation into the details of matter without much worry,
at least for a while. Only in the last part of our mountain ascent, where the requirements
for precision will be even higher and where general relativity will limit the size of phys-
ical systems, trouble will appear again: the impossibility to build precise clocks will then
Vol. VI, page 65 become a central issue.

Why are predictions so difficult, especially of the fu ture?

“
Future: that period of time in which our affairs
prosper, our friends are true, and our happiness

”
is assured.
Ambrose Bierce

Nature limits predictions in four ways:
1. We have seen that quantum theory, through the uncertainty relations, limits the pre-
cision of measurements, and of clocks and time measurements in particular. Thus,
the physics of pleasure 47

the quantum of action makes it hard to determine initial states to full precision –
even for a single particle.
Vol. I, page 239 2. We have seen that high numbers of particles make it difficult to predict the future due
to the often statistical nature of their initial conditions.
3. We have found in our adventure that predictions of the future are made difficult by
Vol. I, page 424 non-linearities and by the divergence from nearby initial conditions.
Vol. II, page 110 4. We have seen that a non-trivial space-time topology can limit predictability. For ex-
ample, we will discover that black hole and horizons can limit predictability due to
their one-way inclusion of energy, mass and signals.
Vol. VI, page 40 5. We will find out in the last pat of our adventure that quantum gravity effects even
make a precise definition of time and space impossible.
Measurements and practical predictability are thus limited. The central reason for this
limitation is the quantum of action. But if the quantum of action makes perfect clocks
impossible, is determinism still the correct description of nature? And does time exist
after all? The answer is clear: yes and no. We learned that all the mentioned limitations

Motion Mountain – The Adventure of Physics
of clocks can be overcome for limited time intervals; in practice, these time intervals can
be made so large that the limitations do not play a role in everyday life. As a result,

⊳ In practice, in all quantum systems both determinism and the concept of
time remain applicable.

This conclusion is valid even though theory says otherwise. Our ability to enjoy the pleas-
ures due to the flow of time remains intact.
However, when extremely large momentum flows or extremely large dimensions need

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
to be taken into account, quantum theory cannot be applied alone; in those cases, gen-
eral relativity needs to be taken into account. The fascinating effects that occur in those
Vol. VI, page 57 situations will be explored in detail later on.

Decay and the golden rule

“ ”
I prefer most of all to remember the future.
Salvador Dalì

All pleasure only makes sense in the face of death. And death is a form of decay. Decay
is any spontaneous change. Like the wave aspect of matter, decay is a process with no
classical counterpart. Of course, any decay – including the emission of light by a lamp,
the triggering of a camera sensor, radioactivity or the ageing of humans – can be observed
classically; however, the origin of decay is a pure quantum effect.
In any decay of unstable systems or particles, the decoherence of superpositions of
Vol. IV, page 143 macroscopically distinct states plays an important role. Indeed, experiments confirm that
the prediction of decay for a specific system, like a scattering of a particle, is only pos-
sible on average, for a large number of particles or systems, and never for a single one.
These observations confirm the quantum origin of decay. In every decay process, the su-
perposition of macroscopically distinct states – in this case those of a decayed and an
undecayed particle – is made to decohere rapidly by the interaction with the environ-
ment. Usually the ‘environment’ vacuum, with its fluctuations of the electromagnetic,
48 1 motion for enjoying life

weak and strong fields, is sufficient to induce decoherence. As usual, the details of the
involved environment states are unknown for a single system and make any prediction
for a specific system impossible.
What is the origin of decay? Decay is always due to tunnelling. With the language of
quantum electrodynamics, we can say that decay is motion induced by the vacuum fluc-
tuations. Vacuum fluctuations are random. The experiment between the plates confirms
the importance of the environment fluctuations for the decay process.
Quantum theory gives a simple description of decay. For a system consisting of a large
number 𝑁 of decaying identical particles, any decay is described by

𝑁 1 2π 󵄨󵄨 󵄨2
𝑁̇ = − where = 󵄨󵄨⟨𝜓initial|𝐻int|𝜓final ⟩󵄨󵄨󵄨 . (6)
𝜏 𝜏 ℏ

This result for 𝑁̇ = d𝑁/d𝑡 was named the golden rule by Fermi,* because it works so well
despite being an approximation whose domain of applicability is not easy to specify. The

Motion Mountain – The Adventure of Physics
Challenge 30 e golden rule leads to

𝑁(𝑡) = 𝑁0 e−𝑡/𝜏 . (7)

Decay is thus predicted to follow an exponential behaviour, independently of the details
of the physical process. In addition, the decay time 𝜏 depends on the interaction and on
the square modulus of the transition matrix element. For almost a century, all experi-
ments confirmed that quantum decay is exponential.
On the other hand, when quantum theory is used to derive the golden rule, it is found

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 30 that decay is exponential only in certain special systems. A calculation that takes into ac-
count higher order terms predicts two deviations from exponential decay for completely
isolated systems: for short times, the decay rate should vanish; for long times, the de-
cay rate should follow an algebraic – not an exponential – dependence on time, in some
Ref. 31 cases even with superimposed oscillations. After an intense experimental search, devi-
ations for short times have been observed. The observation of deviations at long times are
rendered impossible by the ubiquity of thermal noise. In summary, it turns out that decay
is exponential only when the environment is noisy, the system made of many weakly in-
teracting particles, or both. Since this is usually the case, the mathematically exceptional
exponential decrease becomes the (golden) rule in the description of decay.
Can you explain why human life, despite being a quantum effect, is not observed to
Challenge 31 s follow an exponential decay?

The present in quantum theory

“ ”
Utere temporibus.**
Ovidius

* Originally, the golden rule is a statement from the christian bible, (Matthew 7,12) namely the precept ‘Do
to others whatever you would like them to do to you.’
** ‘Use the occasions.’ Tristia 4, 3, 83
the physics of pleasure 49

Many sages advise to enjoy the present. As shown by perception research, what humans
Ref. 32 call ‘present’ has a duration of between 20 and 70 milliseconds. This result on the biolo-
gical present leads us to ask whether the physical present might have a duration as well.
In everyday life, we are used to imagine that shortening the time taken to measure the
position of a point object as much as possible will approach the ideal of a particle fixed
at a given point in space. When Zeno discussed the flight of an arrow, he assumed that
this is possible. However, quantum theory changes the situation.
Can we really say that a moving system is at a given spot at a given time? In order to
find an answer through experiment, we could use a photographic camera whose shutter
time can be reduced at will. What would we find? When the shutter time approaches the
oscillation period of light, the sharpness of the image would decrease; in addition, the
colour of the light would be influenced by the shutter motion. We can increase the energy
of the light used, but the smaller wavelengths only shift the problem, they do not solve it.
Worse, at extremely small wavelengths, matter becomes transparent, and shutters cannot
be realized any more. All such investigations confirm: Whenever we reduce shutter times

Motion Mountain – The Adventure of Physics
as much as possible, observations become unsharp. The lack of sharpness is due to the
quantum of action. Quantum theory thus does not confirm the naive expectation that
shorter shutter times lead to sharper images. In contrast, the quantum aspects of nature
show us that there is no way in principle to approach the limit that Zeno was discussing.
In summary, the indeterminacy relation and the smallest action value prevent that
moving objects are at a fixed position at a given time. Zeno’s discussion was based on
an extrapolation of classical concepts into domains where it is not valid any more. Every
observation, like every photograph, implies a time average:

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Observations average interactions over a given time span.

For a photograph, the duration is given by the shutter time; for a measurement, the aver-
age is defined by the details of the set-up. Whatever this set-up might be, the averaging
time is never zero. * There is no ‘point-like’ instant of time that describes the present. The
observed, physical present is always an average over a non-vanishing interval of time. In
nature, the present has a finite duration. To give a rough value that guides our thought,
in most situations the length of the present will be less than a yoctosecond, so that it can
usually be neglected.

Why can we observe motion?
Zeno of Elea was thus wrong in assuming that motion is a sequence of specific positions
in space. Quantum theory implies that motion is only approximately the change of pos-
ition with time.
Why then can we observe and describe motion in quantum theory? Quantum the-
ory shows that motion is the low energy approximation of quantum evolution. Quantum
evolution assumes that space and time measurements of sufficient precision can be per-
formed. We know that for any given observation energy, we can build clocks and metre
bars with much higher accuracy than required, so that in practice, quantum evolution

Page 50 * Also the discussion of the quantum Zeno effect, below, does not change the conclusions of this section.
50 1 motion for enjoying life

is applicable in all cases. As long as energy and time have no limits, all problems are
avoided, and motion is a time sequence of quantum states.
In summary, we can observe motion because for any known observation energy we
can find a still higher energy and a still longer averaging time that can be used by the
measurement instruments to define space and time with higher precision than for the
Vol. VI, page 46 system under observation. In the final part of our mountain ascent, we will discover that
there is a maximum energy in nature, so that we will need to change our description in
those situations. However, this energy value is so huge that it does not bother us at all at
the present point of our exploration.

R est and the quantum Z eno effect
The quantum of action implies that there is no rest in nature. Rest is thus always either
an approximation or a time average. For example, if an electron is bound in an atom, not
freely moving, the probability cloud, or density distribution, is stationary in time. But
there is another apparent case of rest in quantum theory, the quantum Zeno effect. Usu-

Motion Mountain – The Adventure of Physics
ally, observation changes the state of a system. However, for certain systems, observation
can have the opposite effect, and fix a system.
Quantum mechanics predicts that an unstable particle can be prevented from decay-
ing if it is continuously observed. The reason is that an observation, i.e., the interaction
with the observing device, yields a non-zero probability that the system does not evolve.
If the frequency of observations is increased, the probability that the system does not
decay at all approaches 1. Three research groups – Alan Turing by himself in 1954, the
group of A. Degasperis, L. Fonda and G.C. Ghirardi in 1974, and George Sudarshan and
Baidyanath Misra in 1977 – have independently predicted this effect, today called the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
quantum Zeno effect. In sloppy words, the quantum Zeno effect states: if you look at a
system all the time, nothing happens.
The quantum Zeno effect is a natural consequence of quantum theory; nevertheless,
its strange circumstances make it especially fascinating. After the prediction, the race for
the first observation began. The effect was partially observed by David Wineland and his
Ref. 33 group in 1990, and definitively observed by Mark Raizen and his group in 2001. In the
meantime, other groups have confirmed the measurements. Thus, quantum theory has
been confirmed also in this surprising aspect.
The quantum Zeno effect is also connected to the deviations from exponential decay
– due to the golden rule – that are predicted by quantum theory. Indeed, quantum theory
predicts that every decay is exponential only for intermediate times, and quadratic for
short times and polynomial for extremely long times. These issues are research topics to
this day.
Ref. 34 In a fascinating twist, in 2002, Saverio Pascazio and his team have predicted that the
quantum Zeno effect can be used to realize X-ray tomography of objects with the lowest
radiation levels imaginable.
In summary, the quantum Zeno effect does not contradict the statement that there
is no rest in nature; in situations showing the effect, there is an non-negligible interac-
tion between the system and its environment. The details of the interaction are import-
Ref. 35 ant: in certain cases, frequent observation can actually accelerate the decay or evolution.
Quantum physics still remains a rich source of fascinating effects.
the physics of pleasure 51

C onsciousness – a result of the quantum of action
In the pleasures of life, consciousness plays an essential role.

Vol. III, page 339 ⊳ Consciousness is our ability to observe what is going on in our mind.

This activity, like any type of change, can itself be observed and studied. Though it is
hard and probably impossible to do so by introspection, we can study consciousness in
others. Obviously, consciousness takes place in the brain. If it were not, there would be
no way to keep it connected with a given person. Simply said, we know that each brain
located on Earth moves with over one million kilometres per hour through the cosmic
background radiation; we also observe that consciousness moves along with it.
The brain is a quantum system: it is based on molecules and electrical currents. The
changes in consciousness that appear when matter is taken away from the brain – in
operations or accidents – or when currents are injected into the brain – in accidents, ex-

Motion Mountain – The Adventure of Physics
periments or misguided treatments – have been described in great detail by the medical
profession. Also the observed influence of chemicals on the brain – from alcohol to hard
drugs – makes the same point. The brain is a quantum system.
Modern imaging machines can detect which parts of the brain work when sensing,
remembering or thinking. Not only is sight, noise and thought processed in the brain; we
can follow these processes with measurement apparatus. The best imaging machines are
Page 162 based on magnetic resonance, as described below. Another, more questionable imaging
technique, positron tomography, works by letting people swallow radioactive sugar. Both
techniques confirm the findings on the location of thought and on its dependence on
chemical fuel. In addition, we already know that memory depends on the particle nature

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of matter. All these observations depend on the quantum of action.
Today, we are thus in the same situation as material scientists were a century ago: they
knew that matter is made of charged particles, but they could not say how matter is built
up. Similarly, we know today that consciousness is made from the signal propagation
and signal processing in the brain; we know that consciousness is an electrochemical
process. But we do not know yet the details of how the signals make up consciousness.
Unravelling the workings of this fascinating quantum system is the aim of neurology.
This is one of the great challenges of twenty-first century science.
Can you add a few arguments to the ones given here, showing that consciousness is a
physical process? Can you show in particular that not only the consciousness of others,
but also your own consciousness is a quantum process? Can you show, in addition, that
Challenge 32 s despite being a quantum process, coherence plays no essential role in consciousness?
In short, our consciousness is a consequence of the matter that makes us up. Con-
sciousness and pleasure depend on matter, its interactions and the quantum of action.

Why can we observe motion? – Again
Studying nature can be one of the most intense pleasures of life. All pleasures are based
on our ability to observe or detect motion. And our human condition is central to this
ability. In particular, in our adventure so far we found the following connections: We

experience motion
52 1 motion for enjoying life

— only because we are of finite size, and in particular, because we are large compared
to our quantum mechanical wavelength (so that we do not experience wave effects in
everyday life),
— only because we are large compared to a black hole of our same mass (so that we have
useful interactions with our environment),
— only because we are made of a large but finite number of atoms (to produce memory
and enable observations),
— only because we have a limited memory (so that we can clear it),
— only because we have a finite but moderate temperature (finite so that we have a life-
time, not zero so that we can be working machines),
— only because we are a mixture of liquids and solids (enabling us to move and thus to
experiment),
— only because we are approximately electrically neutral (thus avoiding that our sensors
get swamped),
— only because our brain forces us to approximate space and time by continuous entities

Motion Mountain – The Adventure of Physics
(otherwise we would not form these concepts),
— only because our brain cannot avoid describing nature as made of different parts (oth-
erwise we would not be able to talk or think),
— only because our ancestors reproduced,
— only because we are animals (and thus have a brain),
— only because life evolved here on Earth,
— only because we live in a relatively quiet region of our galaxy (which allowed evolu-
tion), and
— only because the human species evolved long after the big bang (when the conditions

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
were more friendly to life).
If any of these conditions – and many others – were not fulfilled, we would not observe
motion; we would have no fun studying physics. In fact, we can also say: if any of these
conditions were not fulfilled, motion would not exist. In many ways motion is thus an
Vol. I, page 15 illusion, as Zeno of Elea had claimed a long time ago. Of course, motion is a inevitable
illusion, one that is shared by many other animals and machines. To say the least, the
observation and the concept of motion is a result of the properties and limitations of the
human condition. A complete description of motion and nature must take this connec-
tion into account. Before we attempt that in the last volume of this adventure, we explore
a few additional details.

Curiosities and fun challenges ab ou t quantum experience
Most clocks used in everyday life, those built inside the human body and those made by
humans, are electromagnetic. Any clock on the wall, be it mechanical, quartz controlled,
radio or solar controlled, is based on electromagnetic effects. Do you know an exception?
Challenge 33 s

∗∗
The sense of smell is quite complex. For example, the substance that smells most badly to
humans is skatole, also called, with his other name, 3-methylindole. This is the molecule
to which the human nose is most sensitive. Skatole makes faeces smell bad; it is a result of
the physics of pleasure 53

haemoglobin entering the digestive tract through the bile. Skatole does not smell bad to
all animals; in contrast to humans, flies are attracted by its smell. Skatole is also produced
by some plants for this reason.
On the other hand, small levels of skatole do not smell bad to humans. Skatole is also
used by the food industry in small quantities to give smell and taste to vanilla ice cream
– though under the other name.
∗∗
It is worth noting that human senses detect energies of quite different magnitudes. The
eyes can detect light energies of about 1 aJ, whereas the sense of touch can detect only
Challenge 34 s energies as small as about 10 μJ. Is one of the two systems relativistic?
∗∗
The human construction plan is stored in the DNA. The DNA is structured into 20 000
genes, which make up about 2 % of the DNA, and 98 % non-coding DNA, once called

Motion Mountain – The Adventure of Physics
‘junk DNA’. Humans have as many genes as worms; plants have many more. Only
around 2010 it became definitely clear, through the international ‘Encode’ project, what
the additional 98 % of the DNA do: they switch genes on and off. They form the admin-
istration of the genes and work mostly by binding to specific proteins.
Research suggests that most genetic defects, and thus of genetic diseases, are not due
to errors in genes, but in errors of the control switches. All this is an ongoing research
field.
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Even at perfect darkness, the eye does not yield a black impression, but a slightly brighter
one, called eigengrau. This is a result of noise created inside the eye, probably triggered
by spontaneous decay of rhodopsin, or alternatively, by spontaneous release of neuro-
transmitters.
∗∗
The high sensitivity of the ear can be used to hear light. To do this, take an empty 750 ml
jam glass. Keeping its axis horizontal, blacken the upper half of the inside with a candle.
The lower half should remain transparent. After doing this, close the jam glass with its
lid, and drill a 2 to 3 mm hole into it. If you now hold the closed jam glass with the hole
to your ear, keeping the black side up, and shining into it from below with a 50 W light
Challenge 35 s bulb, something strange happens: you hear a 100 Hz sound. Why?
∗∗
Most senses work already before birth. It is well-known since many centuries that playing
the violin to a pregnant mother every day during the pregnancy has an interesting effect.
Even if nothing is told about it to the child, it will become a violin player later on. In fact,
most musicians are ‘made’ in this way.
∗∗
There is ample evidence that not using the senses is damaging. People have studied what
happens when in the first years of life the vestibular sense – the one used for motion
54 1 motion for enjoying life

detection and balance restoration – is not used enough. Lack of rocking is extremely hard
to compensate later in life. Equally dangerous is the lack of use of the sense of touch.
Babies, like all small mammals, that are generally and systematically deprived of these
Ref. 37 experiences tend to violent behaviour during the rest of their life.
∗∗
The importance of ion channels in the human body can not be overstressed. Ion chan-
nels malfunctions are responsible for many infections, for certain types of diabetes and
Ref. 38 for many effects of poisons. But above all, ion channels, and electricity in general, are
essential for life.
∗∗
Our body contains many systems that avoid unpleasant outcomes. For example, it
was discovered in 2006 that saliva contains a strong pain killer, much stronger than
morphine; it is now called opiorphin. It prevents that small bruises inside the mouth

Motion Mountain – The Adventure of Physics
disturb us too much. Opiorphine also acts as an antidepressant. Future research has to
show whether food addiction is related to this chemical.
∗∗
It is still unknown why people – and other mammals – yawn. This is still a topic of re-
search.
∗∗
Nature has invented the senses to increase pleasure and avoid pain. But neurologists have

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
found out that nature has gone even further; there is a dedicated pleasure system in the
brain, shown in Figure 16, whose function is to decide which experiences are pleasurable
and which not. The main parts of the pleasure system are the ventral tegmental area in
the midbrain and the nucleus accumbens in the forebrain. The two parts regulate each
other mainly through dopamine and GABA, two important neurotransmitters. Research
has shown that dopamine is produced whenever pleasure exceeds expectations. Nature
has thus developed a special signal for this situation.
In fact, well-being and pleasure are controlled by a large number of neurotransmitters
and by many additional regulation circuits. Researchers are trying to model the pleasure
system with hundreds of coupled differential equations, with the distant aim being to
understand addiction and depression, for example. On the other side, also simple mod-
els of the pleasure system are possible. One, shown in Figure 15, is the ‘neurochemical
Ref. 39 mobile’ model of the brain. In this model, well-being is achieved whenever the six most
important neurotransmitters* are in relative equilibrium. The different possible depar-
tures from equilibrium, at each joint of the mobile, can be used to describe depression,
schizophrenia, psychosis, the effect of nicotine or alcohol intake, alcohol dependency,
delirium, drug addiction, detoxication, epilepsy and more.
* Neurotransmitters come in many types. They can be grouped into mono- and diamines – such as dopam-
ine, serotonin, histamine, adrenaline – acetylcholine, amino acids – such as glycine, GABA and glutamate
– polypeptides – such as oxytocin, vasopressin, gastrin, the opiods, the neuropetides etc. – gases – such as
NO and CO – and a number of molecules that do not fit into the previous classes – such as anandamide or
NAAG.
the physics of pleasure 55

Ideal state:
Sympathetic brain Parasympathetic
or ergotrope or trophotrope
system system

noradrenaline acetylcholine

serotonin dopamine glutamate GABA
excitatory inhibitory

Psychosis/intoxication: brain

Motion Mountain – The Adventure of Physics
acetylcholine

noradrenaline glutamate

GABA

serotonin dopamine

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 15 The ‘neurochemical mobile’ model of well-being, with one of the way it can get out of
balance.

∗∗
The pleasure system in the brain is not only responsible for addiction. It is also respons-
Ref. 40 ible, as Helen Fisher showed through MRI brain scans, for romantic love. Romantic love,
directed to one single other person, is a state that is created in the ventral tegmental area
and in the nucleus accumbens. Romantic love is thus a part of the reptilian brain; in-
deed, romantic love is found in many animal species. Romantic love is a kind of positive
addiction, and works like cocaine. In short, in life, we can all choose between addiction
and love.
∗∗
An important aspect of life is death. When we die, conserved quantities like our energy,
momentum, angular momentum and several other quantum numbers are redistributed.
They are redistributed because conservation means that nothing is lost. What does all
Challenge 36 s this imply for what happens after death?
∗∗
56 1 motion for enjoying life

Motion Mountain – The Adventure of Physics
F I G U R E 16 The location of the ventral tegmental area (VTA) and of the nucleus accumbens (NAcc) in
the brain. The other regions involved in pleasure and addiction are Amy, the amygdala, Hip, the
hippocampus, Thal, the thalamus, rACC, the rostral anterior cingulate cortex, mPFC, the medial
prefrontal cortex, Ins, the insula, and LN, the lentiform nucleus (courtesy NIH).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
We all know the smell that appears in the open field at the start of a summer rainfall, or
the smell of fresh earth. It is due to a substance called geosmin, a bicyclic alcohol that is
produced by bacteria in the soil. The bacteria produce it when it rains. For reasons not
fully understood, the human nose is especially sensitive to the smell of geosmin: we are
able to smell it at concentrations below 10−10 .
∗∗
Also plants have sensors. Plants can sense light, touch, gravity, chemicals, as well as elec-
tric fields and currents. Many plants grow differently when touched; roots sense and con-
duct electric signals and then grow accordingly; obviously, plants grow against gravity
and towards light. Above all, plants are known to be able to distinguish many different
chemicals in the air, above all, ethene (ethylene), which is also an important plant hor-
mone.
∗∗
Many plants have built-in sensors and clocks that measure the length of the day. For ex-
ample, spinach does not grow in the tropics, because in order to flower, spinach must
sense for at least fourteen days in a row that the day is at least 14 hours long – and this
never happens in the tropics. This property of plants, called photoperiodism, was dis-
the physics of pleasure 57

covered in 1920 by Wightman Garner and Harry Allard while walking across tobacco
fields. They discovered – and proved experimentally – that tobacco plants and soya plants
only flower when the length of the day gets sufficiently short, thus around September.
Garner and Allard found that plants could be divided into species that flower when days
are short – such as chrysanthemum or coffee – others that flower when days are long
– such as carnation or clover – and still others that do not care about the length of the
day at all – such as roses or tomatoes. The measurement precision for the length of the
day is around 10 min. The day length sensor itself was discovered only much later; it is
located in the leaves of the plant, is called the phytochrome system and is based on spe-
cialised proteins. The proteins are able to measure the ratio between bright red and dark
red light and control the moment of flower opening in such plants.

Summary on biolo gy and pleasure
To ensure the successful reproduction of living beings, evolution has pursued miniatur-
ization as much as possible. Molecular motors, including molecular pumps, are the smal-

Motion Mountain – The Adventure of Physics
lest motors known so far; they work as quantum ratchets. Molecular motors are found
in huge numbers in every living cell. In short:

⊳ Every human consists of trillions of machines.

To increase pleasure and avoid pain, evolution has also supplied the human body also
with numerous sensors, sensor mechanisms, and a pleasure system deep inside the brain.
In short, nature has invented pleasure as a guide for behaviour. Neurologists have thus
proven what Epicurus said 23 centuries ago and Sigmund Freud repeated one century

⊳ Pleasure controls human life.*

All biological pleasure sensors and pleasure systems are based on quantum motion, in
particular on chemistry and materials science. We therefore explore both fields in the
following.

* But Epicurus also said: ‘It is impossible to live a pleasant life without living wisely and honourably and
justly, and it is impossible to live wisely and honourably and justly without living pleasantly.’ This is one of
his Principal Doctrines.
Chapter 2

C HA NG I NG T H E WOR L D W I T H
QUA N T UM E F F E C T S

T
he discovery of quantum effects has changed everyday life. It has allowed
he distribution of speech, music and films. The numerous possibilities of
elecommunications and of the internet, the progress in chemistry, materials
science, electronics and medicine would not have been possible without quantum ef-

Motion Mountain – The Adventure of Physics
fects. Many other improvements of our everyday life are due to quantum physics, and
many are still expected. In the following, we give a short overview of this vast field.

chemistry – from atoms to dna

“ ”
Bier macht dumm.**
Albert Einstein

It is an old truth that Schrödinger’s equation contains all of chemistry. With quantum

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
theory, for the first time people were able to calculate the strengths of chemical bonds,
and what is more important, the angle between them. Quantum theory thus explains
the shape of molecules and thus indirectly, the shape of all matter. In fact, the correct
statement is: the Dirac equation contains all of chemistry. The relativistic effects that
Ref. 41 distinguish the two equations are necessary, for example, to understand why gold is yel-
low and does not rust or why mercury is liquid.
To understand molecules and everyday matter, the first step is to understand atoms.
The early quantum theorists, lead by Niels Bohr, dedicated their life to understanding
their atoms and their detailed structure. The main result of their efforts is what you learn
in secondary school: in atoms with more than one electron, the various electron clouds
form spherical layers around the nucleus. The electron layers can be grouped into groups
of related clouds that are called shells. For electrons outside the last fully occupied shell,
the nucleus and the inner shells, the atomic core, can often be approximated as a single
charged entity.
Shells are numbered from the inside out. This principal quantum number, usually writ-
ten 𝑛, is deduced and related to the quantum number that identifies the states in the
hydrogen atom. The relation is shown in Figure 17.
Quantum theory shows that the first atomic shell has room for two electrons, the
second for 8, the third for 18, and the general n-th shell for 2𝑛2 electrons. The (neutral)

** ‘Beer makes stupid.’
chemistry – from atoms to dna 59

Continuum of
Energy ionized states
n=

8
n=3

n=2

n=1

Motion Mountain – The Adventure of Physics
nonrelativistic
(Bohr) levels
F I G U R E 17 The principal quantum numbers in hydrogen.

atom with one electron is hydrogen, the atom with two electrons is called helium. Every
chemical element has a specific number of electrons (and the same number of protons,
as we will see). A way to picture this connection is shown in Figure 18. It is called the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 42 periodic table of the elements. The standard way to show the table is found on page 345
and, more vividly, in Figure 19. (For a periodic table with a video about each element,
see www.periodicvideos.com.)
Experiments show that different atoms that share the same number of electrons in
their outermost shell show similar chemical behaviour. Chemists know that the chemical
behaviour of an element is decided by the ability of its atoms to from bonds. For example,
the elements with one electron in their outer s shell are the alkali metals lithium, sodium,
potassium, rubidium, caesium and francium; hydrogen, the exception, is conjectured to
be metallic at high pressures. The elements with filled outermost shells are the noble gases
helium, neon, argon, krypton, xenon, radon and ununoctium.

Atomic b onds
When two atoms approach each other, their electron clouds are deformed and mixed.
The reason for these changes is the combined influence of the two nuclei. These cloud
changes are highest for the outermost electrons: they form chemical bonds.
Bonds can be pictured, in the simplest approximation, as cloud overlaps that fill the
outermost shell of both atoms. These overlaps lead to a gain in energy. The energy gain is
the reason that fire is hot. In wood fire, chemical reactions between carbon and oxygen
atoms lead to a large release of energy. After the energy has been released, the atomic
bond produces a fixed distance between the atoms, as shown in Figure 20. This distance
is due to an energy minimum: a lower distance would lead to electrostatic repulsion
60 2 changing the world with quantum effects

n=1 1H 2 He

7N 8O
n=2 6C 3 Li 4 Be 9F
5B 10 Ne

25 Mn 26 Fe
24 Cr 15 P 16 S 27 Co
n=3 23 V 14 Si 11 Na 12 Mg 17 Cl 28 Ni
22 Ti 13 Al 18 Ar 29 Cu
21 Sc 30 Zn

64 Gd 65 Tb
n=4 63 Eu 43 Tc 44 Ru 66 Dy
62 Sm 42 Mo 33 As 34 Se 45 Rh 67 Ho
61 Pm 41 Nb 32 Ge 19 K 20 Ca 35 Br 46 Pd 68 Er
60 Nd 40 Zr 31 Ga 36 Kr 47 Ag 69 Tm
59 Pr 39 Y 48 Cd 70 Yb
58 Ce 71 Lu

Motion Mountain – The Adventure of Physics
96 Cm 97 Bk
n=5 95 Am 75 Re 76 Os 98 Cf
94 Pu 74 W 51 Sb 52 Te 77 Ir 99 Es
93 Np 73 Ta 50 Sn 37 Rb 38 Sr 53 I 78 Pt 100 Fm
92 U 72 Hf 49 In 54 Xe 79 Au 101 Md
91 Pa 57 La 80 Hg 102 No
90 Th 103 Lr

107 Bh 108 Hs
106 Sg 83 Bi 84 Po 109 Mt
n=6 105 Db 82 Pb 55 Cs 56 Ba 85 At 110 Ds
104 Rf 81 Tl 86 Rn 111 Rg

115 Mc 116 Lv
n=7 114 Fl 87 Fr 88 Ra 117 Ts
113 Nh 118 Og

n=8 119 Uun 120 Udn

s p d f
shell electron number
shell of last electron increases clockwise
F I G U R E 18 An unusual form of the periodic table of the elements.

between the atomic cores, a higher distance would increase the electron cloud energy.
Many atoms can bind to more than one neighbours. In this case, energy minimization
also leads to specific bond angles, as shown in Figure 21. Maybe you remember those
Ref. 43 funny pictures of school chemistry about orbitals and dangling bonds. Such dangling
bonds can now be measured and observed. Several groups were able to image them using
Ref. 44 scanning force or scanning tunnelling microscopes, as shown in Figure 22.
The repulsion between the clouds of each bond explains why angle values near that of
Challenge 37 e tetrahedral skeletons (2 arctan√2 = 109.47°) are so common in molecules. For example,
chemistry – from atoms to dna 61

Motion Mountain – The Adventure of Physics
F I G U R E 19 A modern periodic table of the elements (© Theodore Gray, for sale at www.theodoregray.
com).

potential energy [in eV≈100 kJ/mol]

4
electron

nucleus 2

-2

-4 nuclear
bond
bond length length separation [pm]
-6
0 100 200

F I G U R E 20 The forming of a chemical bond between two atoms, and the related energy minimum
(left hand image © chemistry4gcms2011.wikispaces.com).

the H–O–H angle in water molecules is 107°.
Atoms can also be connected by multiple bonds. Double bonds appear in carbon di-
oxide, or CO2 , which is therefore often written as O = C = O, triple bonds appear in
carbon monoxide, CO, which is often written as C ≡ O. Both double and triple bonds are
common in organic compounds. (In addition, the well-known hexagonal benzene ring
molecule 𝐶6 𝐻6 , like many other compounds, has a one-and-a-half-fold bond.) Higher
bonds are rare but do exist; quadruple bonds occur among transition metal atoms such
62 2 changing the world with quantum effects

F I G U R E 21 An artistic
illustration of chemical bond
angles when several atoms are
involved: in a water molecule,
with its charge distribution
due to its covalent bonds,
blue colour at the two ends
indicates positive charge, and
the red colour in upper vertex
indicates negative charge. The
central drawing shows a

Motion Mountain – The Adventure of Physics
typical structural rendering of
the water molecule
(© Benjah-bmm27).

as rhenium or tungsten. Research also confirmed that the uranium U2 molecule, among
Ref. 45 others, has a quintuple bond, and that the tungsten W2 molecule has a hextuple bond.

R ib onucleic acid and deox yrib onucleic acid

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Probably the most fascinating molecule of all is human deoxyribonucleic acid, better
known with its abbreviation DNA. The nucleic acids where discovered in 1869 by the
physician Friedrich Miescher (b. 1844 Basel, d. 1895 Davos) in white blood cells. He also
found it in cell nuclei, and thus called the substance ‘Nuklein’. In 1874 he published an
important study showing that the molecule is contained in spermatozoa, and discussed
the question if this substance could be related to heredity. With his work, Miescher paved
the way to a research field that earned many colleagues Nobel Prizes (though not for
himself, as he died before they were established). They changed the name to ‘nucleic
acid’.
DNA is, as shown in Figure 23, a polymer. A polymer is a molecule built of many sim-
ilar units. In fact, DNA is among the longest molecules known. Human DNA molecules,
for example, can be up to 5 cm in length. Inside each human cell there are 46 chromo-
somes. In other words, inside each human cell there are molecules with a total length of
2 m. The way nature keeps them without tangling up and knotting is a fascinating topic
in itself. All DNA molecules consist of a double helix of sugar derivates, to which four
nuclei acids are attached in irregular order. Nowadays, it is possible to make images of
Ref. 46 single DNA molecules; an example is shown in Figure 24.
At the start of the twentieth century it became clear that Desoxyribonukleinsäure
(DNS) – translated as deoxyribonucleic acid (DNA) into English – was precisely what
Erwin Schrödinger had predicted to exist in his book What Is Life? As central part of
the chromosomes contained the cell nuclei, DNA is responsible for the storage and re-
production of the information on the construction and functioning of Eukaryotes. The
chemistry – from atoms to dna 63

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 22 Top two rows: measured chemical bonds in the pentacene molecule, using different
techniques; bottom row: textbook calculations and illustrations of the same experiment (© IBM).

information is coded in the ordering of the four nucleic acids. DNA is the carrier of hered-
itary information. DNA determines in great part how the single cell we all once have been
grows into the complex human machine we are as adults. For example, DNA determines
the hair colour, predisposes for certain illnesses, determines the maximum size one can
grow to, and much more. Of all known molecules, human DNA is thus most intimately
related to human existence. The large size of the molecules is the reason that understand-
ing its full structure and its full contents is a task that will occupy scientists for several
generations to come.
To experience the wonders of DNA, have a look at the animations of DNA copying and
of other molecular processes at the unique website www.wehi.edu.au/education/wehitv.

Curiosities and fun challenges ab ou t chemistry
Among the fascinating topics of chemistry are the studies of substances that influence
humans: toxicology explores poisons, pharmacology explores medicines (pharmaceutical
drugs) and endocrinology explores hormones.
Over 50 000 poisons are known, starting with water (usually kills when drunk in
amounts larger than about 10 l) and table salt (can kill when 100 g are ingested) up to
64 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
F I G U R E 23 Several ways to
picture B-DNA, all in false
colours (© David Deerﬁeld).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
polonium 210 (kills in doses as low as 5 ng, far less than a spec of dust). Most countries
have publicly accessible poison databases; see for example www.gsbl.de.
Can you imagine why ‘toxicology’, the science of poisons, actually means ‘bow sci-
Challenge 38 e ence’ in Greek? In fact, not all poisons are chemical. Paraffin and oil for lamps, for ex-
ample, regularly kill children who taste it because some oil enters the lung and forms
a thin film over the alveoles, preventing oxygen intake. This so-called lipoid pneumonia
can be deadly even when only a single drop of oil is in the mouth and then inhaled by a
child. Paraffin should never be present in homes with children.
In the 1990s, the biologist Binie Ver Lipps discovered a substance, a simple poly-
peptide, that helps against venoms of snakes and other poisonous animals. The medical
industry worldwide refuses to sell the substance – it could save many lives – because it
is too cheap.
∗∗
Whether a substance is a poison depends on the animal ingesting it. Chocolate is poison
for dogs, but not for children. Poisonous mushrooms are edible for snails; bitten mush-
rooms are thus not a sign of edibility.
∗∗
Hormones are internal signalling substances produced by the human body. Chemically,
chemistry – from atoms to dna 65

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 24 Two ways to image single DNA molecules: by holography with electrons emitted from
atomically sharp tips (top) and by ﬂuorescence microscopy, with a commercial optical microscope
(bottom) (© Hans-Werner Fink/Wiley VCH).

they can be peptides, lipids or monoamines. Hormones induce mood swings, organize
the fight, flight or freeze responses, stimulate growth, start puberty, control cell death
and ageing, activate or inhibit the immune system, regulate the reproductive cycle and
activate thirst, hunger and sexual arousal.
∗∗
When one mixes 50 ml of distilled water and 50 ml of ethanol (alcohol), the volume of
Challenge 39 s the mixture has less than 100 ml. Why?
∗∗
Why do organic materials, i.e., materials that contain several carbon atoms, usually burn
at much lower temperature than inorganic materials, such as aluminium or magnesium?
Challenge 40 ny

∗∗
66 2 changing the world with quantum effects

A cube of sugar does not burn. However, if you put some cigarette ash on top of it, it
Challenge 41 ny burns. Why?
∗∗
Sugars are essential for life. One of the simplest sugars is glucose, also called dextrose or
grape sugar. Glucose is a so-called monosaccharide, in contrast to cane sugar, which is a
disaccharide, or starch, which is a polysaccharide.
The digestion of glucose and the burning, or combustion, of glucose follow the same
chemical reaction:

C6 H12 O6 + 6 O2 → 6 CO2 + 6 H2 O + 2808 kJ (8)

This is the simplest and main reaction that fuels the muscle and brain activities in our
body. The reaction is the reason we have to eat even if we do not grow in size any more.
The required oxygen, 𝑂2 , is the reason that we breathe in, and the resulting carbon di-

Motion Mountain – The Adventure of Physics
oxide, 𝐶𝑂2 , is the reason that we breathe out. Life, in contrast to fire, is thus able to
‘burn’ sugar at 37°. That is one of the great wonders of nature. Inside cells, the energy
gained from the digestion of sugars is converted into adenosinetriphosphate (ATP) and
then converted into motion of molecules.
∗∗
Chemical reactions can be slow but still dangerous. Spilling mercury on aluminium will
lead to an amalgam that reduces the strength of the aluminium part after some time. That
is the reason that bringing mercury thermometers on aeroplanes is strictly forbidden.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
Two atoms can form bound states by a number of effects that are weaker than electron
bonds. A famous one is the bound state formed by two sodium atoms at a distance of
Ref. 47 around 60 Bohr radii, thus much larger than usual bond distances. The bond appears
due to the continuous exchange of a photon between the two atoms.
∗∗
What happens if you take the white powder potassium iodide – KJ – and the white
Challenge 42 s powder lead nitrate – Pb(NO3 )2 – and mix them with a masher? (This needs to be done
with proper protection and supervision.)
∗∗
Writing on paper with a pen filled with lemon juice instead of ink produces invisible
writing. Later on, the secret writing can be made visible by carefully heating the paper
on top of a candle flame.
∗∗
How is the concentration of ozone, with the chemical composition O3 , maintained in
the high atmosphere? It took many years of research to show that the coolants used in
refrigerators, the so-called fluoro-chloro-hydrocarbons or FCHCs, slowly destroyed this
important layer. The reduction of ozone has increased the rate of skin cancer all across
materials science 67

the world. By forbidding the most dangerous refrigerator coolants all over the world, it
is hoped that the ozone concentration can recover. The first results are encouraging. In
1995, Paul Crutzen, Mario Molina and Sherwood Rowland received the Nobel Price for
Chemistry for the research that led to these results and policy changes.
∗∗
In 2008, it was shown that perispinal infusion of a single substance, etanercept, reduced
Alzheimer’s symptoms in a patient with late-onset Alzheimer’s disease, within a few
minutes. Curing Alzheimer’s disease is one of the great open challenges for modern
medicine. In 2013, Jens Pahnke found that an extract of Hypericum has positive effects
on the cognition and memory of Alzheimer patients. The extract is already available as
prescription-free medication, for other uses, with the name LAIF900.
∗∗
Cyanoacrylate is a fascinating substance. It is the main ingredient of instant glue, the glue

Motion Mountain – The Adventure of Physics
that starts to harden after a few seconds of exposure to moisture. The smell of evaporating
Ref. 48 cyanoacrylate glue is strong and known to everybody who has used this kind of adhesive.
The vapour also has another use: they make finger prints visible. You can try this at home!
Challenge 43 e

∗∗
Fireworks fascinate many. A great challenge of firework technology is to produce forest
green and greenish blue colours. Producers are still seeking to solve the problem. For
more information about fireworks, see the cc.oulu.fi/~kempmp website.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
materials science

“
Did you know that one cannot use a boiled egg

”
as a toothpick?
Karl Valentin

We mentioned several times that the quantum of action explains all properties of matter.
Many researchers in physics, chemistry, metallurgy, engineering, mathematics and bio-
logy have cooperated in the proof of this statement. In our mountain ascent we have only
a little time to explore this vast but fascinating topic. Let us walk through a selection.

Why d oes the flo or not fall?
We do not fall through the mountain we are walking on. Some interaction keeps us from
falling through. In turn, the continents keep the mountains from falling through them.
Also the liquid magma in the Earth’s interior keeps the continents from sinking. All
these statements can be summarized in two ideas: First, atoms do not penetrate each
other: despite being mostly empty clouds, atoms keep a distance. Secondly, atoms cannot
be compressed in size. Both properties are due to Pauli’s exclusion principle between
Vol. IV, page 136 electrons. The fermion character of electrons avoids that atoms shrink or interpenetrate
– on Earth.
68 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
F I G U R E 25 A comparison of star sizes (© Dave Jarvis).

In fact, not all floors keep up due to the fermion character of electrons. Atoms are

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
not impenetrable at all pressures. At sufficiently large pressures, atoms can collapse, and
form new types of floors. Such floors do not exist on Earth. Some people have spent their
whole life to understand why such other floors, namely surfaces of stars, do not fall, or
when they do, how it happens.
The floors and the sizes of all astronomic objects are due to quantum effects. Fig-
ure 25 illustrates the range of sizes that are found in astronomic objects. In each object,
a quantum effect leads to an internal pressure which fixes the floor, and thus the size of
the object.
In solid or liquid planets, the size is given by the incompressibility of condensed mat-
ter, which in turn is due to Pauli’s exclusion principle. The effective internal pressure of
condensed matter is often called the Pauli pressure. In gaseous planets, such as Jupiter,
and in usual stars, such as in the Sun, the gas pressure takes the role that the incompress-
ibility of solids and liquids has for smaller planets. The gas pressure is due to the heat
stored in them; the heat is usually released by internal nuclear reactions.
Light pressure does play a role in determining the size of red giants, such as Betelgeuse;
but for average stars, light pressure is negligible.
Other quantum effects appear in dense stars. Whenever light pressure, gas pressure
and the electronic Pauli pressure cannot keep atoms from interpenetrating, atoms are
compressed until all electrons are pushed into the protons. Protons then become neut-
rons, and the whole star has the same mass density of atomic nuclei, namely about
2.3 ⋅ 1017 kg/m3 . A drop weighs about 200 000 tons. In these so-called neutron stars, the
floor – or better, the size – is also determined by Pauli pressure; however, it is the Pauli
materials science 69

Page 186 pressure between neutrons, triggered by the nuclear interactions. These neutron stars are
all around 10 km in radius.
If the pressure increases still further, the star becomes a black hole, and never stops
Vol. II, page 262 collapsing. Black holes have no floor at all; they still have a constant size though, determ-
ined by the horizon curvature.
The question whether other star types exist in nature, with other floor forming mech-
anisms – such as the conjectured quark stars – is still a topic of research.

Ro cks and stones
If a geologist takes a stone in his hands, he is usually able to give the age of the stone,
within an error of a few per cent, simply by looking at it. The full story behind this as-
tonishing ability forms a large part of geology, but the general lines should also be known
to every physicist.
Generally speaking, the mass density of the Earth decreases from the centre towards
the surface. The upper mantle, below the solid crust of the Earth, is mostly composed

Motion Mountain – The Adventure of Physics
of peridotite, a dense, obviously igneous rock with a density of around 3.3 g/cm3 . The
oceanic crust, with a thickness between 5 and 10 km, is mainly composed of igneous rocks
such as basalt, diabase and gabbro. These rocks are somewhat less dense, around 3 g/cm3 ,
and are typically 200 million years old. The continental crust has a depth of 30 to 50 km,
consists of lighter rocks, around 2.7 g/cm3 , such as granite. The age of the continental
crust varies strongly; on average it is 2000 million years old, with a range from extremely
young rocks to some older than 4300 million years. The continental crust contains most
of the incompatible elements.
Every stone arrives in our hand through the rock cycle(s). The main rock cycle is a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
process that transforms magma from the interior of the Earth into igneous (or magmatic)
rocks through cooling and crystallization. Igneous rocks, such as basalt, can transform
through erosion, transport and deposition into sedimentary rocks, such as sandstone.
(Sedimentary rocks can also form from biogenic base materials.) Either of these two
rock types can be transformed through high pressures or temperatures into metamorphic
rocks, such as marble. Finally, most rocks are generally – but not always – transformed
back into magma.
The main rock cycle takes around 110 to 170 million years. For this reason, rocks that
are older than this age are less common on Earth. Any stone that we collect during a
walk is the product of erosion of one of the rock types. A geologist can usually tell, just
by looking at the stone, the type of rock it belongs to; if he sees or knows the original
environment, he can also give the age and often tell the story of the formation, without
any laboratory.
In the course of millions of years, minerals float upwards from the mantle or are
pushed down the crust, they are transformed under heat and pressure, they dissolve or
precipitate, and they get enriched in certain locations. These captivating stories about
minerals are explored in detail by geologists. Geologists can tell where to find beaches
with green sand (made of olivine); they can tell how contact between sedimentary lime-
stone with molten igneous rocks leads to marble, ruby and other gemstones, and under
Ref. 49 which precise conditions; they can also tell from small crystals of quartz that enclose
coesite that in earlier times the rock has been under extremely high pressure – either
70 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 26 Igneous rocks (top three rows): gabbro, andesite, permatite, basalt, pumice, porphyry,
obsidian, granite, tuff; sedimentary rocks (centre): clay, limestone, sandstone; and (below) two specimen
of a metamorphic rock: marble (© Siim Sepp at www.sandatlas.org, Wikimedia).
materials science 71

TA B L E 4 The types of rocks and stones.

Type Properties Subtype Example

Igneous rocks formed from volcanic or extrusive basalt (ocean floors,
(magmatites) magma, 95 % of all Giant’s Causeway),
rocks andesite, obsidian
plutonic or intrusive granite, gabbro
Sedimentary rocks often with fossils, a clastic shale, siltstone,
(sedimentites) few % sandstone
biogenic limestone, chalk,
dolostone
precipitate halite, gypsum
Metamorphic rocks transformed by foliated slate, schist, gneiss
(metamorphites) heat and pressure, a (Himalayas)

Motion Mountain – The Adventure of Physics
few %
non-foliated marble, skarn, quartzite
(grandoblastic or
hornfelsic)
Meteorites from the solar rock meteorites
system
iron meteorites

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
because it once was at a depth of the order of 70 km, or because of an asteroid impact, or
because of an atomic bomb explosion.
From the point of view of materials science, rocks are mixtures of minerals. Even
though more than 5000 minerals are known, only about 200 form rocks. These rock-
forming minerals can be grouped in a few general types. The main group are the silica-
based rocks. They contain SiO4 tetrahedra and form around 92 % of all rocks. The re-
maining 8 % of rocks are of different composition, such as carbonates or oxides. Table 5
gives more details. The table covers the minerals found on the Earth’s crust. However, the
most common mineral in absolute is Bridgmanite, a form of MgSiO3 . About one third of
the Earth is made of Bridgmanite, a silicate perovskite; it is formed in the lower mantle,
at temperatures of around 1800°C and pressures above 24 GPa. The mineral never ap-
pears on the Earth’s crust. Recent research suggests that some forms of Bridgmanite may
Ref. 50 even have contributed, when it rose to the surface by convection, to the oxygen in the
atmosphere.
From the point of view of chemistry, rocks are even more uniform. 99 % of all rocks
are made of only nine elements. Table 6 shows the details.
Almost all minerals are crystals. Crystals are solids with a regular arrangement of
atoms and are a fascinating topic by themselves.
72 2 changing the world with quantum effects

TA B L E 5 The mineralogic composition of rocks and stones in the Earth’s crust.

Group Mineral Vo l u m e
fraction
Inosilicates single chain silicates: 11(2) %
pyroxenes, e.g., diopside
double chain silicates: 5(1) %
amphiboles/hornblende,
e.g., tremolite
Phyllosilicates sheet silicates: 10 (2) %
clays, e.g., kaolinite, talc 5(1) %
mica-based minerals, e.g., 5(1) %
biotite, muscovite
Tectosilicates volume silicates: 65(5) %
quartz, tridymite, 11(1) %

Motion Mountain – The Adventure of Physics
cristobalite, coesite
the plagioclase feldspar 43(4) %
series, e.g., albite
the alkali feldspars, e.g., 14(2) %
orthoclase
Other silicates with isolated, double or 3(1) %
cyclic silica groups, e.g.,
olivine, beryl and garnets,
or amorphous silicates,
e.g., opal

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Oxide-based rocks e.g., magnetite, hematite, 5(1) %
bauxite
Carbonate-based e.g., calcite, dolomite
rocks
Sulfate-based rocks e.g., gypsum, anhydrite
Halide-based rocks e.g., rock salt or halite,
fluorite
Other rocks
phosphates, e.g., apatite 1(0.5) %
sulfides, e.g., pyrite
native metals, e.g., gold
borates, e.g.,
and many others.

Crystal formation
Have you ever admired a quartz crystal or some other crystalline material? The beautiful
shape and atomic arrangement has formed spontaneously, as a result of the motion of
atoms under high temperature and pressure, during the time that the material was deep
under the Earth’s surface. The details of crystal formation are complex and interesting.
materials science 73

TA B L E 6 The chemical composition of rocks and stones in the Earth’s crust.

Element Vo l u m e
fraction
Oxygen 46.7(1.0) %
Silicon 27.6(0.6) %
Aluminium 8.1(0.1) %
Iron 5(1) %
Calcium 4.3(0.7) %
Sodium 2.5(0.2) %
Potassium 2.0(0.5) %
Magnesium 2.5(0.4) %
Titanium 0.5(0.1) %
Other elements 0.8(0.8) %

Motion Mountain – The Adventure of Physics
Are regular crystal lattices energetically optimal? This simple question leads to a wealth
of problems. We might start with the much simpler question whether a regular dense
packing of spheres is the most dense packing possible. Its density is π/√18 , i.e., a bit
Challenge 44 s over 74 %. Even though this was conjectured to be the maximum possible value already
Ref. 51 in 1609 by Johannes Kepler, the statement was proven only in 1998 by Tom Hales. The
proof is difficult because in small volumes it is possible to pack spheres up to almost
78 %. To show that over large volumes the lower value is correct is a tricky business.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Next, does a regular crystal of solid spheres, in which the spheres do not touch, really
have the highest possible entropy? This simple problem has been the subject of research
only from the 1990s onwards. Interestingly, for low temperatures, regular sphere arrange-
ments indeed show the largest possible entropy. At low temperatures, spheres in a crystal
can oscillate around their average position and be thus more disordered than if they were
in a liquid; in the liquid state the spheres would block each other’s motion and would
not allow reaching the entropy values of a solid.
Many similar results deduced from the research into these so-called entropic forces
show that the transition from solid to liquid is – at least in part – simply a geometrical
effect. For the same reason, one gets the surprising result that even slightly repulsing
Ref. 52 spheres (or atoms) can form crystals and melt at higher temperatures. These are beautiful
examples of how classical thinking can explain certain material properties, using from
quantum theory only the particle model of matter.
But the energetic side of crystal formation provides other interesting questions.
Quantum theory shows that it is possible that two atoms repel each other, while three
attract each other. This beautiful effect was discovered and explained by Hans-Werner
Ref. 53 Fink in 1984. He studied rhenium atoms on tungsten surfaces and showed, as observed,
that they cannot form dimers – two atoms moving together – but readily form trimers.
This is an example contradicting classical physics; the effect is impossible if one pictures
atoms as immutable spheres, but becomes possible when one remembers that the elec-
tron clouds around the atoms rearrange depending on their environment.
For an exact study of crystal energy, the interactions between all atoms have to be
74 2 changing the world with quantum effects

F I G U R E 27 On tungsten tips, rhenium atoms, visible at the centre of the images, do not form dimers
(left) but do form trimers (right) (© Hans-Werner Fink/APS, from Ref. 53).

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 28 Some snow ﬂakes (© Furukawa Yoshinori).

included. The simplest question about crystal energy is to determine whether a regular
array of alternatively charged spheres has lower energy than some irregular collection.
Already such simple questions are still topic of research; the answer is still open.
The previous topics concerned bulk crystals. The next topic is the face formation in
crystals. Can you confirm that crystal faces are those planes with the slowest growth
Challenge 45 s speed, because all fast growing planes are eliminated? The finer details of the process
Ref. 54 form a complete research field in itself.
However, not always the slowest growing planes win out during crystal growth. Fig-
ure 28 shows some well-known exceptions: snow flakes. Explaining the shapes of snow
flakes is possible today. Furukawa Yoshinori is one of the experts in the field, heading a
Ref. 55 dedicated research team. These explanations also settle the question of symmetry: why
are crystals often symmetric, instead of asymmetric? This is a topic of self-organization,
Vol. I, page 415 as mentioned already in the section of classical physics. It turns out that the symmetry
is an automatic result of the way molecular systems grow under the combined influence
of diffusion and non-linear growth processes. But as usual, the details are still a topic of
research.
materials science 75

S ome interesting crystals
Every crystal, like every structure in nature, is the result of growth. Every crystal is thus
the result of motion. To form a crystal whose regularity is as high as possible and whose
shape is as symmetric as possible, the required motion is a slow growth of facets from the
liquid (or gaseous) basic ingredients. The growth requires a certain pressure, temperat-
ure and temperature gradient for a certain time. For the most impressive crystals, the
gemstones, the conditions are usually quite extreme; this is the reason for their durabil-
ity. The conditions are realized in specific rocks deep inside the Earth, where the growth
process can take thousands of years. Mineral crystals can form in all three types of rocks:
Page 69 igneous (magmatic), metamorphic, and sedimentary. Other crystals can be made in the
laboratory in minutes, hours or days and have led to a dedicated industry. Only a few
crystals grow from liquids at standard conditions; examples are gypsum and several other
sulfates, which can be crystallized at home, potassium bitartrate, which appears in the
making of wine, and the crystals grown inside plants or animals, such as teeth, bones or
magnetosensitive crystallites.

Motion Mountain – The Adventure of Physics
Growing, cutting, treating and polishing crystals is an important industry. Especially
the growth of crystals is a science in itself. Can you show with pencil and paper that
Challenge 46 e only the slowest growing facets are found in crystals? In the following, a few important
crystals are presented.

F I G U R E 29 Quartz found F I G U R E 30 Citrine F I G U R E 31 Amethystine and orange copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
at St. Gotthard, Switzerland, found on quartz found in the Orange River,
picture size 12 cm (© Rob Magaliesberg, Namibia, picture size 6 cm (© Rob
Lavinsky). South Africa, Lavinsky).
crystal height 9 cm
(© Rob Lavinsky).

∗∗

Quartz, amethyst (whose colour is due to radiation and iron Fe4+ impurities), citrine
(whose colour is due to Fe3+ impurities), smoky quartz (with colour centres induced by
radioactivity), agate and onyx are all forms of crystalline silicon dioxide or SiO2 . Quartz
forms in igneous and in magmatic rocks; crystals are also found in many sedimentary
rocks. Quartz crystals can sometimes be larger than humans. By the way, most amethysts
76 2 changing the world with quantum effects

lose their colour with time, so do not waste money buying them.
Quartz is the most common crystal on Earth’s crust and is also grown synthetically
for many high-purity applications. The structure is rombohedral, and the ideal shape is a
six-sided prism with six-sided pyramids at its ends. Quartz melts at 1986 K and is piezo-
and pyroelectric. Its piezoelectricity makes it useful as electric oscillator and filter. A film
Vol. I, page 291 of an oscillating clock quartz is found in the first volume. Quartz is also used for glass
production, in communication fibres, for coating of polymers, in gas lighters, as source
of silicon and for many other applications.

Motion Mountain – The Adventure of Physics
F I G U R E 32 Corundum found in F I G U R E 33 Ruby F I G U R E 34
Laacher See, Germany, picture size found in Jagdalak, Sapphire
4 mm (© Stephan Wolfsried). Afghanistan, picture found in
height 2 cm (© Rob Ratnapura, Sri

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Lavinsky). Lanka, size
1.6 cm (© Rob
Lavinsky).

∗∗
Corundum, ruby and sapphire are crystalline variations of alumina, or Al2 O3 . Corundum
is pure and colourless crystalline alumina, ruby is Cr doped and blue sapphire is Ti or Fe
doped. They have trigonal crystal structure and melt at 2320 K. Natural gems are formed
in metamorphic rocks. Yellow, green, purple, pink, brown, grey and salmon-coloured
sapphires also exist, when doped with other impurities. The colours of natural sapphires,
like that of many other gemstones, are often changed by baking and other treatments.
Corundum, ruby and sapphire are used in jewellery, as heat sink and growth substrate,
and for lasers. Corundum is the second-hardest material known, just after diamond, and
is therefore used as scratch-resistant ‘glass’ in watches and, since a short time, in mobile
phones. Ruby was the first gemstone that was grown synthetically in gem quality, in 1892
by Auguste Verneuil (1856-1913), who made his fortune in this way. Modern synthetic
single crystals of corundum can weigh 30 kg and more. Also alumina ceramics, which
can be white or even transparent, are important in industrial and medical systems.
∗∗
Tourmaline is a frequently found mineral and can be red, green, blue, orange, yellow,
materials science 77

F I G U R E 35 Left: raw crystals, or boules, of synthetic corundum, picture size c. 50 cm. Right: a modern,
115 kg corundum single crystal, size c. 50 cm (© Morion Company, GT Advanced).

Motion Mountain – The Adventure of Physics
pink or black, depending on its composition. The chemical formula is astonishingly com-
plex and varies from type to type. Tourmaline has trigonal structure and usually forms
columnar crystals that have triangular cross-section. It is only used in jewellery. Paraiba
tourmalines, a very rare type of green or blue tourmaline, are among the most beautiful
gemstones and can be, if natural and untreated, more expensive than diamonds.

F I G U R E 36 Natural F I G U R E 37 Cut Paraiba tourmaline from
bicoloured Brazil, picture size 3 cm (© Manfred Fuchs).
tourmaline found in
Paprok, Afghanistan,
picture size 9 cm
(© Rob Lavinsky).

∗∗
Garnets are a family of compounds of the type X2 Y3 (SiO4 )3 . They have cubic crystal
structure. They can have any colour, depending on composition. They show no cleavage
and their common shape is a rhombic dodecahedron. Some rare garnets differ in col-
78 2 changing the world with quantum effects

our when looked at in daylight or in incandescent light. Natural garnets form in meta-
morphic rocks and are used in jewellery, as abrasive and for water filtration. Synthetic
garnets are used in many important laser types.

F I G U R E 38 Red garnet with F I G U R E 39 Green F I G U R E 40 Synthetic Cr,Tm,Ho:YAG, a
smoky quartz found in demantoid, a garnet doped yttrium aluminium garnet, picture
Lechang, China, picture size owing its colour to size 25 cm (© Northrop Grumman).
9 cm (© Rob Lavinsky). chromium doping,
found in Tubussis,

Motion Mountain – The Adventure of Physics
Namibia, picture size
5 cm (© Rob Lavinsky).

∗∗
Alexandrite, a chromium-doped variety of chrysoberyl, is used in jewellery and in lasers.
Its composition is BeAl2 O4 ; the crystal structure is orthorhombic. Chrysoberyl melts at
2140 K. Alexandrite is famous for its colour-changing property: it is green in daylight or
fluorescent light but amethystine in incandescent light, as shown in Figure 41. The effect
is due to its chromium content: the ligand field is just between that of chromium in red

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ruby and that in green emerald. A few other gems also show this effect, in particular the
rare blue garnet and some Paraiba tourmalines.

F I G U R E 41 Alexandrite found in the Setubal river, F I G U R E 42 Synthetic alexandrite, picture size
Brazil, crystal height 1.4 cm, illuminated with 20 cm (© Northrop Grumman).
daylight (left) and with incandescent light (right)
(© Trinity Mineral).

∗∗
Perovskites are a large class of cubic crystals used in jewellery and in tunable lasers. Their
general composition is XYO3 , XYF3 or XYCl3 .
∗∗
materials science 79

F I G U R E 43 Perovskite found in F I G U R E 44 Synthetic PZT, or lead
Hillesheim, Germany. Picture width zirconium titanate, is a perovskite used in
3 mm (© Stephan Wolfsried). numerous products. Picture width 20 cm

Motion Mountain – The Adventure of Physics
(© Ceramtec).

Diamond is a metastable variety of graphite, thus pure carbon. Theory says that graph-
ite is the stable form; practice says that diamond is still more expensive. In contrast to
graphite, diamond has face-centred cubic structure, is a large band gap semiconductor
and typically has octahedral shape. Diamond burns at 1070 K; in the absence of oxygen
it converts to graphite at around 1950 K. Diamond can be formed in magmatic and in
metamorphic rocks. Diamonds can be synthesized in reasonable quality, though gem-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
stones of large size and highest quality are not yet possible. Diamond can be coloured
and be doped to achieve electrical conductivity in a variety of ways. Diamond is mainly
used in jewellery, for hardness measurements and as abrasive.
∗∗
Silicon, Si, is not found in nature in pure form; all crystals are synthetic. The structure
is face-centred cubic, thus diamond-like. It is moderately brittle, and can be cut in thin
wafers which can be further thinned by grinding or chemical etching, even down to
a thickness of 10 μm. Being a semiconductor, the band structure determines its black
colour, its metallic shine and its brittleness. Silicon is widely used for silicon chips and
electronic semiconductors. Today, human-sized silicon crystals can be grown free of dis-
locations and other line defects. (They will still contain some point defects.)
∗∗
Teeth are the structures that allowed animals to be so successful in populating the Earth.
They are composed of several materials; the outer layer, the enamel, is 97 % hydroxylapat-
ite, mixed with a small percentage of two proteins groups, the amelogenins and the
enamelins. The growth of teeth is still not fully understood; neither the molecular level
nor the shape-forming mechanisms are completely clarified. Hydroxylapatite is soluble
in acids; addition of fluorine ions changes the hydroxylapatite to fluorapatite and greatly
reduces the solubility. This is the reason for the use of fluorine in tooth paste.
Hydroxylapatite (or hydroxyapatite) has the chemical formula Ca10 (PO4 )6 (OH)2 , pos-
80 2 changing the world with quantum effects

F I G U R E 45 Natural diamond from F I G U R E 46 Synthetic diamond,
Saha republic, Russia, picture size 4 cm picture size 20 cm (© Diamond
(© Rob Lavinsky). Materials GmbH).

Motion Mountain – The Adventure of Physics
F I G U R E 47 Ophthalmic diamond

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
knife, picture size 1 cm (© Diamatrix
Ltd.).

sesses hexagonal crystal structure, is hard (more than steel) but relatively brittle. It occurs
as mineral in sedimentary rocks (see Figure 49), in bones, renal stones, bladders tones,
bile stones, atheromatic plaque, cartilage arthritis and teeth. Hydroxylapatite is mined
as a phosphorus ore for the chemical industry, is used in genetics to separate single and
double-stranded DNA, and is used to coat implants in bones.
∗∗
Pure metals, such as gold, silver and even copper, are found in nature, usually in magmatic
rocks. But only a few metallic compounds form crystals, such as pyrite. Monocrystal-
line pure metal crystals are all synthetic. Monocrystalline metals, for example iron, alu-
minium, gold or copper, are extremely soft and ductile. Either bending them repeatedly
– a process called cold working – or adding impurities, or forming alloys makes them
hard and strong. Stainless steel, a carbon-rich iron alloy, is an example that uses all three
processes.
∗∗
In 2009, Luca Bindi of the Museum of Natural History in Florence, Italy, made headlines
materials science 81

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 48 A silicon crystal growing machine and two resulting crystals, with a length of c. 2 m
(© www.pvatepla.com).

Ref. 56 across the world with his discovery of the first natural quasicrystal. Quasicrystals are ma-
terials that show non-crystallographic symmetries. Until 2009, only synthetic materials
were known. Then, in 2009, after years of searching, Bindi discovered a specimen in his
collection whose grains clearly show fivefold symmetry.
∗∗
There are about 4000 known mineral types. On the other hand, there are ten times as
many obsolete mineral names, namely around 40 000. An official list can be found in
82 2 changing the world with quantum effects

F I G U R E 49 Hydroxylapatite found in F I G U R E 50 The main and the reserve
Oxsoykollen, Snarum, Norway, length teeth on the jaw bone of a shark, all
65 mm (© Aksel Österlöf). covered in hydroxylapatite, picture size
15 cm (© Peter Doe).

Motion Mountain – The Adventure of Physics
F I G U R E 51 Pyrite, found in F I G U R E 52 Silver from F I G U R E 53 Synthetic copper
Navajún, Spain, picture width Colquechaca, Bolivia, picture single crystal, picture width
5.7 cm (© Rob Lavinsky). width 2.5 cm (© Rob Lavinsky). 30 cm (© Lachlan Cranswick).

F I G U R E 54 The specimen, found in the Koryak Mountains in Russia, is part of a triassic mineral, about
220 million years old; the black material is mostly khatyrkite (CuAl2 ) and cupalite (CuAl2 ) but also
contains quasicrystal grains with composition Al63 Cu24 Fe13 that have ﬁvefold symmetry, as clearly
shown in the X-ray diffraction pattern and in the transmission electron image. (© Luca Bindi).

various places on the internet, including www.mindat.org or www.mineralienatlas.de.
To explore the world of crystal shapes, see the www.smorf.nl website. Around 40 new
minerals are discovered each year. Searching for minerals and collecting them is a fas-
cinating pastime.
materials science 83

How can we lo ok through mat ter?
The quantum of action tells us that all obstacles have only finite potential heights. The
Vol. IV, page 28 quantum of action implies that matter is penetrable. That leads to a question: Is it possible
to look through solid matter? For example, can we see what is hidden inside a mountain?
To achieve this, we need a signal which fulfils two conditions: the signal must be able to
penetrate the mountain, and it must be scattered in a material-dependent way. Indeed,
such signals exist, and various techniques use them. Table 7 gives an overview of the
possibilities.

TA B L E 7 Signals penetrating mountains and other matter.

Signal Penet- Achie- Ma- Use
r at i o n ved terial
depth resolu - de-
in stone tion pend-

Motion Mountain – The Adventure of Physics
ence
Fluid signals
Diffusion of gases, c. 5 km c. 100 m medium exploring vacuum systems and
such as helium tube systems
Diffusion of water c. 5 km c. 100 m medium mapping hydrosystems
or liquid chemicals
Sound signals
Infrasound and 100 000 km 100 km high mapping of Earth crust and
earthquakes mantle

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Sound, explosions, 0.1 − 10 m c. 𝜆/100 high oil and ore search, structure
short seismic waves mapping in rocks, searching for
underwater treasures in sunken
ship with sub-bottom-profilers
Ultrasound 1 mm high medical imaging, acoustic
microscopy, sonar, echo systems
Acousto-optic or 1 mm medium blood stream imaging, mouse
photoacoustic imaging
imaging
Electromagnetic signals
Static magnetic field medium cable search, cable fault
variations localization, search for structures
and metal inside soil, rocks and
the seabed
Electrical currents soil and rock investigations,
search for tooth decay
Electromagnetic soil and rock investigations in
sounding, deep water and on land
0.2 − 5 Hz
Radio waves 10 m 30 m to 1 mm small soil radar (up to 10 MW),
magnetic resonance imaging,
research into solar interior
84 2 changing the world with quantum effects

TA B L E 7 (Continued) Signals penetrating mountains and other matter.

Signal Penet- Achie- Ma- Use
r at i o n ved terial
depth resolu - de-
in stone tion pend-
ence
Ultra-wide band 10 cm 1 mm sufficient searching for wires and tubes in
radio walls, breast cancer detection
THz and mm waves below 1 mm 1 mm see through clothes, envelopes
and teeth Ref. 57
Infrared c. 1 m 0.1 m medium mapping of soil over 100 m
Visible light c. 1 cm 0.1 μm medium imaging of many sorts, including
breast tumour screening
X-rays a few metres 5 μm high medicine, material analysis,

Motion Mountain – The Adventure of Physics
airports, food production check
γ-rays a few metres 1 mm high medicine
Matter particle signals
Neutrons from a up to c. 1 m 1 mm medium tomography of metal structures,
reactor e.g., archaeologic statues or car
engines
Muons created by up to 0.1 m small finding caves in pyramids,
cosmic radiation or c. 300 m imaging interior of trucks
technical sources

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Positrons up to c. 1 m 2 mm high brain tomography
Electrons up to c. 1 μm 10 nm small transmission electron
microscopes
Neutrino beams light years none very weak studies of Sun
Radioactivity 1 mm to 1 m airport security checks
mapping
Gravitation
Variation of 𝑔 50 m low oil and ore search

We see that many signals are able to penetrate a mountain, and even more signals
are able to penetrate other condensed matter. To distinguish different materials, or to
distinguish solids from liquids and from air, sound and radio waves are the best choice.
In addition, any useful method requires a large number of signal sources and of signal
receptors, and thus a large amount of money. Will there ever be a simple method that
allows looking into mountains as precisely as X-rays allow looking into human bodies?
For example, is it possible to map the interior of the pyramids? A motion expert should
Challenge 47 s be able to give a definite answer.
One of the high points of twentieth century physics was the development of the best
method so far to look into matter with dimensions of about a metre or less: magnetic
Page 162 resonance imaging. We will discuss it later on.
Various modern imaging techniques, such as X-rays, ultrasound imaging and several
materials science 85

Motion Mountain – The Adventure of Physics
F I G U R E 55 A switchable Mg-Gd mirror (© Ronald Griessen).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Vol. I, page 313 future ones, are useful in medicine. As mentioned before, the use of ultrasound imaging
Vol. I, page 313 for prenatal diagnostics of embryos is not recommended. Studies have found that ultra-
sound produces extremely high levels of audible sound to the embryo, especially when
the ultrasound is repeatedly switched on and off, and that babies react negatively to this
loud noise.
Looking into the ground is important for another reason. It can help in locating land
mines. Detecting land mines, especially metal-free mines, buried in the ground is a big
technological challenge that is still unsolved. Many technologies have been tested: X-
ray backscatter devices working at 350 to 450 keV, ground-penetrating radar and ultra-
wideband radar, infrared detection, thermal or fast neutron bombardment and analysis,
acoustic and sonar detection, electric impedance tomography, radio-frequency bom-
bardment, nuclear quadrupole resonance, millimetre waves, visual detection, ion mo-
bility spectrometers, using dogs, using rats, and explosive vapour detection with dedic-
ated sensors. (And of course, for metallic mines, magnetometers and metal detectors are
used.) But so far, the is still no solution in sight. Can you find one? If you do, get in touch
with www.gichd.org.

What is necessary to make mat ter invisible?
You might have already imagined what adventures would be possible if you could be
invisible for a while. In 1996, a team of Dutch scientists found a material, yttrium hydride
or YH3 , that can be switched from mirror mode to transparent mode using an electrical
86 2 changing the world with quantum effects

Ref. 58 signal. A number of other materials were also discovered. An example of the effect for
Mg-Gd layers is shown in Figure 55.
Switchable mirrors might seem a first step to realize the dream to become invisible
and visible at will. In 2006, and repeatedly since then, researchers made the headlines in
the popular press by claiming that they could build a cloak of invisibility. This is a blatant
lie. This lie is frequently used to get funding from gullible people, such as buyers of bad
science fiction books or the military. For example, it is often claimed that objects can be
made invisible by covering them with metamaterials. The impossibility of this aim has
Vol. III, page 169 been already shown earlier on. But we now can say more.
Nature shows us how to realize invisibility. An object is invisible if it has no surface, no
absorption and small size. In short, invisible objects are either small clouds or composed
of them. Most atoms and molecules are examples. Homogeneous non-absorbing gases
also realize these conditions. That is the reason that air is (usually) invisible. When air is
not homogeneous, it can be visible, e.g. above hot surfaces.
In contrast to gases, solids or liquids do have surfaces. Surfaces are usually visible,

Motion Mountain – The Adventure of Physics
even if the body is transparent, because the refractive index changes there. For example,
quartz can be made so transparent that one can look through 1 000 km of it; pure quartz
is thus more transparent than usual air. Still, objects made of pure quartz are visible to
the eye, due to their refractive index. Quartz can be invisible only when submerged in
liquids with the same refractive index.
In short, anything that has a shape cannot be invisible. If we want to become invisible,
we must transform ourselves into a diffuse gas cloud of non-absorbing atoms. On the way
to become invisible, we would lose all memory and all genes, in short, we would lose all
our individuality, because an individual cannot be made of gas. An individual is defined

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
through its boundary. There is no way that we can be invisible and alive at the same time;
a way to switch back to visibility is even less likely. In summary, quantum theory shows
that only the dead can be invisible. Quantum theory has a reassuring side: we already
Vol. IV, page 136 found that quantum theory forbids ghosts; we now find that it also forbids any invisible
beings.

What moves inside mat ter?
All matter properties are due to the motion of the components of matter. Therefore, we
can correctly argue that understanding the motion of electrons and nuclei implies under-
standing all properties of matter. Sometimes, however, it is more practical to explore the
motion of collections of electrons or nuclei as a whole. Here is a selection of such col-
lective motions. Collective motions that appear to behave like single particles are called
quasiparticles.
In crystalline solids, sound waves can be described as the motion of phonons. For ex-
ample, transverse phonons and longitudinal phonons describe many processes in semi-
conductors, in solid state lasers and in ultrasound systems. Phonons approximately be-
have as bosons.
Page 298 In metals, the motion of crystal defects, the so-called dislocations and disclinations, is
central to understand their hardening and their breaking.
Also in metals, the charge waves of the conductive electron plasma, can be seen as
composed of so-called plasmons. Plasmons are also important in the behaviour of high-
materials science 87

speed electronics.
In magnetic materials, the motion of spin orientation is often best described with
the help of magnons. Understanding the motion of magnons and that of magnetic do-
main walls is useful to understand the magnetic properties of magnetic material, e.g.,
in permanent magnets, magnetic storage devices, or electric motors. Magnons behave
approximately like bosons.
In semiconductors and insulators, the motion of conduction electrons and electron
holes, is central for the description and design of most electronic devices. They behave
as fermions with spin 1/2, elementary electric charge, and a mass that depends on the
material, on the specific conduction band and on the specific direction of motion. The
bound system of a conduction electron and a hole is called an exciton. It can have spin 0
or spin 1.
In polar materials, the motion of light through the material is often best described
in terms of polaritons, i.e., the coupled motion of photons and dipole carrying material
excitations. Polaritons are approximate bosons.

Motion Mountain – The Adventure of Physics
In dielectric crystals, such as in many inorganic ionic crystals, the motion of an elec-
tron is often best described in terms of polarons, the coupled motion of the electron with
the coupled polarization region that surrounds it. Polarons are fermions.
In fluids, the motion of vortices is central in understanding turbulence or air hoses.
Especially in superfluids, vortex motion is quantized in terms of rotons which determ-
ines flow properties. Also in fluids, bubble motion is often useful to describe mixing
processes.
In superconductors, not only the motion of Cooper pairs, but also the motion of mag-
netic flux tubes determines the temperature behaviour. Especially in thin and flat super-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
conductors – so-called ‘two-dimensional’ systems – such tubes have particle-like prop-
erties.
In all condensed matter systems, the motion of surface states – such as surface
magnons, surface phonons, surface plasmons, surface vortices – has also to be taken
into account.
Many other, more exotic quasiparticles exist in matter. Each quasiparticle in itself is an
important research field where quantum physics and material science come together. To
clarify the concepts, we mention that a soliton is not, in general, a quasiparticle. ‘Soliton’
is a mathematical concept; it applies to macroscopic waves with only one crest that re-
Vol. I, page 316 main unaltered after collisions. Many domain walls can be seen as solitons. But quasi-
particles are concepts that describe physical observations similar to quantum particles.
In summary, all the mentioned examples of collective motion inside matter, both mac-
roscopic and quantized, are of importance in electronics, photonics, engineering and
medical applications. Many are quantized and their motion can be studied like the mo-
tion of real quantum particles.

Curiosities and fun challenges ab ou t materials science
What is the maximum height of a mountain? This question is of course of interest to all
Ref. 59 climbers. Many effects limit the height. The most important is that under heavy pressure,
solids become liquid. For example, on Earth this happens for a mountain with a height
Challenge 48 ny of about 27 km. This is quite a bit more than the highest mountain known, which is
88 2 changing the world with quantum effects

the volcano Mauna Kea in Hawaii, whose top is about 9.45 km above the base. On Mars
gravity is weaker, so that mountains can be higher. Indeed the highest mountain on Mars,
Olympus mons, is 80 km high. Can you find a few other effects limiting mountain height?
Challenge 49 s

∗∗
Do you want to become rich? Just invent something that can be produced in the factory, is
cheap and can fully substitute duck feathers in bed covers, sleeping bags or in badminton
Challenge 50 r shuttlecocks. Another industrial challenge is to find an artificial substitute for latex, and a
third one is to find a substitute for a material that is rapidly disappearing due to pollution:
cork.
∗∗
Materials differ in density, in elasticity, in strength, stiffness, toughness, melting temper-
ature, heat insulation, electric resistivity, and many other parameters. To get an overview,

Motion Mountain – The Adventure of Physics
the so-called Ashby charts are most useful, of which Figure 56 shows an example. The race
to find materials that are lighter and stiffer than wood, in particular balsa wood, is still
ongoing.
∗∗
How much does the Eiffel tower change in height over a year due to thermal expansion
Challenge 51 s and contraction?
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
What is the difference between the makers of bronze age knifes and the builders of the
Eiffel tower? Only their control of defect distributions. The main defects in metals are
disclinations and dislocations. Disclinations are crystal defects in form of surfaces; they
are the microscopic aspect of grain boundaries. Dislocations are crystal defects in form
of curved lines; above all, their distribution and their motion in a metal determines the
Page 298 stiffness. For a picture of dislocations, see below.
∗∗
Challenge 52 e What is the difference between solids, liquids and gases?
∗∗
One subject of materials science is the way a solid object breaks. The main distinction
is between brittle fraction and ductile fraction. In brittle fraction, as in a breaking of
a glass pane, the resulting edges are sharp and irregular; in ductile fraction, as occurs
in hot glass, the edges are rounded and regular. The two fraction types also differ in
their mechanisms, i.e., in the motion of the involved defects and atoms. This difference
is important: when a car accident occurs at night, looking at the shapes of the fraction
surface of the tungsten wire inside the car lamps with a microscope, it is easy to decide
whether the car lamps were on or off at the time of the accident.
∗∗
Material science can also help to make erased information visible again. Many laborat-
materials science 89

1000
Diamond Engineering WC-Co
Modulus - Density B SIC Sl 2N 4 ceramics

Youngs modulus E Be Aluminas
Mo W-Alloys
(G = 3E/8 ; K = E) Sialons ZrO 2 Alloys
Si BeO Ni Alloys
MFA : 88-91 CFRP Glasses Steels
UNI-P LY 3e Cu Alloys
Pottery Ti Alloys
100 Zn Alloys
KFRP Al Alloys
GFRP
CFRP Rock, stone Tin Alloys
12
Engineering Laminates Cement, concrete
) )
E
ρ (m/s) composites GFRP Lead alloys
Young’s Modulus E (GPa)

KFRP
Mg
104 Ash
Oak Alloys
Porous
Pine ceramics
Fir
10 Parallel
Engineering
to grain MEL alloys
PC
Balsa Epoxies
Wood PS
products
PMMA
PVC

Motion Mountain – The Adventure of Physics
Woods Nylon
Engineering
3
3x10 Ash PP Polyesters polymers
Oak
1.0 Pine
Fir
HDPE
Lower E limit Perpendicular
for true solids to grain Guide lines for
PTFE minimum weight
Spruce
LDPE E
Balsa ρ =C design
Plasticised
PVC
103
0.1
Hard Elastomers

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
BUTYL PU
2 1
3x10 E 2
ρ =C
Cork Silicone
Soft 1
Polymer foams E 3
BUTYL
ρ =C
0.01
0.1 0.3 1.0 3 10 30
Density ρ (Mg /m 3)

F I G U R E 56 An overview of the elastic modulus and the density of materials. For structures that need to
be light and stiff, a high ratio E / 𝜌3 is required; the graph shows that wood is well optimized for this
task. (© Carol Livermore/Michael Ashby).

ories are now able to recover data from erased magnetic hard disks. Other laboratories
can make erased serial numbers in cars bodies visible again, either by heating the metal
part or by using magnetic microscopy.
∗∗
Challenge 53 s Quantum theory shows that tight walls do not exist. Every material is penetrable. Why?
∗∗
Quantum theory shows that even if tight walls would exist, the lid of a box made of such
90 2 changing the world with quantum effects

F I G U R E 57 Insects and geckos stick to glass and other surfaces using the van der Waals force at the
ends of a high number of spatulae (© Max Planck Gesellschaft).

Motion Mountain – The Adventure of Physics
Challenge 54 s walls can never be tightly shut. Can you provide the argument?
∗∗
In 1936, Henry Eyring proposed that the shear viscosity of a liquid 𝜂 obeys

𝜂 ⩾ 𝜌ℏ , (9)

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 55 ny where 𝜌 is the density of the fluid. Is the lower limit valid?
∗∗
Heat can flow. Like for all flows, quantum theory predicts that heat transport quantized.
This implied that thermal conductance is quantized. And indeed, in the year 2000, ex-
Ref. 60 periments have confirmed the prediction. Can you guess the smallest unit of thermal
Challenge 56 s conductance?
∗∗
Ref. 61 Robert Full has shown that van der Waals forces are responsible for the way that geckos
Vol. I, page 91 walk on walls and ceilings. (A picture is found in Figure 56.) The gecko, a small reptile
with a mass of about 100 g, uses an elaborate structure on its feet to perform the trick.
Each foot has 500 000 hairs or setae, each split in up to 1000 small spatulae, and each
spatula uses the van der Waals force (or alternatively, capillary forces) to stick to the
surface. As a result of these 500 million sticking points, the gecko can walk on vertical
glass walls or even on glass ceilings; the sticking force can be as high as 100 N per foot.
The adhesion forces are so high that detaching the foot requires a special technique. The
internet has slow-motion videos showing how geckos perform the feat, in each step they
take.
Hairy feet as adhesion method are also used by jumping spiders (Salticidae). For ex-
ample, Evarcha arcuata have hairs at their feet which are covered by hundred of thou-
materials science 91

Ref. 62 sands of setulae. Again, the van der Waals force in each setula helps the spider to stick
on surfaces. Also many insects use small hairs for the same aim. Figure 57 shows a
Ref. 63 comparison. Researchers have shown that the hairs – or setae – are finer the more
massive the animal is. Eduard Arzt likes to explain that small flies and beetles have
simple, spherical setae with a diameter of a few micrometers whereas the considerably
bigger and heavier geckos have branched nanohairs with diameters of 200 nm.
Researchers have copied the hairy adhesion mechanism for the first time in 2003,
using microlithography on polyimide, and they hope to make durable sticky materials –
without using any glue – in the future.
∗∗
One of the most fascinating materials in nature are bones. Bones are light, stiff, and can
Ref. 64 heal after fractures. If you are interested in composite materials, read more about bones:
their structure, shown in Figure 58, and their material properties are fascinating and
complex, and so are their healing and growth mechanisms. All these aspects are still

Motion Mountain – The Adventure of Physics
subject of research.
∗∗
A cereal stalk has a height-to-width ratio of about 300. No human-built tower or mast
Challenge 57 s achieves this. Why?
∗∗
Millimetre waves or terahertz waves are emitted by all bodies at room temperature. Mod-
ern camera systems allow producing images with them. In this way, it is possible to see

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
through clothes, as shown by Figure 59. (Caution is needed; there is the widespread sus-
picion that the image is a fake produced to receive more development funding.) This
ability could be used in future to detect hidden weapons in airports. But the develop-
ment of a practical and affordable detector which can be handled as easily as a binocular
is still under way. The waves can also be used to see through paper, thus making it unne-
cessary to open letters in order to read them. Secret services are exploiting this technique.
A third application of terahertz waves might be in medical diagnostic, for example for the
search of tooth decay. Terahertz waves are almost without side effects, and thus superior
to X-rays. The lack of low-priced quality sources is still an obstacle to their application.
∗∗
Does the melting point of water depend on the magnetic field? This surprising claim was
Ref. 65 made in 2004 by Inaba Hideaki and colleagues. They found a change of 0.9 mK/T. It is
known that the refractive index and the near infrared spectrum of water is affected by
magnetic fields. Indeed, not everything about water might be known yet.
∗∗
Plasmas, or ionized gases, are useful for many applications. A few are shown in Figure 60.
Not only can plasmas be used for heating or cooking and generated by chemical means
(such plasmas are variously called fire or flames) but they can also be generated electric-
ally and used for lighting or deposition of materials. Electrically generated plasmas are
even being studied for the disinfection of dental cavities.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
2 changing the world with quantum effects

F I G U R E 58 The structure of bones, shown for a human vertebra (© Peter Fratzl and Physik Journal).
92
materials science 93

F I G U R E 59 An alleged image acquired
with terahertz waves. Can you explain
why it is a fake? (© Jefferson Lab)

Motion Mountain – The Adventure of Physics
∗∗
It is known that the concentration of CO2 in the atmosphere between 1800 and 2005
Ref. 66 has increased from 280 to 380 parts per million, as shown in Figure 61. (In 2016, the
Challenge 58 s value was already 403 ppm. How would you measure this?) It is known without doubt
that this increase is due to human burning of fossil fuels, and not to natural sources such
as the oceans or volcanoes. There are three arguments. First of all, there was a parallel
decline of the 14 C/12 C ratio. Second, there was a parallel decline of the 13 C/12 C ratio.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Finally, there was a parallel decline of the oxygen concentration. All three measurements
independently imply that the CO2 increase is due to the burning of fuels, which are low
in 14 C and in 13 C, and at the same time decrease the oxygen ratio. Natural sources do
not have these three effects. Since CO2 is a major greenhouse gas, the data implies that
humans are also responsible for a large part of the temperature increase during the same
period. Global warming exists and is mainly due to humans. On average, the Earth has
cooled over the past 10 million years; since a few thousand years, the temperature has,
however, slowly risen; together with the fast rise during the last decades the temperature
Ref. 67 is now at the same level as 3 million years ago. How do the decade global warming trend,
the thousand year warming trend and the million year cooling trend interact? This is a
topic of intense research.
∗∗
Making crystals can make one rich. The first man who did so, the Frenchman Auguste
Verneuil (b. 1856 Dunkerque, d. 1913 Paris), sold rubies grown in his laboratory for many
years without telling anybody. Many companies now produce synthetic gems with ma-
chines that are kept secret. An example is given in Figure 62.
Synthetic diamonds have now displaced natural diamonds in almost all applications.
In the last years, methods to produce large, white, jewel-quality diamonds of ten carats
Ref. 68 and more are being developed. These advances will lead to a big change in all the domains
that depend on these stones, such as the production of the special surgical knives used
in eye lens operation.
94 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 60 Some plasma machines: a machine for coating metal parts, a machine for cleaning polymer
parts and a device for healing wounds (© www.cemecon.de, www.diener.de, Max Planck Gesellschaft).

∗∗
The technologies to produce perfect crystals, without grain boundaries or dislocations,
are an important part of modern industry. Perfectly regular crystals are at the basis of
the integrated circuits used in electronic appliances, are central to many laser and tele-
communication systems and are used to produce synthetic jewels.
∗∗
Challenge 59 s How can a small plant pierce through tarmac?
∗∗
materials science 95

380
360
340
320
CO2 (ppmv)

300
280
260
240

Temperature relative to 1900-2000 (°C)
220
200 6
180 4
2
0
-2
-4

Motion Mountain – The Adventure of Physics
-6
-8
-1 0

800 700 600 500 400 300 200 100 0

Age (1000 years before present)

F I G U R E 61 The concentration of CO2 and the change of average atmospheric temperature in the past
0.8 million years (© Dieter Lüthi).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
If you like abstract colour images, do not miss looking at liquid crystals through a mi-
croscope. You will discover a wonderful world. The best introduction is the text by Ingo
Ref. 69 Dierking.
∗∗
The Lorentz force leads to an interesting effect inside materials. If a current flows along
a conducting strip that is in a (non-parallel) magnetic field, a voltage builds up between
two edges of the conductor, because the charge carriers are deflected in their flow. This ef-
fect is called the (classical) Hall effect after the US-American physicist Edwin Hall (b. 1855
Great Falls, d. 1938 Cambridge), who discovered it in 1879, during his PhD. The effect,
shown in Figure 63, is regularly used, in so-called Hall probes, to measure magnetic fields;
the effect is also used to read data from magnetic storage media or to measure electric
currents (of the order of 1 A or more) in a wire without cutting it. Typical Hall probes
have sizes of around 1 cm down to 1 μm and less. The Hall voltage 𝑉 turns out to be given
by
𝐼𝐵
𝑉= , (10)
𝑛𝑒𝑑
where 𝑛 is the electron number density, 𝑒 the electron charge, and 𝑑 is the thickness of
the probe, as shown in Figure 63. Deducing the equation is a secondary school exercise.
Challenge 60 e The Hall effect is a material effect, and the material parameter 𝑛 determines the Hall
voltage. The sign of the voltage also tells whether the material has positive or negative
96 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 62 Four crystal and synthetic gemstone growing methods. Top: the hydrothermal technique,
used to grow emeralds, quartz, rock crystal and amethyst, and Czochralski’s pulling technique, used for
growing ruby, sapphire, spinel, yttrium-aluminium-garnet, gadolinium-gallium-garnet and alexandrite.
Bottom: Verneuil’s ﬂame fusion technique, used for growing corundum, sapphire, ruby and spinel
boules, and the ﬂux process, used for chrysoberyl (© Ivan Golota).

charge carriers; indeed, for metal strips the voltage polarity is opposite to the one shown
Challenge 61 e in the figure.
Many variations of the Hall effect have been studied. For example, the quantum Hall
Page 107 effect and the fractional quantum Hall effect will be explored below.
In 1998, Geert Rikken and his coworkers found that in certain materials photons can
Ref. 70 also be deflected by a magnetic field; this is the photonic Hall effect.
In 2005, again Geert Rikken and his coworkers found a material, a terbium gallium
garnet, in which a flow of phonons in a magnetic field leads to temperature difference on
materials science 97

Lorentz deflection of probe
magnetic current (if due to positive
field B carriers, as in certain
semiconductors)

probe of -
-
thickness d -
- +
wire with - +
- +
probe current I + resulting
+
(opposite to + edge charges
electron flow) lead to a Hall
voltage V

Motion Mountain – The Adventure of Physics
F I G U R E 63
Top: the
(classical) Hall
effect. Bottom:
a modern
miniature Hall
probe using the
effect to
measure
magnetic ﬁelds

Ref. 71 the two sides. They called this the phonon Hall effect.
∗∗
Do magnetic fields influence the crystallization of calcium carbonate in water? This issue
Ref. 72 is topic of intense debates. It might be, or it might not be, that magnetic fields change the
crystallization seeds from calcite to aragonite, thus influencing whether water tubes are
covered on the inside with carbonates or not. The industrial consequences of reduction
in scaling, as this process is called, would be enormous. But the issue is still open, as are
convincing data sets.
∗∗
Ref. 73 It has recently become possible to make the thinnest possible sheets of graphite and other
materials (such as BN, MoS2 , NbSe2 , Bi2 Sr2 CaCu2 Ox ): these crystal sheets are precisely
one atom thick! The production of graphene – that is the name of a monoatomic graphite
layer – is extremely complicated: you need graphite from a pencil and a roll of adhesive
tape. That is probably why it was necessary to wait until 2004 for the development of the
technique. (In fact, the stability of monoatomic sheets was questioned for many years be-
fore that. Some issues in physics cannot be decided with paper and pencil; sometimes you
98 2 changing the world with quantum effects

1μm 1μm

F I G U R E 64 Single atom sheets,
mapped by atomic force microscopy,
of a: NbSe2 , b: of graphite or
graphene, d: a single atom sheet of
MoS2 imaged by optical microscopy,
and c: a single atom sheet of

Motion Mountain – The Adventure of Physics
Bi2 Sr2 CaCu2 Ox on a holey carbon
ﬁlm imaged by scanning electron
1μm 1μm microscopy (from Ref. 73, © 2005
National Academy of Sciences).

10 μm

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
air one two
graphene graphene
mono- mono-
layer layers F I G U R E 65 A microscope photograph shows the
absorption of a single and of a double layer of
graphene – and thus provides a way to see the
ﬁne structure constant. (© Andre Geim).

need adhesive tape as well.) Graphene and the other so-called two-dimensional crystals
(this is, of course, a tabloid-style exaggeration) are studied for their electronic and mech-
anical properties; in the future, they might even have applications in high-performance
batteries.
∗∗
A monolayer of graphene has an astonishing optical property. Its optical absorption over
the full optical spectrum is π𝛼, where 𝛼 is the fine structure constant. (The exact expres-
materials science 99

F I G U R E 66 The beauty of materials science: the surface of a lotus leaf leads to almost spherical water
droplets; plasma-deposited PTFE, or teﬂon, on cotton leads to the same effect for the coloured water
droplets on it (© tapperboy, Diener Electronics).

Motion Mountain – The Adventure of Physics
sion for the absorption is 𝐴 = 1 − (1 + π𝛼/2)−2 .) The expression for the absorption is the
consequence of the electric conductivity 𝐺 = 𝑒2 /4ℏ for every monolayer of graphene. The
numeric value of the absorption is about 2.3 %. This value is visible by the naked eye, as
Ref. 74 shown in Figure 65. Graphene thus yields a way to ‘see’ the fine structure constant.
∗∗
Gold absorbs light. Therefore it is used, in expensive books, to colour the edges of pages.
Apart from protecting the book from dust, it also prevents that sunlight lets the pages
turn yellow near the edges.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
Like trees, crystals can have growth rings. Smoke quartz is known for these so-called
phantoms, but also fluorite and calcite.
∗∗
The science and art of surface treatment is still in full swing, as Figure 66 shows. Mak-
ing hydrophobic surfaces is an important part of modern materials science, that copies
what the lotus, Nelumbo nucifera, does in nature. Hydrophobic surfaces allow that wa-
Vol. I, page 40 ter droplets bounce on them, like table tennis balls on a table. The lotus surface uses
this property to clean itself, hence the name lotus effect. This is also the reason that lotus
plants have become a symbol of purity.
∗∗
Sometimes research produces bizarre materials. An example are the so-called aerogels,
highly porous solids, shown in Figure 67. Aerogels have a density of a few g/l, thus a
few hundred times lower than water and only a few times that of air. Like any porous
material, aerogels are good insulators; however, they are easily destroyed and therefore
have not found important applications up to now.
∗∗
Where do the minerals in the Amazonian rainforest come from? The Amazonas river
100 2 changing the world with quantum effects

F I G U R E 67 A piece of aerogel, a solid that is so porous
that it is translucent (courtesy NASA).

washes many nutrient minerals into the Atlantic Ocean. How does the rainforest get
its minerals back? It was a long search until it became clear that the largest supply of

Motion Mountain – The Adventure of Physics
minerals is airborne: from the Sahara. Winds blow dust from the Sahara desert to the
Amazonas basin, across the Atlantic Ocean. It is estimated that 40 million tons of dust
are moved from the Sahara to the Amazonas rainforest every year.
∗∗
Some materials undergo almost unbelievable transformations. What is the final state of
moss? Large amounts of moss often become peat (turf). Old turf becomes lignite, or
brown coal. Old lignite becomes black coal (bituminous coal). Old black coal can be-
come diamond. In short, diamonds can be the final state of moss.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
In materials science, there is a dream: to make a material that is harder than diamond. It
is not clear whether this dream can be realized. The coming years will tell.

quantum technol o gy

“
I were better to be eaten to death with a rust
than to be scoured to nothing with perpetual

”
motion.
William Shakespeare King Henry IV.

Quantum effects do not appear only in microscopic systems or in material properties.
Also applied quantum effects are important in modern life: technologies such as tran-
sistors, lasers, superconductivity and other effects and systems have deeply affected our
civilisation.

Transistors
Transistors are found in almost all devices that improve health, as well as in almost all
devices for telecommunication. A transistor, shown in Figure 68 is a device that allows
controlling a large electric current with the help of a small one; therefore it can play
quantum technology 101

collector

N
base
P

emitter

collector

Motion Mountain – The Adventure of Physics
base

emitter

collector

emitter

F I G U R E 68 Top: examples of packaged single transistors. Right: the basic semiconductor structure, the
equivalent water structure, and the technical drawing of an NPN transistor. Bottom: a typical integrated
circuit for smart cards incorporating a large number of transistors. (© Benedikt Seidl, blog.ioactive.com)

the role of an electrically controlled switch or of an amplifier. Transistors are made from
silicon and can be as small as a 2 by 2 μm and as large as 10 by 10 cm. Transistors are used
to control the signals in pacemakers for the heart and the current of electric train engines.
Amplifying transistors are central to the transmitter in mobile phones and switching
transistors are central to computers and their displays.
Transistors are (almost exclusively) based on semiconductors, i.e., on materials where
the electrons that are responsible for electric conductivity are almost free. The devices
102 2 changing the world with quantum effects

MOSFET Bipolar transistor

Off state

water hose

On state

Motion Mountain – The Adventure of Physics
water hose

Off state

+
N N +

P
F I G U R E 69 The working and construction of a metal-oxide silicon ﬁeld effect transistor (left) and of a
bipolar transistor (right). The ‘off’ and ‘on’ states are shown (© Leiﬁ Physik, Wikimedia).

are built in such a way that applying an electric signal changes the conductivity. Every
explanation of a transistor makes use of potentials and tunnelling; transistors are applied
quantum devices.
quantum technology 103

The transistor is just one of a family of semiconductor devices that includes the field-
effect transistor (FET), the metal-oxide-silicon field-effect transistor (MOSFET), the
junction gate field-effect transistor (JFET), the insulated gate bipolar transistor (IGBT)
and the unijunction transistor (UJT), but also the memristors, diode, the PIN diode, the
Zener diode, the avalanche diode, the light-emitting diode (LED), the photodiode, the
photovoltaic cell, the diac, the triac, the thyristor and finally, the integrated circuit (IC).
These are important in industrial applications: the semiconductor industry has at least
300 thousand million Euro sales every year (2010 value) and employ millions of people
across the world.

Motion withou t friction – superconductivit y and superfluidity
We are used to thinking that friction is inevitable. We even learned that friction was an
inevitable result of the particle structure of matter. It should come to the surprise of every
physicist that motion without friction is indeed possible.
In 1911 Gilles Holst and Heike Kamerlingh Onnes discovered that at low temperatures,

Motion Mountain – The Adventure of Physics
electric currents can flow with no resistance, i.e., with no friction, through lead. The
observation is called superconductivity. In the century after that, many metals, alloys and
ceramics have been found to show the same behaviour.
The condition for the observation of motion without friction is that quantum effects
play an essential role. To ensure this, low temperature is usually needed. Despite a large
amount of data, it took over 40 years to reach a full understanding of superconductivity.
Ref. 75 This happened in 1957, when Bardeen, Cooper and Schrieffer published their results. At
low temperatures, electron behaviour in certain materials is dominated by an attractive
interaction that makes them form pairs. These so-called Cooper pairs are effective bosons.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
And bosons can all be in the same state, and can thus effectively move without friction.
In superconductivity, the attractive interaction between electrons is due to the de-
formation of the lattice. At low temperature, two electrons attract each other in the same
way as two masses attract each other due to deformation of the space-time mattress.
However, in the case of solids, these deformations are quantized. With this approach,
Bardeen, Cooper and Schrieffer explained the lack of electric resistance of supercon-
ducting materials, their complete diamagnetism (𝜇𝑟 = 0), the existence of an energy gap,
the second-order transition to normal conductivity at a specific temperature, and the de-
pendence of this temperature on the mass of the isotopes. As a result, they received the
Nobel Prize in 1972.*
Another type of motion without friction is superfluidity. In 1937, Pyotr Kapitsa had
understood that usual liquid helium, i.e., 4 He, below a transition observed at the tem-
perature of 2.17 K, is a superfluid: the liquid effectively moves without friction through

* For John Bardeen (b. 1908 Madison, d. 1991 Boston), this was his second, after he had got the first Nobel
Prize in Physics in 1956, shared with William Shockley and Walter Brattain, for the discovery of the tran-
sistor. The first Nobel Prize was a problem for Bardeen, as he needed time to work on superconductivity.
In an example to many, he reduced the tam-tam around himself to a minimum, so that he could work as
much as possible on the problem of superconductivity. By the way, Bardeen is topped by Frederick Sanger
and by Marie Curie. Sanger first won a Nobel Prize in Chemistry in 1958 by himself and then won a second
one shared with Walter Gilbert in 1980; Marie Curie first won one with her husband and a second one by
herself, though in two different fields.
104 2 changing the world with quantum effects

The situation before
the heat is switched on:

heater

helium

Motion Mountain – The Adventure of Physics
plug with
tiny pores,
or `superleak’

helium reservoir

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 70 The superﬂuidity of helium 4 can be used to produce the fountain effect above a disc with
very small pores, through which superﬂuid helium can pass, but normal ﬂuid cannot. Superﬂuid helium
4 has a large thermal conductivity and ﬂows towards a heated region trying to cool it down again,
whereas the normal liquid cannot return back through the pores. This thermomechanical effect leads to
the fountain (© Paciﬁc Institute of Theoretical Physics).

devices, tubes, etc. More precisely, liquid helium remains a mixture of a superfluid com-
ponent and a normal component; only the superfluid component moves without friction.
Superfluid helium is even able, after an initial kick, to flow over obstacles, such as glass
walls, or to flow out of bottles. A well-known effect of superfluidity is shown in Figure 70.
Superfluidity occurs because the 4 He atom is a boson. Therefore no pairing is necessary
for it to move without friction. This research earned Kapitsa a Nobel Prize in 1978.
In 1972, Richardson, Lee and Osheroff found that even 3 He is superfluid, provided that
the temperature is lowered below 2.7 mK. 3 He is a fermion, and requires pairing to be-
come superfluid. In fact, below 2.2 mK, 3 He is even superfluid in two different ways; one
speaks of phase A and phase B. They received the Nobel Prize in 1996 for this discovery.
In the case of 3 He, the theoreticians had been faster than the experimentalists. The
theory for superconductivity through pairing had been adapted to superfluids already
in 1958 – before any data were available – by Bohr, Mottelson and Pines. This theory
was then adapted and expanded by Anthony Leggett.* The attractive interaction between

* Aage Bohr, son of Niels Bohr, and Ben Mottelson received the Nobel Prize in 1975, Anthony Leggett in
quantum technology 105

Motion Mountain – The Adventure of Physics
F I G U R E 71 A vortex lattice in cold lithium gas, showing their quantized structure (© Andre Schirotzek).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
He atoms, the basic mechanism that leads to superfluidity, turns out to be the spin-spin
interaction.
Superfluidity has also been observed in a number of gases, though at much lower tem-
peratures. Studying the behaviour of gases at lowest temperatures has become popular
in recent years. When the temperature is so low that the de Broglie wavelength is com-
parable to the atom-atom distance, bosonic gases form a Bose–Einstein condensate. The
first such states were realized in 1995 by several groups; the group around Eric Cornell
and Carl Wieman used 87 Rb, Rand Hulet and his group used 7 Li and Wolfgang Ketterle
and his group used 23 Na. For fermionic gases, the first degenerate gas, 40 K, was observed
in 1999 by the group around Deborah Jin. In 2004, the same group observed the first
gaseous Fermi condensate, after the potassium atoms paired up. All these condensates
show superfluidity.
Superfluids are fascinating substances. Vortices also exist in them. But in superfluids,
be they gases or liquids, vortices have properties that do not appear in normal fluids.
In the superfluid 3 He-B phase, vortices are quantized: vortices only exist in integer mul-
tiples of the elementary circulation ℎ/2𝑚3 He . (This is also the case in superconductors.)
Vortices in superfluids have quantized angular momentum. An effect of the quantiza-
tion can be seen in Figure 71. In superfluids, these quantized vortices flow forever! Like
in ordinary fluids, also in superfluids one can distinguish between laminar and turbulent
flow. The transition between the two regimes is mediated by the behaviour the vortices
Ref. 76 in the fluid. Present research is studying how these vortices behave and how they induce
the transition.
106 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 72 The quantum Hall effect (above) and the fractional quantum Hall effect (below): each graph
yielded a Nobel prize. The graphs show how the Hall resistance and the Ohmic resistance vary with the
applied magnetic ﬁeld at very low temperature. The step height is quantized in integer or simple
fractions of ℎ/𝑒2 = 25.812 807 557(18) kΩ. Quantum Hall experiments allow the most precise
determination known to date of this constant of nature.
quantum technology 107

The fractional quantum Hall effect
The fractional quantum Hall effect is one of the most intriguing discoveries of materials
science, and possibly, of physics as a whole. The effect concerns the flow of electrons in
Ref. 77 a two-dimensional surface. In 1982, Robert Laughlin predicted that in this system one
should be able to observe objects with electrical charge 𝑒/3. This strange and fascinating
Ref. 78 prediction was indeed verified in 1997.
Page 97 We encountered the (classical) Hall effect above. The story continues with the dis-
covery by Klaus von Klitzing of the quantum Hall effect. In 1980, Klitzing and his
Ref. 79 collaborators found that in two-dimensional systems at low temperatures – about 1 K –
the electrical conductance 𝑆, also called the Hall conductance, is quantized in multiples
of the quantum of conductance
𝑒2
𝑆=𝑛 . (11)
ℎ
The explanation is straightforward: it is the quantum analogue of the classical Hall effect,

Motion Mountain – The Adventure of Physics
which describes how conductance varies with applied magnetic field. The corresponding
resistance values are
1ℎ 1
𝑅= 2
= 25, 812 807 557(18) kΩ . (12)
𝑛𝑒 𝑛
The values are independent of material, temperature, or magnetic field. They are con-
stants of nature. Von Klitzing received the Nobel Prize in Physics for the discovery, be-
cause the effect was unexpected, allows a highly precise measurement of the fine struc-
ture constant, and also allows one to build detectors for the smallest voltage variations
measurable so far. His discovery started a large wave of subsequent research.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Only two years later, in 1982, it was found that in extremely strong magnetic fields
and at extremely low temperatures, the conductance could vary in steps one third that
Ref. 80 size. Shortly afterwards, even stranger numerical fractions were also found. In fact, all
fractions of the form 𝑚/(2𝑚 + 1) or of the form (𝑚 + 1)/(2𝑚 + 1), 𝑚 being an integer, are
possible. This is the fractional quantum Hall effect. In a landmark paper, Robert Laughlin
Ref. 77 explained all these results by assuming that the electron gas could form collective states
showing quasiparticle excitations with a charge 𝑒/3. This was confirmed experimentally
15 years later and earned him a Nobel Prize as well. We have seen in several occasions
that quantization is best discovered through noise measurements; also in this case, the
clearest confirmation came from electrical current noise measurements.
Subsequent experiments confirmed Laughlin’s deduction. He had predicted the ap-
pearance of a new form of a composite quasi-particle, built of electrons and of one or
several magnetic flux quanta. If an electron bonds with an even number of quanta, the
composite is a fermion, and leads to Klitzing’s integral quantum Hall effect. If the elec-
tron bonds with an odd number of quanta, the composite is a boson, and the fractional
quantum Hall effect appears. The experimental and theoretical details of these quasi-
particles might well be the most complex and fascinating aspects of physics, but explor-
ing them would lead us too far from the aim of our adventure.
Ref. 81 In 2007, a new chapter in the story was opened by Andre Geim and his team, and a

2003.
108 2 changing the world with quantum effects

TA B L E 8 Matter at lowest temperatures.

Phase Type L o w t e m p e r at u r e Example
b e h av i o u r
Solid conductor superconductivity lead, MgB2 (40 K)
antiferromagnet chromium, MnO
ferromagnet iron
insulator diamagnet
4
Liquid bosonic Bose–Einstein condensation, i.e., He
superfluidity
fermionic pairing, then BEC, i.e., superfluidity 3 He
87
Gas bosonic Bose–Einstein condensation Rb, 7 Li, 23 Na, H, 4 He, 41 K
40
fermionic pairing, then Bose–Einstein K, 6 Li
condensation

Motion Mountain – The Adventure of Physics
second team, when they discovered a new type of quantum Hall effect at room temper-
Page 98 ature. They used graphene, i.e., single-atom layers of graphite, and found a relativistic
analogue of the quantum Hall effect. This effect was even more unexpected than the pre-
vious ones, is equally interesting, and can be performed on a table top. The groups are
good candidates for a trip to Stockholm.*
What do we learn from these results? Systems in two dimensions have states which
follow different rules than systems in three dimensions. The fractional charges in super-
conductors have no relation to quarks. Quarks, the constituents of protons and neutrons,
have charges 𝑒/3 and 2𝑒/3. Might the quarks have something to do with a mechanism

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
similar to superconductivity? At this point we need to stand the suspense, as no answer
is possible; we come back to this issue in the last part of this adventure.

How d oes mat ter behave at the lowest temperatures?
The low-temperature behaviour of matter has numerous experimental and theoretical
aspects. The first issue is whether matter is always solid at low temperatures. The answer
is no: all phases exist at low temperatures, as shown in Table 8.
Concerning the electric properties of matter at lowest temperatures, the present status
is that matter is either insulating or superconducting. Finally, one can ask about the mag-
netic properties of matter at low temperatures. We know already that matter can not be
paramagnetic at lowest temperatures. It seems that matter is either ferromagnetic, dia-
magnetic or antiferromagnetic at lowest temperatures.

L asers and other spin-one vector b oson launchers
Photons are vector bosons; a lamp is thus a vector boson launcher. All existing lamps fall
Ref. 82 into one of three classes. Incandescent lamps use emission from a hot solid, gas discharge
lamps use excitation of atoms, ions or molecules through collision, and recombination
lamps generate (cold) light through recombination of charges in semiconductors or li-

* This prediction from the December 2008 edition became reality in December 2010.
quantum technology 109

Motion Mountain – The Adventure of Physics
F I G U R E 73 The beauty of lasers: the ﬁne mesh created by a green laser delay line (© Laser Zentrum
Hannover).

quids. The latter are the only lamp types found in living systems. The other main sources
of light are lasers. All light sources are based on quantum effects, but for lasers the con-
nection is especially obvious. The following table gives an overview of the main types
and their uses.
TA B L E 9 A selection of lamps and lasers.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
L a m p t y p e , a p p l i c a t i o n Wa v e - Bright - Cost Life-
length ness or time
power
Incandescent lamps
Oil lamps, candles, for illumination white up to 500 lm 1 cent/lm 5h
Tungsten wire light bulbs, halogen 300 to 800 nm 5 to 25 lm/W 0.1 cent/lm 700 h
lamps, for illumination
Stars, for production of heavy full spectrum up to 1044 W free up to
elements thousands
of millions
of years
Gas discharge lamps
Neon lamps, for advertising red up to 30 kh
Mercury lamps, for illumination UV plus 45 to 0.05 cent/lm 3000 to
spectrum 110 lm/W 24 000 h
Metal halogenide lamps (ScI3 or white 110 lm/W 1 cent/lm up to 20 kh
‘xenon light’, NaI, DyI3 , HoI3 , TmI5 )
for car headlights and illumination
Sodium low pressure lamps for 589 nm yellow 200 lm/W 0.2 cent/lm up to 18 kh
street illumination
110 2 changing the world with quantum effects

TA B L E 9 A selection of lamps and lasers (continued).

L a m p t y p e , a p p l i c a t i o n Wa v e - Bright - Cost Life-
length ness or time
power
Sodium high pressure lamps for broad yellow 120 lm/W 0.2 cent/lm up to 24 kh
street illumination
Xenon arc lamps, for cinemas white 30 to 100 to
150 lm/W, up 2500 h
to 15 kW
Stars, for production of heavy many lines up to 1020 W free up to
elements thousands
of millions
of years
Recombination lamps

Motion Mountain – The Adventure of Physics
Foxfire in forests, e.g. due to green just visible free years
Armillaria mellea, Neonothopanus
gardneri or other bioluminescent
fungi
Firefly, to attract mates green-yellow free c. 10 h
Large deep sea squid, Taningia red c. 1 W free years
danae, producing light flashes, to
confuse prey
Deep-sea fish, such as angler fish, to white c. 1 μW free years

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
attract prey or find mates
Deep-sea medusae, to produce blue and all free years
attention so that predators of other colours
predators are attracted
Light emitting diodes, for red, green, up to 10 cent/lm 15k to
measurement, illumination and blue, UV 150 lm/W, up 100 kh
communication to 5 W
Synchrotron radiation sources
Electron synchroton source X-rays to radio pulsed many MEuro years
waves
Maybe some stars broad free thousands
spectrum of years
Ideal white lamp or laser visible c. 300 lm/W 0 ∞
Ideal coloured lamp or laser green 683 lm/W 0 ∞
Gas lasers
He-Ne laser (obsolete), for school 632.8 nm 550 lm/W 2000 cent/lm 300 h
experiments
Argon laser, for pumping and laser several blue up to 100 W 10 kEuro
shows, now obsolete and green lines
quantum technology 111

TA B L E 9 A selection of lamps and lasers (continued).

L a m p t y p e , a p p l i c a t i o n Wa v e - Bright - Cost Life-
length ness or time
power
Krypton laser, for pumping and several blue, 50 W
laser shows, now obsolete green, red lines
Xenon laser many lines in 20 W
the IR, visible
and near UV
Nitrogen (or ‘air’) laser, for 337.1 nm pulsed up to down to a few limited by
pumping of other lasers, for 1 MW hundred Euro metal
hobbyists electrode
lifetime
Water vapour laser, for research, many lines CW 0.5 W, a few kEuro
now obsolete between 7 and pulsed much

Motion Mountain – The Adventure of Physics
220 μm, often higher
118 μm
CO2 laser, for cutting, welding, glass 10.6 μm CW up to c. 100 Euro/W 1500 h
welding and surgery 100 kW, pulsed
up to 10 TW
Excimer laser, for lithography in 193 nm (ArF), 100 W 10 to 500 kEuroyears
silicon chip manufacturing, eye 248 nm (KrF),
surgery, laser pumping, psoriasis 308 nm (XeCl),
treatment, laser deposition 353 nm (XeF)

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Metal vapour lasers (Cu, Cd, Se, Ca,
Ag, Au, Mn, Tl, In, Hg)
Copper vapour laser, for pumping, 248 nm, pulses up to 10 kEuro 1 khour
photography, dermatology, laser 511 nm and 5 MW
cutting, hobby constructions and 578 nm
explorative research
Cadmium vapour laser, for printing, 325 nm and up to 200 mW 12 kEuro 10 kh
typesetting and recognition of 442 nm
forged US dollar notes
Gold vapour laser, for explorative 627 nm pulses up to from a few
research, dermatology 1 MW hundred Euro
upwards
Chemical gas lasers
HF, DF and oxygen-iodine laser, 1.3 to 4.2 μm up to MW in over 10 MEuro unknown
used as weapons, pumped by CW mode
chemical reactions, all obsolete
Liquid dye lasers
112 2 changing the world with quantum effects

TA B L E 9 A selection of lamps and lasers (continued).

L a m p t y p e , a p p l i c a t i o n Wa v e - Bright - Cost Life-
length ness or time
power

Rhodamine, stilbene, coumarin tunable, range up to 10 W 10 kEuro dye-
etc. lasers, for spectroscopy and depends on dependent
medical uses dye in 300 to
1100 nm range
Beer, vodka, whiskey, diluted IR, visible usually mW 1 kEuro a few
marmelade and many other liquids minutes
work as laser material
Solid state lasers
Ruby laser (obsolete), for 694 nm 1 kEuro

Motion Mountain – The Adventure of Physics
holography and tattoo removal
Nd:YAG (neodymium:yttrium 1064 nm CW 10 kW, 50 to 1000 h
aluminium granate) laser, for pulsed 500 kEuro
material processing, surgery, 300 MW
pumping, range finding,
velocimetry, also used with doubled
frequency (532 nm), with tripled
frequency (355 nm) and with
quadrupled frequency (266 nm),
also used as slab laser

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Er:YAG laser, for dermatology 2940 nm
Ti:sapphire laser, for ultrashort 650 to 1200 nm CW 1 W, from 5 kEuro
pulses for spectroscopy, LIDAR, and pulsed 300 TW upwards
research
Alexandrite laser, for laser 700 to 840 nm
machining, dermatology, LIDAR
Cr:LiSAF laser pulsed 10 TW,
down to 30 fs
Cr:YAG laser 1.35 to 1.6 μm pulsed, down
to 100 fs
Cr:Forsterite laser, optical 1200 to pulsed, below
tomography 1300 nm 100 fs
Erbium doped glass fibre laser, used 1.53 to 1.56 μm years
in optical communications
(undersea cables) and optical
amplifiers
Perovskite laser, such as Co:KZnF3 , NIR tunable, 100 mW 2 kEuro
for research 1650 to
2070 nm
F-centre laser, for spectroscopy tuning ranges 100 mW 20 kEuro
(NaCl:OH-, KI:Li, LiF) between 1.2
and 6 μm
quantum technology 113

TA B L E 9 A selection of lamps and lasers (continued).

L a m p t y p e , a p p l i c a t i o n Wa v e - Bright - Cost Life-
length ness or time
power

Semiconductor lasers
GaN laser diode, for optical 355 to 500 nm, up to 150 mW a few Euro to 5 c. 10 000 h
recording depending on kEuro
doping
AlGaAs laser diode, for optical 620 to 900 nm, up to 1 W below 1 Euro to c. 10 000 h
recording, pointers, data depending on 100 Euro
communication, laser fences, bar doping
code readers (normal or vertical
cavity)
InGaAsP laser diode, for fiberoptic 1 to 2.5 μm up to 100 W below 1 Euro up to

Motion Mountain – The Adventure of Physics
communication, laser pumping, up to a few k 20 000 h
material processing, medical uses Euro
(normal and vertical cavity or
VCSEL)
Lead salt (PbS/PbSe) laser diode, for 3 to 25 μm 0.1 W a few 100 Euro
spectroscopy and gas detection
Quantum cascade laser, for research 2.7 to 350 μm up to 4 W c. 10 kEuro c. 1 000 h
and spectroscopy
Hybrid silicon lasers, for research IR nW 0.1 MEuro

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Free electron lasers
Used for materials science 5 nm to 1 mm CW 20 kW, 10 MEuro years
pulsed in GW
range
Nuclear-reaction pumped lasers
Have uses only in science fiction and for getting money from gullible military

From lamps to lasers
Most solid state lamps are light emitting diodes. The large progress in brightness of light
emitting diodes could lead to a drastic reduction in future energy consumption, if their
cost is lowered sufficiently. Many engineers are working on this task. Since the cost is a
good estimate for the energy needed for production, can you estimate which lamp is the
Challenge 62 s most friendly to the environment?
Nobody thought much about lamps, until Albert Einstein and a few other great physi-
cists came along, such as Theodore Maiman and Hermann Haken. Many other research-
ers later received Nobel Prizes by building on their work. In 1916, Einstein showed that
there are two types of sources of light – or of electromagnetic radiation in general – both
of which actually ‘create’ light. He showed that every lamp whose brightness is turned
up high enough will change behaviour when a certain intensity threshold is passed. The
114 2 changing the world with quantum effects

main mechanism of light emission then changes from spontaneous emission to stimu-
lated emission. Nowadays such a special lamp is called a laser. (The letters ‘se’ in laser are
an abbreviation of ‘stimulated emission’.) After a passionate worldwide research race, in
1960 Maiman was the first to build a laser emitting visible light. (So-called masers emit-
ting microwaves were already known for several decades.) In summary, Einstein and
the other physicists showed that whenever a lamp is sufficiently turned up, it becomes
a laser. Lasers consist of some light producing and amplifying material together with a
mechanism to pump energy into it. The material can be a gas, a liquid or a solid; the
pumping process can use electrical current or light. Usually, the material is put between
two mirrors, in order to improve the efficiency of the light production. Common lasers
are semiconductor lasers (essentially strongly pumped LEDs or light emitting diodes),
He–Ne lasers (strongly pumped neon lamps), liquid lasers (essentially strongly pumped
fire flies) and ruby lasers (strongly pumped luminescent crystals). Most materials can be
used to make lasers for fun, including water, beer and vodka.
Lasers produce radiation in the range from microwaves and extreme ultraviolet. They

Motion Mountain – The Adventure of Physics
have the special property of emitting coherent light, usually in a collimated beam. There-
fore lasers achieve much higher light intensities than lamps, allowing their use as tools.
In modern lasers, the coherence length, i.e., the length over which interference can be
observed, can be thousands of kilometres. Such high quality light is used e.g. in gravita-
tional wave detectors.
People have become pretty good at building lasers. Lasers are used to cut metal sheets
up to 10 cm thickness, others are used instead of knives in surgery, others increase surface
hardness of metals or clean stones from car exhaust pollution. Other lasers drill holes in
teeth, measure distances, image biological tissue or grab living cells.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Some materials amplify light so much that end mirrors are not necessary. This is the
case for nitrogen lasers, in which nitrogen, or simply air, is used to produce a UV beam.
Even a laser made of a single atom (and two mirrors) has been built; in this example, only
Ref. 83 eleven photons on average were moving between the two mirrors. Quite a small lamp.
Also lasers emitting light in two dimensions have been built. They produce a light plane
instead of a light beam.

The three lightbulb scams
In the 1990s, all major light bulb producers in the world were fined large sums because
they had agreed to keep the lifetimes of light bulbs constant. It is no technical problem to
make light bulbs that last 2000 hours; however, the producers agreed not to increase the
lifetime above 700 hours, thus effectively making every lightbulb three times as expensive
as it should. This was the first world-wide light bulb scam.
Despite the fines, the crooks in the light bulb industry did not give up. In 2012, a large
German light bulb maker explained in its advertising that its new light sources were
much longer living than its conventional light bulbs, which, they explained on their ads,
lasted only 500 hours. In other words, not only did the fines not help, the light bulb
industry even reduced the lifetimes of the light bulbs from the 1990s to 2012. This was
the second light bulb scam.
Parallel to the second scam, in the years around 2000, the light bulb industry star-
ted lobbying politics with the false statement that light bulbs were expensive and would
quantum technology 115

waste energy. As a result of the false data provided from the other two scams, light bulbs
were forbidden in Europe, with the result that consumers in Europe are now forced to
buy other, much more expensive means of illumination. On top of this, many of these
more expensive light sources are bad for the eyes. Indeed, flickering mercury or flick-
ering LED lamps, together with their reduced colour spectrum, force the human visual
system in overload mode, a situation that does not occur with the constantly glowing
light bulbs. In other words, with this third scam, the light bulb industry increased their
profits even more, while ruining the health of consumers at the same time. One day,
maybe, parliaments will be less corrupt and more sensible. The situation will then again
improve.

Applications of lasers
As shown in Figure 74, lasers can be used to make beautiful parts – including good violins
and personalized bicycle parts – via sintering of polymer or metal powders. Lasers are
used in rapid prototyping machines and to build architectural models. Lasers can cut

Motion Mountain – The Adventure of Physics
paper, metal, plastics and flesh.
Lasers are used to read out data from compact discs (CDs) and digital versatile discs
(DVDs), are used in the production of silicon integrated circuits and for the transport
telephone signals through optical fibres. In our adventure, we already encountered lasers
Vol. I, page 408 that work as loudspeakers. Important advances in recent years came from the applica-
tions of femtosecond laser pulses. Femtosecond pulses generate high-temperature plas-
mas in the materials they propagate through; this happens even in air, if the pulses are
focused. The effect has been used to create luminous three-dimensional displays floating
in mid-air, as shown in Figure 74, or in liquids. Such short pulses can also be used to

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
cut material without heating it, for example to cut bones in skull operations. The lack of
heating is so complete that femtosecond lasers can be used to engrave matches and even
dynamite without triggering a reaction. Femtosecond lasers have been used to make high
resolution holograms of human heads within a single flash. Recently such lasers have
been used to guide lightning along a predetermined path; they seem promising candid-
Ref. 84 ates for laser ligtning rods. A curious demonstration application of femtosecond lasers
is the storage of information in fingernails (up to 5 Mbit for a few months), in a way not
Ref. 85 unlike that used in recordable compact discs (CD-R).
Lasers are used in ophthalmology, with a technique called optical coherence tomo-
graphy, to diagnose eye and heart illnesses. Around 2025, there will finally be laser-based
breast screening devices that use laser light to search for cancer without any danger to
the patient. The race to produce the first working system is already ongoing since the
1990s. Additional medical laser applications will appear in the coming years.
Lasers have been used in recent demonstrations, together with image processing soft-
ware, to kill mosquitos in flight; other lasers are burning weeds while the laser is moved
over a field of crops. One day, such combined laser and vision systems will be used to
evaporate falling rain drops one by one; as soon as the first such laser umbrella will be
Ref. 86 available, it will be presented here. The feat should be possible before the year 2022.
116 2 changing the world with quantum effects

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 74 Some laser applications. Top: a violin, with excellent sound quality, made of a single piece
of polymer (except for the chords and the black parts) through laser sintering of PEEK by EOS from
Krailling, in Germany. Bottom: a display ﬂoating in mid-air produced with a galvanometer scanner and a
fast focus shifter (© Franz Aichinger, Burton).

Challenges, dreams and curiosities ab ou t quantum technolo gy
Nowadays, we carry many electronic devices in our jacket or trousers. Almost all use bat-
teries. In the future, there is a high chance that some of these devices will extract energy
from the human body. There are several options. One can extract thermal energy with
thermoelements, or one can extract vibrational energy with piezoelectric, electrostatic or
electromagnetic transducers. The challenge is to make these elements small and cheap.
It will be interesting to find out which technology will arrive to the market first.
∗∗
quantum technology 117

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 75 The most expensive laser pointer: a yellow 10 W laser that is frequency-stabilized at the
wavelength of sodium lamps allows astronomers to improve the image quality of terrestrial telescopes.
By exciting sodium atoms found at a height of 80 to 90 km, the laser provides an artiﬁcial guide star
that is used to compensate for atmospheric turbulence, using the adaptive optics built into the
telescope. (© ESO/Babak Tafreshi).

In 2007, Humphrey Maris and his student Wei Guo performed an astonishing experi-
Ref. 87 ment: they filmed single electrons with a video camera. Actually the truth is a bit more
complicated, but it is not a lie to summarize it in this way.
Maris is an expert on superfluid helium. For many years he knew that free electrons
in superfluid helium repel helium atoms, and can move, surrounded by a small vacuum
bubble, about 2 nm across, through the fluid. He also discovered that under negative
pressure, these bubbles can grow and finally explode. When they explode, they are able
to scatter light. With his student Wei Guo, he then injected electrons into superfluid
helium through a tungsten needle under negative voltage, produced negative pressure
by focussing waves from two piezoelectric transducers in the bulk of the helium, and
118 2 changing the world with quantum effects

QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.

F I G U R E 76 How to image single
electrons with a video camera:
isolated electrons surrounded by
bubbles that explode in liquid
helium under negative pressure
produce white spots (mpg ﬁlm ©
Humphrey Maris).

Motion Mountain – The Adventure of Physics
shone light through the helium. When the pressure became negative enough they saw
the explosions of the bubbles. Figure 76 shows the video. The experiment is one of the
highlights of experimental physics in the last decade.
∗∗
Is it possible to make A4-size flexible colour displays for an affordable price and with
Challenge 63 d print-like quality?

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
Will there ever be rechargeable batteries with an energy content per mass that is com-
parable to diesel oil? How long will it take, from 2014 onwards, until the last company
producing electric cars powered by batteries stops production?
∗∗
How many companies promising free energy, engineers promising cars powered by wa-
ter, politicians promising fusion energy or quacks promising food additives or sugar pills
that cure cancer will we see every year?
∗∗
Challenge 64 r Will there ever be room-temperature superconductivity?
∗∗
Challenge 65 r Will there ever be desktop laser engravers for 1000 euro?
∗∗
Challenge 66 s Will there ever be teleportation of everyday objects?
∗∗
One process that quantum physics does not allow is telepathy. An unnamed space agency
quantum technology 119

pickup loop
z Ω
λ N
S
x β
inlet port y
weak link

parallel coil

membrane

1 cm
Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 77 The interior of a gyroscope that uses superﬂuid helium (© Eric Varoquaux).

found this out during the Apollo 14 mission, when, during the flight to the moon, cosmo-
naut Edgar Mitchell tested telepathy as communication means. Unsurprisingly, he found
Ref. 88 that telepathy was useless. (This not a joke.) It is unclear why the space agency spent so
much money for a useless experiment – an experiment that could have been performed,
at a cost of a phone call, also down here on earth.
∗∗
Challenge 67 d Will there ever be applied quantum cryptology?
∗∗
Will there ever be printable polymer electronic circuits, instead of lithographically pat-
Challenge 68 d terned silicon electronics as is common now?
∗∗
Challenge 69 r Will there ever be radio-controlled flying toys in the size of insects?
∗∗
Ref. 107 By shining an invisible and harmless laser onto cars driving by, it is now possible to
120 2 changing the world with quantum effects

F I G U R E 78 These colours were
produced on steel using just an
infrared laser shining on it.
(© Trotec Laser at www.
troteclaser.com)

Motion Mountain – The Adventure of Physics
detect whether the persons inside have drunk alcohol. Will this method ever become
Challenge 70 s widespread?
∗∗
In 1997, Eric Varoquaux and his group built a quantum version of the Foucault pendu-
Ref. 108 lum, using the superfluidity of helium. In this beautiful piece of research, they cooled
a small ring of fluid helium below the temperature of 0.28 K, below which the helium
moves without friction. In such situations helium can behave like a Foucault pendulum.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
With a clever arrangement, shown in Figure 77, they were able to measure the rotation
of the helium in the ring using phonon signals, and, finally, to detect the rotation of the
Earth.
∗∗
Lasers are quantum devices that can be used for many applications. Figure 78 shows
a way to produce colours on steel by scanning a focused infrared laser beam over the
Challenge 71 e surface. Why and how do the colours appear?

Summary on changing the world with quantum effects
Atoms form bonds. Quantum effects thus produce molecules, gases, liquids and solids,
as well as all effects and properties of all materials. In the past, quantum effects have been
used to develop numerous materials with desired properties, such as new steel types, new
carbon fibre composites, new colourants, new magnetic materials and new polymers.
Quantum effects have been used to develop modern electronics, lasers, light detect-
ors, data storage devices, superconducting magnets, new measurement systems and new
production machines. Magnetic resonance imaging, computers, polymers, telecommu-
nication and the internet resulted from applying quantum effects to technology.
Quantum effects will continue to be used to design new materials and systems: new
nanoparticles to deliver drugs inside the body, new polymers, new crystals, new envir-
onmentally friendly production processes and new medical devices, among others.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
121
quantum technology
Chapter 3

QUA N T UM E L E C T RODY NA M IC S
– T H E OR IG I N OF V I RT UA L R E A L I T Y

T
he central concept that quantum field theory adds to the description of nature is
he idea of virtual particles. Virtual particles are short-lived particles; they owe
heir existence exclusively to the quantum of action. Because of the quantum of
action, they do not need to follow the energy-mass relation that special relativity re-

Motion Mountain – The Adventure of Physics
quires of usual, real particles. Virtual particles can move faster than light and can move
backward in time. Despite these strange properties, they have many observable effects.
We explore the most spectacular ones.

Ships, mirrors and the C asimir effect
When two parallel ships roll in a big swell, without even the slightest wind blowing, they
will attract each other. The situation is illustrated in Figure 79. It might be that this effect
was known before the nineteenth century, when many places still lacked harbours.**
Waves induce oscillations of ships because a ship absorbs energy from the waves.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
When oscillating, the ship also emits waves. This happens mainly towards the two sides
of the ship. As a result, for a single ship, the wave emission has no net effect on its pos-
ition. Now imagine that two parallel ships oscillate in a long swell, with a wavelength
much larger than the distance between the ships. Due to the long wavelength, the two
ships will oscillate in phase. The ships will thus not be able to absorb energy from each
other. As a result, the energy they radiate towards the outside will push them towards
each other.
The effect is not difficult to calculate. The energy of a rolling ship is

𝐸 = 𝑚𝑔ℎ 𝛼2 /2 (13)

where 𝛼 is the roll angle amplitude, 𝑚 the mass of the ship and 𝑔 = 9, 8 m/s2 the acce-
leration due to gravity. The metacentric height ℎ is the main parameter characterizing a
ship, especially a sailing ship; it tells with what torque the ship returns to the vertical
when inclined by an angle 𝛼. Typically, one has ℎ =1.5 m.
When a ship is inclined, it will return to the vertical by a damped oscillation. A
damped oscillation is characterized by a period 𝑇 and a quality factor 𝑄. The quality
factor is the number of oscillations the system takes to reduce its amplitude by a factor

** Sipko Boersma published a paper in which he gave his reading of shipping manuals, advising captains to
Ref. 89 let the ships be pulled apart using a well-manned rowing boat. This reading has been put into question by
Ref. 90 subsequent research, however.
the origin of virtual reality 123

Left: parallel objects Right: parallel objects
in quiet evironment surrounded by waves
or noise
ships on water
in a harbour
waves on
water
lead to
attraction
of the ships

rigid plates in air

noise or

Motion Mountain – The Adventure of Physics
sound
in air
air air
leads to
interaction
of the
plates

mirrors in vacuum

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
electro-
magnetic
fluctuations
not possible in vacuum
lead to
attractive
force between
mirrors

F I G U R E 79 The analogy between ships in a harbour, metal plates in air and metal mirrors in vacuum.

𝑒 = 2.718. If the quality factor 𝑄 of an oscillating ship and its oscillation period 𝑇 are
given, the radiated power 𝑊 is
𝐸
𝑊 = 2π . (14)
𝑄𝑇

Vol. III, page 120 We saw above that radiation force (radiation pressure times area) is 𝑊/𝑐, where 𝑐 is
the wave propagation velocity. For gravity water waves in deep water, we have the well-
Vol. III, page 295 known relation
𝑔𝑇
𝑐= . (15)
2π
124 3 quantum electrodynamics

Assuming that for two nearby ships each one completely absorbs the power emitted from
the other, we find that the two ships are attracted towards each other following

ℎ𝛼2
𝑚𝑎 = 𝑚2π2 . (16)
𝑄𝑇2

Inserting typical values such as 𝑄 = 2.5, 𝑇 =10 s, 𝛼 =0.14 rad and a ship mass of 700
tons, we get about 1.9 kN. Long swells thus make ships attract each other. The strength
of the attraction is comparatively small and could be overcome with a rowing boat. On
the other hand, even the slightest wind will damp the oscillation amplitude and have
other effects that will hide or overshadow this attraction.
Sound waves or noise in air show the same effect. It is sufficient to suspend two metal
Ref. 91 plates in air and surround them by loudspeakers. The sound will induce attraction (or
repulsion) of the plates, depending on whether the sound wavelength cannot (or can) be
taken up by the other plate.

Motion Mountain – The Adventure of Physics
In 1948, the Dutch physicist Hendrik Casimir made one of the most spectacular pre-
dictions of quantum theory: he predicted a similar effect for metal plates in vacuum.
Casimir, who worked at the Dutch Electronics company Philips, wanted to understand
why it was so difficult to build television tubes. The light-emitting surface in a cath-
ode ray tube – or today, in a plasma display – of a television, the phosphor, is made
by deposing small neutral, but conductive particles on glass. Casimir observed that the
particles somehow attracted each other. Casimir got interested in understanding how
neutral particles interact. During these theoretical studies he discovered that two neutral
metal plates (or metal mirrors) would attract each other even in complete vacuum. This

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
is the famous Casimir effect. Casimir also determined the attraction strength between a
sphere and a plate, and between two spheres. In fact, all conducting neutral bodies attract
Ref. 92, Ref. 93 each other in vacuum, with a force depending on their geometry.
In all these situations, the role of the sea is taken by the zero-point fluctuations of
the electromagnetic field, the role of the ships by the conducting bodies. Casimir under-
stood that the space between two parallel conducting mirrors, due to the geometrical
constraints, had different zero-point fluctuations than the free vacuum. Like in the case
of two ships, the result would be the attraction of the two mirrors.
Casimir predicted that the attraction for two mirrors of mass 𝑚 and surface 𝐴at dis-
tance 𝑑 is given by
𝑚𝑎 π3 ℏ𝑐
= . (17)
𝐴 120 𝑑4
The effect is a pure quantum effect; in classical electrodynamics, two neutral bodies
do not attract. The effect is small; it takes some dexterity to detect it. The first exper-
Ref. 94 imental confirmation was by Derjaguin, Abrikosova and Lifshitz in 1956; the second
experimental confirmation was by Marcus Sparnaay, Casimir’s colleague at Philips, in
Ref. 95 1958. Two beautiful high-precision measurements of the Casimir effect were performed
Ref. 96 in 1997 by Lamoreaux and in 1998 by Mohideen and Roy; they confirmed Casimir’s pre-
diction with a precision of 5 % and 1 % respectively. (Note that at very small distances,
Ref. 97 the dependence is not 1/𝑑4 , but 1/𝑑3 .) In summary, uncharged bodies attract through
electromagnetic field fluctuations.
the origin of virtual reality 125

The Casimir effect thus confirms the existence of the zero-point fluctuations of the
electromagnetic field. It confirms that quantum theory is valid also for electromagnetism.
The Casimir effect between two spheres is proportional to 1/𝑟7 and thus is much
weaker than between two parallel plates. Despite this strange dependence, the fascin-
ation of the Casimir effect led many amateur scientists to speculate that a mechanism
similar to the Casimir effect might explain gravitational attraction. Can you give at least
three arguments why this is impossible, even if the effect had the correct distance de-
Challenge 72 s pendence?
Like the case of sound, the Casimir effect can also produce repulsion instead of at-
traction. It is sufficient that one of the two materials be perfectly permeable, the other a
perfect conductor. Such combinations repel each other, as Timothy Boyer discovered in
Ref. 98 1974.
In a cavity, spontaneous emission is suppressed, if it is smaller than the wavelength
of the emitted light! This effect has also been observed. It confirms that spontaneous
emission is emission stimulated by the zero point fluctuations.

Motion Mountain – The Adventure of Physics
Ref. 99 The Casimir effect bears another surprise: between two metal plates, the speed of light
changes and can be larger than 𝑐. Can you imagine what exactly is meant by ‘speed of
Challenge 73 s light’ in this context?
In 2006, the Casimir effect provided another surprise. The ship story just presented
is beautiful, interesting and helps understanding the effect; but it seems that the story
is based on a misunderstanding. Alas, the interpretation of the old naval text given by
Ref. 90 Sipko Boersma seems to be wishful thinking. There might be such an effect for ships, but
it has never been observed nor put into writing by seamen, as Fabrizio Pinto has pointed
out after carefully researching naval sources. As an analogy however, it remains valid.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The L amb shift
In the old days, it was common that a person receives the Nobel Prize in Physics for
observing the colour of a lamp – if the observation was sufficiently careful. In 1947, Willis
Lamb (b. 1913 Los Angeles, d. 2008 Tucson) performed such a careful measurement of
the spectrum of hydrogen. He found that the 2𝑆1/2 energy level in atomic hydrogen lies
slightly above the 2𝑃1/2 level. This observation is in contrast to the calculation performed
Vol. IV, page 188 earlier on, where the two levels are predicted to have the same energy. In contrast, the
measured energy difference is 1057.864 MHz, or 4.3 μeV. This discovery had important
consequences for the description of quantum theory and yielded Lamb a share of the
1955 Nobel Prize in Physics. Why?
The reason for contrast between calculation and observation is an approximation per-
formed in the relativistic calculation of the hydrogen levels that took over twenty years to
clarify. There are two equivalent explanations. One explanation is to say that the relativ-
istic calculation neglects the coupling terms between the Dirac equation and the Maxwell
equations. This explanation lead to the first calculations of the Lamb shift, around the
year 1950. The other, equivalent explanation is to say that the calculation neglects virtual
particles. In particular, the calculation neglects the virtual photons emitted and absorbed
during the motion of the electron around the nucleus. This second explanation is in line
with the modern vocabulary of quantum electrodynamics. Quantum electrodynamics,
or QED, is the (perturbative) approach to solve the coupled Dirac and Maxwell equations.
126 3 quantum electrodynamics

In short, Lamb discovered the first effect due to virtual particles. In fact, Lamb used
microwaves for his experiments; only in the 1970 it became possible to see the Lamb shift
with optical means. For this and similar feats Arthur Schawlow received the Nobel Prize
in Physics in 1981.

The QED L agrangian and its symmetries
In simplified terms, quantum electrodynamics is the description of electron motion. This
implies that the description is fixed by the effects of mass and charge, and by the quantum
of action. The QED Lagrangian density is given by:

LQED = 𝜓(𝑖ℏ𝑐∂/ − 𝑐2 𝑚𝑘 )𝜓 } the matter term
1 𝐹 𝐹𝜇𝜈
− 4𝜇 } the electromagnetic field term
𝜇𝜈 (18)
0
+ 𝑒ℏ𝑐𝐴 𝜇 𝜓𝛾𝜇 𝜓 . } the electromagnetic interaction
term

Motion Mountain – The Adventure of Physics
We know the matter term from the Dirac equation for free particles; it describes the
kinetic energy of free electrons. We know the term of the electromagnetic field from the
Maxwell’s equations; it describes the kinetic energy of photons. The interaction term is
the term that encodes the gauge symmetry of electromagnetism, also called ‘minimal
coupling’; it encodes the potential energy. In other words, the Lagrangian describes the
motion of electrons and photons.
All experiments ever performed agree with the prediction by this Lagrangian. In other
words, this Lagrangian is the final and correct description of the motion of electrons and

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
photons. In particular, the Lagrangian describes the size, shape and colour of atoms, the
size, shape and colour of molecules, as well as all interactions of molecules. In short, the
Lagrangian describes all of materials science, all of chemistry and all of biology. Exag-
gerating a bit, this is the Lagrangian that describes life. (In fact, the description of atomic
Page 162 nuclei must be added; we will explore it below.)
All electromagnetic effects, including the growth of the coloured spots on butterfly
wings, the functioning of the transistor or the cutting of paper with scissors, are com-
pletely described by the QED Lagrangian. In fact, the Lagrangian also describes the mo-
tion of muons, tau leptons and all other charged particles. Since the Lagrangian is part
of the final description of motion, it is worth thinking about it in more detail.
Which requirements are necessary to deduce the QED Lagrangian? This issue has been
explored in great detail. The answer is given by the following list:
— compliance with the observer-invariant quantum of action for the motion of elec-
trons and photons,
— symmetry under the permutation group among many electrons, i.e., fermion beha-
viour of electrons,
— compliance with the invariance of the speed of light, i.e., symmetry under transform-
ations of special relativity,
— symmetry under U(1) gauge transformations for the motion of photons and of
charged electrons,
— symmetry under renormalization group,
the origin of virtual reality 127

— low-energy interaction strength described by the fine structure constant, the electro-
magnetic coupling constant, 𝛼 ≈ 1/137.036.
The last two points require some comments. As in all cases of motion, the action is the
time-volume integral of the Lagrangian density. All fields, be they matter and radiation,
move in such a way that this action remains minimal. In fact there are no known differ-
ences between the prediction of the least action principle based on the QED Lagrangian
density and observations. Even though the Lagrangian density is known since 1926, it
took another twenty years to learn how to calculate with it. Only in the years around 1947
it became clear, through the method of renormalization, that the Lagrangian density of
QED is the final description of all motion of matter due to electromagnetic interaction in
flat space-time. The details were developed independently by Julian Schwinger, Freeman
Dyson, Richard Feynman and Tomonaga Shin’ichiro, four among the smartest physicists
ever. *
The QED Lagrangian density contains the strength of the electromagnetic interaction
in the form of the fine structure constant 𝛼 = 𝑒2 /(4π𝜀0 ℏ𝑐) ≈ 1/137.036(1). This number

Motion Mountain – The Adventure of Physics
is part of the Lagrangian; no explanation for its value is given, and the explanation was
still unknown in the year 2016. It is one of the hardest puzzles of physics. Also the U(1)
gauge group is specific to electromagnetism. All others requirements are valid for every
type of interaction. Indeed, the search for the Lagrangians of the two nuclear interactions
became really focused and finally successful only when the necessary requirements were
clearly spelled out, as we will discover in the rest of this volume.
Vol. I, page 435 The Lagrangian density retains all symmetries that we know from classical physics.
Challenge 74 e Motion is continuous, it conserves energy–momentum and angular momentum, it is
relative, it is right–left symmetric, it is reversible, i.e., symmetric under change of velo-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
city sign, and it is lazy, i.e., it minimizes action. In short, within the limits given by the
quantum of action, also motion due to QED remains predictable.

Interactions and virtual particles
The electromagnetic interaction is exchange of virtual photons. So how can the interac-
tion be attractive? At first sight, any exchange of virtual photons should drive the elec-
trons from each other. However, this is not correct. The momentum of virtual photons
does not have to be in the direction of its energy flow; it can also be in opposite direc-
tion.** Obviously, this is only possible within the limits provided by the indeterminacy
relation.
But virtual particles have also other surprising properties: virtual photons for ex-
ample, cannot be counted.

* Tomonaga Shin’ichiro (b. 1906 Tokio, d. 1979 Tokio) developed quantum electrodynamics and won the
1965 Nobel Prize in Physics together with Feynman and Schwinger. Later he became an important figure of
science politics; together with his class mate from secondary school and fellow physics Nobel Prize winner,
Yukawa Hidei, he was an example to many scientists in Japan.
** One of the most beautiful booklets on quantum electrodynamics which makes this point remains the
text by Richard Feynman, QED: the Strange Theory of Light and Matter, Penguin Books, 1990.
128 3 quantum electrodynamics

Vacuum energy: infinite or zero?
The strangest result of quantum field theory is the energy density of the vacuum. On one
side, the vacuum has, to an excellent approximation, no mass and no energy content. The
vacuum energy of vacuum is thus measured and expected to be zero (or at least extremely
small).*
On the other side, the energy density of the zero-point fluctuations of the electromag-
netic field is given by
𝐸 4πℎ ∞ 3
= 3 ∫ 𝜈 d𝜈 . (19)
𝑉 𝑐 0

The result of this integration is infinite. Quantum field theory thus predicts an infinite
energy density of the vacuum.
We can try to moderate the problem in the following way. As we will discover in the
Vol. VI, page 40 last part of our adventure, there are good arguments that a smallest measurable distance
exists in nature; this smallest length appears when gravity is taken into account. The

Motion Mountain – The Adventure of Physics
minimal distance is of the order of the Planck length

𝑙Pl = √ℏ𝐺/𝑐3 ≈ 1.6 ⋅ 10−35 m . (20)

Vol. VI, page 40 A minimal distance leads to a maximum cut-off frequency. But even in this case the
vacuum density that follows is still a huge number, and is much larger than observed
by over 100 orders of magnitude. In other words, QED seems to predict an infinite, or,
when gravity is taken into account, a huge vacuum energy. But measurements show a
tiny value. What exactly is wrong in this simple calculation? The answer cannot be given

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
at this point; it will become clear in the last volume of our adventure.

Moving mirrors
Mirrors also work when they or the light source is in motion. In contrast, walls, i.e.,
sound mirrors, do not produce echoes for every sound source or for every wall speed.
For example, experiments show that walls do not produce echoes if the wall or the sound
source moves faster than sound. Walls do not produce echoes even if the sound source
moves with them, if both objects move faster than sound. On the other hand, light mir-
rors always produce an image, whatever the involved speed of the light source or the
mirror may be. These observations confirm that the speed of light is the same for all ob-
Challenge 75 s servers: it is invariant and a limit speed. (Can you detail the argument?) In contrast, the
speed of sound in air depends on the observer; it is not invariant.
Light mirrors also differ from tennis rackets. (Rackets are tennis ball mirrors, to con-
Vol. II, page 22 tinue the previous analogy.) We have seen that light mirrors cannot be used to change
the speed of the light they hit, in contrast to what tennis rackets can do with balls. This
observation shows that the speed of light is a limit speed. In short, the simple existence
of mirrors and of their properties are sufficient to derive special relativity.

* In 1998, this side of the issue was confused even further. Astrophysical measurements, confirmed in the
subsequent years, have found that the vacuum energy has a small, but non-zero value, of the order of
0.5 nJ/m3 . The reason for this value is not yet understood, and is one of the open issues of modern physics.
the origin of virtual reality 129

wall
sound
source

no echo
from wall

faster than slower or
sound faster than
sound

Motion Mountain – The Adventure of Physics
mirror
light
source

mirror image
always appears

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
any speed any speed

F I G U R E 80 A fast wall does not produce an echo; a fast mirror does.

But there are more interesting things to be learned from mirrors. We only have to
ask whether mirrors work when they undergo accelerated motion. This issue yields a
surprising result.
In the 1970s, quite a number of researchers independently found that there is no va-
cuum for accelerated observers. This effect is called Fulling–Davies–Unruh effect. (The
incorrect and rarely used term dynamical Casimir effect has been abandoned.) For an ac-
celerated observer, the vacuum is full of heat radiation. We will discuss this below. This
fact has an interesting consequence for accelerated mirrors: a mirror in accelerated mo-
tion reflects the heat radiation it encounters. In short, an accelerated mirror emits light!
Unfortunately, the intensity of this so-called Unruh radiation is so weak that it has not
Page 146 been measured directly, up to now. We will explore the issue in more detail below. (Can
Challenge 76 s you explain why accelerated mirrors emit light, but not matter?)
130 3 quantum electrodynamics

Photons hit ting photons
Usually, light can cross light undisturbed: interference is the proof and the result of this
basic property of light. But there is an exception. When virtual particles are taken into
account, light beams can ‘bang’ onto each other – though only slightly. This result is in
full contrast to classical electrodynamics.
Indeed, QED shows that the appearance of virtual electron-positron pairs allow
photons to hit each other. And such pairs are found in any light beam. However, the
cross-section for photons banging onto each other is small. In other words, the bang is
extremely weak. When two light beams cross, most photons will pass undisturbed. The
cross-section 𝐴 is approximately

973 ℏ 2 ℏ𝜔 6
𝐴≈ 𝛼4 ( ) ( ) (21)
10 125π 𝑚e 𝑐 𝑚e 𝑐2

Motion Mountain – The Adventure of Physics
for the everyday case that the energy ℏ𝜔 of the photon is much smaller than the rest en-
ergy 𝑚e 𝑐2 of the electron. This low-energy value is about 18 orders of magnitude smaller
than what was measurable in 1999; the future will show whether the effect will ever be
observable for visible light. However, for high energy photons these effects are observed
daily in particle accelerators. In these settings one observes not only interaction through
virtual electron–antielectron pairs, but also through virtual muon–antimuon pairs, vir-
tual quark–antiquark pairs, and much more.
Everybody who consumes science fiction knows that matter and antimatter annihilate
and transform into pure light. More precisely, a matter particle and an antimatter particle
annihilate into two or more photons. Interestingly, quantum theory predicts that the op-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
posite process is also possible: photons hitting photons can produce matter! In 1997, this
Ref. 100 prediction was also confirmed experimentally.
At the Stanford particle accelerator, photons from a high energy laser pulse were
bounced off very fast electrons. In this way, the reflected photons acquired a large energy,
when seen in the inertial frame of the experimenter. The green laser pulse, of 527 nm
wavelength or 2.4 eV photon energy, had a peak power density of 1022 W/m2 , about the
highest achievable so far. That is a photon density of 1034 /m3 and an electric field of
1012 V/m, both of which were record values at the time. When this green laser pulse was
reflected off a 46.6 GeV electron beam, the returning photons had an energy of 29.2 GeV
Challenge 77 e and thus had become high-energy gamma rays. These gamma rays then collided with
other, still incoming green photons and produced electron–positron pairs through the
reaction
γ29.2GeV + 𝑛 γgreen → e+ + e− (22)

for which both final particles were detected by special apparatuses. The experiment thus
showed that light can hit light in nature, and above all, that doing so can produce matter.
This is the nearest we can get to the science fiction fantasy of light swords or of laser
swords banging onto each other.
the origin of virtual reality 131

Is the vacuum a bath?
If the vacuum is a sea of virtual photons and particle–antiparticle pairs, vacuum could
be suspected to act as a bath. In general, the answer is negative. Quantum field theory
works because the vacuum is not a bath for single particles. However, there is always an
exception. For dissipative systems made of many particles, such as electrical conductors,
Ref. 101 the vacuum can act as a viscous fluid. Irregularly shaped, neutral, but conducting bodies
can emit photons when accelerated, thus damping such type of motion. This is due to the
Fulling–Davies–Unruh effect, as described above. The damping depends on the shape
and thus also on the direction of the body’s motion.
Vol. I, page 391 In 1998, Gour and Sriramkumar even predicted that Brownian motion should also
appear for an imperfect, i.e., partly absorbing mirror placed in vacuum. The fluctuations
Ref. 102 of the vacuum should produce a mean square displacement

ℏ
⟨𝑑2 ⟩ = 𝑡 (23)

Motion Mountain – The Adventure of Physics
𝑚
that increases linearly with time; however, the extremely small displacement produced
in this way is out of experimental reach so far. But the result is not a surprise. Are you
Challenge 78 ny able to give another, less complicated explanation for it?

R enormalization – why is an electron so light?
In classical physics, the field energy of a point-like charged particle, and hence its mass,
Vol. III, page 243 was predicted to be infinite. QED effectively smears out the charge of the electron over its
Compton wavelength; as a result, the field energy contributes only a small correction to

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 79 s its total mass. Can you confirm this?
QED is a perturbative description. This means, that any predicted result 𝑅 is found as
a Taylor series of powers of a small parameter:

𝑅 = 𝑅0 + 𝑅1 𝛼 + 𝑅2 𝛼2 + 𝑅3 𝛼3 + 𝑅4 𝛼4 + ... (24)

In QED, the small parameter is the fine structure constant 𝛼 = 1/137.036(1). With the
help of the perturbation series, the exact result 𝑅 is approximated more and more pre-
cisely.
Now, in QED, many intermediate results in the perturbation expansion are divergent
integrals, i.e., integrals with infinite value. The divergence is due to the assumption that
infinitely small distances are possible in nature. The divergences thus are artefacts that
can be eliminated; the elimination procedure is called renormalization.
Sometimes it is claimed that the infinities appearing in quantum electrodynamics in
the intermediate steps of the calculation show that the theory is incomplete or wrong.
However, this type of statement would imply that classical physics is also incomplete or
wrong, on the ground that in the definition of the velocity 𝑣 with space 𝑥 and time 𝑡,
namely
d𝑥 Δ𝑥 1
𝑣= = lim = lim Δ𝑥 , (25)
d𝑡 Δ𝑡→0 Δ𝑡 Δ𝑡→0 Δ𝑡
132 3 quantum electrodynamics

cards
or
bricks l

table

F I G U R E 81 What is the maximum possible value of h/l?

one gets an infinity as intermediate step. Indeed, d𝑡 being vanishingly small, one could

Motion Mountain – The Adventure of Physics
argue that one is dividing by zero. Both arguments show the difficulty to accept that the
result of a limit process can be a finite quantity even if infinite quantities appear in the
calculation. The parallel between electron mass and velocity is closer than it seems; both
intermediate ‘infinities’ stem from the assumption that space-time is continuous, i.e., in-
finitely divisible. The infinities necessary in limit processes for the definition of differen-
tiation, integration or for renormalization appear only when space-time is approximated,
as physicists say, as a ‘continuous’ set, or as mathematicians say, as a ‘complete’ set.
On the other hand, the conviction that the appearance of an infinity might be a sign
of incompleteness of a theory was an interesting development in physics. It shows how

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
uncomfortable many physicists had become with the use of infinity in our description
Ref. 103 of nature. Notably, this was the case for Paul Dirac himself, who, after having laid in his
youth the basis of quantum electrodynamics, has tried for the rest of his life to find a way,
without success, to change the theory so that intermediate infinities are avoided.
Renormalization is a procedure that follows from the requirement that continuous
space-time and gauge theories must work together. In particular, renormalization fol-
lows form the requirement that the particle concept is consistent, i.e., that perturbation
expansions are possible. Intermediate infinities are not an issue. In a bizarre twist, a few
decades after Dirac’s death, his wish has been fulfilled after all, although in a different
manner than he envisaged. The final part of this mountain ascent will show the way out
Vol. VI, page 37 of the issue.

Curiosities and fun challenges of quantum electrodynamics
Motion is an interesting topic, and when a curious person asks a question about it, most
of the time quantum electrodynamics is needed for the answer. Together with gravity,
quantum electrodynamics explains almost all of our everyday experience, including nu-
merous surprises. Let us have a look at some of them.
∗∗
A famous riddle, illustrated in Figure 81, asks how far the last card (or the last brick) of
a stack can hang over the edge of a table. Of course, only gravity, no glue nor any other
the origin of virtual reality 133

means is allowed to keep the cards on the table. After you solved the riddle, can you give
Challenge 80 s the solution in case that the quantum of action is taken into account?
∗∗
Quantum electrodynamics explains why there are only a finite number of different
Ref. 104 atom types. In fact, it takes only two lines to prove that pair production of electron–
antielectron pairs make it impossible that a nucleus has more than about 137 protons.
Challenge 81 s Can you show this? In short, the fine structure constant limits the number of chemical
elements in nature. The effect at the basis of this limit, the polarization of the vacuum,
Page 153 also plays a role in much larger systems, such as charged black holes, as we will see shortly.
∗∗
Stripping 91 of the 92 electrons off an uranium atom allows researchers to check with
high precision whether the innermost electron still is described by QED. The electric field
near the uranium nucleus, 1 EV/m, is the highest achievable in the laboratory; the field

Motion Mountain – The Adventure of Physics
value is near the threshold for spontaneous pair production. The field is the highest con-
stant field producible in the laboratory, and an ideal testing ground for precision QED
experiments. The effect of virtual photons is to produce a Lamb shift; but even for these
Ref. 105 extremely high fields, the value matches the calculation.
∗∗
Is there a critical magnetic field in nature, like there is a critical electric field, limited by
Challenge 82 ny spontaneous pair production?
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Microscopic evolution can be pretty slow. Light, especially when emitted by single atoms,
is always emitted by some metastable state. Usually, the decay times, being induced by
the vacuum fluctuations, are much shorter than a microsecond. However, there are meta-
stable atomic states with a lifetime of ten years: for example, an ytterbium ion in the 2 𝐹7/2
state achieves this value, because the emission of light requires an octupole transition, in
Ref. 106 which the angular momentum changes by 3ℏ; this is an extremely unlikely process.
In radioactive decay, the slowness record is held by 209 Bi, with over 1019 years of half-
Page 342 life.
∗∗
Microscopic evolution can be pretty fast. Can you imagine how to deduce or to measure
Challenge 83 s the speed of electrons inside atoms? And inside metals?
∗∗
If an electrical wire is sufficiently narrow, its electrical conductance is quantized in steps
of 2𝑒2 /ℏ. The wider the wire, the more such steps are added to its conductance. Can you
Challenge 84 s explain the effect? By the way, quantized conductance has also been observed for light
Ref. 109 and for phonons.
∗∗
134 3 quantum electrodynamics

The Casimir effect, as well as other experiments, imply that there is a specific and finite
energy density that can be ascribed to the vacuum. Does this mean that we can apply the
Challenge 85 d Banach–Tarski effect to pieces of vacuum?
∗∗
Challenge 86 s Can you explain why mud is not clear?
∗∗
The instability of the vacuum also yields a (trivial) limit on the fine structure constant.
The fine structure constant value of around 1/137.036 cannot be explained by quantum
Ref. 201 electrodynamics. However, it can be deduced that it must be lower than 1 to lead to
a consistent theory. Indeed, if its value were larger than 1, the vacuum would become
unstable and would spontaneously generate electron-positron pairs.
∗∗

Motion Mountain – The Adventure of Physics
Challenge 87 s Can the universe ever have been smaller than its own Compton wavelength?
∗∗
In the past, the description of motion with formulae was taken rather seriously. Before
computers appeared, only those examples of motion were studied that could be described
with simple formulae. But this narrow-minded approach turns out to be too restrictive.
Indeed, mathematicians showed that Galilean mechanics cannot solve the three-body
problem, special relativity cannot solve the two-body problem, general relativity the one-
body problem and quantum field theory the zero-body problem. It took some time to the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
community of physicists to appreciate that understanding motion does not depend on
the description by formulae, but on the description by clear equations based on space
and time.
∗∗
In fact, quantum electrodynamics, or QED, provides a vast number of curiosities and
every year there is at least one interesting new discovery. We now conclude the theme
with a more general approach.

How can one move on perfect ice? – The ultimate physics test
In our quest, we have encountered motion of many sorts. Therefore, the following test –
not to be taken too seriously – is the ultimate physics test, allowing you to check your
understanding and to compare it with that of others.
Imagine that you are on a perfectly frictionless surface and that you want to move
to its border. How many methods can you find to achieve this? Any method, so tiny its
effect may be, is allowed.
Classical physics provided quite a number of methods. We saw that for rotating
ourselves, we just need to turn our arm above the head. For translation motion, throwing
a shoe or inhaling vertically and exhaling horizontally are the simplest possibilities. Can
Challenge 88 s you list at least six additional methods, maybe some making use of the location of the
surface on Earth? What would you do in space?
the origin of virtual reality 135

Electrodynamics and thermodynamics taught us that in vacuum, heating one side of
the body more than the other will work as motor; the imbalance of heat radiation will
push you, albeit rather slowly. Are you able to find at least four other methods from these
Challenge 89 s two domains?
General relativity showed that turning one arm will emit gravitational radiation un-
Challenge 90 s symmetrically, leading to motion as well. Can you find at least two better methods?
Quantum theory offers a wealth of methods. Of course, quantum mechanics shows
that we actually are always moving, since the indeterminacy relation makes rest an im-
possibility. However, the average motion can be zero even if the spread increases with
time. Are you able to find at least four methods of moving on perfect ice due to quantum
Challenge 91 s effects?
Materials science, geophysics, atmospheric physics and astrophysics also provide ways
Challenge 92 s to move, such as cosmic rays or solar neutrinos. Can you find four additional methods?
Self-organization, chaos theory and biophysics also provide ways to move, when the
inner workings of the human body are taken into account. Can you find at least two

Motion Mountain – The Adventure of Physics
Challenge 93 s methods?
Assuming that you read already the section following the present one, on the effects
of semiclassical quantum gravity, here is an additional puzzle: is it possible to move by
Challenge 94 s accelerating a pocket mirror, using the emitted Unruh radiation? Can you find at least
two other methods to move yourself using quantum gravity effects? Can you find one
from string theory?
If you want points for the test, the marking is simple. For students, every working
method gives one point. Eight points is ok, twelve points is good, sixteen points is very
good, and twenty points or more is excellent. For graduated physicists, the point is given

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
only when a back-of-the-envelope estimate for the ensuing momentum or acceleration
is provided.

A summary of quantum electrodynamics
The shortest possible summary of quantum electrodynamics is the following:

⊳ Everyday matter is made of charged elementary particles that interact
through photon exchange in the way described by Figure 82.

No additional information is necessary. In a bit more detail, quantum electrodynamics
starts with elementary particles – characterized by their mass, spin, charge, and parities –
and with the vacuum, essentially a sea of virtual particle–antiparticle pairs. Interactions
between charged particles are described as the exchange of virtual photons, and electro-
magnetic decay is described as the interaction with the virtual photons of the vacuum.

⊳ The Feynman diagram of Figure 82 provides an exact description of all elec-
tromagnetic phenomena and processes.

No contradiction between observation and calculation are known. In particular, the
Page 126 Feynman diagram is equivalent to the QED Lagrangian of equation (18). Because QED
is a perturbative theory, the Feynman diagram directly describes the first order effects;
136 3 quantum electrodynamics

γ (photon), i.e.
el.m. radiation:
uncharged,
massless,
charged spin S=1
matter, i.e.
charged
lepton or
quark:
spin S = 1/2,
m>0 interaction:
√α = 1/11.7062...
1/α = 137.0359...
Σ E = const
Σ p = const
Σ S = const F I G U R E 82 The basis of QED; more

Motion Mountain – The Adventure of Physics
Σ q = const precisely, the fundamental diagram
of QED as a perturbation theory in
space-time.

its composite diagrams describe effects of higher order. QED is a perturbative theory.
QED describes all everyday properties of matter and radiation. It describes the divis-
ibility down to the smallest constituents, the isolability from the environment and the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
impenetrability of matter. It also describes the penetrability of radiation. All these prop-
erties are due to electromagnetic interactions of constituents and follow from Figure 82.
Matter is divisible because the interactions are of finite strength, matter is also divisible
because the interactions are of finite range, and matter is impenetrable because inter-
actions among the constituents increase in intensity when they approach each other, in
particular because matter constituents are fermions. Radiation is divisible into photons,
and is penetrable because photons are bosons and first order photon-photon interactions
do not exist.
Both matter and radiation are made of elementary constituents. These elementary
constituents, whether bosons or fermions, are indivisible, isolable, indistinguishable, and
point-like.
It is necessary to use quantum electrodynamics in all those situations for which the
characteristic dimensions 𝑑 are of the order of the Compton wavelength

ℎ
𝑑 ≈ 𝜆C = . (26)
𝑚𝑐
In situations where the dimensions are of the order of the de Broglie wavelength, or equi-
valently, where the action is of the order of the Planck value, simple quantum mechanics
is sufficient:
ℎ
𝑑 ≈ 𝜆 dB = . (27)
𝑚𝑣
the origin of virtual reality 137

For even larger dimensions, classical physics will do.
Together with gravity, quantum electrodynamics explains almost all observations of
motion on Earth; QED unifies the description of matter and electromagnetic radiation
in daily life. All everyday objects and all images are described, including their prop-
erties, their shape, their transformations and their other changes. This includes self-
organization and chemical or biological processes. In other words, QED gives us full grasp
of the effects and the variety of motion due to electromagnetism.

Open questions in QED
Even though QED describes motion due to electromagnetism without any discrepancy
from experiment, that does not mean that we understand every detail of every example
of such motion. For example, nobody has described the motion of an animal with QED
yet.* In fact, there is beautiful and fascinating research going on in many branches of
electromagnetism.
Atmospheric physics still provides many puzzles and regularly delivers new, previ-

Motion Mountain – The Adventure of Physics
Ref. 111 ously unknown phenomena. For example, the detailed mechanisms at the origin of au-
rorae are still controversial; and the recent unexplained discoveries of discharges above
Ref. 112 clouds should not make one forget that even the precise mechanism of charge separation
inside clouds, which leads to lightning, is not completely clarified. In fact, all examples
of electrification, such as the charging of amber through rubbing, the experiment which
gave electricity its name, are still poorly understood.
Materials science in all its breadth, including the study of solids, fluids, and plasmas,
as well as biology and medicine, still provides many topics of research. In particular, the
twenty-first century will undoubtedly be the century of the life sciences.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The study of the interaction of atoms with intense light is an example of present re-
search in atomic physics. Strong lasers can strip atoms of many of their electrons; for
such phenomena, there are not yet precise descriptions, since they do not comply to the
weak field approximations usually assumed in physical experiments. In strong fields, new
Ref. 113 effects take place, such as the so-called Coulomb explosion.
But also the skies have their mysteries. In the topic of cosmic rays, it is still not clear
Ref. 114 how rays with energies of 1022 eV are produced outside the galaxy. Researchers are in-
tensely trying to locate the electromagnetic fields necessary for their acceleration and to
understand their origin and mechanisms.
In the theory of quantum electrodynamics, discoveries are expected by all those who
Ref. 115 study it in sufficient detail. For example, Dirk Kreimer has discovered that higher order
interaction diagrams built using the fundamental diagram of Figure 82 contain relations
to the theory of knots. This research topic will provide even more interesting results in the
near future. Relations to knot theory appear because QED is a perturbative description,
with the vast richness of its non-perturbative effects still hidden. Studies of QED at high
energies, where perturbation is not a good approximation and where particle numbers
are not conserved, promise a wealth of new insights.

* On the other hand, outside QED, there is beautiful work going on how humans move their limbs; it seems
that any general human movement is constructed in the brain by combining a small set of fundamental
Ref. 110 movements.
138 3 quantum electrodynamics

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 83 The rainbow can only be explained fully if the ﬁne structure constant 𝛼 can be calculated
(© ed g2s, Christophe Afonso).

If we want to be very strict, we need to add that we do not fully understand any colour,
because we still do not know the origin of the fine structure constant. In particular, the
Ref. 116 fine structure constant determines the refractive index of water, and thus the formation
of a rainbow, as pictured in Figure 83.
Many other open issues of more practical nature have not been mentioned. Indeed,
by far the largest numbers of physicists get paid for some form of applied QED. In our
the origin of virtual reality 139

adventure however, our quest is the description of the fundamentals of motion. And so
far, we have not achieved it. In particular, we still need to understand motion in the realm
of atomic nuclei and the effect of the quantum of action in the domain of gravitation. We
start with the latter topic.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Chapter 4

QUA N T UM M E C HA N IC S W I T H
G R AV I TAT ION – F I R ST ST E P S

G
ravitation is a weak effect. Indeed, every seaman knows that storms, not
ravity, cause the worst accidents. Despite its weakness, the inclusion of
ravity into quantum theory raises a number of issues. We must solve them
all in order to complete our mountain ascent.

Motion Mountain – The Adventure of Physics
Gravity acts on quantum systems: in the chapter on general relativity we already men-
tioned that light frequency changes with height. Thus gravity has a simple and measur-
able effect on photons. But gravity also acts on all other quantum systems, such as atoms
and neutrons, as we will see. And the quantum of action plays an important role in the
behaviour of black holes. We explore these topics now.

Falling atoms
In 2004 it finally became possible to repeat Galileo’s leaning tower experiment with
single atoms instead of steel balls. This is not an easy experiment because even the smal-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 117 lest effects disturb the motion. The result is as expected: single atoms do fall like stones.
In particular, atoms of different mass fall with the same acceleration, within the experi-
mental precision of one part in 6 million.
The experiment was difficult to perform, but the result is not surprising, because all
falling everyday objects are made of atoms. Indeed, Galileo himself had predicted that
atoms fall like stones, because parts of a body have to fall with the same acceleration
Vol. I, page 202 as the complete body. But what is the precise effect of gravity on wave functions? This
question is best explored with the help of neutrons.

Playing table tennis with neu trons
The gravitational potential also has directly measurable effects on quantum particles.
Classically, a table tennis ball follows a parabolic path when bouncing over a table tennis
table, as long as friction can be neglected. The general layout of the experiment is shown
in Figure 84. How does a quantum particle behave in the same setting?
In the gravitational field, a bouncing quantum particle is still described by a wave
function. In contrast to the classical case however, the possible energy values of a falling
quantum particle are discrete. Indeed, the quantization of the action implies that for a
4 quantum mechanics with gravitation – first steps 141

can be
sticky ceiling / neutron absorber lowered

path of table tennis ball / neutron

Motion Mountain – The Adventure of Physics
table tennis table / neutron mirror
(wooden plate / silicon crystal)

Figure to be completed

F I G U R E 84 Table tennis and neutrons.

Challenge 95 e bounce of energy 𝐸𝑛 and duration 𝑡𝑛 ,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝐸3/2
𝑛ℏ ∼ 𝐸𝑛 𝑡𝑛 ∼ 𝑛1/2 . (28)
𝑔𝑚

In other words, only discrete bounce heights, distinguished by the number 𝑛, are pos-
sible in the quantum case. The discreteness leads to an expected probability density that
changes with height in discrete steps, as shown in Figure 84.
The best way to realize the experiment with quantum particles is to produce an in-
tense beam of neutral particles, because neutral particles are not affected by the stray
electromagnetic fields that are present in every laboratory. Neutrons are ideal, as they are
produced in large quantities by nuclear reactors. The experiment was first performed in
Ref. 118 2002, by Hartmut Abele and his group, after years of preparations. Using several clever
tricks, they managed to slow down neutrons from a nuclear reactor to the incredibly
small value of 8 m/s, comparable to the speed of a table tennis ball. (The equivalent tem-
perature of these ultracold neutrons is 1 mK, or 100 neV.) They then directed the neut-
rons onto a neutron mirror made of polished glass – the analogue of the table tennis table
– and observed the neutrons bouncing back up. To detect the bouncing, they lowered an
absorber – the equivalent of a sticky ceiling – towards the table tennis table, i.e., towards
the neutron mirror, and measured how many neutrons still reached the other end of the
table. (Both the absorber and the mirror were about 20 cm in length.)
Why did the experiment take so many years of work? The lowest energy levels for
neutrons due to gravity are 2.3 ⋅ 10−31 J, or 1.4 peV, followed by 2.5 peV,3.3 peV, 4.1 peV,
142 4 quantum mechanics with gravitation – first steps

length l

beam II
height h

beam I

neutron silicon silicon
beam beam mirror
splitter

F I G U R E 85 The weakness of gravitation. A neutron interferometer made of a silicon single crystal (with
the two neutron beams I and II) can be used to detect the effects of gravitation on the phase of wave
functions (photo © Helmut Rauch and Erwin Seidl).

Motion Mountain – The Adventure of Physics
and so forth. To get an impression of the smallness of these values, we can compare it
to the value of 2.2 ⋅ 10−18 J or 13.6 eV for the lowest state in the hydrogen atom. Despite
these small energy values, the team managed to measure the first few discrete energy
levels. The results confirmed the prediction of the Schrödinger equation, with the grav-
itational potential included, to the achievable measurement precision.
In short, gravity influences wave functions. In particular, gravity changes the phase of
wave functions, and does so as expected.

The gravitational phase of wave functions

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Not only does gravity change the shape of wave functions; it also changes their phase. Can
Challenge 96 s you imagine why? The prediction was first confirmed in 1975, using a device invented
Ref. 119 by Helmut Rauch and his team. Rauch had developed neutron interferometers based
on single silicon crystals, shown in Figure 85, in which a neutron beam – again from a
nuclear reactor – is split into two beams and the two beams are then recombined and
brought to interference.
By rotating the interferometer mainly around the horizontal axis, Samuel Werner and
his group let the two neutron beams interfere after having climbed a small height ℎ at
Ref. 120 two different locations. The experiment is shown schematically on the right of Figure 85.
The neutron beam is split; the two beams are deflected upwards, one directly, one a few
centimetres further on, and then recombined.
For such a experiment in gravity, quantum theory predicts a phase difference Δ𝜑
Challenge 97 ny between the two beams given by

𝑚𝑔ℎ𝑙
Δ𝜑 = , (29)
ℏ𝑣
where 𝑙 is the horizontal distance between the two climbs and 𝑣 and 𝑚 are the speed and
mass of the neutrons. All experiments – together with several others of similar simple
elegance – have confirmed the prediction by quantum theory within experimental errors.
Ref. 121 In the 1990s, similar experiments have even been performed with complete atoms.
4 quantum mechanics with gravitation – first steps 143

These atom interferometers are so sensitive that local gravity 𝑔 can be measured with a
precision of more than eight significant digits.
In short, neutrons, atoms and photons show no surprises in gravitational fields. Grav-
ity can be included into all quantum systems of everyday life. By including gravity in the
potential, the Schrödinger and Dirac equations can thus be used, for example, to describe
the growth and the processes inside trees. Trees can mostly be described with quantum
electrodynamics in weak gravity.

The gravitational B ohr atom
Can gravity lead to bound quantum systems? A short calculation shows that an electron
circling a proton due to gravity alone, without electrostatic attraction, would do so at a
Challenge 98 ny gravitational Bohr radius of

ℏ2
𝑟gr.B. = = 1.1 ⋅ 1029 m (30)

Motion Mountain – The Adventure of Physics
𝐺 𝑚2e 𝑚p

which is about a thousand times the distance to the cosmic horizon. A gravitational Bohr
atom would be larger than the universe. This enormous size is the reason that in a nor-
mal hydrogen atom there is not a single way to measure gravitational effects between its
Challenge 99 e components. (Are you able to confirm this?)
But why is gravity so weak? Or equivalently, why are the universe and normal atoms
so much smaller than a gravitational Bohr atom? At the present point of our quest these
questions cannot be answered. Worse, the weakness of gravity even means that with
high probability, future experiments will provide little additional data helping to decide

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
among competing answers. The only help is careful thought.
We might conclude from all this that gravity does not require a quantum description.
Indeed, we stumbled onto quantum effects because classical electrodynamics implies,
in stark contrast with reality, that atoms decay in about 0.1 ns. Classically, an orbiting
electron would emit radiation until it falls into the nucleus. Quantum theory is thus ne-
cessary to explain the existence of matter.
When the same stability calculation is performed for the emission of gravitational
Challenge 100 ny radiation by orbiting electrons, one finds a decay time of around 1037 s. (True?) This
extremely large value, trillions of times longer than the age of the universe, is a result of
Vol. II, page 179 the low emission of gravitational radiation by rotating masses. Therefore, the existence
of normal atoms does not require a quantum theory of gravity.

Curiosities ab ou t quantum theory and gravit y
Due to the influence of gravity on phases of wave functions, some people who do not
believe in bath induced decoherence have even studied the influence of gravity on the
Ref. 145 decoherence process of usual quantum systems in flat space-time. Predictably, the calcu-
lated results do not reproduce experiments.
∗∗
Despite its weakness, gravitation provides many puzzles. Most famous are a number of
144 4 quantum mechanics with gravitation – first steps

curious coincidences that can be found when quantum mechanics and gravitation are
combined. They are usually called ‘large number hypotheses’ because they usually in-
volve large dimensionless numbers. A pretty, but less well known version connects the
Ref. 146 Planck length, the cosmic horizon 𝑅0 , and the number of baryons 𝑁b :

3 𝑅0 4 𝑡0 4
(𝑁b ) ≈ ( ) = ( ) ≈ 10244 (31)
𝑙Pl 𝑡Pl

in which 𝑁b = 1081 and 𝑡0 = 1.2 ⋅ 1010 a were used. There is no known reason why the
number of baryons and the horizon size 𝑅0 should be related in this way. This coincid-
ence is equivalent to the one originally stated by Dirac,* namely

ℏ2
𝑚3p ≈ . (33)
𝐺𝑐𝑡0

Motion Mountain – The Adventure of Physics
where 𝑚p is the proton mass. This approximate equality seems to suggest that certain
microscopic properties, namely the mass of the proton, is connected to some general
properties of the universe as a whole. This has lead to numerous speculations, especially
since the time dependence of the two sides differs. Some people even speculate whether
relations (31) or (33) express some long-sought relation between local and global topo-
Ref. 148 logical properties of nature. Up to this day, the only correct statement seems to be that
they are coincidences connected to the time at which we happen to live, and that they
should not be taken too seriously.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
Photons not travelling parallel to each other attract each other through gravitation and
thus deflect each other. Could two such photons form a bound state, a sort of atom of
light, in which they would circle each other, provided there were enough empty space
Challenge 101 s for this to happen?

In summary, quantum gravity is unnecessary in every single domain of everyday life.
However, we will see now that quantum gravity is necessary in domains which are more
remote, but also more fascinating.

Ref. 147 * The equivalence can be deduced using 𝐺𝑛b 𝑚p = 1/𝑡20 , which, as Weinberg explains, is required by several
Vol. VI, page 104 cosmological models. Indeed, this can be rewritten simply as

𝑚20 /𝑅20 ≈ 𝑚2Pl /𝑅2Pl = 𝑐4 /𝐺2 . (32)

Together with the definition of the baryon density 𝑛b = 𝑁b /𝑅30 one gets Dirac’s large number hypothesis,
substituting protons for pions. Note that the Planck time and length are defined as √ℏ𝐺/𝑐5 and √ℏ𝐺/𝑐3
and are the natural units of length and time. We will study them in detail in the last part of the mountain
ascent.
4 quantum mechanics with gravitation – first steps 145

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 86 A simpliﬁed simulated image – not a photograph – of how a black hole of ten solar masses,
with Schwarzschild radius of 30 km, seen from a constant distance of 600 km, will distort an image of
the Milky Way in the background (image © Ute Kraus at www.tempolimit-lichtgeschwindigkeit.de).

Gravitation and limits to disorder

“
Die Energie der Welt ist constant.

”
Die Entropie der Welt strebt einem Maximum zu.*
Rudolph Clausius

We have already encountered the famous statement by Clausius, the father of the term
Vol. I, page 256 ‘entropy’. We have also found that the Boltzmann constant 𝑘 is the smallest entropy value
found in nature.
What is the influence of gravitation on entropy, and on thermodynamics in general?
For a long time, nobody was interested in this question. In parallel, for many decades
nobody asked whether there also exists a theoretical maximum for entropy. The situ-
ations changed dramatically in 1973, when Jacob Bekenstein discovered that the two is-
sues are related.
Ref. 122 Bekenstein was investigating the consequences gravity has for quantum physics. He

* ‘The energy of the universe is constant. Its entropy tends towards a maximum.’
146 4 quantum mechanics with gravitation – first steps

found that the entropy 𝑆 of an object of energy 𝐸 and size 𝐿 is bound by

𝑘π
𝑆 ⩽ 𝐸𝐿 (34)
ℏ𝑐
for all physical systems, where 𝑘 is the Boltzmann constant. In particular, he deduced that
(nonrotating) black holes saturate the bound. We recall that black holes are the densest
Vol. II, page 262 systems for a given mass. They occur when matter collapses completely. Figure 86 shows
an artist’s impression.
Challenge 102 s Bekenstein found that black holes have an entropy given by

𝑘𝑐3 4π𝑘𝐺
𝑆=𝐴 = 𝑀2 (35)
4𝐺ℏ ℏ𝑐

where 𝐴 is now the area of the horizon of the black hole. It is given by 𝐴 = 4π𝑅2 =

Motion Mountain – The Adventure of Physics
4π(2𝐺𝑀/𝑐2 )2 . In particular, the result implies that every black hole has an entropy. Black
holes are thus disordered systems described by thermodynamics. In fact, black holes are
the most disordered systems known.*
As an interesting note, the maximum entropy also implies an upper memory limit for
Challenge 103 s memory chips. Can you find out how?
Black hole entropy is somewhat mysterious. What are the different microstates leading
to this macroscopic entropy? It took many years to convince physicists that the micro-
states are due to the various possible states of the black hole horizon itself, and that they
Ref. 124 are somehow due to the diffeomorphism invariance at this boundary. As Gerard ’t Hooft
explains, the entropy expression implies that the number of degrees of freedom of a black

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 104 s hole is about (but not exactly) one per Planck area of the horizon.
If black holes have entropy, they must have a temperature. What does this temper-
ature mean? In fact, nobody believed this conclusion until two unrelated developments
confirmed it within a short time.
Measuring acceleration with a thermometer:
Fulling–Davies–Unruh radiation
Independently, Stephen Fulling in 1973, Paul Davies in 1975 and William Unruh in 1976
Ref. 125 made the same theoretical discovery while studying quantum theory: if an inertial ob-
server observes that he is surrounded by vacuum, a second observer accelerated with
respect to the first does not: he observes black body radiation. The appearance of radi-
ation for an accelerated observer in vacuum is called the Fulling–Davies–Unruh effect. All

* The precise discussion that black holes are the most disordered systems in nature is quite subtle. The issue
Ref. 123 is summarized by Bousso. Bousso claims that the area appearing in the maximum entropy formula cannot
be taken naively as the area at a given time, and gives four arguments why this should be not allowed.
However, all four arguments are wrong in some way, in particular because they assume that lengths smaller
than the Planck length or larger than the universe’s size can be measured. Ironically, he brushes aside some
of the arguments himself later in the paper, and then deduces an improved formula, which is exactly the
same as the one he criticizes first, just with a different interpretation of the area 𝐴. Later in his career, Bousso
revised his conclusions; he now supports the maximum entropy bound. In short, the expression of black
hole entropy is indeed the maximum entropy for a physical system with surface 𝐴.
4 quantum mechanics with gravitation – first steps 147

these results about black holes were waiting to be discovered since the 1930s; incredibly,
nobody had thought about them for the subsequent 40 years.
The radiation has a spectrum corresponding to the temperature

ℏ
𝑇= 𝑎, (36)
2π𝑘𝑐
where 𝑎 is the magnitude of the acceleration. The result means that there is no vacuum
on Earth, because any observer on its surface can maintain that he is accelerated with
9.8 m/s2 , thus leading to 𝑇 = 40 zK! We can thus measure gravity, at least in principle,
using a thermometer. However, even for the largest practical accelerations the temperat-
ure values are so small that it is questionable whether the effect will ever be confirmed
Ref. 126 experimentally in this precise way. But if it will, it will be a beautiful experimental result.
When this effect was predicted, people explored all possible aspects of the argument.
For example, also an observer in rotational motion detects radiation following expression

Motion Mountain – The Adventure of Physics
(36). But that was not all. It was found that the simple acceleration of a mirror leads to
radiation emission! Mirrors are thus harder to accelerate than other bodies of the same
mass.
When the acceleration is high enough, also matter particles can be emitted and de-
tected. If a particle counter is accelerated sufficiently strongly across the vacuum, it will
start counting particles! We see that the difference between vacuum and matter becomes
fuzzy at large accelerations. This result will play an important role in the search for uni-
Vol. VI, page 65 fication, as we will discover later on.
Surprisingly, at the end of the twentieth century it became clear that the Fulling–
Davies–Unruh effect possibly had already been observed before it was predicted! The

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Fulling–Davies–Unruh effect turned out be related to a well-established observation: the
Ref. 127 so-called Sokolov–Ternov effect. In 1963, the Russian physicist Igor Ternov, together with
Arsenji Sokolov, had used the Dirac equation to predict that electrons in circular accel-
erators and in storage rings that circulate at high energy would automatically polarize.
The prediction was first confirmed by experiments at the Russian Budker Institute of
Nuclear Physics in 1971, and then confirmed by experiments in Orsay, in Stanford and
in Hamburg. Nowadays, the effect is used routinely in many accelerator experiments. In
the 1980s, Bell and Leinaas realized that the Sokolov–Ternov effect is the same effect as
Ref. 127 the Fulling–Davies–Unruh effect, but seen from a different reference frame! The equival-
ence is somewhat surprising. In charges moving in a storage ring, the emitted radiation
is not thermal, so that the analogy is not obvious or simple. But the effect that polarizes
the beam – namely the difference in photon emission for spins that are parallel and anti-
parallel to the magnetic field – is the same as the Fulling–Davies–Unruh effect. We thus
have another case of a theoretical discovery that was made much later than necessary. In
Ref. 127 2006 however, this equivalence was put into question again. The issue is not closed.

Black holes aren ’ t black
In 1973 and 1974, Jacob Bekenstein, and independently, Stephen Hawking, famous for the
intensity with which he fights a disease which forces him into the wheelchair, surprised
the world of general relativity with a fundamental theoretical discovery. They found that
148 4 quantum mechanics with gravitation – first steps

space station
with dynamo

rope
mirror
box filled
with light

black horizon
hole

support
shell

Motion Mountain – The Adventure of Physics
F I G U R E 87 A thought experiment
allowing you to deduce the existence
of black hole radiation.

if a virtual particle–antiparticle pair appeared in the vacuum near the horizon, there is
a finite chance that one particle escapes as a real particle, while the virtual antiparticle is
captured by the black hole. The virtual antiparticle is thus of negative energy, and reduces

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the mass of the black hole. The mechanism applies both to fermions and bosons. From far
away this effect looks like the emission of a particle. A detailed investigation showed that
the effect is most pronounced for photon emission. In short, Bekenstein and Hawking
showed that black holes radiate as black bodies.
Black hole radiation confirms both the result on black hole entropy by Bekenstein
and the effect for observers accelerated in vacuum found by Fulling, Davies and Unruh.
When all this became clear, a beautiful thought experiment was published by William
Ref. 128 Unruh and Robert Wald, showing that the whole result could have been deduced already
50 years earlier!
Shameful as this delay of the discovery is for the community of theoretical physicists,
the story itself remains beautiful. It starts in the early 1970s, when Robert Geroch stud-
ied the issue shown in Figure 87. Imagine a mirror box full of heat radiation, thus full of
light. The mass of the box is assumed to be negligible, such as a box made of thin alu-
minium paper. We lower the box, with all its contained radiation, from a space station
towards a black hole. On the space station, lowering the weight of the heat radiation al-
lows generating energy. Obviously, when the box reaches the black hole horizon, the heat
radiation is red-shifted to infinite wavelength. At that point, the full amount of energy
originally contained in the heat radiation has been provided to the space station. We can
now do the following: we can open the box on the horizon, let drop out whatever is still
inside, and wind the empty and massless box back up again. As a result, we have com-
pletely converted heat radiation into mechanical energy. Nothing else has changed: the
black hole has the same mass as beforehand.
4 quantum mechanics with gravitation – first steps 149

But the lack of change contradicts the second principle of thermodynamics! Geroch
concluded that something must be wrong. We must have forgotten an effect which makes
this process impossible.
In the 1980s, William Unruh and Robert Wald showed that black hole radiation is
precisely the forgotten effect that puts everything right. Because of black hole radiation,
the box feels buoyancy, so that it cannot be lowered down to the horizon completely.
The box floats somewhat above the horizon, so that the heat radiation inside the box
has not yet zero energy when it falls out of the opened box. As a result, the black hole
does increase in mass and thus in entropy when the box is opened. In summary, when
the empty box is pulled up again, the final situation is thus the following: only part of
the energy of the heat radiation has been converted into mechanical energy, part of the
energy went into the increase of mass and thus of entropy of the black hole. The second
principle of thermodynamics is saved.
Well, the second principle of thermodynamics is only saved if the heat radiation has
precisely the right energy density at the horizon and above. Let us have a look. The centre

Motion Mountain – The Adventure of Physics
of the box can only be lowered up to a hovering distance 𝑑 above the horizon. At the ho-
rizon, the acceleration due to gravity is 𝑔surf = 𝑐4 /4𝐺𝑀. The energy 𝐸 gained by lowering
the box is
𝑑 𝑑𝑐2
𝐸 = 𝑐2 𝑚 − 𝑚𝑔surf = 𝑐2 𝑚 (1 − ) . (37)
2 8𝐺𝑀

The efficiency of the process is 𝜂 = 𝐸/𝑐2 𝑚. To be consistent with the second principle of
thermodynamics, this efficiency must obey

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝐸 𝑇
𝜂= = 1 − BH , (38)
𝑐2 𝑚 𝑇
where 𝑇 is the temperature of the radiation inside the box. We thus find a black hole
temperature 𝑇BH that is determined by the hovering distance 𝑑. The hovering distance is
roughly given by the size of the box. The box size in turn must be at least the wavelength
of the thermal radiation; in first approximation, Wien’s relation gives 𝑑 ≈ ℏ𝑐/𝑘𝑇. A
precise calculation introduces a factor π, giving the result

ℏ𝑐3 ℏ𝑐 1 ℏ 𝑐4
𝑇BH = = = 𝑔 with 𝑔surf = , (39)
8π𝑘𝐺𝑀 4π𝑘 𝑅 2π𝑘𝑐 surf 4𝐺𝑀
where 𝑅 and 𝑀 are the radius and the mass of the black hole. The quantity 𝑇BH is either
called the black-hole temperature or the Bekenstein–Hawking temperature. As an example,
a black hole with the mass of the Sun would have the rather small temperature of 62 nK,
whereas a smaller black hole with the mass of a mountain, say 1012 kg, would have a
temperature of 123 GK. That would make quite a good oven. All known black hole can-
didates have masses in the range from a few to a few million solar masses. The radiation
is thus extremely weak – much too weak to be detectable.
The reason for the weakness of black hole radiation is that the emitted wavelength is
Challenge 105 ny of the order of the black hole radius, as you might want to check. The radiation emitted
Ref. 129 by black holes is often also called Bekenstein–Hawking radiation.
150 4 quantum mechanics with gravitation – first steps

TA B L E 10 The principles of thermodynamics and those of horizon mechanics.

Principle Thermody nam i c s Horizons

Zeroth principle the temperature 𝑇 is the the surface gravity 𝑎 is the
same across a body at equi- same across the horizon
librium
First principle energy is conserved: d𝐸 =energy is con-
𝑇d𝑆 − 𝑝d𝑉 + 𝜇d𝑁 served: d(𝑐2 𝑚) =
𝑎𝑐2
8π𝐺
d𝐴 + Ωd𝐽 + Φd𝑞
Second principle entropy never decreases: surface area never de-
d𝑆 ⩾ 0 creases: d𝐴 ⩾ 0 (except for
black hole radiation)
Third principle 𝑇 = 0 cannot be achieved 𝑎 = 0 cannot be achieved

Motion Mountain – The Adventure of Physics
All thermodynamic principles are valid for black holes, of course. A summary of the
meaning of each thermodynamic principle in the case of black holes is given in Table 10.
Black hole radiation is thus so weak that we must speak of an academic effect! It leads
Challenge 106 ny to a luminosity that increases with decreasing mass or size as

1 1 𝑛 𝑐6 ℏ
𝐿∼ 2
∼ 2 or 𝐿 = 𝑛𝐴𝜎𝑇4 = (40)
𝑀 𝑅 15 ⋅ 211 π 𝐺2 𝑀2

Vol. III, page 239 where 𝜎 is the Stefan–Boltzmann or black body radiation constant, 𝑛 is the number of

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
particle degrees of freedom that can be radiated; as long as only photons are radiated –
Ref. 130 the only case of practical importance – we have 𝑛 = 2.
Black holes thus shine, and the more the smaller they are. For example, a solar-mass
black holes emits less than 0.1 xW. This is a genuine quantum effect, since classically,
black holes, as the name says, cannot emit any light at all. Even though the effect is aca-
demically weak, it will be of importance later on. In actual systems, many other effects
around black holes increase the luminosity far above the Bekenstein–Hawking value; in-
deed, black holes are usually brighter than normal stars, due to the radiation emitted by
the matter falling into them. But that is another story. Here we are treating isolated black
holes, surrounded only by pure vacuum.

The lifetime of black holes
Due to the emitted radiation, black holes gradually lose mass. Therefore their theoretical
Challenge 107 ny lifetime is finite. A calculation shows that the lifetime is given by

20 480 π 𝐺2
𝑡 = 𝑀3 ≈ 𝑀3 3.4 ⋅ 10−16 s/kg3 (41)
ℏ𝑐4
as function of their initial mass 𝑀. For example, a black hole with mass of 1 g would
have a lifetime of 3.4 ⋅ 10−25 s, whereas a black hole of the mass of the Sun, 2.0 ⋅ 1030 kg,
would have a lifetime of about 1068 years. Again, these numbers are purely academic.
4 quantum mechanics with gravitation – first steps 151

The important point is that black holes evaporate. However, this extremely slow process
for usual black holes determines their lifetime only if no other, faster process comes into
play. We will present a few such processes shortly. Bekenstein–Hawking radiation is the
weakest of all known effects. It is not masked by stronger effects only if the black hole is
non-rotating, electrically neutral and with no matter falling into it from the surround-
ings.
So far, none of these quantum gravity effects has been confirmed experimentally, as
the values are much too small to be detected. However, the deduction of a Hawking tem-
Ref. 131 perature has been beautifully confirmed by a theoretical discovery of William Unruh,
who found that there are configurations of fluids in which sound waves cannot escape,
so-called ‘silent holes’. Consequently, these silent holes radiate sound waves with a tem-
perature satisfying the same formula as real black holes. A second type of analogue sys-
Ref. 132 tem, namely optical black holes, are also being investigated.

Black holes are all over the place

Motion Mountain – The Adventure of Physics
Around the year 2000, astronomers amassed a large body of evidence that showed some-
thing surprising: there seems to be a supermassive black hole at the centre of almost all
galaxies. The most famous of all is of course the black hole at the centre of our own
galaxy. Also quasars, active galactic nuclei and gamma-ray bursters seem to be due to
supermassive black holes at the centre of galaxies. The masses of these black holes are
typically higher than a million solar masses.
Astronomers also think that many other, smaller astrophysical objects contain black
holes: ultraluminous X-ray sources and x-ray binary stars are candidates for black holes
of intermediate mass.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Finally, one candidate explanation for dark matter on the outskirts of galaxies is a
hypothetical cloud of small black holes.
In short, black holes seem to be quite common across the universe. Whenever astro-
nomers observe a new class of objects, two questions arise directly: how do the objects
form? And how do they disappear? We have seen that quantum mechanics puts an upper
limit to the life time of a black hole. The upper limit is academic, but that is not import-
ant. The main point is that it exists. Indeed, astronomers think that most black holes
disappear in other ways, and much before the Bekenstein–Hawking limit, for example
through mergers. All this is still a topic of research. The detectors of gravitational waves
Vol. II, page 174 might clarify these processes in the future.
How are black holes born? It turns out that the birth of black holes can actually be
observed.

Fascinating gamma-ray bursts
Nuclear explosions produce flashes of γ rays, or gamma rays. In the 1960s, several coun-
tries thought that detecting γ ray flashes, or better, their absence, using satellites, would
be the best way to ensure that nobody was detonating nuclear bombs above ground. But
when the military sent satellites into the sky to check for such flashes, they discovered
something surprising. They observed about two γ flashes every day. For fear of being
laughed at, the military kept this result secret for many years.
It took the military six years to understand what an astronomer could have told
152 4 quantum mechanics with gravitation – first steps

+90

+180 -180

-90

10 –10 10 –9 10 –8 10 –7
Fluence, 50-300 keV ( J m -2 )

F I G U R E 88 The location and energy of the 2704 γ ray bursts observed in the sky between 1991 and
2000 by the BATSE experiment on board of the Compton Gamma Ray Observatory, a large satellite
deployed by the space shuttle after over 20 years of planning and construction. The Milky Way is
located around the horizontal line running from +180 to −180 (NASA).

Motion Mountain – The Adventure of Physics
them in five minutes: the flashes, today called gamma-ray bursts, were coming from
Ref. 134 outer space. Finally, the results were published; this is probably the only discovery about
nature that was made by the military. Another satellite, this time built by scientists, the
Compton Gamma Ray Observatory, confirmed that the bursts were extragalactic in ori-
gin, as proven by the map of Figure 88.
Measurements of gamma-ray bursts are done by satellites because most gamma rays
do not penetrate the atmosphere. In 1996, the Italian-Dutch BeppoSAX satellite started
mapping and measuring gamma-ray bursts systematically. It discovered that they were

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
followed by an afterglow in the X-ray domain, lasting many hours, sometimes even days.
In 1997, afterglow was discovered also in the optical domain. The satellite also allowed
researchers to find the corresponding X-ray, optical and radio sources for each burst.
These measurements in turn allowed determining the distance of the burst sources; red-
Ref. 135 shifts between 0.0085 and 4.5 were measured. In 1999 it finally became possible to detect
the optical bursts corresponding to gamma-ray bursts.*
All this data together showed that gamma-ray bursts have durations between milli-
seconds and about an hour. Gamma-ray bursts seem to fall into (at least) two classes:
the short bursts, usually below 3 s in duration and emitted from closer sources, and the
long bursts, emitted from distant galaxies, typically with a duration of 30 s and more, and
Ref. 137 with a softer energy spectrum. The long bursts produce luminosities estimated to be up
to 1045 W. This is about one hundredth of the brightness all stars of the whole visible
Challenge 108 s universe taken together! Put differently, it is the same amount of energy that is released
when converting several solar masses into radiation within a few seconds.
In fact, the measured luminosity of long bursts is near the theoretical maximum lu-
Vol. II, page 108 minosity a body can have. This limit is given by

𝑐5
𝐿 < 𝐿 Pl = = 0.9 ⋅ 1052 W , (42)
4𝐺

* For more detail about this fascinating topic, see the www.aip.de/~jcg/grb.html website by Jochen Greiner.
4 quantum mechanics with gravitation – first steps 153

Challenge 109 e as you might want to check yourself. In short, the sources of gamma ray bursts are the
biggest bombs found in the universe. The are explosions of almost unimaginable pro-
portions. Recent research seems to suggest that long gamma-ray bursts are not isotropic,
but that they are beamed, so that the huge luminosity values just mentioned might need
to be divided by a factor of 1000.
However, the mechanism that leads to the emission of gamma rays is still unclear.
It is often speculated that short bursts are due to merging neutron stars, whereas long
Ref. 135 bursts are emitted when a black hole is formed in a supernova or hypernova explosion. In
this case, long gamma-ray bursts would be ‘primal screams’ of black holes in formation.
However, a competing explanation states that long gamma-ray bursts are due to the death
of black holes.
Indeed, already 1975, a powerful radiation emission mechanism was predicted for dy-
Ref. 133 ing charged black holes by Damour and Ruffini. Charged black holes have a much shorter
lifetime than neutral black holes, because during their formation a second process takes
place. In a region surrounding them, the electric field is larger than the so-called vacuum

Motion Mountain – The Adventure of Physics
polarization value, so that large numbers of electron-positron pairs are produced, which
then almost all annihilate. This process effectively reduces the charge of the black hole to
a value for which the field is below critical everywhere, while emitting large amounts of
high energy light. It turns out that the mass is reduced by up to 30 % in a time of the order
of seconds. That is quite shorter than the 1068 years predicted by Bekenstein–Hawking
radiation! This process thus produces an extremely intense gamma-ray burst.
Ruffini took up his 1975 model again in 1997 and with his collaborators showed that
the gamma-ray bursts generated by the annihilation of electron-positrons pairs created
by vacuum polarization, in the region they called the dyadosphere, have a luminosity and

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
a duration exactly as measured, if a black hole of about a few up to 30 solar masses is as-
sumed. Charged black holes therefore reduce their charge and mass through the vacuum
polarization and electron positron pair creation process. (The process reduces the mass
because it is one of the few processes which is reversible; in contrast, most other attempts
to reduce charge on a black hole, e.g. by throwing in a particle with the opposite charge,
increase the mass of the black hole and are thus irreversible.) The left over remnant then
can lose energy in various ways and also turns out to be responsible for the afterglow
discovered by the BeppoSAX satellite. Among others, Ruffini’s team speculates that the
remnants are the sources for the high energy cosmic rays, whose origin had not been
Ref. 136 localized so far. All these exciting studies are still ongoing.
Understanding long gamma-ray bursts is one of the most fascinating open questions
in astrophysics. The relation to black holes is generally accepted. But many processes
leading to emission of radiation from black holes are possible. Examples are matter fall-
ing into the black hole and heating up, or matter being ejected from rotating black holes
Vol. II, page 271 through the Penrose process, or charged particles falling into a black hole. These mech-
anisms are known; they are at the origin of quasars, the extremely bright quasi-stellar
sources found all over the sky. They are assumed to be black holes surrounded by mat-
ter, in the development stage following gamma-ray bursters. But even the details of what
happens in quasars, the enormous voltages (up to 1020 V) and magnetic fields generated,
as well as their effects on the surrounding matter are still object of intense research in
astrophysics.
154 4 quantum mechanics with gravitation – first steps

Material properties of black holes
Once the concept of entropy of a black hole was established, people started to think about
black holes like about any other material object. For example, black holes have a matter
density, which can be defined by relating their mass to a fictitious volume defined by
4π𝑅3 /3, where 𝑅 is their radius. This density is then given by

1 3𝑐6
𝜌= (43)
𝑀2 32π𝐺3
and can be quite low for large black holes. For the largest black holes known, with 1000
Challenge 110 e million solar masses or more, the density is of the order of the density of air. Nevertheless,
even in this case, the density is the highest possible in nature for that mass.
By the way, the gravitational acceleration at the horizon is still appreciable, as it is
given by
1 𝑐4 𝑐2

Motion Mountain – The Adventure of Physics
𝑔surf = = (44)
𝑀 4𝐺 2𝑅

Challenge 111 e which is still 15 km/s2 for an air density black hole.
Obviously, the black hole temperature is related to the entropy 𝑆 by its usual definition

1 ∂𝑆 󵄨󵄨󵄨󵄨 ∂𝑆 󵄨󵄨󵄨󵄨
= 󵄨󵄨 = 󵄨 (45)
𝑇 ∂𝐸 󵄨󵄨𝜌 ∂(𝑐2 𝑀) 󵄨󵄨󵄨𝜌

All other thermal properties can be deduced by the standard relations from thermostat-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ics.
In particular, black holes are the systems in nature with the largest possible entropy.
Challenge 112 e Can you confirm this statement?
It also turns out that black holes have a negative heat capacity: when heat is added,
they cool down. In other words, black holes cannot achieve equilibrium with a bath.
This is not a real surprise, since any gravitationally bound material system has negative
specific heat. Indeed, it takes only a bit of thinking to see that any gas or matter system
Challenge 113 ny collapsing under gravity follows 𝑑𝐸/𝑑𝑅 > 0 and 𝑑𝑆/𝑑𝑅 > 0. That means that while
collapsing, the energy and the entropy of the system shrink. (Can you find out where
Challenge 114 s they go?) Since temperature is defined as 1/𝑇 = 𝑑𝑆/𝑑𝐸, temperature is always positive;
from the temperature increase 𝑑𝑇/𝑑𝑅 < 0 during collapse one deduces that the specific
Ref. 138 heat 𝑑𝐸/𝑑𝑇 is negative.
Black holes, like any object, oscillate when slightly perturbed. These vibrations have
Ref. 139 also been studied; their frequency is proportional to the mass of the black hole.
Nonrotating black holes have no magnetic field, as was established already in the 1960s
Ref. 130 by Russian physicists. On the other hand, black holes have something akin to a finite elec-
trical conductivity and a finite viscosity. Some of these properties can be understood if
Ref. 140 the horizon is described as a membrane, even though this model is not always applicable.
In any case, we can study and describe isolated macroscopic black holes like any other
macroscopic material body. The topic is not closed.
4 quantum mechanics with gravitation – first steps 155

How d o black holes evaporate?
When a nonrotating and uncharged black hole loses mass by radiating Hawking radi-
ation, eventually its mass reaches values approaching the Planck mass, namely a few mi-
crograms. Expression (41) for the lifetime, applied to a black hole of Planck mass, yields
a value of over sixty thousand Planck times. A surprising large value. What happens in
those last instants of evaporation?
A black hole approaching the Planck mass at some time will get smaller than its own
Compton wavelength; that means that it behaves like an elementary particle, and in par-
ticular, that quantum effects have to be taken into account. It is still unknown how these
final evaporation steps take place, whether the mass continues to diminish smoothly or
in steps (e.g. with mass values decreasing as √𝑛 when 𝑛 approaches zero), how its in-
ternal structure changes, whether a stationary black hole starts to rotate (as the author
predicts), or how the emitted radiation deviates from black body radiation. There is still
enough to study. However, one important issue has been settled.

Motion Mountain – The Adventure of Physics
The information parad ox of black holes
When the thermal radiation of black holes was discovered, one question was hotly de-
bated for many years. The matter forming a black hole can contain lots of information;
e.g., we can imagine the black hole being formed by a large number of books collapsing
onto each other. On the other hand, a black hole radiates thermally until it evaporates.
Since thermal radiation carries no information, it seems that information somehow dis-
appears, or equivalently, that entropy increases.
An incredible number of papers have been written about this problem, some even

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
claiming that this example shows that physics as we know it is incorrect and needs to
be changed. As usual, to settle the issue, we need to look at it with precision, laying all
prejudice aside. Three intermediate questions can help us finding the answer.
— What happens when a book is thrown into the Sun? When and how is the information
radiated away?
— How precise is the sentence that black hole radiate thermal radiation? Could there be
a slight deviation?
— Could the deviation be measured? In what way would black holes radiate informa-
tion?
Challenge 115 e You might want to make up your own mind before reading on.
Let us walk through a short summary. When a book or any other highly complex – or
low entropy – object is thrown into the Sun, the information contained is radiated away.
The information is contained in some slight deviations from black hole radiation, namely
in slight correlations between the emitted radiation emitted over the burning time of the
Sun. A short calculation, comparing the entropy of a room temperature book and the
information contained in it, shows that these effects are extremely small and difficult to
measure.
Ref. 141 A clear exposition of the topic was given by Don Page. He calculated what information
would be measured in the radiation if the system of black hole and radiation together
would be in a pure state, i.e., a state containing specific information. The result is simple.
Even if a system is large – consisting of many degrees of freedom – and in pure state,
any smaller subsystem nevertheless looks almost perfectly thermal. More specifically, if
156 4 quantum mechanics with gravitation – first steps

a total system has a Hilbert space dimension 𝑁 = 𝑛𝑚, where 𝑛 and 𝑚 ⩽ 𝑛 are the
dimensions of two subsystems, and if the total system is in a pure state, the subsystem 𝑚
Challenge 116 ny would have an entropy 𝑆𝑚 given by

𝑚𝑛
1−𝑚 1
𝑆𝑚 = + ∑ (46)
2𝑛 𝑘=𝑛+1
𝑘

which is approximately given by
𝑚
𝑆𝑚 = ln 𝑚 − for 𝑚 ≫ 1 . (47)
2𝑛
To discuss the result, let us think of 𝑛 and 𝑚 as counting degrees of freedom, instead of
Hilbert space dimensions. The first term in equation (47) is the usual entropy of a mixed
state. The second term is a small deviation and describes the amount of specific informa-

Motion Mountain – The Adventure of Physics
tion contained in the original pure state; inserting numbers, one finds that it is extremely
small compared to the first. In other words, the subsystem 𝑚 is almost indistinguishable
from a mixed state; it looks like a thermal system even though it is not.
A calculation shows that the second, small term on the right of equation (47) is indeed
sufficient to radiate away, during the lifetime of the black hole, any information contained
in it. Page then goes on to show that the second term is so small that not only it is lost in
measurements; it is also lost in the usual, perturbative calculations for physical systems.
The question whether any radiated information could be measured can now be
answered directly. As Don Page showed, even measuring half of the system only gives
about one half of a bit of the radiated information. It is thus necessary to measure al-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 142
most the complete radiation to obtain a sizeable chunk of the radiated information. In
other words, it is extremely hard to determine the information contained in black hole
radiation.
In summary, at any given instant, the amount of information radiated by a black
hole is negligible when compared with the total black hole radiation; it is practically
impossible to obtain valuable information through measurements or even through cal-
culations that use usual approximations.

More parad oxes
A black hole is a macroscopic object, similar to a star. Like all objects, it can interact with
its environment. It has the special property to swallow everything that falls into them.
This immediately leads us to ask if we can use this property to cheat around the usual
everyday ‘laws’ of nature. Some attempts have been studied in the section on general
Vol. II, page 275 relativity and above; here we explore a few additional ones.
∗∗
Apart from the questions of entropy, we can look for methods to cheat around conser-
Challenge 117 ny vation of energy, angular momentum, or charge. But every thought experiment comes
to the same conclusions. No cheats are possible. Every reasoning confirms that the max-
imum number of degrees of freedom in a region is proportional to the surface area of
4 quantum mechanics with gravitation – first steps 157

the region, and not to its volume. This intriguing result will keep us busy for quite some
time.
∗∗
A black hole transforms matter into antimatter with a certain efficiency. Indeed, a black
hole formed by collapsing matter also radiates antimatter. Thus one might look for de-
Challenge 118 ny partures from particle number conservation. Are you able to find an example?
∗∗
Black holes deflect light. Is the effect polarization dependent? Gravity itself makes no
difference of polarization; however, if virtual particle effects of QED are included, the
Ref. 143 story might change. First calculations seem to show that such an effect exists, so that
gravitation might produce rainbows. Stay tuned.
∗∗

Motion Mountain – The Adventure of Physics
If lightweight boxes made of mirrors can float in radiation, one might deduce a strange
consequence: such a box could self-accelerate in free space. In a sense, an accelerated box
could float on the Fulling–Davies–Unruh radiation it creates by its own acceleration. Are
you able to show the that this situation is impossible because of a small but significant
Challenge 119 ny difference between gravity and acceleration, namely the absence of tidal effects? (Other
reasons, such as the lack of perfect mirrors, also make the effect impossible.)
∗∗
In 2003, Michael Kuchiev has made the spectacular prediction that matter and radiation

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
with a wavelength larger than the diameter of a black hole is partly reflected when it hits
Ref. 144 a black hole. The longer the wavelength, the more efficient the reflection would be. For
stellar or even larger black holes, he predicts that only photons or gravitons are reflected.
Black holes would thus not be complete trash cans. Is the effect real? The discussion is
still ongoing.

Q uantum mechanics of gravitation
Let us take a conceptual step at this stage. So far, we looked at quantum theory with
gravitation; now we have a glimpse at quantum theory of gravitation.
If we bring to our mind the similarity between the electromagnetic field and the grav-
itational ‘field,’ our next step should be to find the quantum description of the gravit-
ational field. However, despite attempts by many brilliant minds for almost a century,
this search was not successful. Indeed, modern searches take another direction, as will
be explained in the last part of our adventure. But let us see what was achieved and why
the results are not sufficient.

Do gravitons exist?
Quantum theory says that everything that moves is made of particles. What kind of
particles are gravitational waves made of? If the gravitational field is to be treated
quantum mechanically like the electromagnetic field, its waves should be quantized.
Most properties of these quanta of gravitation can be derived in a straightforward way.
158 4 quantum mechanics with gravitation – first steps

The 1/𝑟2 dependence of universal gravity, like that of electricity, implies that the
quanta of the gravitational field have vanishing mass and move at light speed. The in-
dependence of gravity from electromagnetic effects implies a vanishing electric charge.
We observe that gravity is always attractive and never repulsive. This means that the
field quanta have integer and even spin. Vanishing spin is ruled out, since it implies no
Ref. 149 coupling to energy. To comply with the property that ‘all energy has gravity’, spin 𝑆 = 2
is needed. In fact, it can be shown that only the exchange of a massless spin 2 particle
leads, in the classical limit, to general relativity.
The coupling strength of gravity, corresponding to the fine structure constant of elec-
tromagnetism, is given either by

𝐺 𝐺𝑚𝑚 𝑚 2 𝐸 2
𝛼G1 = = 2.2 ⋅ 10−15 kg−2 or by 𝛼G2 = =( ) =( ) . (48)
ℏ𝑐 ℏ𝑐 𝑚Pl 𝐸Pl

However, the first expression is not a pure number; the second expression is, but de-

Motion Mountain – The Adventure of Physics
pends on the mass we insert. These difficulties reflect the fact that gravity is not properly
speaking an interaction, as became clear in the section on general relativity. It is often
argued that 𝑚 should be taken as the value corresponding to the energy of the system
in question. For everyday life, typical energies are 1 eV, leading to a value 𝛼G2 ≈ 1/1056 .
Gravity is indeed weak compared to electromagnetism, for which 𝛼em = 1/137.036.
If all this is correct, virtual field quanta would also have to exist, to explain static grav-
itational fields. However, up to this day, the so-called graviton has not yet been detected,
and there is in fact little hope that it ever will. On the experimental side, nobody knows
Challenge 120 s yet how to build a graviton detector. Just try! On the theoretical side, the problems with

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the coupling constant probably make it impossible to construct a renormalizable theory
of gravity; the lack of renormalization means the impossibility to define a perturbation
expansion, and thus to define particles, including the graviton. It might thus be that re-
lations such as 𝐸 = ℏ𝜔 or 𝑝 = ℏ/2π𝜆 are not applicable to gravitational waves. In short,
it may be that the particle concept has to be changed before applying quantum theory to
gravity. The issue is still open at this point.

Space-time foam
The indeterminacy relation for momentum and position also applies to the gravitational
field. As a result, it leads to an expression for the indeterminacy of the metric tensor 𝑔
in a region of size 𝑙, which is given by

𝑙Pl2
Δ𝑔 ≈ 2 , (49)
𝑙2

Challenge 121 ny where 𝑙Pl = √ℏ𝐺/𝑐3 is the Planck length. Can you deduce the result? Quantum theory
thus shows that like the momentum or the position of a particle, also the metric tensor
𝑔 is a fuzzy observable.
But that is not all. Quantum theory is based on the principle that actions below ℏ
cannot be observed. This implies that the observable values for the metric 𝑔 in a region
4 quantum mechanics with gravitation – first steps 159

of size 𝐿 are bound by
2ℏ𝐺 1
𝑔⩾ . (50)
𝑐3 𝐿2
Can you confirm this? The result has far-reaching consequences. A minimum value for
the metric depending inversely on the region size implies that it is impossible to say
what happens to the shape of space-time at extremely small dimensions. In other words,
at extremely high energies, the concept of space-time itself becomes fuzzy. John Wheeler
introduced the term space-time foam to describe this situation. The term makes clear
that space-time is not continuous nor a manifold in those domains. But this was the
basis on which we built our description of nature so far! We are forced to deduce that
our description of nature is built on sand. This issue will be essential in the last volume
Vol. VI, page 59 of our mountain ascent.

Decoherence of space-time

Motion Mountain – The Adventure of Physics
General relativity taught us that the gravitational field and space-time are the same. If
the gravitational field evolves like a quantum system, we may ask why no superpositions
of different macroscopic space-times are observed.
Ref. 150 The discussion is simplified for the simplest case of all, namely the superposition, in a
vacuum region of size 𝑙, of a homogeneous gravitational field with value 𝑔 and one with
Vol. IV, page 148 value 𝑔󸀠 . As in the case of a superposition of macroscopic distinct wave functions, such
a superposition decays. In particular, it decays when particles cross the volume. A short
Challenge 122 ny calculation yields a decay time given by

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
2𝑘𝑇 3/2 𝑛𝑙4
𝑡d = ( ) , (51)
π𝑚 (𝑔 − 𝑔󸀠 )2

where 𝑛 is the particle number density, 𝑘𝑇 their kinetic energy and 𝑚 their mass. In-
serting typical numbers, we find that the variations in gravitational field strength are
Challenge 123 e extremely small. In fact, the numbers are so small that we can deduce that the gravita-
tional field is the first variable which behaves classically in the history of the universe.
Quantum gravity effects for space-time will thus be extremely hard to detect.
In short, matter not only tells space-time how to curve, it also tells it to behave with
class.

Q uantum theory as the enemy of science fiction
How does quantum theory change our ideas of space-time? The end of the twentieth cen-
tury has brought several unexpected but strong results in semiclassical quantum gravity.
Ref. 151 In 1995 Ford and Roman found that worm holes, which are imaginable in general re-
lativity, cannot exist if quantum effects are taken into account. They showed that macro-
scopic worm holes require unrealistically large negative energies. (For microscopic worm
holes the issue is still unclear.)
Ref. 152 In 1996 Kay, Radzikowski and Wald showed that closed time-like curves do not exist
in semiclassical quantum gravity; there are thus no time machines in nature.
Ref. 153 In 1997 Pfenning and Ford showed that warp drive situations, which are also imagin-
160 4 quantum mechanics with gravitation – first steps

Motion Mountain – The Adventure of Physics
F I G U R E 89 Every tree, such as this
beautiful Madagascar baobab (Adansonia
grandidieri), shows that nature, in contrast
to physicists, is able to combine quantum
theory and gravity (© Bernard Gagnon).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
able in general relativity, cannot exist if quantum effects are taken into account. Such
situations require unrealistically large negative energies.
In short, the inclusion of quantum effects destroys all those fantasies which were star-
ted by general relativity.

No vacuum means no particles
Gravity has an important consequence for quantum theory. To count and define
Vol. IV, page 115 particles, quantum theory needs a defined vacuum state. However, the vacuum state can-
not be defined when the curvature radius of space-time, instead of being larger than
the Compton wavelength, becomes comparable to it. In such highly curved space-times,
particles cannot be defined. The reason is the impossibility to distinguish the environ-
ment from the particle in these situations: in the presence of strong curvatures, the va-
cuum is full of spontaneously generated matter, as black holes show. Now we just saw that
at small dimensions, space-time fluctuates wildly; in other words, space-time is highly
curved at small dimensions or high energies. In other words, strictly speaking particles
cannot be defined; the particle concept is only a low energy approximation! We will ex-
plore this strange conclusion in more detail in the final part of our mountain ascent.
4 quantum mechanics with gravitation – first steps 161

Summary on quantum theory and gravit y
Every tree tells us: everyday, weak gravitational fields can be included in quantum the-
ory. Weak gravitational fields have measurable and predictable effects on wave functions:
quantum particles fall and their phases change in gravitational fields. Conversely, the in-
clusion of quantum effects into general relativity leads to space-time foam, space-time
superpositions and gravitons. The inclusion of quantum effects into gravity prevents the
existence of wormholes, time-like curves and negative energy regions.
The inclusion of strong gravitational fields into quantum theory works for practical
situations but leads to problems with the particle concept. Conversely, the inclusion of
quantum effects into situations with high space-time curvature leads to problems with
the concept of space-time.
In summary, the combination of quantum theory and gravitation leads to problems
with both the particle concept and the space-time concept. The combination of quantum
theory and general relativity puts into question the foundations of the description of
nature that we used so far. As shown in Figure 89, nature is smarter than we are.

Motion Mountain – The Adventure of Physics
In fact, up to now we hid a simple fact: conceptually, quantum theory and general re-
lativity contradict each other. This contradiction was one of the reasons that we stepped
back to special relativity before we started exploring quantum theory. By stepping back
we avoided many problems, because quantum theory does not contradict special relativ-
ity, but only general relativity. The issues are dramatic, changing everything from the
basis of classical physics to the results of quantum theory. There will be surprising con-
sequences for the nature of space-time, for the nature of particles, and for motion itself.
Vol. VI, page 17 Before we study these issues, however, we complete the theme of the present, quantum
part of the mountain ascent, namely exploring motion inside matter, and in particular

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the motion of and in nuclei.
Chapter 5

T H E ST RU C T U R E OF T H E N U C L E U S
– T H E DE N SE ST C LOU DS

N
uclear physics was born in 1896 in France, but is now a small activity.
ot many open issues are left. But in its golden past, researchers produced
ew ways for medical doctors to dramatically improve the healing rate of pa-
Ref. 154 tients. Researchers also discovered why stars shine, how powerful bombs work, and how

Motion Mountain – The Adventure of Physics
cosmic evolution produced the atoms we are made of. We will explore these topics now.
A fascinating spin-off of nuclear physics, high energy particle physics, will keep us busy
later on.

“ ”
Nuclear physics is just low-density astrophysics.
Anonymous

A physical wonder – magnetic resonance imaging
Arguably, the most spectacular tool that physical research produced in the twentieth cen-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tury was magnetic resonance imaging, or MRI for short. This imaging technique allows
one to image the interior of human bodies with a high resolution and with no damage
or danger to the patient, in strong contrast to X-ray imaging. The technique is based on
moving atomic nuclei. Though the machines are still expensive – costing up to several
million euro – there is hope that they will become cheaper in the future. Such an MRI
machine, shown in Figure 90, consists essentially of a large magnetic coil, a radio trans-
mitter and a computer. Some results of putting part of a person into the coil are shown
in Figure 91. The images allow detecting problems in bones, in the spine, in the beating
heart and in general tissue. Many people owe their life and health to these machines; in
many cases the machines allow making precise diagnoses and thus choosing the appro-
priate treatment for patients.
In MRI machines, a radio transmitter emits radio waves that are absorbed because
hydrogen nuclei – protons – are small spinning magnets. The magnets can be parallel
or antiparallel to the magnetic field produced by the coil. The transition energy 𝐸 is ab-
sorbed from a radio wave whose frequency 𝜔 is tuned to the magnetic field 𝐵. The energy
absorbed by a single hydrogen nucleus is given by

𝐸 = ℏ𝜔 = ℏ𝛾𝐵 (52)

The material constant 𝛾/2π has a value of 42.6 MHz/T for hydrogen nuclei; it results from
the non-vanishing spin of the proton. The absorption of the radio wave is a pure quantum
effect, as shown by the appearance of the quantum of action ℏ. Using some cleverly ap-
5 the densest clouds 163

F I G U R E 90 A commercial MRI

Motion Mountain – The Adventure of Physics
machine (© Royal Philips
Electronics).

F I G U R E 91 Sagittal images of the head and the spine (used with permission from Joseph P. Hornak,
The Basics of MRI, www.cis.rit.edu/htbooks/mri, Copyright 2003).

plied magnetic fields, typically with a strength between 0.3 and 7 T for commercial and
up to 21 T for experimental machines, the absorption for each volume element can be
measured separately. Interestingly, the precise absorption level depends on the chemical
compound the nucleus is built into. Thus the absorption value will depend on the chem-
ical substance. When the intensity of the absorption is plotted as grey scale, an image is
formed that retraces the different chemical compositions. Figure 91 shows two examples.
Using additional tricks, modern machines can picture blood flow in the heart or air flow
164 5 the structure of the nucleus

F I G U R E 92 An image of the
ﬁrst magnetic resonance
video of a human birth (© C.
Bamberg).

Motion Mountain – The Adventure of Physics
Ref. 155 in lungs; they now routinely make films of the heart beat. Other techniques show how
the location of sugar metabolism in the brain depends on what you are thinking about.*
Magnetic resonance imaging can even image the great wonders of nature. The first
Ref. 156 video of a human birth, taken in 2010 and published in 2012, is shown in Figure 92. MRI
scans are loud, but otherwise harmless for unborn children. The first scan of a married
Ref. 157 couple making love has been taken by Willibrord Weijmar Schultz and his group in 1999.
It is shown on page 399.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Every magnetic resonance image thus proves that

⊳ Many – but not all – atoms have nuclei that spin.

Like any other object, nuclei have size, shape, colour, composition and interactions. Let
us explore them.

The size of nuclei and the discovery of radioactivity
The magnetic resonance signal shows that hydrogen nuclei spin with high speed. Thus
they must be small. Indeed, the 𝑔-factor of protons, defined using the magnetic moment
𝜇, their mass 𝑚 and charge 𝑒, is found to be
𝑚
𝑔 = 𝜇4 ≈ 5.6 . (53)
𝑒ℏ

Vol. IV, page 107 This is a small value. Using the expression that relates the 𝑔-factor and the radius of a
composite object, we deduce that the radius of the proton is about 0.9 fm; this value

* The website www.cis.rit.edu/htbooks/mri by Joseph P. Hornak gives an excellent introduction to magnetic
resonance imaging, both in English and Russian, including the physical basis, the working of the machines,
and numerous beautiful pictures. The method of studying nuclei by putting them at the same time into
magnetic and radio fields is also called nuclear magnetic resonance.
5 the densest clouds 165

F I G U R E 93 Henri Becquerel (1852–1908)

Motion Mountain – The Adventure of Physics
F I G U R E 94 Marie Curie (1867 –1934)

is confirmed by many experiments and other measurement methods. Protons are thus

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
about 30 000 times smaller than hydrogen atoms, whose radius is about 30 pm. The pro-
ton is the smallest of all nuclei; the largest known nuclei have radii about 7 times as large.
The small size of nuclei is no news. It is known since the beginning of the twentieth
century. The story starts on the first of March in 1896, when Henri Becquerel* discovered
a puzzling phenomenon: minerals of uranium potassium sulphate blacken photographic
plates. Becquerel had heard that the material is strongly fluorescent; he conjectured that
fluorescence might have some connection to the X-rays discovered by Conrad Rönt-
gen the year before. His conjecture was wrong; nevertheless it led him to an important
new discovery. Investigating the reason for the effect of uranium on photographic plates,
Becquerel found that these minerals emit an undiscovered type of radiation, different
from anything known at that time; in addition, the radiation is emitted by any substance
containing uranium. In 1898, Gustave Bémont named the property of these minerals ra-
dioactivity.
Radioactive rays are also emitted from many elements other than uranium. This ra-
diation can be ‘seen’: it can be detected by the tiny flashes of light that appear when the
rays hit a scintillation screen. The light flashes are tiny even at a distance of several metres
from the source; thus the rays must be emitted from point-like sources. In short, radio-

* Henri Becquerel (b. 1852 Paris, d. 1908 Le Croisic), important physicist; his primary research topic was the
study of radioactivity. He was the thesis adviser of Marie Curie, the wife of Pierre Curie, and was central to
bringing her to fame. The SI unit for radioactivity is named after Becquerel. For his discovery of radioactivity
he received the 1903 Nobel Prize in Physics; he shared it with the Curies.
166 5 the structure of the nucleus

TA B L E 11 The main types of radioactivity and rays emitted by matter.

Type Pa r t - Example Range Dan- Use
icle ger Shield
235
𝛼 rays helium U, 238 U, 238 Pu, a few cm in when any thickness
238
3 to 10 MeV nuclei Pu, 241 Am air eaten, material, measurement
inhaled, e.g. paper
touched
14
𝛽 rays electrons C, 40 K, 3 H, < 1 mm in serious metals cancer
101
0 to 5 MeV and Tc metal treatment
antineu- light years none none research
trinos
𝛽+ rays positrons 40
K, 11 C, 11 C, less than β medium any tomography
13
and N, 15 O material
neutrinos light years none none research

Motion Mountain – The Adventure of Physics
110
𝛾 rays high Ag several m in high thick lead preservation
energy air of herbs,
photons disinfection
252
n reactions neutrons Cf, Po-Li many m in high 0.3 m of nuclear
c. 1 MeV (α,n), 38 Cl-Be air paraffin power,
(γ,n) quantum
gravity
experiments

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
9
n emission neutrons He, 24 N, 254 Cf many m in high 0.3 m of research
typ. 40 MeV air paraffin experiments
5
p emission protons Be, 161 Re like α rays small solids
typ. 20 MeV
232
spontaneous nuclei Cm, 263 Rf like α rays small solids detection
fission of new
typ. 100 MeV elements

activity has to be emitted from single atoms. Thus radioactivity confirmed unambigu-
ously that atoms do exist. In fact, radioactivity even allows counting atoms: in a diluted
radioactive substance, the flashes can be counted, either with the help of a photographic
Vol. IV, page 42 film or with a photon counting system.
The intensity of radioactivity cannot be influenced by magnetic or electric fields; and
it does not depend on temperature or light irradiation. In short, radioactivity does not
depend on electromagnetism and is not related to it. Also the high energy of the emitted
radiation cannot be explained by electromagnetic effects. Radioactivity must thus be due
to another, new type of force. In fact, it took 30 years and a dozen of Nobel Prizes to fully
understand the details. It turns out that several types of radioactivity exist; the types
of emitted radiation behave differently when they fly through a magnetic field or when
they encounter matter. The types of radiation are listed in Table 11. In the meantime, all
5 the densest clouds 167

Fluorescent screen
(or rotating particle detector)

Particle Slit
beam
Gold foil

Drilled lead Undeflected
shielding block α - particles
with a radioactive
substance inside

Backward scattered, Forward scattered
`reflected’ particle particle

Motion Mountain – The Adventure of Physics
F I G U R E 95 The schematics of the Rutherford–Geiger scattering experiment. The gold foil is about a
square centimetre in size.

these rays have been studied in great detail, with the aim to understand the nature of the
emitted entity and its interaction with matter.
In 1909, radioactivity inspired the 37 year old physicist Ernest Rutherford,* who had
won the Nobel Prize just the year before, to another of his smart experiments. He asked
his collaborator Hans Geiger to take an emitter of α radiation – a type of radioactivity
which Rutherford had identified and named 10 years earlier – and to point the radiation

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
at a thin metal foil. The quest was to find out where the α rays would end up. The ex-
periment is shown in Figure 95. The research group followed the path of the particles by
using scintillation screens; later on they used an invention by Charles Wilson: the cloud
chamber. A cloud chamber, like its successor, the bubble chamber, produces white traces
along the path of charged particles; the mechanism is the same as the one than leads to
the white lines in the sky when an aeroplane flies by. Both cloud chambers and bubble
chambers thus allow seeing radioactivity, as shown in the examples of Figure 96.
The radiation detectors around the thin gold foil give a consistent, but strange result:
most α particles pass through the foil undisturbed, whereas a few are scattered and a few
are reflected. In addition, those few which are reflected are not reflected by the surface,
Challenge 124 s but in the inside of the foil. (Can you imagine how to show this?) Rutherford and Geiger
deduced from their scattering experiment that first of all, the atoms in the metal foil are
mainly transparent. Only transparency of atoms explains why most α particles pass the

* Ernest Rutherford (b. 1871 Brightwater, d. 1937 Cambridge), important physicist. He emigrated to Britain
and became professor at the University of Manchester. He coined the terms α particle, β particle, proton
and neutron. A gifted experimentalist, he discovered that radioactivity transmutes the elements, explained
the nature of α rays, discovered the nucleus, measured its size and performed the first nuclear reactions.
Ironically, in 1908 he received the Nobel Prize in Chemistry, much to the amusement of himself and of
the world-wide physics community; this was necessary as it was impossible to give enough physics prizes
to the numerous discoverers of the time. He founded a successful research school of nuclear physics and
many famous physicists spent some time at his institute. Ever an experimentalist, Rutherford deeply disliked
quantum theory, even though it was and is the only possible explanation for his discoveries.
168 5 the structure of the nucleus

F I G U R E 96 The ‘Big European Bubble Chamber’ – the biggest bubble chamber ever built – and an

Motion Mountain – The Adventure of Physics
example of tracks of relativistic particles it produced, with the momentum values deduced from the
photograph (© CERN).

Illustrating a free atom in its ground state

(1) in an acceptable way: (2) in an unacceptable way:

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
electron
nucleus nucleus

Correct: the electron Correct: almost nothing!
cloud has a spherical
and blurred shape. Wrong: nuclei are ten to one hundred
thousand times smaller than atoms,
Wrong: the cloud and electrons do not move on paths, electrons
the nucleus have no are not extended, free atoms are not flat
visible colour, nucleus but always spherical, neither atoms nor
is still too large by far. nucleons have a sharp border, no particle
involved has a visible colour.

F I G U R E 97 A reasonably realistic (left) and a misleading illustration of an atom (right) as is regularly
found in school books. Atoms in the ground state are spherical electron clouds with a tiny nucleus, itself a
cloud, at its centre. Interacting atoms, chemically bound atoms and some, but not all excited atoms have
electron clouds of different shapes.

foil without disturbance, even though it was over 2000 atoms thick. But some particles
were scattered by large angles or even reflected. Rutherford showed that the reflections
must be due to a single scattering point. By counting the particles that were reflected
(about 1 in 20000 for his 0.4 μm gold foil), Rutherford was also able to deduce the size
5 the densest clouds 169

F I G U R E 98 Left: a modern Wilson cloud chamber, diameter c. 100 mm. Right: one of the ﬁrst pictures of
α rays taken with a cloud chamber in the 1920s by Patrick Blackett, showing also a collision with an
atom in the chamber (© Wiemann Lehrmittel, Royal Society)

Motion Mountain – The Adventure of Physics
of the reflecting entity and to estimate its mass. (This calculation is now a standard exer-
cise in the study of physics at universities.) He found that the reflecting entity contains
practically all of the mass of the atom in a diameter of a few fm. Rutherford named this
concentrated mass the atomic nucleus.
Using the knowledge that atoms contain electrons, Rutherford then deduced from this
experiment that atoms consist of an electron cloud that determines the size of atoms –
of the order of 0.1 nm – and of a tiny but heavy nucleus at the centre. If an atom had the
size of a basketball, its nucleus would have the size of a dust particle, yet contain 99.9 %

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of the basketball’s mass. Thus

⊳ Atoms resemble candy floss around a heavy dust particle.

Even though the candy floss – the electron cloud – around the nucleus is extremely thin
and light, it is strong enough to avoid that two atoms interpenetrate. In solids, the candy
floss, i.e., the electron cloud, keeps the neighbouring nuclei at constant distance. Fig-
ure 97 shows the more and less correct ways to picture an atom. Candy floss explains the
Rutherford–Geiger experiment: for the tiny and massive α particles however, the candy
floss is essentially empty space, so that they simply fly through the electron clouds until
they either exit on the other side of the foil or hit a nucleus.
The density of the nucleus is impressive: about 5.5 ⋅ 1017 kg/m3 . At that density, the
mass of the Earth would fit in a sphere of 137 m radius and a grain of sand would have a
Challenge 125 e mass larger than the largest oil tanker. (Can you confirm this?)

“ ”
I now know how an atom looks like!
Ernest Rutherford

Nuclei are composed
Magnetic resonance images also show that nuclei are composed. Indeed, images can also
be taken using heavier nuclei instead of hydrogen, such as certain fluorine or oxygen
170 5 the structure of the nucleus

Illustrating an atomic nucleus

(1) in an acceptable way (2) in an misleading way

Correct: blurred and usually Correct: only the composition.
ellipsoidal shape of nucleus.
Wrong: nucleons are not at fixed positions
Wrong: nucleus does not have with respect to each other, nucleons have
visible colour; some nuclei no sharp borders, nucleons do not have
have other shapes. visible colours.

Motion Mountain – The Adventure of Physics
F I G U R E 99 A reasonably realistic (left) and a misleading illustration of a nucleus (right) as is regularly
found in school books. Nuclei are spherical nucleon clouds.

nuclei. Also the 𝑔-factors of these nuclei depart from the value 2 characteristic of point
particles; the more massive the nuclei are, the bigger the departure. Therefore, all nuclei
Vol. IV, page 107 have a finite size. The size of nuclei can actually be measured; the Rutherford–Geiger
experiment and many other scattering experiments allow to do so. The measured values

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
confirm the values predicted by the 𝑔-factor. In short, both the values of the 𝑔-factor and
the non-vanishing sizes show that nuclei are composed.
Interestingly, the idea that nuclei are composed is older than the concept of nucleus
itself. Already in 1815, after the first mass measurements of atoms by John Dalton and
others, researchers noted that the mass of the various chemical elements seem to be al-
most perfect multiples of the weight of the hydrogen atom. William Prout then formu-
lated the hypothesis that all elements are composed of hydrogen. When the nucleus was
discovered, knowing that it contains almost all mass of the atom, it was therefore first
thought that all nuclei are made of hydrogen nuclei. Being at the origin of the list of con-
stituents, the hydrogen nucleus was named proton, from the greek term for ‘first’ and
reminding the name of Prout at the same time. Protons carry a positive unit of electric
charge, just the opposite of that of electrons, but are about 1836 times as heavy. More
details on the proton are listed in Table 12.
However, the charge multiples and the mass multiples for the heavier nuclei do not
match. On average, a nucleus that has 𝑛 times the charge of a proton, has a mass that is
about 2.6 𝑛 times than of the proton. Additional experiments confirmed an idea formu-
lated by Werner Heisenberg: all nuclei heavier than hydrogen nuclei are made of pos-
itively charged protons and of neutral neutrons. Neutrons are particles a tiny bit more
massive than protons (the difference is less than a part in 700, as shown in Table 12), but
without any electrical charge. Since the mass is almost the same, the mass of nuclei –
and thus that of atoms – is still an (almost perfect) integer multiple of the proton mass.
But since neutrons are neutral, the mass and the charge number of nuclei differ. Being
5 the densest clouds 171

TA B L E 12 The properties of the nucleons: proton and neutron (source: pdg.web.
cern.ch).

Propert y Proton Neutron

Mass 1.672 621 777(74) ⋅ 10−27 kg 1.674 927 351(74) ⋅ 10−27 kg
0.150 327 7484(66) nJ 0.150 534 9631(66) nJ
938, 272 046(21) MeV 939, 565 379(21) MeV
1.007 276 466 812(90) u 1.008 664 916 00(43) u
1836.152 6675(39)⋅ 𝑚e 1838.683 6605(11)⋅ 𝑚e
Spin 1/2 1/2
P parity +1 +1
Antiparticle antiproton p antineutron n
Quark content uud udd
Electric charge 1e 0
Charge radius 0.88(1) f m 0.12(1) f m2

Motion Mountain – The Adventure of Physics
Electric dipole < 5.4 ⋅ 10−26 e ⋅ m < 2.9 ⋅ 10−28 e ⋅ m
moment
Electric 1.20(6) ⋅ 10−3 f m3 1.16(15) ⋅ 10−3 f m3
polarizability
Magnetic 1.410 606 743(33) ⋅ 10−26 J/T −0.966 236 47(23) ⋅ 10−26 J/T
moment
g-factor 5.585 694 713(46) −3.826 085 45(90)
2.792 847 356 (23) ⋅ μN −1.913 042 72(45) ⋅ μN
Gyromagnetic 0.267 522 2005(63) 1/nsT

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ratio
Magnetic 1.9(5) ⋅ 10−4 f m3 3.7(2.0) ⋅ 10−4 f m3
polarizability
Mean life (free > 2.1 ⋅ 1029 a 880.1(1.1) s
particle)
Shape oblate oblate
(quadrupole
moment)
Excited states more than ten more than ten

neutral, neutrons do not leave tracks in clouds chambers and are more difficult to de-
tect than protons, charged hadrons or charged leptons. For this reason, the neutron was
discovered later than several other, more exotic subatomic particles.
Today, it is possible to keep single neutrons suspended between suitably shaped coils,
with the aid of teflon ‘windows’. Such traps were proposed in 1951 by Wolfgang Paul.
They work because neutrons, though they have no charge, do have a small magnetic
moment. (By the way, this implies that neutrons are themselves composed of charged
particles.) With a suitable arrangement of magnetic fields, neutrons can be kept in place,
in other words, they can be levitated. Obviously, a trap only makes sense if the trapped
particle can be observed. In case of neutrons, this is achieved by the radio waves absorbed
172 5 the structure of the nucleus

when the magnetic moment switches direction with respect to an applied magnetic field.
The result of these experiments is simple: the lifetime of free neutrons is 885.7(8) s. Nev-
ertheless, we all know that inside most nuclei we are made of, neutrons do not decay for
millions of years, because the decay products do not lead to a state of lower energy. (Why
Challenge 126 s not?)
Magnetic resonance images also show that most elements have different types of
atoms. These elements have atoms with the same number of protons, but with differ-
ent numbers of neutrons. One says that these elements have several isotopes.* This result
also explains why some elements radiate with a mixture of different decay times. Though
chemically isotopes are (almost) indistinguishable, they can differ strongly in their nuc-
lear properties. Some elements, such as tin, caesium, or polonium, have over thirty iso-
topes each. Together, the 118 known elements have over 2000 isotopes. They are shown
in Figure 100. (Isotopes without electrons, i.e., specific nuclei with a given number of
neutrons and protons, are called nuclides.)
Because nuclei are so extremely dense despite containing numerous positively

Motion Mountain – The Adventure of Physics
charged protons, there must be a force that keeps everything together against the
electrostatic repulsion. We saw that the force is not influenced by electromagnetic or
gravitational fields; it must be something different. The force must be short range; oth-
erwise nuclei would not decay by emitting high energy α rays. The additional force is
Page 219 called the strong nuclear interaction. We shall study it in detail shortly.
The strong nuclear interaction binds protons and neutrons in the nucleus. It is essen-
tial to recall that inside a nucleus, the protons and neutrons – they are often collectively
called nucleons – move in a similar way to the electrons moving in atoms. Figure 99 il-
lustrates this. The motion of protons and neutrons inside nuclei allows us to understand

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the shape, the spin and the magnetic moment of nuclei.

Nuclei can move alone – cosmic rays
In everyday life, nuclei are mostly found inside atoms. But in some situations, they move
all by themselves, without surrounding electron clouds. The first to discover an example
was Rutherford; with a clever experiment he showed that the α particles emitted by many
radioactive substance are helium nuclei. Like all nuclei, α particles are small, so that they
are quite useful as projectiles.
Then, in 1912, Viktor Heß** made a completely unexpected discovery. Heß was in-
Vol. III, page 23 trigued by electroscopes (also called electrometers). These are the simplest possible de-
tectors of electric charge. They mainly consist of two hanging, thin metal foils, such
as two strips of aluminium foil taken from a chocolate bar. When the electroscope is

* The name is derived from the Greek words for ‘same’ and ‘spot’, as the atoms are on the same spot in the
periodic table of the elements.
** Viktor Franz Heß, (b. 1883 Waldstein, d. 1964 Mount Vernon), nuclear physicist, received the Nobel Prize
in Physics in 1936 for his discovery of cosmic radiation. Heß was one of the pioneers of research into radio-
activity. Heß’ discovery also explained why the atmosphere is always somewhat charged, a result important
for the formation and behaviour of clouds. Twenty years after the discovery of cosmic radiation, in 1932
Carl Anderson discovered the first antiparticle, the positron, in cosmic radiation; in 1937 Seth Nedder-
meyer and Carl Anderson discovered the muon; in 1947 a team led by Cecil Powell discovered the pion; in
1951, the Λ0 and the kaon 𝐾0 are discovered. All discoveries used cosmic rays and most of these discoveries
led to Nobel Prizes.
5 the densest clouds 173

Half-life

> 10+15 s 10-1 s
10+10 s 10-2 s
10+7 s 10-3 s
10+5 s 10-4 s
10+4 s 10-5 s
10+3 s 10-6 s
10+2 s 10-7 s
10+1 s 10-15 s
10+0 s <10-15 s
unknown

Motion Mountain – The Adventure of Physics
Decay type

stable
el. capture (beta+)
beta - emission
alpha emission

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
proton emission
neutron emission
spontaneous fission
unknown

F I G U R E 100 All known nuclides with their lifetimes (above) and main decay modes (below). The data
are from www.nndc.bnl.gov/nudat2.

metal wire
(e.g. paper clip)

F I G U R E 101 An
thin electroscope (or
aluminium electrometer)
foils (© Harald Chmela) and
its charged (middle)
and uncharged state
(right).

charged, the strips repel each other and move apart, as shown in Figure 101. (You can
174 5 the structure of the nucleus

F I G U R E 102 Viktor Heß (1883–1964)

build one easily yourself by covering an empty glass with some transparent cellophane
foil and suspending a paper clip and the aluminium strips from the foil. You can charge
Challenge 127 e the electroscope with the help of a rubber balloon and a woollen pullover.) An electro-

Motion Mountain – The Adventure of Physics
scope thus measures electrical charge. Like many before him, Heß noted that even for a
completely isolated electroscope, the charge disappears after a while. He asked: why? By
careful study he eliminated one explanation after the other. Heß (and others) were left
with only one possibility: that the discharge could be due to charged rays, such as those
of the recently discovered radioactivity, emitted from the environment. To increase the
distance to the environment, Heß prepared a sensitive electrometer and took it with him
on a balloon flight.
As expected, the balloon flight showed that the discharge effect diminished with
height, due to the larger distance from the radioactive substances on the Earth’s surface.
But above about 1000 m of height, the discharge effect increased again, and the higher he

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
flew, the stronger it became. Risking his health and life, he continued upwards to more
than 5000 m; there the discharge was several times faster than on the surface of the Earth.
This result is exactly what is expected from a radiation coming from outer space and ab-
sorbed by the atmosphere. In one of his most important flights, performed during an
(almost total) solar eclipse, Heß showed that most of the ‘height radiation’ did not come
from the Sun, but from further away. He thus called the radiation cosmic rays. One also
speaks of cosmic radiation. During the last few centuries, many people have drunk from
a glass and eaten chocolate covered by aluminium foil; but only Heß combined these
activities with such careful observation and deduction that he earned a Nobel Prize.*
Today, the most common detectors for cosmic rays are Geiger–Müller counters and
spark chambers. Both share the same idea; a high voltage is applied between two metal
parts kept in a thin and suitably chosen gas (a wire and a cylindrical mesh for the Geiger-
Müller counter, two plates or wire meshes in the spark chambers). When a high energy
ionizing particle crosses the counter, a spark is generated, which can either be observed
through the generated spark (as you can do yourself by watching the spark chamber in
the entrance hall of the CERN main building), or detected by the sudden current flow. His-
torically, the current was first amplified and sent to a loudspeaker, so that the particles
can be heard by a ‘click’ noise. In short, with a Geiger counter one cannot see ions or
particles, but one can hear them. Later on, with the advances in electronics, ionized

* In fact, Hess used gold foils in his electrometer, not aluminium foils.
5 the densest clouds 175

TA B L E 13 The main types of cosmic radiation.

Pa r t i c l e Energy Origin Detector Shield

At high altitude, the primary particles:
Protons (90 %) 109 to 1022 eV stars, supernovae, scintillator in mines
extragalactic,
unknown
α rays (9 %) typ. 5 ⋅ 106 eV stars, galaxy ZnS, counters 1 mm of any
material
Other nuclei, such 109 to 1019 eV stars, novae counters, films 1 mm of any
as Le, Be, B, Fe material
(1 %)
Neutrinos MeV, GeV Sun, stars chlorine, none
gallium, water

Motion Mountain – The Adventure of Physics
Electrons (0.1 %) 106 to supernova remnants
> 1012 eV
Gammas (10−6 ) 1 eV to 50 TeV stars, pulsars, galactic, semiconductor in mines
extragalactic detectors
At sea level, secondary particles are produced in the atmosphere:
Muons 3 GeV, protons hit drift chamber, 15 m of
150/ m2 s atmosphere, produce bubble water or
pions which decay into chamber, 2.5 m of soil

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
muons scintillation
detector
Oxygen, varies e.g., 𝑛 + 16 O → p + 14 C counters soil
radiocarbon and
other nuclei
Positrons varies counters soil
Neutrons varies reaction product when counters soil
proton hits 16 O
nucleus
Pions varies reaction product when counters soil
proton hits 16 O
nucleus
In addition, there are slowed down primary beam particles.

atoms or particles could be counted.
Finding the right gas mixture for a Geiger–Müller counter is tricky; it is the reason
that the counter has a double name. One needs a gas that extinguishes the spark after a
while, to make the detector ready for the next particle. Müller was Geiger’s assistant; he
made the best counters by adding the right percentage of alcohol to the gas in the cham-
ber. Nasty rumours maintained that this was discovered when another assistant tried,
176 5 the structure of the nucleus

cylindrical metal mesh

gas
central
metal
wire

typ.
5kV
high current
voltage meter
source (and often,
a beeper)

Motion Mountain – The Adventure of Physics
F I G U R E 103 A Geiger–Müller counter with the detachable detection tube, the connection cable to the
counter electronics, and, for this model, the built-in music player (© Joseph Reinhardt).

F I G U R E 104 A modern spark chamber showing the cosmic rays that constantly arrive on Earth
(QuickTime ﬁlm © Wolfgang Rueckner).

without success, to build counters while Müller was absent. When Müller, supposedly a
heavy drinker, came back, everything worked again. However, the story is apocryphal.
Today, Geiger–Müller counters are used around the world to detect radioactivity; the
5 the densest clouds 177

F I G U R E 105 The cosmic ray moon shadow, observed with the L3 detector at CERN. The shadow is
shifted with respect of the position of the moon, indicated by a white circle, because the Earth’s

Motion Mountain – The Adventure of Physics
magnetic ﬁeld deﬂects the charged particles making up cosmic rays (© CERN Courier).

smallest versions fit in mobile phones and inside wrist watches. An example is shown in
Figure 103.
If you can ever watch a working spark chamber, do so. The one in the CERN entrance
hall is about 0.5 m3 in size. A few times per minute, you can see the pink sparks showing
the traces of cosmic rays. The rays appear in groups, called showers. And they hit us all
the time.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Various particle detectors also allow measuring the energy of particles. The particle
energy in cosmic rays spans a range between 103 eV and at least 1020 eV; the latter is the
same energy as a tennis ball after serve, but for a single ion. This is a huge range in energy.
Understanding the origin of cosmic rays is a research field on its own. Some cosmic rays
are galactic in origin, some are extragalactic. For most energies, supernova remnants –
pulsars and the like – seem the best candidates. However, the source of the highest energy
particles is still unknown; black holes might be involved in their formation.
Cosmic rays are probably the only type of radiation discovered without the help of
shadows. But in the meantime, such shadows have been found. In a beautiful experi-
ment performed in 1994, the shadow thrown by the Moon on high energy cosmic rays
(about 10 TeV) was measured, as shown in Figure 105. When the position of the shadow
is compared with the actual position of the Moon, a shift is found. And indeed, due to
the magnetic field of the Earth, the cosmic ray Moon shadow is expected to be shifted
westwards for protons and eastwards for antiprotons. The data are consistent with a ratio
Ref. 158 of antiprotons in cosmic rays between 0 % and 30 %. By studying the shadow’s position,
the experiment thus showed that high energy cosmic rays are mainly positively charged
and thus consist mainly of matter, and only in small part, if at all, of antimatter.
Detailed observations showed that cosmic rays arrive on the surface of the Earth as
Page 175 a mixture of many types of particles, as shown in Table 13. They arrive from outside
the atmosphere as a mixture of which the largest fraction are protons, followed by α
particles, iron and other nuclei. And, as mentioned above, most rays do not originate
from the Sun. In other words, nuclei can thus travel alone over large distances. In fact,
178 5 the structure of the nucleus

F I G U R E 106 An aurora
borealis, produced by

Motion Mountain – The Adventure of Physics
charged particles in the
night sky (© Jan Curtis).

the distribution of the incoming direction of cosmic rays shows that many rays must be
extragalactic in origin. Indeed, the typical nuclei of cosmic radiation are ejected from
stars and accelerated by supernova explosions. When they arrive on Earth, they interact
with the atmosphere before they reach the surface of the Earth. The detailed acceleration
mechanisms at the origin of cosmic rays are still a topic of research.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The flux of charged cosmic rays arriving at the surface of the Earth depends on their
energy. At the lowest energies, charged cosmic rays hit the human body many times a
second. Measurements also show that the rays arrive in irregular groups, called showers.
In fact, the neutrino flux is many orders of magnitude higher than the flux of charged
Page 256 rays, but does not have any effect on human bodies.
Vol. III, page 218 Cosmic rays have several effects on everyday life. Through the charges they pro-
duce in the atmosphere, they are probably responsible for the start and for the jagged,
non-straight propagation of lightning. (Lightning advances in pulses, alternating fast
propagation for about 30 m with slow propagation, until they hit connect. The direction
they take at the slow spots depends on the wind and the charge distribution in the at-
mosphere.) Cosmic rays are also important in the creation of rain drops and ice particles
inside clouds, and thus indirectly in the charging of the clouds. Cosmic rays, together
Vol. III, page 19 with ambient radioactivity, also start the Kelvin generator.
If the magnetic field of the Earth would not exist, we might get sick from cosmic rays.
The magnetic field diverts most rays towards the magnetic poles. Also both the upper and
lower atmosphere help animal life to survive, by shielding life from the harmful effects of
cosmic rays. Indeed, aeroplane pilots and airline employees have a strong radiation ex-
posure that is not favourable to their health. Cosmic rays are also one of several reasons
that long space travel, such as a trip to Mars, is not an option for humans. When cosmo-
nauts get too much radiation exposure, the body weakens and eventually they die. Space
heroes, including those of science fiction, would not survive much longer than two or
three years.
5 the densest clouds 179

F I G U R E 107 Two aurorae australes on Earth, seen from space (a composed image with superimposed
UV intensity, and a view in the X-ray domain) and a double aurora on Saturn (all NASA).

Cosmic rays also produce beautifully coloured flashes inside the eyes of cosmonauts;

Motion Mountain – The Adventure of Physics
they regularly enjoy these events in their trips. (And they all develop cataracts as a con-
sequence.) But cosmic rays are not only dangerous and beautiful. They are also useful. If
cosmic rays would not exist, we would not exist either. Cosmic rays are responsible for
mutations of life forms and thus are one of the causes of biological evolution. Today, this
effect is even used artificially; putting cells into a radioactive environment yields new
strains. Breeders regularly derive new mutants in this way.
Cosmic rays cannot be seen directly, but their cousins, the ‘solar’ rays, can. This is
most spectacular when they arrive in high numbers. In such cases, the particles are in-
evitably deviated to the poles by the magnetic field of the Earth and form a so-called au-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
rora borealis (at the North Pole) or an aurora australis (at the South pole). These slowly
moving and variously coloured curtains of light belong to the most spectacular effects
in the night sky. (See Figure 106 or www.nasa.gov/mov/105423main_FUV_2005-01_v01.
mov.) Visible light and X-rays are emitted at altitudes between 60 and 1000 km. Seen
from space, the aurora curtains typically form a circle with a few thousand kilometres
diameter around the magnetic poles. Aurorae are also seen in the rest of the solar sys-
tem. Aurorae due to core magnetic fields have been observed on Jupiter, Saturn, Uranus,
Neptune, Earth, Io and Ganymede. For an example, see Figure 107. Aurorae due to other
mechanisms have been seen on Venus and Mars.
Cosmic rays are mainly free nuclei. With time, researchers found that nuclei appear
without electron clouds also in other situations. In fact, the vast majority of nuclei in the
universe have no electron clouds at all: in the inside of stars, no nucleus is surrounded
by bound electrons; similarly, a large part of intergalactic matter is made of protons. It
is known today that most of the matter in the universe is found as protons or α particles
inside stars and as thin gas between the galaxies. In other words, in contrast to what the
Greeks said, matter is not usually made of atoms; it is mostly made of bare nuclei. Our
everyday environment is an exception when seen on cosmic scales. In nature, atoms are
rare, bare nuclei are common.
Incidentally, nuclei are in no way forced to move; nuclei can also be stored with almost
no motion. There are methods – now commonly used in research groups – to superpose
electric and magnetic fields in such a way that a single nucleus can be kept floating in
Vol. III, page 226 mid-air; we discussed this possibility in the section on levitation earlier on.
180 5 the structure of the nucleus

Nuclei decay – more on radioactivit y
Not all nuclei are stable over time. The first measurement that provided a hint was the
decrease of radioactivity with time. It is observed that the number 𝑁 of emitted rays
decreases. More precisely, radioactivity follows an exponential decay with time 𝑡:

𝑁(𝑡) = 𝑁(0) e−𝑡/𝜏 . (54)

The parameter 𝜏, the so-called life time or decay time, depends on the type of nucleus
emitting the rays. Life times can vary from far less than a microsecond to millions of mil-
lions of years. The expression has been checked for as long as 34 multiples of the duration
𝜏; its validity and precision is well-established by experiments. Obviously, formula (54)
is an approximation for large numbers of atoms, as it assumes that 𝑁(𝑡) is a continuous
variable. Despite this approximation, deriving this expression from quantum theory is
Page 48 not a simple exercise, as we saw above. Though in principle, the quantum Zeno effect

Motion Mountain – The Adventure of Physics
could appear for small times 𝑡, for the case of radioactivity it has not yet been observed.
Instead of the life-time, often the half-life is used. The half-life is the time during which
radioactivity decreases to half the starting value. Can you deduce how the two times are
Challenge 128 s related?
Radioactivity is the decay of unstable nuclei. Most of all, radioactivity allows us to
count the number of atoms in a given mass of material. Imagine to have measured the
mass of radioactive material at the beginning of your experiment; you have chosen an
element that has a lifetime of about a day. Then you put the material inside a scintillation
box. After a few weeks the number of flashes has become so low that you can count them;
using expression (54) you can then determine how many atoms have been in the mass

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
to begin with. Radioactivity thus allows us to determine the number of atoms, and thus
their size, in addition to the size of nuclei.
The exponential decay (54) and the release of energy is typical of metastable systems.
In 1903, Rutherford and Soddy discovered what the state of lower energy is for α and
β emitters. In these cases, radioactivity changes the emitting atom; it is a spontaneous
transmutation of the atom. An atom emitting α or β rays changes its chemical nature.
Radioactivity thus implies, for the case of nuclei, the same result that statistical mech-
Vol. I, page 406 anics of gases implies for the case of atoms: they are quantum particles with a structure
that can change over time.

— In α decay – or alpha decay – the radiating nucleus emits a (doubly charged) helium
nucleus, also called an α particle. The kinetic energy is typically a handful of MeV.
After the emission, the nucleus has changed to a nucleus situated two places earlier
in the periodic system of the elements. α decay occurs mainly for nuclei that are rich
in protons. An example of α decay is the decay of the 238 U isotope of uranium.
— In β decay – or beta decay – a neutron transforms itself into a proton, emitting an elec-
tron – also called a β particle – and an antineutrino. Also β decay changes the chemical
nature of the atom, but to the place following the original atom in the periodic table
of the elements. Example of β emitters are radiocarbon, 14 C, 38 Cl, and 137 Cs, the iso-
Page 240 tope expelled by damaged nuclear reactors. We will explore β decay below. A variant
is the β+ decay, in which a proton changes into a neutron and emits a neutrino and
5 the densest clouds 181

a positron. It occurs in proton-rich nuclei. An example is 22 Na. Another variant is
electron capture; a nucleus sometimes captures an orbital electron, a proton is trans-
formed into a neutron and a neutrino is emitted. This happens in 7 Be. Also bound β
decay, as seen in 187 Re, is a variant of β decay.
— In γ decay – or gamma decay – the nucleus changes from an excited to a lower energy
state by emitting a high energy photon, or γ particle. In this case, the chemical nature
is not changed. Typical energies are in the MeV range. Due to the high energy, γ rays
ionize the material they encounter; since they are not charged, they are not well ab-
sorbed by matter and penetrate deep into materials. γ radiation is thus by far the most
dangerous type of (environmental) radioactivity. An example of γ decay is 99𝑚 Tc. A
variant of γ decay is isomeric transition. Still another variant is internal conversion,
observed, for example, in 137𝑚 Ba.
— In neutron emission the nucleus emits a neutron. The decay is rare on Earth, but
occurs in the stellar explosions. Most neutron emitters have half-lives below a few
seconds. Examples of neutron emitters are 5 He and 17 N.

Motion Mountain – The Adventure of Physics
— The process of spontaneous fission was discovered in 1940. The decay products vary,
even for the same starting nucleus. But 239 Pu and 235 U can decay through spontan-
eous fission, though with a small probability.
— In proton emission the nucleus emits a proton. This decay is comparatively rare, and
occurs only for about a hundred nuclides, for example for 53𝑚 Co and 4 Li. The first
example was discovered only in 1970. Around 2000, the simultaneous emission of
two protons was also observed for the first time.
— In 1984, cluster emission or heavy ion emission was discovered. A small fraction of
223
Ra nuclei decay by emitting a 14 C nucleus. This decay occurs for half a dozen nuc-
lides. Emission of 18 O has also been observed.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 159 Many combined and mixed decays also exist. These decays are studied by nuclear phys-
icists. Radioactivity is a common process. As an example, in every human body about
nine thousand radioactive decays take place every second, mainly 4.5 kBq (0.2 mSv/a)
Challenge 129 s from 40 K and 4 kBq from 14 C (0.01 mSv/a). Why is this not dangerous?
All radioactivity is accompanied by emission of energy. The energy emitted by an
atom through radioactive decay or reactions is regularly a million times larger than that
emitted by a chemical process. More than a decay, a radioactive process is thus a micro-
scopic explosion. A highly radioactive material thus emits a large amount of energy. That
is the reason for the danger of nuclear weapons.
What distinguishes those atoms that decay from those which do not? An exponential
Challenge 130 e decay law implies that the probability of decay is independent of the age of the atom. Age
Vol. IV, page 113 or time plays no role. We also know from thermodynamics, that all atoms have exactly
identical properties. So how is the decaying atom singled out? It took around 30 years
to discover that radioactive decays, like all decays, are quantum effects. All decays are
triggered by the statistical fluctuations of the vacuum, more precisely, by the quantum
fluctuations of the vacuum. Indeed, radioactivity is one of the clearest observations that
classical physics is not sufficient to describe nature.
Radioactivity, like all decays, is a pure quantum effect. Only a finite quantum of action
makes it possible that a system remains unchanged until it suddenly decays. Indeed, in
1928 George Gamow explained α decay with the tunnelling effect. He found that the
182 5 the structure of the nucleus

Motion Mountain – The Adventure of Physics
F I G U R E 108 A modern accelerator mass spectrometer for radiocarbon dating, at the Hungarian
Academy of Sciences (© HAS).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tunnelling effect explains the relation between the lifetime and the range of the rays,
as well as the measured variation of lifetimes – between 10 ns and 1017 years – as the
consequence of the varying potentials to be overcome in different nuclei.

R adiometric dating
As a result of the chemical effects of radioactivity, the composition ratio of certain ele-
ments in minerals allows us to determine the age of the mineral. Using radioactive decay
to deduce the age of a sample is called radiometric dating. With this technique, geologists
determined the age of mountains, the age of sediments and the age of the continents.
They determined the time that continents moved apart, the time that mountains formed
when the continents collided and the time when igneous rocks were formed. Where there
surprises? No. The times found with radiometric dating are consistent with the relative
time scale that geologists had defined independently for centuries before the technique
appeared. Radiometric dating confirmed what had been deduced before.
Ref. 161 Radiometric dating is a science of its own. An overview of the isotopes used, together
Page 183 with their specific applications in dating of specimen, is given in Table 14. The table shows
how the technique of radiometric dating has deeply impacted astronomy, geology, evol-
utionary biology, archaeology and history. (And it has reduced the number of violent
Ref. 160 believers.) Radioactive life times can usually be measured to within one or two per cent
5 the densest clouds 183

TA B L E 14 The main natural isotopes used in radiometric dating.

Isotope D e c ay Half- life Method using it Examples
product
147 143
Sm Nd 106 Ga samarium–neodymium rocks, lunar soil,
method meteorites
87 87
Rb Sr 48.8 Ga rubidium–strontium rocks, lunar soil,
method meteorites
187 187
Re Os 42 Ga rhenium–osmium rocks, lunar soil,
method meteorites
176 176
Lu Hf 37 Ga lutetium–hafnium rocks, lunar soil,
method meteorites
40 40
K Ar 1.25 Ga potassium–argon & rocks, lunar soil,
argon–argon method meteorites
40 40
K Ca 1.25 Ga potassium–calcium granite dating, not

Motion Mountain – The Adventure of Physics
method precise
232 208
Th Pb 14 Ga thorium–lead method, rocks, lunar soil,
lead–lead method meteorites
238 206
U Pb 4.5 Ga uranium–lead method, rocks, lunar soil,
lead–lead method meteorites
235 207
U Pb 0.7 Ga uranium–lead method, rocks, lunar soil,
lead–lead method meteorites
234 230
U Th 248 ka uranium–thorium corals, stalagmites,
method bones, teeth
230 226
Th Ra 75.4 ka thorium-radon method plant dating

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
26 26
Al Mg 0.72 Ma supernova debris dating,
checking that
cosmogenic nucleosynthesis still
takes place in the galaxy
10 10
Be B 1.52 Ma cosmogenic radiometric ice cores
dating
60 60
Fe Ni 2.6 Ma (not 1.5 supernova debris dating deep sea crust
Ma)
36 36
Cl Ar 0.3 Ma cosmogenic radiometric ice cores
dating
53 53
Mn Cr 3.7 Ma cosmogenic radiometric meteorites, K/T
dating boundary
182 182
Hf W 9 Ma cosmogenic radiometric meteorites, sediments
dating
14 14
C N 5730 a radiocarbon method, wood, clothing, bones,
cosmogenic organic material, wine
137 137
Cs Ba 30 a γ-ray counting dating food and wine
after nuclear accidents
210
Pb 22 a γ-ray counting dating wine
3 3
H He 12.3 a γ-ray counting dating wine
184 5 the structure of the nucleus

of accuracy, and they are known both experimentally and theoretically not to change
over geological time scales. As a result, radiometric dating methods can be surprisingly
precise. Can you imagine how to measure half-lives of thousands of millions of years to
Challenge 131 s high precision?
Radiometric dating was even more successful in the field of ancient history. With
the radiocarbon dating method historians determined the age of civilizations and the age
of human artefacts.* Many false beliefs were shattered. In some belief communities the
shock is still not over, even though over hundred years have passed since these results
Ref. 160 became known.
Radiocarbon dating uses the β decay of the radioactive carbon isotope 14 C, which has
a decay time of 5730 a. This isotope is continually created in the atmosphere through the
influence of cosmic rays. This happens through the reaction 14 N + n → p + 14 C. As a res-
ult, the concentration of radiocarbon in air is relatively constant over time. Inside living
plants, the metabolism thus (unknowingly) maintains the same concentration. In dead
plants, the decay sets in. The life time value of a few thousand years is particularly useful

Motion Mountain – The Adventure of Physics
to date historic material. Therefore, radiocarbon dating has been used to determine the
age of mummies, the age of prehistoric tools and the age of religious relics. The original
version of the technique measured the radiocarbon content through its radioactive decay
and the scintillations it produced. A quality jump was achieved when accelerator mass
spectroscopy became commonplace. It was not necessary any more to wait for decays:
it is now possible to determine the 14 C content directly. As a result, only a tiny amount
of carbon, as low as 0.2 mg, is necessary for a precise dating. Such small amounts can
be detached from most specimen without big damage. Accelerator mass spectroscopy
showed that numerous religious relics are forgeries, such as a cloth in Turin, and that, in

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
addition, several of their wardens are crooks.
Researchers have even developed an additional method to date stones that uses ra-
dioactivity. Whenever an α ray is emitted, the emitting atom gets a recoil. If the atom is
part of a crystal, the crystal is damaged by the recoil. In many materials, the damage can
be seen under the microscope. By counting the damaged regions it is possible to date the
time at which rocks have been crystallized. In this way it has been possible to determine
when the liquid material from volcanic eruptions has become rock.
With the advent of radiometric dating, for the first time it became possible to reliably
date the age of rocks, to compare it with the age of meteorites and, when space travel
became fashionable, with the age of the Moon. The result was beyond all previous estim-
ates and expectations: the oldest rocks and the oldest meteorites, studied independently
Ref. 162 using different dating methods, are 4570(10) million years old. From this data, the age of
the Earth is estimated to be 4540(50) million years. The Earth is indeed old.
But if the Earth is so old, why did it not cool down in its core in the meantime?

Why is hell hot?
The lava seas and streams found in and around volcanoes are the origin of the imagery
that many cultures ascribe to hell: fire and suffering. Because of the high temperature of
lava, hell is inevitably depicted as a hot place located at the centre of the Earth. A striking
* In 1960, the developer of the radiocarbon dating technique, Willard Libby, received the Nobel Prize in
Chemistry.
5 the densest clouds 185

Motion Mountain – The Adventure of Physics
F I G U R E 109 The lava sea in the volcano Erta Ale in Ethiopia (© Marco Fulle).

example is the volcano Erta Ale, shown in Figure 109. But why is lava still hot, after so

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
many million years?
A straightforward calculation shows that if the Earth had been a hot sphere in the
Challenge 132 ny beginning, it should have cooled down and solidified already long time ago. The Earth
should be a solid object, like the moon: the Earth should not contain any magma nor
eject any lava; hell would not be hot.
The solution to the riddle is provided by radioactivity: the centre of the Earth contains
an oven that is fuelled with an estimated 8 to 10 TW by radioactive uranium 235 U and
238
U, with 8 to 10 TW by radioactive thorium 232 Th and with around 4 TW by radio-
Ref. 163 active potassium 40 K. The radioactivity of these elements, and a few others to a minor
degree, keeps the centre of the Earth glowing. More precise investigations, taking into
Page 183 account the decay times and measured material concentrations, show that this mechan-
ism indeed explains the internal heat of the Earth. (In addition, the decay of radioactive
potassium is the origin for the 1 % of argon found in the Earth’s atmosphere.)
In short, radioactivity keeps lava hot. Radioactivity is the reason that we depict hell
as hot. This brings up a challenge: why is the radioactivity of lava and of the Earth in
Challenge 133 s general not dangerous to humans?

Nuclei can form composites
Nuclei are highly unstable when they contain more than about 280 nucleons. Nuclei with
higher number of nucleons inevitably decay into smaller fragments. In short, heavy nuc-
lei are unstable. But when the mass is above 1057 nucleons, they are stable again: such sys-
186 5 the structure of the nucleus

tems are called neutron stars. This is the most extreme example of pure nuclear matter
found in nature. Neutron stars are left overs of (type II) supernova explosions. They do
not run any fusion reactions any more, as other stars do; in first approximation neutron
stars are simply large nuclei.
Neutron stars are made of degenerate matter. Their density of 1018 kg/m3 is a few
times that of a nucleus, as gravity compresses the star. This density value means that a
tea spoon of such a star has a mass of several hundred million tons. Neutron stars are
about 10 km in diameter. They are never much smaller, as such smaller stars are unstable.
They are never much larger, because much larger neutron stars turn into black holes.

Nuclei have colours and shapes
In everyday life, the colour of objects is determined by the wavelength of light that is
least absorbed, or, if they shine, by the wavelength that is emitted. Also nuclei can ab-
sorb photons of suitably tuned energies and get into an excited state. In this case, the
photon energy is converted into a higher energy of one or several of the nucleons whirl-

Motion Mountain – The Adventure of Physics
ing around inside the nucleus. Many radioactive nuclei also emit high energy photons,
which then are called γ rays, in the range between 1 keV (or 0.2 fJ) and more than 20 MeV
(or 3.3 pJ). The emission of γ rays by nuclei is similar to the emission of light by electrons
in atoms. From the energy, the number and the lifetime of the excited states – they range
from 1 ps to 300 d – researchers can deduce how the nucleons move inside the nucleus.
In short, the energies of the emitted and absorbed γ ray photons define the ‘colour’ of
the nucleus. The γ ray spectrum can be used, like all colours, to distinguish nuclei from
each other and to study their motion. In particular, the spectrum of the γ rays emitted by
excited nuclei can be used to determine the chemical composition of a piece of matter.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Some of these transition lines are so narrow that they can been used to study the change
due to the chemical environment of the nucleus, to measure nuclear motion inside solids
or to detect the gravitational Doppler effect.
The study of γ-rays also allows us to determine the shape of nuclei. Many nuclei are
spherical; but many are prolate or oblate ellipsoids. Ellipsoids are favoured if the reduc-
tion in average electrostatic repulsion is larger than the increase in surface energy. All
nuclei – except the lightest ones such as helium, lithium and beryllium – have a constant
mass density at their centre, given by about 0.17 nucleons per fm3 , and a skin thickness
of about 2.4 fm, where their density decreases. Nuclei are thus small clouds, as illustrated
in Figure 110.
We know that molecules can be of extremely involved shape. In contrast, nuclei are
mostly spheres, ellipsoids or small variations of these. The reason is the short range, or
better, the fast spatial decay of nuclear interactions. To get interesting shapes like in mo-
lecules, one needs, apart from nearest neighbour interactions, also next neighbour inter-
actions and next next neighbour interactions. The strong nuclear interaction is too short
ranged to make this possible. Or does it? It might be that future studies will discover that
some nuclei are of more unusual shape, such as smoothed pyramids. Some predictions
Ref. 164 have been made in this direction; however, the experiments have not been performed
yet.
The shape of nuclei does not have to be fixed; nuclei can also oscillate in shape. Such
oscillations have been studied in great detail. The two simplest cases, the quadrupole and
5 the densest clouds 187

Fixed nuclear shapes Oscillating nuclear shapes

6Li 17O, 2H, 64Zn 122Te
28Si, 20Ne,
36Ar, 57Fe,
63Cu, 59Co,
115Sb, 161Dy,

Motion Mountain – The Adventure of Physics
129I, 177Lu
209Bi

F I G U R E 110 Various nuclear shapes – ﬁxed: spherical, oblate, prolate (left) and oscillating (right), shown
realistically as clouds (above) and simpliﬁed as geometric shapes (below).

octupole oscillations, are shown in Figure 110. In addition, non-spherical nuclei can also
rotate. Several rapidly spinning nuclei, with a spin of up to 60ℏ and more, are known.
They usually slow down step by step, emitting a photon and reducing their angular mo-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
mentum at each step. Recently it was even discovered that nuclei can also have bulges
Ref. 165 that rotate around a fixed core, a bit like the tides that rotate around the Earth.

The four t ypes of motion in the nuclear d omain
Nuclei are small because the nuclear interactions are short-ranged. Due to this short
range, nuclear interactions play a role only in four types of motion:
— scattering,
— bound motion,
— decay and
— a combination of these three called nuclear reactions.
The history of nuclear physics has shown that the whole range of observed phenomena
can be reduced to these four fundamental processes. Each process is a type of motion.
And in each process, the main interest is the comparison of the start and the end situ-
ations; the intermediate situations are less interesting. Nuclear interactions thus lack the
complex types of motion which characterize everyday life. That is also the main reason
for the shortness of this chapter.
Scattering is performed in all accelerator experiments. Such experiments repeat for
nuclei what we do when we look at an object. Eye observation, or seeing something, is a
scattering experiment, as eye observation is the detection of scattered light. Scattering of
X-rays was used to see atoms for the first time; scattering of high energy alpha particles
188 5 the structure of the nucleus

was used to discover and study the nucleus, and later the scattering of electrons with
even higher energy was used to discover and study the components of the proton.
Bound motion is the motion of protons and neutrons inside nuclei or the motion of
quarks inside mesons and baryons. In particular, bound motion determines shape and
changes of shape of compounds: hadrons and nuclei.
Decay is obviously the basis of radioactivity. Nuclear decay can be due to the elec-
tromagnetic, the strong or the weak nuclear interaction. Decay allows studying the con-
served quantities of nuclear interactions.
Nuclear reactions are combinations of scattering, decay and possibly bound motion.
Nuclear reactions are for nuclei what the touching of objects is in everyday life. Touch-
ing an object we can take it apart, break it, solder two objects together, throw it away,
and much more. The same can be done with nuclei. In particular, nuclear reactions are
responsible for the burning of the Sun and the other stars; they also tell the history of the
nuclei inside our bodies.
Quantum theory showed that all four types of nuclear motion can be described in the

Motion Mountain – The Adventure of Physics
same way. Each type of motion is due to the exchange of virtual particles. For example,
scattering due to charge repulsion is due to exchange of virtual photons, the bound mo-
tion inside nuclei due to the strong nuclear interaction is due to exchange of virtual
gluons, β decay is due to the exchange of virtual W bosons, and neutrino reactions are
due to the exchange of virtual Z bosons. The rest of this chapter explains these mechan-
isms in more details.

Nuclei react
The first man thought to have made transuranic elements, the physics genius Enrico

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Vol. IV, page 118 Fermi, received the Nobel Prize in Physics for the discovery. Shortly afterwards, Otto
Hahn and his collaborators Lise Meitner and Fritz Strassmann showed that Fermi was
wrong, and that his prize was based on a mistake. Fermi was allowed to keep his prize,
the Nobel committee gave Hahn and Strassmann the Nobel Prize as well, and to make
the matter unclear to everybody and to women physicists in particular, the prize was not
given to Lise Meitner. (After her death though, a new chemical element was named after
her.)
When protons or neutrons are shot into nuclei, they usually remained stuck inside
them, and usually lead to the transformation of an element into a heavier one. After hav-
ing done this with all elements, Fermi used uranium; he found that bombarding it with
neutrons, a new element appeared, and concluded that he had created a transuranic ele-
ment. Alas, Hahn and his collaborators found that the element formed was well-known:
it was barium, a nucleus with less than half the mass of uranium. Instead of remaining
stuck as in the previous 91 elements, the neutrons had split the uranium nucleus. In short,
Fermi, Hahn, Meitner and Strassmann had observed reactions such as:
235
U + n → 143 Ba + 90 Kr + 3𝑛 + 170 MeV . (55)

Meitner called the splitting process nuclear fission. The amount of energy liberated in
fission is unusually large, millions of times larger than in a chemical interaction of an
atom. In addition, several neutrons are emitted, which in turn can lead to the same pro-
5 the densest clouds 189

cess; fission can thus start a chain reaction. Later, and (of course) against the will of the
team, the discovery would be used to make nuclear bombs.
Nuclear reactions are typically triggered by neutrons, protons, deuterons or γ
particles. Apart from triggering fission, neutrons are used to transform lithium into
tritium, which is used as (one type of) fuel in fusion reactors; and neutrons from
(secondary) cosmic rays produce radiocarbon from the nitrogen in the atmosphere.
Deuterons impinging on tritium produce helium in fusion reactors. Protons can trigger
the transformation of lithium into beryllium. Photons can knock alpha particles or
neutrons out of nuclei.
All nuclear reactions and decays are transformations. In each transformation, already
the ancient Greek taught us to search, first of all, for conserved quantities. Besides the
well-known cases of energy, momentum, electric charge and angular momentum conser-
vation, the results of nuclear physics lead to several new conserved quantities. The beha-
viour is quite constrained. Quantum field theory implies that particles and antiparticles
(commonly denoted by a bar) must behave in compatible ways. Both experiment and

Motion Mountain – The Adventure of Physics
quantum field theory show for example that every reaction of the type

A+B→ C+D (56)

implies that the reactions
A+C →B+D (57)

or
C+D →A+B (58)

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
or, if energy is sufficient,
A→C+D+B, (59)

are also possible. Particles thus behave like conserved mathematical entities.
Experiments show that antineutrinos differ from neutrinos. In fact, all reactions con-
firm that the so-called lepton number is conserved in nature. The lepton number 𝐿 is zero
for nucleons or quarks, is 1 for the electron and the neutrino, and is −1 for the positron
and the antineutrino.
In addition, all reactions conserve the so-called baryon number. The baryon number
𝐵 for protons and neutrons is 1 (and 1/3 for quarks), and −1 for antiprotons and anti-
neutrons (and thus −1/3 for antiquarks). So far, no process with baryon number viola-
tion has ever been observed. Baryon number conservation is one reason for the danger
of radioactivity, fission and fusion. The concept of baryon number was introduced by
Ernst Stückelberg (b. 1905 Basel, d. 1984 Geneva), an important physicist who discovered
several other concepts of particle physics, including Feynman diagrams before Feyn-
man himself. Baryon number was renamed when Abraham Pais (b. 1918 Amsterdam,
d. 2000 Copenhagen) introduced the terms ‘lepton’ and ‘baryon’.
190 5 the structure of the nucleus

F I G U R E 111 The destruction of four nuclear reactors in 2011 in Fukushima, in Japan, which rendered

Motion Mountain – The Adventure of Physics
life impossible at a distance of 30 km around it (courtesy Digital Globe).

F I G U R E 112 The explosion of a nuclear bomb: an involved method of killing many children in the
country where it explodes and ruining the economic future of many children in the country that built it.
5 the densest clouds 191

B ombs and nuclear reactors
Uranium fission is triggered by a neutron, liberates energy and produces several addi-
tional neutrons. Therefore, uranium fission can trigger a chain reaction that can lead
either to an explosion or to a controlled generation of heat. Once upon a time, in the
middle of the twentieth century, these processes were studied by quite a number of re-
searchers. Most of them were interested in making weapons or in using nuclear energy,
despite the high toll these activities place on the economy, on human health and on the
environment.
Most stories around the development of nuclear weapons are almost incredibly ab-
surd. The first such weapons were built during the Second World War, with the help of
the smartest physicists that could be found. Everything was ready, including the most
complex physical models, several huge factories and an organization of incredible size.
There was just one little problem: there was no uranium of sufficient quality. The mighty
United States thus had to go around the world to shop for good uranium. They found it
in the (then) Belgian colony of Congo, in central Africa. In short, without the support

Motion Mountain – The Adventure of Physics
of Belgium, which sold the Congolese uranium to the USA, there would have been no
nuclear bomb, no early war end and no superpower status.
Congo paid a high price for this important status. It was ruled by a long chain of
military dictators up to this day. But the highest price was paid by the countries that
actually built nuclear weapons. Some went bankrupt, others remained underdeveloped;
even the richest countries have amassed huge debts and a large underprivileged popula-
tion. There is no exception. The price of nuclear weapons has also been that some regions
of our planet became uninhabitable, such as numerous islands, deserts, rivers, lakes and
marine environments. But it could have been worse. When the most violent physicist

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ever, Edward Teller, made his first calculations about the hydrogen bomb, he predicted
that the bomb would set the atmosphere into fire. Nobel Prize winner Hans Bethe* cor-
rected the mistake and showed that nothing of this sort would happen. Nevertheless, the
military preferred to explode the hydrogen bomb in the Bikini atoll, the most distant
Ref. 166 place from their homeland they could find. The place is so radioactive that today it is
even dangerous simply to fly over that island!
It was then noticed that nuclear test explosions increased ambient radioactivity in the
atmosphere all over the world. Of the produced radioactive elements, 3 H is absorbed by
humans in drinking water, 14 C and 90 Sr through food, and 137 Cs in both ways. Fortu-
nately, in the meantime, all countries have agreed to perform their nuclear tests under-
ground.
Radioactivity is dangerous to humans, because it disrupts the processes inside living
cells. Details on how radioactivity is measured and what effects on health it produces are
Page 194 provided below.
Not only nuclear bombs, also peaceful nuclear reactors are dangerous. The reason was

* Hans Bethe (b. 1906 Strasbourg, d. 2005) was one of the great physicists of the twentieth century, even
though he was head of the theory department that lead to the construction of the first atomic bombs. He
worked on nuclear physics and astrophysics, helped Richard Feynman in developing quantum electrody-
namics, and worked on solid state physics. When he got older and wiser, he became a strong advocate of
arms control; he also was essential in persuading the world to stop atmospheric nuclear test explosions and
saved many humans from cancer in doing so.
192 5 the structure of the nucleus

TA B L E 15 Some radioactivity measurements.

M at e r i a l Activity in
B q/kg
Air c. 10−2
Sea water 101
Human body c. 102
Cow milk max. 103
Pure 238 U metal c. 107
Highly radioactive α emitters > 107
Radiocarbon: 14 C (β emitter) 108
Highly radioactive β and γ emitters > 109
Main nuclear fallout: 137 Cs, 90 Sr (α emitter) 2 ⋅ 109
Polonium, one of the most radioactive materials (α) 1024

Motion Mountain – The Adventure of Physics
discovered in 1934 by Frédéric Joliot and his wife Irène, the daughter of Pierre and Marie
Curie: artificial radioactivity. The Joliot–Curies discovered that materials irradiated by
α rays become radioactive in turn. They found that α rays transformed aluminium into
radioactive phosphorus:
27 4 30 1
13 Al + 2 𝛼 → 15 P + 0 n . (60)

In fact, almost all materials become radioactive when irradiated with alpha particles,
neutrons or γ rays. As a result, radioactivity itself can only be contained with difficulty.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
After a time span that depends on the material and the radiation, any box that contains
radioactive material has itself become radioactive. The ‘contagion’ stops only for very
small amounts of radioactive material.
The dangers of natural and artificial radioactivity are the reason for the high costs of
nuclear reactors. After about thirty years of operation, reactors have to be dismantled.
The radioactive pieces have to be stored in specially chosen, inaccessible places, and at
the same time the workers’ health must not be put in danger. The world over, many
reactors now need to be dismantled. The companies performing the job sell the service
at high price. All operate in a region not far from the border to criminal activity, and
since radioactivity cannot be detected by the human senses, many companies cross that
border.
In fact, one important nuclear reactor is (usually) not dangerous to humans: the Sun.
Page 200 We explore it shortly.

Curiosities and challenges on nuclei and radioactivity
Nowadays, nuclear magnetic resonance is also used to check the quality of food. For
example, modern machines can detect whether orange juice is contaminated with juice
from other fruit and can check whether the fruit were ripe when pressed. Other machines
can check whether wine was made from the correct grapes and how it aged.
∗∗
5 the densest clouds 193

Motion Mountain – The Adventure of Physics
F I G U R E 113 A machine to test fruit quality with
the help of nuclear magnetic resonance
(© Bruker).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Magnetic resonance machines pose no danger; but they do have some biological effects,
Ref. 171 as Peter Mansfield, one of the inventors of the technique, explains. The first effect is due
to the conductivity of blood. When blood in the aorta passes through a magnetic field,
a voltage is induced. This effect has been measured and it might interfere with cardiac
functioning at 7 T; usual machines have 1.5 T and pose no risk. The second effect is due to
the switching of the magnetic field. Some people sense the switching in the thorax and in
the shoulders. Not much is known about the details of such peripheral nerve stimulation
yet.
∗∗
The amount of radioactive radiation is called the dose. The unit for the radioactive dose
is one gray: it is the amount of radioactivity that deposits the energy 1 J on 1 kg of matter:
1 Gy = 1 J/kg. A sievert, or 1 Sv, is the unit of radioactive dose equivalent; it is adjusted
to humans by weighting each type of human tissue with a factor representing the impact
of radiation deposition on it. Three to five sievert are a lethal dose to humans. In com-
parison, the natural radioactivity present inside human bodies leads to a dose of 0.2 mSv
Ref. 167 per year. An average X-ray image implies an irradiation of 1 mSv; a CAT scan 8 mSv. For
other measurement examples, see Table 15.
The amount of radioactive material is measured by the number of nuclear decays
per second. One decay per second is called one becquerel, or 1 Bq. An adult human
body typically contains 9 kBq, the European limit for food, in 2011, varies between 370
194 5 the structure of the nucleus

TA B L E 16 Human exposure to radioactivity and the corresponding doses.

Exposure Dose

Daily human exposure:
Average exposure to cosmic radiation in Europe
at sea level 0.3 mSv/a
at a height of 3 km 1.2 mSv/a
Average (and maximum) exposure from the soil, 0.4 mSv/a (2 mSv/a)
not counting radon effects
Average (and maximum) inhalation of radon 1 mSv/a (100 mSv/a)
Average exposure due to internal radionuclides 0.3 mSv/a
natural content of 40 K in human muscles 10−4 Gy and 4500 Bq
natural content of Ra in human bones 2 ⋅ 10−5 Gy and 4000 Bq
natural content of 14 C in humans 10−5 Gy

Motion Mountain – The Adventure of Physics
Total average (and maximum) human exposure 2 mSv/a (100 mSv/a)
Common situations:
Dental X-ray c. 10 mSv equivalent dose
Lung X-ray c. 0.5 mSv equivalent dose
Short one hour flight (see www.gsf.de/epcard) c. 1 μSv
Transatlantic flight c. 0.04 mSv
Maximum allowed dose at work 30 mSv/a
Smoking 60 cigarettes a day 26 to 120 mSv/a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Deadly exposures:
Ionization 0.05 C/kg can be deadly
Dose 100 Gy=100 J/kg is deadly in 1 to 3
days
Equivalent dose more than 3 Sv leads to death for
50 % of untreated patients

and 600 Bq/kg. The amount released by the Hiroshima bomb is estimated to have been
between 4 PBq and 60 PBq, the amount released by the Chernobyl disaster was between
2 and 12 EBq, thus between 200 and 500 times larger. The numbers for the various Rus-
sian radioactive disasters in the 1960s and 1970 are similarly high. The release for the
Fukushima reactor disaster in March 2011 is estimated to have been 370 to 630 PBq,
which would put it at somewhere between 10 and 90 Hiroshima bombs.
The SI units for radioactivity are now common around the world; in the old days,
1 sievert was called 100 rem or ‘Röntgen equivalent man’; the SI unit for dose, 1 gray,
replaces what used to be called 100 rd or Rad. The SI unit for exposure, 1 C/kg, replaces
the older unit ‘röntgen’, with the relation 1 R = 2.58 ⋅ 10−4 C/kg. The SI unit becquerel
replaces the curie (Ci), for which 1 Ci = 37 GBq.
∗∗
5 the densest clouds 195

Motion Mountain – The Adventure of Physics
F I G U R E 114 A dated image of Lake Karachay and the nuclear plant that was ﬁlling it with radioactivity
(© Unknown).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Not all γ-rays are due to radioactivity. In the year 2000, an Italian group discovered that
Ref. 168 thunderstorms also emit γ rays, of energies up to 10 MeV. The mechanisms are still being
investigated; they seem to be related to the formation process of lightning.
∗∗
Chain reactions are quite common in nature, and are not limited to the nuclear domain.
Fire is a chemical chain reaction, as are exploding fireworks. In both cases, material needs
heat to burn; this heat is supplied by a neighbouring region that is already burning.
∗∗
Radioactivity can be extremely dangerous to humans. The best example is plutonium.
Only 1 μg of this α emitter inside the human body are sufficient to cause lung cancer. An-
other example is polonium. Polonium 210 is present in tobacco leaves that were grown
with artificial fertilizers. In addition, tobacco leaves filter other radioactive substances
from the air. Polonium, lead, potassium and the other radioactive nuclei found in to-
bacco are the main reason that smoking produces cancer. Table 16 shows that the dose is
considerable and that it is by far the largest dose absorbed in everyday life.
∗∗
Why is nuclear power a dangerous endeavour? The best argument is Lake Karachay near
Mayak, in the Urals in Russia. In less than a decade, the nuclear plants of the region have
transformed it into the most radioactive place on Earth. In the 1970s, walking on the
196 5 the structure of the nucleus

shore of the lake for an hour led to death on the shore. The radioactive material in the lake
was distributed over large areas in several catastrophic explosions in the 1950s and 1960s,
leading to widespread death and illness. Several of these accidents were comparable to
the Chernobyl accident of 1986; they were kept secret. Today, in contrast to Figure 114,
Ref. 169 the lake is partly filled with concrete – but not covered with it, as is often assumed.
∗∗
All lead is slightly radioactive, because it contains the 210 Pb isotope, a β emitter. This lead
isotope is produced by the uranium and thorium contained in the rock from where the
lead is extracted. For sensitive experiments, such as for neutrino experiments, one needs
radioactivity shields. The best shield material is lead, but obviously it has to be lead with
a low radioactivity level. Since the isotope 210 Pb has a half-life of 22 years, one way to
do it is to use old lead. In a precision neutrino experiment in the Gran Sasso in Italy,
the research team uses lead mined during Roman times, thus 2000 years old, in order to
reduce spurious signals.

Motion Mountain – The Adventure of Physics
∗∗
Not all nuclear reactors are human made. The occurrence of natural nuclear reactors
have been predicted in 1956 by Paul Kuroda. In 1972, the first such example was found.
In Oklo, in the African country of Gabon, there is a now famous geological formation
where uranium is so common that two thousand million years ago a natural nuclear
reactor has formed spontaneously – albeit a small one, with an estimated power gener-
ation of 100 kW. It has been burning for over 150 000 years, during the time when the
uranium 235 percentage was 3 % or more, as required for chain reaction. (Nowadays,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the uranium 235 content on Earth is 0.7 %.) The water of a nearby river was periodically
heated to steam during an estimated 30 minutes; then the reactor cooled down again for
an estimated 2.5 hours, since water is necessary to moderate the neutrons and sustain the
chain reaction. The system has been studied in great detail, from its geological history
up to the statements it makes about the constancy of the ‘laws’ of nature. The studies
showed that 2000 million years ago the mechanisms were the same as those used today.
∗∗
Nuclear reactors exist in many sizes. The largest are used in power plants and can produce
over 1000 MW in electrical power; the smallest are used in satellites, and usually produce
around 10 kW. All work without refuelling for between one and thirty years.
∗∗
Radioactivity also has forensic uses. On many surfaces, it is hard to make finger prints
visible. One method is to put the object in question in an atmosphere of radioactive
iodine or radioactive sulphur dioxide. The gases react with the substances in finger prints.
The fingerprints have thus become radioactive. Looking at the scintillation signals of the
prints – a method called autoradiography – then allows imaging the fingerprint simply
by laying a photographic film or an equivalent detector over the object in question.
∗∗
In contrast to massive particles, massless particles cannot decay at all. There is a simple
5 the densest clouds 197

reason for it: massless particles do not experience time, as their paths are ‘null’. A particle
that does not experience time cannot have a half-life. (Can you find another argument?)
Challenge 134 s

∗∗
High energy radiation is dangerous to humans. In the 1950s, when nuclear tests were still
made above ground by the large armies in the world, the generals overruled the orders
of the medical doctors. They positioned many soldiers nearby to watch the explosion,
and worse, even ordered them to walk to the explosion site as soon as possible after the
explosion. One does not need to comment on the orders of these generals. Several of
these unlucky soldiers made a strange observation: during the flash of the explosion,
Challenge 135 s they were able to see the bones in their own hand and arms. How can this be?
∗∗
In 1958, six nuclear bombs were made to explode in the stratosphere by a vast group

Motion Mountain – The Adventure of Physics
of criminals. A competing criminal group performed similar experiments in 1961, fol-
lowed by even more explosions by both groups in 1962. (For reports and films, see en.
wikipedia.org/wiki/High_altitude_nuclear_explosion.) As a result of most of these ex-
plosions, an artificial aurora was triggered the night following each of them. In addi-
tion, the electromagnetic pulse from the blasts destroyed satellites, destroyed electronics
on Earth, disturbed radio communications, injured people on the surface of the Earth,
caused problems with power plants, and distributed large amounts of radioactive mater-
ial over the Earth – during at least 14 years following the blasts. The van Allen radiation
belts around the Earth were strongly affected; it is expected that the lower van Allen belt

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
will recover from the blasts only in a few hundred years. Fortunately for the human race,
after 1962, this activity was stopped by international treaties.
∗∗
Nuclear bombs are terrible weapons. To experience their violence but also the criminal
Ref. 170 actions of many military people during the tests, have a look at the pictures of explosions.
In the 1950s and 60s, nuclear tests were performed by generals who refused to listen to
doctors and scientists. Generals ordered to explode these weapons in the air, making the
complete atmosphere of the world radioactive, hurting all mankind in doing so; worse,
they even obliged soldiers to visit the radioactive explosion site a few minutes after the
explosion, thus doing their best to let their own soldiers die from cancer and leukaemia.
Generals are people to avoid.
∗∗
Several radioactive dating methods are used to date wine, and more are in development.
Page 183 A few are included in Table 14.
∗∗
A few rare radioactive decay times can be changed by external influence. Electron cap-
ture, as observed in beryllium-7, is one of the rare examples were the decay time can
change, by up to 1.5 %, depending on the chemical environment. The decay time for the
same isotope has also been found to change by a fraction of a per cent under pressures
198 5 the structure of the nucleus

Motion Mountain – The Adventure of Physics
F I G U R E 115 Potatoes being irradiated at the Shihorocho Agricultural Cooperative Isotope Irradiation
Center in Japan. Good appetite!

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of 27 GPa. On the other hand, these effects are predicted (and measured) to be negligible
Ref. 159 for nuclei of larger mass. A few additional nuclides show similar, but smaller effects.
The most interesting effect on nuclei is laser-induced fissioning of 238 U, which occurs
Page 259 for very high laser intensities.
∗∗
Both γ radiation and neutron radiation can be used to image objects without destroying
them. γ rays have been used to image the interior of the Tutankhamun mask. Neutron
radiation, which penetrates metals as easily as other materials, has been used to image,
even at film speed, the processes inside car engines.
∗∗
γ rays are used in Asia to irradiate food. This is forbidden other countries, such as Ger-
many. For example, γ rays are used to irradiate potatoes, in order to prevent sprouting.
An example is given in Figure 115. It is better not to work there. In fact, over 30 countries
allowed the food industry to irradiate food. For example, almost all spice in the world are
treated with γ rays, to increase their shelf life. However, the consumer is rarely informed
about such treatments.
∗∗
β rays with 10 MeV and γ rays are used in many large factories across the world to sterilize
5 the densest clouds 199

medical equipment, medical devices, toys, furniture and also to kill moulds in books and
in animal food. (See www.bgs.eu for an example.)
∗∗
The non-radioactive isotopes 2 H – often written simply D – and 18 O can be used for
measuring energy production in humans in an easy way. Give a person a glass of doubly
labelled water to drink and collect his urine samples for a few weeks. Using a mass
spectrometer one can determine his energy consumption. Why? Doubly labelled wa-
ter 2 H2 18 O is processed by the body in three main ways. The oxygen isotope is expired
as C18 O2 or eliminated as H2 18 O; the hydrogen isotope is eliminated as 2 H2 O. Meas-
urements on the urine allow one to determine carbon dioxide production, therefore to
determine how much has food been metabolized, and thus to determine energy produc-
tion.
Human energy consumption is usually given in joule per day. Measurements showed
that high altitude climbers with 20 000 kJ/d and bicycle riders with up to 30 000 kJ/d are

Motion Mountain – The Adventure of Physics
the most extreme sportsmen. Average humans produce 6 000 kJ/d.
∗∗
18
The percentage of the O isotope in the water of the Earth’s oceans can be used to deduce
Vol. I, page 370 where the water came from. This was told in the first volume of our adventure.
∗∗
Many nuclei oscillate in shape. The calculation of these shape oscillations is a research
subject in itself. For example, when a spherical nucleus oscillates, it can do so in three

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
mutually orthogonal axes. A spherical nucleus, when oscillating at small amplitudes, thus
behaves like a three-dimensional harmonic oscillator. Interestingly, the symmetry of the
three-dimensional harmonic oscillator is SU(3), the same symmetry that characterizes
the strong nuclear interaction. However, the two symmetries are unrelated – at least fol-
Vol. VI, page 275 lowing present knowledge. A relation might appear in the future, though.

Summary on nuclei
Atomic nuclei are composed of protons and neutrons. Their diameter is between one
and a few femtometres, and they have angular momentum. Their angular momentum, if
larger than zero, allows us to produce magnetic resonance images. Nuclei can be spher-
ical or ellipsoidal, they can be excited to higher energy states, and they can oscillate in
shape. Nuclei have colours that are determined by their spectra. Nuclei can decay, can
scatter, can break up and can react among each other. Nuclear reactions can be used
to make bombs, power plants, generate biological mutations and to explore the human
body. And as we will discover in the following, nuclear reactions are at the basis of the
working of the Sun and of our own existence.
Chapter 6

T H E SU N , T H E STA R S A N D T H E
BI RT H OF M AT T E R

“ ”
Lernen ist Vorfreude auf sich selbst.**
Peter Sloterdijk

N
uclear physics is the most violent part of physics. But despite this bad image,

Motion Mountain – The Adventure of Physics
uclear physics has also something fascinating to offer: by exploring
uclei, we learn to understand the Sun, the stars, the early universe, the birth
Ref. 172 of matter and our own history.
Nuclei consist of protons and neutrons. Since protons are positively charged, they re-
pel each other. Inside nuclei, protons must be bound by a force strong enough to keep
them together against their electromagnetic repulsion. This is the strong nuclear inter-
action; it is needed to avoid that nuclei explode. The strong nuclear interaction is the
strongest of the four interactions in nature – the others being gravitation, electromag-
netism and the weak nuclear interaction. Despite its strength, we do not experience the
strong nuclear interaction in everyday life, because its range is limited to distances of a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
few femtometres, i.e., to a few proton diameters. Despite this limitation, the strong in-
teraction tells a good story about the burning of the Sun and about the flesh and blood
we are made of.

The Sun
At present, the Sun emits 385 YW of light. The amount was first measured by Claude
Pouillet at the start of the nineteenth century. The power would be sufficient to melt
Challenge 136 e away, every year, a volume of ice 500 times larger than the volume of the Earth.
Where does the huge energy emitted by the Sun come from? If it came from burning
coal, the Sun would stop burning after a few thousands of years. When radioactivity was
discovered, researchers tested the possibility that this process might be at the heart of
the Sun’s shining. However, even though radioactivity – or the process of fission that was
discovered later – is able to release more energy than chemical burning, the composition
of the Sun – mostly hydrogen and helium – makes this impossible.
The origin of the energy radiated by the Sun was clarified in 1929 by Fritz Houtermans,
Ref. 173 Carl Friedrich von Weiszäcker, and Hans Bethe: the Sun burns by hydrogen fusion. Fusion
is the composition of a large nucleus from smaller ones. In the Sun, the central fusion
reaction
4 1 H → 4 He + 2 𝑒+ + 2 𝜈 + 4.4 pJ (61)

** ‘Learning is anticipated joy about yourself.’
and the birth of matter 201

Motion Mountain – The Adventure of Physics
F I G U R E 116 The Sun emits radiation at different wavelengths. Note that almost all images are shown in
a single, false colour selected just for visual appeal. The collections does not show the radio wave

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
images, which also show the solar spots, but with much lower resolution. (Courtesy NASA)

converts hydrogen nuclei into helium nuclei. The reaction is called the hydrogen–
hydrogen cycle or p–p cycle. The hydrogen cycle is the result of a continuous cycle of
three separate nuclear reactions:

p + p → d + 𝑒+ + 𝜈 (a weak nuclear reaction involving the deuteron)
3
d + p → He + 𝛾 (a strong nuclear reaction)
3
He + 3 He → 𝛼 + 2 p + 𝛾 . (62)

We can also write the p-p cycle as
1
H + 1 H → 2 H + 𝑒+ + 𝜈 (a weak nuclear reaction)
2 1 3
H + H → He + 𝛾 (a strong nuclear reaction)
3
He + 3 He → 4 He + 2 1 H + 𝛾 . (63)

In total, four protons are thus fused to one helium nucleus, or alpha particle; if we include
the electrons, we can say that four hydrogen atoms are fused to one helium atom. The
fusion process emits neutrinos and light with a total energy of 4.4 pJ (26.7 MeV). This is
202 6 the sun, the stars

F I G U R E 117 The Sun also emits
neutrinos. Their intensities are
shown here in a false colour
image, taken through the whole
Earth from an underground
experiment, with a 503-day

Motion Mountain – The Adventure of Physics
exposure, at energies from 7 to
25 MeV. However, due to
scattering processes, the bright
spot is several times the size of
the Sun. (© Robert Svoboda)

the energy that makes the Sun shine. Most of the energy is emitted as light; around 10 %
is carried away by neutrinos. The latter part is illustrated in Figure 117.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The first of the three reactions of equation (62) is due to the weak nuclear interaction.
This transmutation and the normal β decay have the same first-order Feynman diagram.
Challenge 137 e The weak interaction avoids that fusion happens too rapidly and ensures that the Sun
will shine still for some time. Indeed, in the Sun, with a luminosity of 385 YW, there are
Ref. 174 thus only 1038 fusions per second. This allows us to deduce that the Sun will last another
handful of Ga (Gigayears) before it runs out of fuel.
The simplicity of the hydrogen-hydrogen cycle does not fully purvey the fascination
of the process. On average, protons in the Sun’s centre move with 600 km/s. Only if
they hit each other precisely head-on can a nuclear reaction occur; in all other cases, the
electrostatic repulsion between the protons keeps them apart. For an average proton, a
head-on collision happens once every 7 thousand million years! Nevertheless, there are
so many proton collisions in the Sun that every second four million tons of hydrogen are
burned to helium. The second reaction of the proton cycle takes a few seconds and the
third about one million years.
The fusion reaction (62) takes place in the centre of the Sun, in the so-called core.
Fortunately for us, the high energy γ photons generated in the Sun’s centre are ‘slowed’
down by the outer layers of the Sun, namely the radiation zone, the convection zone with
its involved internal motion, and the so-called photosphere, the thin layer we actually see.
The last layer, the atmosphere, is not visible during a day, but only during an eclipse, as
shown in Figure 120. More precisely, the solar atmosphere consists of the temperature
minimum, the chromosphere, the transition region, the corona and the heliosphere.
During the elaborate slowing-down process inside the Sun, the γ photons are pro-
and the birth of matter 203

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 118 A photograph of the Sun at a visible wavelength of around 677 nm, by the SOHO space
probe, showing a few sunspots (ESA and NASA).

gressively converted to visible photons, mainly through scattering. Scattering takes time.
In the Sun, it takes along time: the sunlight of today was in fact generated at the time of
Ref. 175 the Neandertals: a typical estimate is about 170 000 years ago. In other words, the average
effective speed of light inside the Sun is estimated to be around 300 km/year! After these
one hundred and seventy thousand years, the photons take another 8.3 minutes to reach
the Earth and to sustain the life of all plants and animals.

Motion in and on the Sun
Vol. III, page 148 In its core, the Sun has a temperature of around 15 MK. At its surface, the temperature
Challenge 138 e is around 5.8 kK. (Why is it cooler?) Since the Sun is cooler on its surface than in its
centre, the Sun is not a homogeneous ball, but an inhomogeneous structure. If you want
to experience the majestic beauty of the Sun, watch the stunning video www.youtube.
204 6 the sun, the stars

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 119 A photograph of the Sun at the extreme ultraviolet wavelength of 30.4 nm, thus in false
colour, again by the SOHO space probe, showing solar prominences (ESA and NASA).

com/watch?v=ipvfwPqh3V4 that shows the Sun’s surface over a two-week period. The
inhomogeneity of the Sun’s structure and surface is due to the convection processes in-
duced by the temperature gradient. The convection, together with the rotation of the Sun
around its axis, leads to fascinating structures that are shown in Figure 119 and the fol-
lowing ones: solar eruptions, including flares and coronal mass ejections, and solar spots.
In short, the Sun is not a static object. The matter in the Sun is in constant motion. An
impressive way to experience the violent processes it contains is to watch the film shown
in Figure 122, which shows the evolution of a so-called solar flare. Many solar eruptions,
such as those shown in the lower left corner in Figure 119 or in Figure 123, eject matter
far into space. When this matter reaches the Earth,* after being diluted by the journey, it
affects our everyday environment. Such solar storms can deplete the higher atmosphere

* It might also be that the planets affect the solar wind; the issue is not settled and is still under study.
and the birth of matter 205

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 120 The complex details of the corona of the Sun during the 2008 eclipse in Bor Udzuur in
Mongolia and the 2009 eclipse on the Marshall Islands. The images are digital compositions of several
dozen photographs chosen to reproduce the experience of looking at the eclipse through a small
telescope. The structures also allow to locate the solar poles. The top image includes protuberances.
(Top image © Miloslav Druckmüller, Martin Dietzel, Peter Aniol, Vojtech Rušin; bottom image © Miloslav
Druckmüller, Peter Aniol, Vojtech Rušin, L’ubomír Klocok, Karel Martišek and Martin Dietzel)
206 6 the sun, the stars

Motion Mountain – The Adventure of Physics
F I G U R E 121 A drawing of the interior of the Sun (courtesy NASA).

F I G U R E 122 The evolution of a solar ﬂare observed by the TRACE satellite (QuickTime ﬁlm courtesy
NASA).

and can thus possibly even trigger usual Earth storms. Other effects of solar storms are
the formation of auroras and the loss of orientation of birds during their migration; this
happens during exceptionally strong solar storms, because the magnetic field of the Earth
and the birth of matter 207

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 123 A spectacular coronal mass ejection observed on June 7, 2011 by the Solar Dynamic
Observatory satellite (QuickTime ﬁlm courtesy NASA).
208 6 the sun, the stars

is disturbed in these situations. A famous effect of a solar storm was the loss of electricity
in large parts of Canada in March of 1989. The flow of charged solar particles triggered
large induced currents in the power lines, blew fuses and destroyed parts of the network,
shutting down the power system. Millions of Canadians had no electricity, and in the
most remote places it took two weeks to restore the electricity supply. Due to the coldness
of the winter and a train accident resulting from the power loss, over 80 people died. In
the meantime, the power network has been redesigned to withstand such events.
How can the Sun’s surface have a temperature of 6 kK, whereas the Sun’s corona –
the thin gas emanating from and surrounding the Sun that is visible during a total solar
eclipse, as shown in Figure 120 – reaches one to three million Kelvin on average, with
localized peaks inside a flare of up to 100 MK? In the latter part of the twentieth century
it was shown, using satellites, that the magnetic field of the Sun is the cause; through
the violent flows in the Sun’s matter, magnetic energy is transferred to the corona in
those places were flux tubes form knots, above the bright spots in the left of Figure 119 or
above the dark spots in Figure 118. As a result, the particles of the corona are accelerated

Motion Mountain – The Adventure of Physics
and heat the corona to temperatures that are a thousand times higher than those at the
surface of the Sun.

Why d o the stars shine?

“
Don’t the stars shine beautifully? I am the only

”
person in the world who knows why they do.
Friedrich (Fritz) Houtermans *

All stars shine because of fusion. When two light nuclei are fused to a heavier one, some
energy is set free, as the average nucleon is bound more strongly. This energy gain is

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
possible until nuclei of iron 56 Fe are produced. For nuclei beyond this nucleus, as shown
in Figure 124, the binding energies per nucleon then decrease again; thus fusion is not
energetically possible. It turns out that the heavier nuclei found on Earth and across the
universe were formed through neutron capture. In short, nuclei below iron are made
through fusion, nuclei above iron are made through neutron capture. And for the same
reason, nuclei release energy through fusion when the result is lighter than iron, and
release energy through fission when the starting point is above iron.
The different stars observed in the sky** can be distinguished by the type of fusion
nuclear reaction that dominates them. Most stars, in particular young or light stars, run
hydrogen fusion. But that is not all. There are several types of hydrogen fusion: the direct
hydrogen–hydrogen (p–p) cycle, as found in the Sun and in many other stars, and the
various CNO cycle(s) or Bethe-Weizsäcker cycle(s).
Page 200 The hydrogen cycle was described above and can be summarized as

4 1 H → 4 He + 2 𝑒+ + 2 𝜈 + 4.4 pJ (64)

* Friedrich Houtermans (1903–1966) was one of the most colourful physicists of his time. He lived in Aus-
tria, England, the Soviet Union, Germany and the United States. He was analyzed by Sigmund Freud, im-
prisoned and tortured by the NKWD in Russia, then imprisoned by the Gestapo in Germany, then worked
on nuclear fission. He worked with George Gamow and Robert Atkinson.
** To find out which stars are in the sky above you at present, see the www.surveyor.in-berlin.de/himmel
website.
and the birth of matter 209

Motion Mountain – The Adventure of Physics
F I G U R E 124 Measured values of the binding energy per nucleon in nuclei. The region on the left of the
maximum, located at 58 Fe, is the region where fusion is energetically possible; the right region is where
ﬁssion is possible (© Max Planck Institute for Gravitational Physics).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
or, equivalently,
4 p → 𝛼 + 2 𝑒+ + 2 𝜈 + 26.7 MeV . (65)

But this is not the only way for a star to burn. If a star has heavier elements inside it,
the hydrogen fusion uses these elements as catalysts. This happens through the so-called
Bethe-Weizsäcker cycle or CNO cycle, which runs as
12
C + 1 H → 13 N + 𝛾
13
N → 13 C + e+ + 𝜈
13
C + 1 H → 14 N + 𝛾
14
N + 1 H → 15 O + 𝛾
15
O → 15 N + e+ + 𝜈
15
N + 1 H → 12 C + 4 He (66)

The end result of the cycle is the same as that of the hydrogen cycle, both in nuclei and in
energy. The Bethe-Weizsäcker cycle is faster than hydrogen fusion, but requires higher
temperatures, as the protons must overcome a higher energy barrier before reacting with
Challenge 139 s carbon or nitrogen than when they react with another proton. (Why?) Inside the Sun,
due to the comparatively low temperature of a few tens of million kelvin, the Bethe-
Weizsäcker cycle (and its variations) is not as important as the hydrogen cycle.
210 6 the sun, the stars

The proton cycle and the Bethe-Weizsäcker cycle are not the only options for the burn-
ing of stars. Heavier and older stars than the Sun can also shine through other fusion re-
actions. In particular, when no hydrogen is available any more, stars run helium burning:

3 4 He → 12 C . (67)

This fusion reaction, also called the triple-α process, is of low probability, since it depends
on three particles being at the same point in space at the same time. In addition, small
amounts of carbon disappear rapidly via the reaction 𝛼 + 12 C → 16 O. Nevertheless, since
8
Be is unstable, the reaction with 3 alpha particles is the only way for the universe to
produce carbon. All these negative odds are countered only by one feature: carbon has an
excited state at 7.65 MeV, which is 0.3 MeV above the sum of the alpha particle masses;
the excited state resonantly enhances the low probability of the three particle reaction.
Only in this way the universe is able to produce the atoms necessary for apes, pigs and
people. The prediction of this resonance by Fred Hoyle is one of the few predictions in

Motion Mountain – The Adventure of Physics
physics made from the simple experimental observation that humans exist. The story
Vol. III, page 337 has lead to a huge outflow of metaphysical speculations, most of which are unworthy of
being even mentioned.
The studies of star burning processes also explain why the Sun and the stars do not
collapse. In fact, the Sun and most stars are balls of hot gas, and the gas pressure due to
the high temperature of its constituents prevents their concentration into a small volume.
For other types of stars – especially those of high mass such as red giants – the radiation
pressure of the emitted photons prevents collapse; for still other stars, such as neutron
stars, the role is taken by the Pauli pressure.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The nuclear reaction rates at the interior of a star are extremely sensitive to its tem-
perature 𝑇. The carbon cycle reaction rate is proportional to between 𝑇13 for hot massive
O stars and 𝑇20 for stars like the Sun. In red giants and supergiants, the triple-α reaction
rate is proportional to 𝑇40 ; these strong dependencies imply that stars usually shine with
constancy over long times, often thousands and millions of years, because any change in
temperature would be damped by a very efficient feedback mechanism. Of course, there
are exceptions: variable stars get brighter and darker with periods of a few days; some
stars change in brightness every few years. And even the Sun shows such effects. In the
1960s, it was discovered that the Sun pulsates with a frequency of 5 minutes. The amp-
litude is small, only 3 kilometres out of 1.4 million; nevertheless, it is measurable. In the
meantime, helioseismologists have discovered numerous additional oscillations of the
Sun, and in 1993, even on other stars. Such oscillations allow studying what is happening
inside stars, even separately in each of the layers they consist of.
By the way, it is still not clear how much the radiation of the Sun changes over long
time scales. There is an 11 year periodicity, the famous solar cycle, but the long term trend
is still unknown. Precise measurements cover only the years from 1978 onwards, which
makes only about 3 cycles. A possible variation of the intensity of the Suns, the so-called
solar constant might have important consequences for climate research; however, the is-
sue is still open.
and the birth of matter 211

F I G U R E 125 A simpliﬁed drawing of

Motion Mountain – The Adventure of Physics
the Joint European Torus in
operation at Culham, showing the
large toroidal chamber and the
magnets for the plasma conﬁnement
(© EFDA-JET).

Why are fusion reactors not common yet?
Across the world, for over 50 years, a large number of physicists and engineers have tried
to build fusion reactors. Fusion reactors try to copy the mechanism of energy release

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
used by the Sun. The first machine that realized macroscopic energy production through
fusion was, in 1991, the Joint European Torus* (JET for short) located in Culham in the
United Kingdom. Despite this success, the produced power was still somewhat smaller
than the power needed for heating.
Ref. 176 The idea of JET is to produce an extremely hot plasma that is as dense as possible.
At high enough temperature and density, fusion takes place; the energy is released as a
particle flux that is transformed (like in a fission reactor) into heat and then into elec-
tricity. To achieve ignition, JET used the fusion between deuterium and tritium, because
this reaction has the largest cross section and energy gain:

D + T → He4 + n + 17.6 MeV . (68)

Because tritium is radioactive, most research experiments are performed with the far less
efficient deuterium–deuterium reactions, which have a lower cross section and a lower
energy gain:

D + D → T + H + 4 MeV
3
D + D → He + n + 3.3 MeV . (69)

* See www.jet.edfa.org.
212 6 the sun, the stars

Fusion takes place when deuterium and tritium (or deuterium) collide at high energy.
The high energy is necessary to overcome the electrostatic repulsion of the nuclei. In
other words, the material has to be hot. To release energy from deuterium and tritium,
one therefore first needs energy to heat it up. This is akin to the ignition of wood: in order
to use wood as a fuel, one first has to heat it with a match.
Following the so-called Lawson criterion, rediscovered in 1957 by the English engineer
Ref. 177 John Lawson, after its discovery by Russian researchers, a fusion reaction releases energy
only if the triple product of density 𝑛, reaction (or containment) time 𝜏 and temperature
𝑇 exceeds a certain value. Nowadays this criterion is written as

𝑛𝜏𝑇 > 3 ⋅ 1028 s K/m3 . (70)

In order to realize the Lawson criterion, JET uses temperatures of 100 to 200 MK, particle
densities of 2 to 3 ⋅ 1020 m−3 , and confinement times of 1 s. The temperature in JET is
thus much higher than the 15 MK at the centre of the Sun, because the densities and the

Motion Mountain – The Adventure of Physics
confinement times are much lower.
Matter at these temperatures is in form of a plasma: nuclei and electrons are com-
pletely separated. Obviously, it is impossible to pour a 100 MK plasma into a container:
the walls would instantaneously evaporate. The only option is to make the plasma float in
a vacuum, and to avoid that the plasma touches the container wall. The main challenge of
fusion research in the past has been to find a way to keep a hot gas mixture of deuterium
and tritium suspended in a chamber so that the gas never touches the chamber walls. The
best way is to suspend the gas using a magnetic field. This works because in the fusion
plasma, charges are separated, so that they react to magnetic fields. The most success-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ful geometric arrangement was invented by the famous Russian physicists Igor Tamm
and Andrei Sakharov: the tokamak. Of the numerous tokamaks around the world, JET
is the largest and most successful. Its concrete realization is shown in Figure 125. JET
manages to keep the plasma from touching the walls for about a second; then the situ-
ation becomes unstable: the plasma touches the wall and is absorbed there. After such
a disruption, the cycle consisting of gas injection, plasma heating and fusion has to be
restarted. As mentioned, JET has already achieved ignition, that is the state were more
energy is released than is added for plasma heating. However, so far, no sustained com-
mercial energy production is planned or possible, because JET has no attached electrical
power generator.
The successor project, ITER, an international tokamak built with European, Japanese,
US-American and Russian funding, aims to pave the way for commercial energy gen-
eration. Its linear reactor size will be twice that of JET; more importantly, ITER plans
to achieve 30 s containment time. ITER will use superconducting magnets, so that it
will have extremely cold matter at 4 K only a few metres from extremely hot matter at
100 MK. In other words, ITER will be a high point of engineering. The facility is being
built in Cadarache in France. Due to its lack of economic sense, ITER has a good chance
to be a modern version of the tower of Babylon; but maybe one day, it will start operation.
Like many large projects, fusion started with a dream: scientists spread the idea that
fusion energy is safe, clean and inexhaustible. These three statements are still found on
every fusion website across the world. In particular, it is stated that fusion reactors are not
dangerous, produce much lower radioactive contamination than fission reactors, and use
and the birth of matter 213

water as basic fuel. ‘Solar fusion energy would be as clean, safe and limitless as the Sun.’
In reality, the only reason that we do not feel the radioactivity of the Sun is that we are far
away from it. Fusion reactors, like the Sun, are highly radioactive. The management of
radioactive fusion reactors is much more complex than the management of radioactive
fission reactors.
It is true that fusion fuels are almost inexhaustible: deuterium is extracted from water
and the tritium – a short-lived radioactive element not found in nature in large quantit-
ies – is produced from lithium. The lithium must be enriched, but since material is not
radioactive, this is not problematic. However, the production of tritium from lithium is
a dirty process that produces large amounts of radioactivity. Fusion energy is thus inex-
haustible, but not safe and clean.
In summary, of all technical projects ever started by mankind, fusion is by far the
most challenging and ambitious. Whether fusion will ever be successful – or whether it
ever should be successful – is another issue.

Motion Mountain – The Adventure of Physics
Where d o our atoms come from?

“
The elements were made in less time than you

”
could cook a dish of duck and roast potatoes.
George Gamow

People consist of electrons and various nuclei. Where did the nucleosynthesis take place?
Ref. 178 Many researchers contributed to answering this question.
About three minutes after the big bang, when temperature was around 0.1 MeV, pro-
tons and neutrons formed. About seven times as many protons as neutrons were formed,
mainly due to their mass difference. Due to the high densities, the neutrons were cap-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tured, through the intermediate step of deuterium nuclei, in α particles. The process
stopped around 20 minutes after the big bang, when temperatures became too low to al-
low fusion. After these seventeen minutes, the mass of the universe was split into 75 % of
hydrogen, 25 % of helium (both percentages result from the factor 7 between the number
of protons and neutrons), and traces of deuterium, lithium and beryllium. This process
is called primordial nucleosynthesis. No heavier elements were formed, because the tem-
perature fall prevented their accumulation in measurable quantities, and because there
are no stable nuclei with 5 or 8 nucleons.
Simulations of primordial nucleosynthesis agree well with the element abundances
found in extremely distant, thus extremely old stars. The abundances are deduced from
the spectra of these stars. In short, hydrogen, helium, lithium and beryllium nuclei are
Vol. II, page 246 formed shortly after (‘during’) the big bang. These are the so-called primordial elements.
All other nuclei are formed many millions of years after the big bang. In particu-
lar, other light nuclei are formed in stars. Young stars run hydrogen burning or helium
Ref. 179 burning; heavier and older stars run neon-burning or even silicon-burning. These latter
processes require high temperatures and pressures, which are found only in stars with a
mass at least eight times that of the Sun. All these fusion processes are limited by photo-
dissociation and thus will only lead to nuclei up to 56 Fe.
Nuclei heavier than iron can only be made by neutron capture. There are two main
neutron capture processes. The first process is the so-called s-process – for ‘slow’. The
process occurs inside stars, and gradually builds up heavy elements – including the most
214 6 the sun, the stars

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 126 Two examples of how exploding stars shoot matter into interstellar space: the Crab nebula
M1 and the Dumbbell nebula M27 (courtesy NASA and ESA, © Bill Snyder).
and the birth of matter 215

ce n m in
rs n
sta ai
en g i ced
se urn r o d u

by
e c by
ted
qu in
H is p

aff d
cy n d uce
He
b

ing ed
od

c
CN ning pr

bu odu
e
a
cle
the bur ar

n
r
dO ep
r
O

r
He nd

Sa
a

C, and
C

du e
an

d
rn pro th

a p re , a r e
ce
Ne

bu are und
by Si

n c ptu ak
,
Mg

oto ca pe

e
r
by pea ts a

tu r
ing

pr on Fe
en

by eutr the
k
m

Motion Mountain – The Adventure of Physics
a r t b y n ond
Ele

Si
Fe

ey
a n u c e ts b

ly
p r men
dp d
Ele
o

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 127 The measured nuclide abundances in the solar system and their main production
processes (from Ref. 180).

heavy stable nucleus, lead – from neutrons flying around. The second neutron capture
process is the rapid r-process. For a long time, its was unknown where it occurred; for
Ref. 181 many decades, it was thought that it takes place in stellar explosions. Recent research
points to neutron star mergers as the more likely place for the r-process. Such collisions
emit material into space. The high neutron flux produces heavy elements. For example,
it seems that most gold nuclei are synthesized in this way. (The first clear neutron star
merger was observed in 2013 with the Hubble space telescope, after it had been detected
as a gamma ray burst with the name GRB 130603B. Such an event is also called a kilonova,
because the emitted energy is between that of a nova and of a supernova. In October 2017,
a further, well-publicized neutron star merger was observed with the help of gravitational
wave detectors, of gamma ray burst detectors and of over 70 optical telescopes. It took
place at a distance of 130 million light years.) The abundances of the heavy elements in the
solar system can be measured with precision, a shown in Figure 127. These data points
correspond well with what is expected from the material synthesized by neutron star
216 6 the sun, the stars

100

Solar 1.35-1.35M NS
10- o
1.20-1.50M NS
10- o
Mass fraction
10-

10-

Motion Mountain – The Adventure of Physics
0 50 100 150 200 250
A
F I G U R E 128 The comparison between measured nuclide abundances (dotted circles) in the solar
system and the calculated values (red squares, blue diamonds) predicted by neutron star mergers (from
reference Ref. 181).

mergers, the most likely candidate for the r-process at present, as illustrated by Figure 128.
A number of other processes, such as proton capture and the so-called equilibrium

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
process, contributed to the formation of the elements.
In summary, the electrons and protons in our body were made during the big bang;
the lighter nuclei, such as carbon or oxygen, in stars; the heavier nuclei in star explosions
and neutron star mergers. But how did those nuclei arrive on Earth?
At a certain stage of their life, many stars explode. An exploding star is called a su-
pernova. Such a supernova has an important effect: it distributes most of the matter of
the star, such as carbon, nitrogen or oxygen, into space. This happens mostly in the form
of neutral atoms. (Some elements are also synthesized during the explosion.) Exploding
supernovae are thus essential for distributing material into space.
The Sun is a second generation star – as so-called ‘population I’ star. and the solar
system formed from the remnants of a supernova, as did, somewhat later, life on Earth.
We all are made of recycled atoms.
We are recycled stardust. This is the short summary of the extended study by astro-
physicists of all the types of stars found in the universe, including their birth, growth,
Vol. II, page 211 mergers and explosions. The exploration of how stars evolve and then move in galaxies
is a fascinating research field, and many aspects are still unknown.

Curiosities ab ou t the Sun and the stars
What would happen if the Sun suddenly stopped shining? Obviously, temperatures
would fall by several tens of degrees within a few hours. It would rain, and then all water
would freeze. After four or five days, all animal life would stop. After a few weeks, the
and the birth of matter 217

oceans would freeze; after a few months, air would liquefy. Fortunately, this will never
happen.
∗∗
Not everything about the Sun is known. For example, the neutrino flux from the Sun
oscillates with a period of 28.4 days. That is the same period with which the magnetic
field of the Sun oscillates. The connection is still being explored.
∗∗
The Sun is a fusion reactor. But its effects are numerous. If the Sun were less brighter
than it is, evolution would have taken a different course. We would not have eyelids, we
would still have more hair, and would have a brighter skin. Can you find more examples?
Challenge 140 e

∗∗

Motion Mountain – The Adventure of Physics
Some stars shine like a police siren: their luminosity increases and decreases regularly.
Such stars, called Cepheids, are important because their period depends on their average
(absolute) brightness. Therefore, measuring their period and their brightness on Earth
thus allows astronomers to determine their distance.
∗∗
The first human-made hydrogen bomb explosion took place the Bikini atoll. Fortunately,
none has ever been used on people.
But nature is much better at building bombs. The most powerful nuclear explosions

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
known take place on the surface of neutron stars in X-ray binaries. The matter falling
into such a neutron star from the companion star, mostly hydrogen, will heat up until
the temperature allows fusion. The resulting explosions can be observed in telescopes as
light or X-ray flashes of about 10 s duration; the explosions are millions of times more
powerful that those of human-made hydrogen bombs.
∗∗
In the 1960s, it was discovered that surface of the Sun oscillates. The surface is covered
with standing waves. The amplitude is a few hundred kilometres, the wavelength can be
hundred times larger; the typical frequency of the famous p-modes, or trapped acoustic
waves, is between 2 and 4 mHz, thus roughly between 8 and 3 minutes. The oscillations
are also visible as diameter oscillations of the Sun. This research field is now called heli-
oseismology.
∗∗
Lithium, beryllium and boron are rare inside stars, because they like to capture protons,
and thus change identity. For the same reason, these elements are rare on Earth.
∗∗
By chance, the composition ratios between carbon, nitrogen and oxygen inside the Sun
are the same as inside the human body.
218 6 the sun, the stars

∗∗
Nucleosynthesis is mainly regulated by the strong interaction. However, if the electro-
magnetic interaction would be much stronger or much weaker, stars would either pro-
duce too little oxygen or too little carbon, and we would not exist. This famous argument
Challenge 141 d is due to Fred Hoyle. Can you fill in the details?

Summary on stars and nucleosynthesis

“
All humans are brothers. We came from the

”
same supernova.
Allan Sandage

Stars and the Sun burn because of nuclear fusion. The energy liberated in nuclear fusion
is due to the strong nuclear interaction that acts between nucleons. When stars have used
up their nuclear fuel, they usually explode. In such a supernova explosion, they distribute
nuclei into space in the form of dust. Already in the distant past, such dust recollected

Motion Mountain – The Adventure of Physics
because of gravity and then formed the Sun, the Earth and, later on, humans.
The nuclear reaction processes behind nucleosynthesis have been studied in great de-
tail. Nucleosynthesis during the big bang formed hydrogen and helium, nucleosynthesis
in stars formed the light nuclei, and nucleosynthesis in neutron star mergers and super-
novae explosions formed the heavy nuclei.

T H E ST RONG I N T E R AC T ION
– I N SI DE N U C L E I A N D N U C L E ON S

B
oth radioactivity and medical images show that nuclei are composed systems.
ut quantum theory predicts even more: also protons and neutrons must
e composed. There are two reasons: first, nucleons have a finite size, and second,
their magnetic moments do not match the value predicted for point particles.

Motion Mountain – The Adventure of Physics
The prediction of components inside protons was confirmed in the late 1960s when
Ref. 182 Kendall, Friedman and Taylor shot high energy electrons into hydrogen atoms. They
found that a proton contains three constituents with spin 1/2. The experiment was able
to ‘see’ the constituents through large angle scattering of electrons, in the same way that
we see objects through large angle scattering of photons. These constituents correspond
Ref. 183 in number and (most) properties to the so-called quarks predicted in 1964 by George
Zweig and also by Murray Gell-Mann.**
Why are there three quarks inside a proton? And how do they interact? The answers
are deep and fascinating.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The feeble side of the strong interaction
The mentioned deep inelastic scattering experiments show that the interaction keeping
the protons together in a nucleus, which was first described by Yukawa Hideki,*** is

** The physicist George Zweig (b. 1937 Moscow ) proposed the quark idea – he called them aces – in 1963,
with more clarity than Gell-Mann. Zweig stressed the reality of aces, whereas Gell-Mann, in the beginning,
did not believe in the existence of quarks. Zweig later moved on to a more difficult field: neurobiology.
Murray Gell-Mann (b. 1929 New York ) received the Nobel Prize in Physics in 1969. He is the originator
of the term ‘quark’. The term has two origins: officially, it is said to be taken from Finnegans Wake, a novel
by James Joyce; in reality, Gell-Mann took it from a Yiddish and German term meaning ‘lean soft cheese’
Ref. 184 and used figuratively in those languages to mean ‘silly idea’.
Gell-Mann was the central figure of particle physics in the 20th century; he introduced the concept of
strangeness, the renormalization group, the flavour SU(3) symmetry and quantum chromodynamics itself.
A disturbing story is that he took the idea, the data, the knowledge, the concepts and even the name of the
V−A theory of the weak interaction from the bright physics student George Sudarshan and published it,
together with Richard Feynman, as his own. The wrong attribution is still found in many textbooks.
Gell-Mann is also known for his constant battle with Feynman about who deserved to be called the most
arrogant physicist of their university. A famous anecdote is the following. Newton’s once used a common
saying of his time in a letter to Hooke: ‘If I have seen further than you and Descartes, it is by standing upon
the shoulders of giants.’ Gell-Mann is known for saying: ‘If I have seen further than others, it is because I
am surrounded by dwarfs.’
*** Yukawa Hideki (b. 1907 Azabu, d. 1981 Kyoto), important physicist specialized in nuclear and particle
physics. He received the 1949 Nobel Prize in Physics for his theory of mesons. Yukawa founded the journal
220 7 the strong interaction

The proton :

Motion Mountain – The Adventure of Physics
F I G U R E 129 Top: SLAC, the electron linear collider, and the detectors used for the deep inelastic
electron scattering experiment. Bottom: an artistic illustration of the ﬁnal result, showing the three
scattering centres observed inside the proton.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
only a feeble shadow of the interaction that keeps quarks together in a proton. Both
interactions are called by the same name. The two cases correspond somewhat to the
two cases of electromagnetism found in atomic matter. The clearest example is provided
by neon atoms: the strongest and ‘purest’ aspect of electromagnetism is responsible for
the attraction of the electrons to the neon nuclei; its feeble ‘shadow’, the Van-der-Waals
interaction, is responsible for the attraction of neon atoms in liquid neon and for pro-
cesses like its evaporation and condensation. Both attractions are electromagnetic, but
the strengths differ markedly. Similarly, the strongest and ‘purest’ aspect of the strong
interaction leads to the formation of the proton and the neutron through the binding of
quarks; the feeble, ‘shadow’ aspect leads to the formation of nuclei and to α decay. Ob-
viously, most information can be gathered by studying the strongest and ‘purest’ aspect.

B ound motion, the particle zo o and the quark model
Deep electron scattering showed that protons are made of interacting constituents. How
can one study these constituents?
Physicists are simple people. To understand the constituents of matter, and of protons
in particular, they had no better idea than to take all particles they could get hold of and
Progress of Theoretical Physics and together with his class mate Tomonaga Shin’ichiro, who also won the
prize, he was an example to many scientists in Japan.
7 inside nuclei and nucleons 221

Motion Mountain – The Adventure of Physics
F I G U R E 130 A typical experiment used to study the quark model: the Proton Synchroton at CERN in
Geneva (© CERN).

Spin 1/2 baryons Spin 3/2 baryons

Ω ++
ccc Omccpp3
(b)
Xiccp
+ +

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ξ cc dcc ucc Ξ cc
Xiccpp
++ Ξ cc dcc ucc
++
Ξ cc Xiccpp3
scc Omccp scc
+ +
(a)
Ωcc Ωcc
Σ c+
LcpScp
Sco
Λ+c, Σ c+ Σc 0
Σc Σ c++
0 Scpp Scpp3
ddc udc uuc Σc++ ddc udc uuc
0 dsc usc
dsc
Xico
Ωc0 usc Xicp Ξc Ξ c+
Ξ c0 n ssc Ξ c+ p
Δ− Δ0
ssc
Ω 0
c
+
Δ ++
Dpp3
Δ
n p ddd udd uud uuu
udd
Σ − Sndds
uud Charm C
uds
Λ ,Σ 0 Σ−
dds uds Σ 0 uus
Σ+
dss uss
uus ΣSp+ dss uss
− LSo Hypercharge
Ξ − sss Ξ 0
Ξ Xin Ξ
0 Xio Y= - C/3 + S + B
−
Ω
Isospin I

F I G U R E 131 The family diagrams for the least massive baryons that can be built as qqq composites of
the ﬁrst four quark types (from Ref. 186).

Ref. 185 to smash them into each other. Many researchers played this game for decades. Obvi-
ously, this is a facetious comment; in fact, quantum theory forbids any other method.
Challenge 142 s Can you explain why?
Understanding the structure of particles by smashing them into each other is not
simple. Imagine that you want to study how cars are built just by crashing them into
222 7 the strong interaction

Spin 0 pseudoscalar mesons Spin 1 vector mesons
𝐷+s (cs) 𝐷∗+
s (cs)

𝐷0 (cu) 𝐷∗0 (cu)
𝐷+ (cd) 𝐷∗+ (cd)

𝐾0 (ds) 𝐾+ (us) 𝐾∗0 (ds) 𝐾∗+ (us)

π0 etal 𝜌0 Aetal

π− (du) etanc etabar 𝜌− (du) Aetanc Aetabar
π+ (ud) 𝜌+ (ud)
𝐾− (us) 0
𝐾 (ds) 𝐾∗− (us) 𝐾
∗0
(ds)
Charm C
𝐷− (cd) 0
𝐷 (cu) 𝐷∗− (cd) 𝐷
∗0
(cu)
Hypercharge

Motion Mountain – The Adventure of Physics
Y= - C/3 + S + B
𝐷−s (cs) 𝐷∗−
s (cs)

Isospin I

F I G U R E 132 The family diagram for the least massive pseudoscalar and vector mesons that can be
built as 𝑞𝑞 ̄ composites of the ﬁrst four quark ﬂavours.

each other. Before you get a list of all components, you must perform and study a non-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
negligible number of crashes. Most give the same result, and if you are looking for a
particular part, you might have to wait for a long time. If the part is tightly attached to
others, the crashes have to be especially energetic. In addition, the part most likely will
be deformed. Compared to car crashes, quantum theory adds the possibility for debris
to transform, to react, to bind and to get excited. Therefore the required diligence and
patience is even greater for particle crashes than for car crashes. Despite these difficulties,
for many decades, researchers have collected an ever increasing number of proton debris,
Page 342 also called hadrons. The list, a small part of which is given in Appendix B, is overwhelm-
ingly long; the official full list, several hundred pages of fine print, is found at pdg.web.
cern.ch and contains hundreds of hadrons. Hadrons come in two main types: integer
spin hadrons are called mesons, half-integer spin hadrons are called baryons. The proton
and the neutron themselves are thus baryons.
Then came the quark model. Using the ingenuity of many experimentalists and the-
oreticians, the quark model explained the whole meson and baryon catalogue as a con-
sequence of only 6 types of bound quarks. Typically, a large part of the catalogue can be
structured in graphs such as the ones given in Figure 132 and Figure 131. These graphs
were the beginning of the end of high energy physics. The quark model explained all
quantum numbers of the debris, and allowed understanding their mass ratios as well as
their decays.
The quark model explained why debris come into two types: all mesons consist of a
quark and an antiquark and thus have integer spin; all baryons consist of three quarks,
and thus have half-integer spin. In particular, the proton and the neutron are seen as
7 inside nuclei and nucleons 223

TA B L E 17 The quarks.

Q ua rk M as s 𝑚 Spin 𝐽 Possible C h a r g e 𝑄, L epton
(see text) pa r i t y c o l o u r s ; i s o s p i n 𝐼, number
𝑃 possible s t r a n g eness 𝑆, 𝐿,
weak be - charm 𝐶, beauty baryon
h av i o u r 𝐵󸀠 , topness 𝑇 number
𝐵
1+
Down 𝑑 4.5 to 5.5 MeV/𝑐2 2
red, green, − 13 , − 21 , 0, 0, 0, 0 0, 13
blue; singlet,
doublet
1+
Up 𝑢 1.8 to 3.0 MeV/𝑐2 2
red, green, + 23 , + 21 , 0, 0, 0, 0 0, 13
blue; singlet,
doublet
1+
Strange 𝑠 95(5) MeV/𝑐2 2
red, green, − 13 , 0, −1, 0, 0, 0 0, 13

Motion Mountain – The Adventure of Physics
blue; singlet,
doublet
1+
Charm 𝑐 1.275(25) GeV/𝑐2 2
red, green, + 23 , 0, 0, +1, 0, 0 0, 13
blue; singlet,
doublet
1+
Bottom 𝑏 4.18(3) GeV/𝑐2 2
red, green, − 13 , 0, 0, 0, −1, 0 0, 13
blue; singlet,
doublet
1+
Top 𝑡 173.5(1.4) GeV/𝑐2 2
red, green, + 23 , 0, 0, 0, 0, +1 0, 13
blue; singlet,

combinations of two quark types, called up (u) and down (d): the proton is a 𝑢𝑢𝑑 state,
the neutron a 𝑢𝑑𝑑 state. The discovery of other hadrons lead to the addition of four
additional types of quarks. The quark names are somewhat confusing: they are called
strange (s), charm (c), bottom (b) – also called ‘beauty’ in the old days – and top (t) –
called ‘truth’ in the past. The quark types are called flavours; in total, there are thus 6
quark flavours in nature.
All quarks have spin one half; they are fermions. Their electric charges are multiples
of 1/3 of the electron charge. In addition, quarks carry a strong charge, called, again
confusingly, colour. In contrast to electromagnetism, which has only positive, negative,
and neutral charges, the strong interaction has red, blue, green quarks on one side, and
anti-red, anti-blue and anti-green on the other. The neutral state is called ‘white’. All
baryons, including proton and neutrons, and all mesons are white, in the same way that
all atoms are neutral.

The essence of quantum chromodynamics
The theory describing the bound states of quarks is called quantum chromodynamics, or
Ref. 187 QCD. It was formulated in its final form in 1973 by Fritzsch, Gell-Mann and Leutwyler. In
the same way that in atoms, electrons and protons are held together by the exchange of
224 7 the strong interaction

gluon gluon gluon
(e.g. red- (e.g. green- (e.g. red-
antiblue) antired) antired)
quark (e.g. green)

gluon
(e.g. green- gluon (e.g.
gs antired) red-antigreen)
gs
gs

gluon
(e.g. green- quark (e.g. red)
antiblue) gluon gluon
(e.g. green- (e.g. blue-
antiblue) antired)

Motion Mountain – The Adventure of Physics
F I G U R E 133 The essence of the QCD Lagrangian: the Feynman diagrams of the strong interaction.

virtual photons, in protons, quarks are held together by the exchange of virtual gluons.
Gluons are the quanta of the strong interaction, and correspond to photons, the quanta

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of the electromagnetic interactions.
Ref. 188 Quantum chromodynamics describes all motion due to the strong interaction with the
three fundamental processes shown in Figure 133: two gluons can scatter, a gluon can
emit or absorb another, and a quark can emit or absorb a gluon. In electrodynamics,
only the last diagram is possible; in the strong interaction, the first two appear as well.
Among others, the first two diagrams are responsible for the confinement of quarks, and
thus for the lack of free quarks in nature.
QCD is a gauge theory: the fields of the strong interaction show gauge invariance
under the Lie group SU(3). We recall that in the case of electrodynamics, the gauge
group is U(1), and Abelian, or commutative. In contrast, SU(3) is non-Abelian; QCD is a
non-Abelian gauge theory. Non-Abelian gauge theory was invented and popularized by
Wolfgang Pauli. It is often incorrectly called Yang–Mills theory after the first two physi-
cists who wrote down Pauli’s ideas.
Due to the SU(3) gauge symmetry, there are 8 gluons; they are called red-antigreen,
blue-antired, etc. Since SU(3) is non-Abelian, gluons interact among themselves, as
shown in the first two processes in Figure 133. Out of the three combinations red-antired,
blue-antiblue and green-antigreen, only two gluons are linearly independent, thus giving
a total of 32 − 1 = 8 gluons.
The coupling strength of the strong interaction, its three fundamental processes in
Figure 133, together with its SU(3) gauge symmetry and the observed number of six
quarks, completely determine the behaviour of the strong interaction. In particular, they
completely determine its Lagrangian density.
7 inside nuclei and nucleons 225

The L agrangian of quantum chromodynamics*
The Lagrangian density of the strong interaction can be seen as a complicated formula-
tion of the Feynman diagrams of Figure 133. Indeed, the Lagrangian density of quantum
chromodynamics is

(𝑎) (𝑎)𝜇𝜈 𝑘 𝑘
L𝑄𝐶𝐷 = − 14 𝐹𝜇𝜈 𝐹 − 𝑐2 ∑ 𝑚𝑞 𝜓𝑞 𝜓𝑞𝑘 + 𝑖ℏ𝑐 ∑ 𝜓𝑞 𝛾𝜇 (𝐷𝜇 )𝑘𝑙 𝜓𝑞𝑙 (71)
𝑞 𝑞

where the gluon field strength and the gauge covariant derivative are
(𝑎)
𝐹𝜇𝜈 = ∂𝜇 𝐴𝑎𝜈 − ∂𝜈 𝐴𝑎𝜇 + 𝑔s 𝑓𝑎𝑏𝑐𝐴𝑏𝜇 𝐴𝑐𝜈
𝑔
(𝐷𝜇 )𝑘𝑙 = 𝛿𝑘𝑙 ∂𝜇 − 𝑖 s ∑ 𝜆𝑎𝑘,𝑙 𝐴𝑎𝜇 .
2 𝑎

We remember from the section on the principle of least action that Lagrangians are al-
Vol. I, page 279 ways sums of scalar products; this is clearly seen in expression (71). The index 𝑎 = 1 . . . 8

Motion Mountain – The Adventure of Physics
numbers the eight types of gluons and the index 𝑘 = 1, 2, 3 numbers the three colours, all
due to SU(3). The index 𝑞 = 1 . . . 6 numbers the six quark flavours. The fields 𝐴𝑎𝜇 (𝑥) are
the eight gluon fields, represented by the coiled lines in Figure 133. The fields 𝜓𝑞𝑘 (𝑥) are
those of the quarks of flavour 𝑞 and colour 𝑘, represented by the straight line in the figure.
The six times three quark fields, like those of any elementary fermion, are 4-component
Dirac spinors with masses 𝑚𝑞 .**
The Lagrangian (71) is that of a local field theory: observables are functions of position.
In other words, QCD is similar to quantum electrodynamics and can be compared to
experiment in the same way.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The first term of the Lagrangian (71) represents the kinetic energy of the radiation (the
gluons), the second or mass term the kinetic energy of the matter particles (the quarks)
and the third term the interaction between the two.
The mass term in the Lagrangian is the only term that spoils or breaks flavour sym-
metry, i.e., the symmetry under exchange of quark types. (In particle physics, this sym-
metry is also called chiral symmetry, for historical reasons.) Obviously, the mass term
also breaks space-time conformal symmetry.
The interaction term in the Lagrangian thus corresponds to the third diagram in Fig-
ure 133. The strength of the strong interaction is described by the strong coupling con-
stant 𝑔s . The constant is independent of flavour and colour, as observed in experiment.
The Interaction term does not mix different quarks; as observed in experiments, flavour
is conserved in the strong interaction, as is baryon number. The strong interaction also
conserves spatial parity P and charge conjugation parity C. The strong interaction does
not transform matter.

* This section can be skipped at first reading.
** In their simplest form, the matrices 𝛾𝜇 can be written as

𝐼 0 0 𝜎𝑖
𝛾0 = ( ) and 𝛾𝑛 = ( ) for 𝑛 = 1, 2, 3 (72)
0 −𝐼 −𝜎𝑖 0

Vol. IV, page 231 where the 𝜎𝑖 are the Pauli spin matrices.
226 7 the strong interaction

In QCD, the eight gluons are massless; also this property is taken from experiment.
Therefore no gluon mass term appears in the Lagrangian. It is easy to see that massive
Challenge 143 ny gluons would spoil gauge invariance. As mentioned above, in contrast to electromagnet-
ism, where the gauge group U(1) is Abelian, the gauge group SU(3) of the strong inter-
actions is non-Abelian. As a consequence, the colour field itself is charged, i.e., carries
colour, and thus the index 𝑎 appears on the fields 𝐴 and 𝐹. As a result, gluons can inter-
act with each other, in contrast to photons, which pass each other undisturbed. The first
two diagrams of Figure 133 are thus reflected in the somewhat complicated definition
(𝑎)
of the field 𝐹𝜇𝜈 . In contrast to electrodynamics, the definition has an extra term that is
quadratic in the fields 𝐴; it is described by the so-called structure constants 𝑓𝑎𝑏𝑐 and the
Page 361 interaction strength 𝑔s . The numbers 𝑓𝑎𝑏𝑐 are the structure constants of the SU(3).
The behaviour of the gauge transformations and of the gluon field is described by
the eight matrices 𝜆𝑎𝑘,𝑙 . They are a fundamental, 3-dimensional representation of the
generators of the SU(3) algebra and correspond to the eight gluon types. The matrices
𝜆 𝑎 , 𝑎 = 1...8, and the structure constants 𝑓𝑎𝑏𝑐 obey the relations

Motion Mountain – The Adventure of Physics
[𝜆 𝑎 , 𝜆 𝑏 ] = 2𝑖𝑓𝑎𝑏𝑐𝜆 𝑐
{𝜆 𝑎 , 𝜆 𝑏 } = 4/3𝛿𝑎𝑏 𝐼 + 2𝑑𝑎𝑏𝑐𝜆 𝑐 (73)

where 𝐼 is the unit matrix. The structure constants 𝑓𝑎𝑏𝑐 of SU(3), which are odd under
permutation of any pair of indices, and 𝑑𝑎𝑏𝑐, which are even, have the values

𝑎𝑏𝑐 𝑓𝑎𝑏𝑐 𝑎𝑏𝑐 𝑑𝑎𝑏𝑐 𝑎𝑏𝑐 𝑑𝑎𝑏𝑐

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
123 1 118 1/√3 355 1/2
147 1/2 146 1/2 366 −1/2
156 −1/2 157 1/2 377 −1/2
246 1/2 228 1/√3 448 −1/(2√3 ) (74)
257 1/2 247 −1/2 558 −1/(2√3 )
345 1/2 256 1/2 668 −1/(2√3 )
367 −1/2 338 1/√3 778 −1/(2√3 )
458 √3 /2 344 1/2 888 −1/√3
678 √3 /2

All other elements vanish. Physically, the structure constants of SU(3) describe the details
of the interaction between quarks and gluons and of the interaction between the gluons
themselves.
A fundamental 3-dimensional representation of the eight generators 𝜆 𝑎 – correspond-
7 inside nuclei and nucleons 227

ing to the eight gluon types – is given, for example, by the set of the Gell-Mann matrices

0 1 0 0 −𝑖 0 1 0 0
𝜆 1 = (1 0 0 ) 𝜆2 = (𝑖 0 0 ) 𝜆 3 = (0 −1 0)
0 0 0 0 0 0 0 0 0
0 0 1 0 0 −𝑖 0 0 0
𝜆 4 = (0 0 0 ) 𝜆5 = (0 0 0 ) 𝜆 6 = (0 0 1)
1 0 0 𝑖 0 0 0 1 0
0 0 0 1 0 0
1
𝜆 7 = (0 0 −𝑖) 𝜆 8 = (0 1 0) . (75)
0 𝑖 0 √3 0 0 −2

There are eight matrices, one for each gluon type, with 3×3 elements, due to the 3 colours
of the strong interaction. There is no ninth gluon, because that gluon would be colourless,
or ‘white’.

Motion Mountain – The Adventure of Physics
The Lagrangian is complete only when the 6 quark masses and the coupling constant
𝑔s are included. These values, like the symmetry group SU(3), are not explained by QCD,
of course.
Only quarks and gluons appear in the Lagrangian of QCD, because only quarks and
gluons interact via the strong force. This can be also expressed by saying that only quarks
and gluons carry colour; colour is the source of the strong force in the same way that elec-
tric charge is the source of the electromagnetic field. In the same way as electric charge,
colour charge is conserved in all interactions. Electric charge comes in two types, pos-
itive and negative; in contrast, colour comes in three types, called red, green and blue.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
The neutral state, with no colour charge, is called white. Protons and neutrons, but also
electrons or neutrinos, are thus ‘white’, thus neutral for the strong interaction.
In summary, the six quark types interact by exchanging eight gluon types. The in-
teraction is described by the Feynman diagrams of Figure 133, or, equivalently, by the
Lagrangian (71). Both descriptions follow from the requirements that the gauge group
is SU(3) and that the masses and coupling constants are given. It was a huge amount
of work to confirm that all experiments indeed agree with the QCD Lagrangian; various
competing descriptions were discarded.

Experimental consequences of the quark model
How can we pretend that quarks and gluons exist, even though they are never found
alone? There are a number of arguments in favour.
∗∗
The quark model explains the non-vanishing magnetic moment of the neutron and ex-
plains the magnetic moments 𝜇 of the baryons. By describing the proton as a 𝑢𝑢𝑑 state
and the neutron a 𝑢𝑑𝑑 state with no orbital angular momentum and using the precise
Challenge 144 e wave functions, we get

𝜇𝑢 = 15 (4𝜇𝑝 + 𝜇𝑛) and 𝜇𝑑 = 15 (4𝜇𝑛 + 𝜇𝑝 ) . (76)
228 7 the strong interaction

Assuming that 𝑚𝑢 = 𝑚𝑑 and that the quark magnetic moment is proportional to their
charge, the quark model predicts a ratio of the magnetic moments of the proton and the
neutron of
𝜇𝑝 3
=− . (77)
𝜇𝑛 2

This prediction differs from measurements only by 3 %. Furthermore, using the same val-
ues for the magnetic moment of the quarks, magnetic moment values of over half a dozen
of other baryons can be predicted. The results typically deviate from measurements only
by around 10 %. In particular, the sign of the resulting baryon magnetic moment is al-
ways correctly calculated.
∗∗
The quark model describes all quantum numbers of mesons and baryons. P-parity, C-
parity, and the absence of certain meson parities are all reproduced. The observed con-

Motion Mountain – The Adventure of Physics
servation of electric charge, baryon number, isospin, strangeness etc. is reproduced. Had-
ron family diagrams such as those shown in Figure 131 and in Figure 132 describe all
existing hadron states (of lowest angular momentum) completely; the states not listed are
not observed. The quark model thus produces a complete and correct classification of all
hadrons as bound states of quarks.
∗∗
The quark model also explains the mass spectrum of hadrons. The best predictions are
made by QCD lattice calculations. With months of computer time, researchers were able
Ref. 189 to reproduce the masses of proton and neutron to within a few per cent. Interestingly,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
if one sets the 𝑢 and 𝑑 quark masses to zero, the resulting proton and neutron mass
Ref. 190 differ from experimental values only by 10 %. The mass of protons and neutrons is almost
Page 233 completely due to the binding, not to the constituents. More details are given below.
∗∗
The number of colours of quarks must be taken into account to get correspondence of
theory and calculation. For example, the measured decay time of the neutral pion is 83 as.
The calculation without colour gives 750 as; if each quark is assumed to appear in 3 col-
ours the value must be divided by 9, and then matches the measurement.
∗∗
In particle colliders, collisions of electrons and positrons sometimes lead to the produc-
tion of hadrons. The calculated production rates also fit experiments only if quarks have
three colours. In more detail, if one compares the ratio of muon–antimuon production
Challenge 145 s and of hadron production, a simple estimate relates them to their charges:

∑ 𝑞hadrons
𝑅= (78)
∑ 𝑞muons

Between 2 and 4 GeV, when only three quarks can appear, this argument thus predicts
𝑅 = 2 if colours exist, or 𝑅 = 2/3 if they don’t. Experiments yield a value of 𝑅 = 2.2, thus
7 inside nuclei and nucleons 229

potential V [GeV]

2 − 34 𝛼sc𝑟ℏ𝑐 + 𝑘𝑟
spin
1
Δ hadrons linear potential
19/2
above 1 fm
15/2 0

11/2 inverse
-1 distance
7/2 potential
below
3/2 1 fm
-2
separation r [fm]
0 5 10
-3

Motion Mountain – The Adventure of Physics
m2 [GeV2] 0 1 2

A baryon approximated A meson approximated
as three quarks connected as a bag containing
by elastic strings: a quark and an antiquark:

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 134 Top left: a Regge trajectory, or Chew–Frautschi plot, due to the conﬁnement of quarks.
Top right: the quark conﬁnement potential. Bottom: two approximate ways to describe quark
conﬁnement: the string model and the bag model of hadrons.

confirming the number of colours. Many other such branching ratios can be calculated
in this way. They agree with experiments only if the number of colours is three.

C onfinement of quarks – and elephants
Many of the observed hadrons are not part of the diagrams of Figure 131 and Figure 132;
these additional hadrons can be explained as rotational excitations of the fundamental
mesons from those diagrams. As shown by Tullio Regge in 1957, the idea of rotational
excitations leads to quantitative predictions. Regge assumed that mesons and baryons are
quarks connected by strings, like rubber bands – illustrated in Figure 134 and Figure 135
– and that the force or tension 𝑘 between the quarks is thus constant over distance.
We assume that the strings, whose length we call 2𝑟0 , rotate around their centre of
mass as rapidly as possible, as shown in Figure 135. Then we have
𝑟
𝑣(𝑟) = 𝑐 . (79)
𝑟0
230 7 the strong interaction

An excited meson approximated as two
quarks rotating around each other
connected by elastic strings:

v (r0) = c

r0 0 r0
F I G U R E 135 Calculating masses of excited
hadrons.

Motion Mountain – The Adventure of Physics
The quark masses are assumed negligible. For the total energy this implies the relation
𝑟0
𝑘
𝐸 = 𝑐2 𝑚 = 2 ∫ d𝑟 = 𝑘𝑟0 π (80)
0 √1 − 𝑣(𝑟)/𝑐2

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
and for angular momentum the relation

2 𝑟0 𝑘𝑟𝑣(𝑟) 𝑘𝑟02
𝐽= 2∫ d𝑟 = . (81)
ℏ𝑐 0 √ 2ℏ𝑐
1 − 𝑣(𝑟)/𝑐2

Including the spin of the quarks, we thus get

𝑐3
𝐽 = 𝛼0 + 𝛼󸀠 𝑚2 where 𝛼󸀠 = . (82)
2π𝑘ℏ
Regge thus deduced a simple expression that relates the mass 𝑚 of excited hadrons to
their total spin 𝐽. For bizarre historical reasons, this relation is called a Regge trajectory.
The value of the constant 𝛼󸀠 is predicted to be independent of the quark–antiquark
pairing. A few years later, as shown in Figure 134, such linear relations were found in
experiments: the Chew-Frautschi plots. For example, the three lowest lying states of Δ
are the spin 3/2 Δ(1232) with 𝑚2 of 1.5 GeV2 , the spin 7/2 Δ(1950) with 𝑚2 of 3.8 GeV2 ,
and the spin 11/2 Δ(2420) with 𝑚2 of 5.9 GeV2 . The value of the constant 𝛼󸀠 is found
experimentally to be around 0.93 GeV−2 for almost all mesons and baryons, whereas the
Ref. 188 value for 𝛼0 varies from particle to particle. The quark string tension is thus found to be

𝑘 = 0.87 GeV/fm = 0.14 MN . (83)
7 inside nuclei and nucleons 231

In other words, two quarks in a hadron attract each other with a force equal to the weight
of two elephants: about 14 tons.
Experiments are thus clear: the observed Chew-Frautschi plots, as well as several
other observations not discussed here, are best described by a quark–quark potential that
grows, above 1 fm, linearly with distance. The slope of the linear potential, the force, has
a value equal to the force with which the Earth attracts two elephants. As a result, quarks
never appear as free particles: quarks are always confined in hadrons. This situation is
in contrast with QED, where the force between charges goes to zero for large distances;
electric charges are thus not confined, but can exist as free particles. At large distances,
the electric potential decreases in the well-known way, with the inverse of the distance.
Ref. 185 In contrast, for the strong interaction, experiments lead to a quark potential given by

4 𝛼sc ℏ𝑐
𝑉=− + 𝑘𝑟 (84)
3 𝑟

Motion Mountain – The Adventure of Physics
where 𝑘 is the mentioned 0.87 GeV/fm, 𝛼sc is 0.2, and ℏ𝑐 is 0.1975 GeV/fm. The quark
potential is illustrated in Figure 134.
Even though experiments are clear, theoreticians face a problem. So far, neither
the quark-quark potential nor the quark bound states can be deduced from the QCD
Lagrangian with a simple approximation method. Nevertheless, complicated non-
perturbative calculations show that the QCD Lagrangian does predict a force between
two coloured particles that levels off at a constant value (corresponding to a linearly
increasing potential). These calculations show that the old empirical approximations
of hadrons as quarks connected by strings or a quarks in bags, shown in Figure 134,
can indeed be deduced from the QCD Lagrangian. However, the calculations are too

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
complex to be summarized in a few lines. Independently, the constant force value has
also been reproduced in computer calculations in which one simplifies space-time to a
lattice and then approximates QCD by so-called lattice QCD or lattice gauge theory. Lat-
tice calculations have further reproduced the masses of most mesons and baryons with
reasonable accuracy. Using the most powerful computers available, these calculations
have given predictions of the mass of the proton and other baryons within a few per
Ref. 192 cent. Discussing these complex and fascinating calculations lies outside the scope of this
text, however.
In fact, the challenge of explaining confinement in simple terms is so difficult that
the brightest minds have been unable to solve it yet. This is not a surprise, as its solu-
tion probably requires the unification of the interactions and, most probably, also the
unification with gravity. We therefore leave this issue for the last part of our adventure.

Asymptotic freed om
QCD has another property that sets it apart form QED: the behaviour of its coupling with
energy. In fact, there are three equivalent ways to describe the strong coupling strength.
The first way is the quantity appearing in the QCD Lagrangian, 𝑔s . The second way is
often used to define the equivalent quantity 𝛼𝑠 = 𝑔s2 /4π. Both 𝛼𝑠 and 𝑔s depend on the
energy 𝑄 of the experiment. If they are known for one energy, they are known for all of
them. Presently, the best experimental value is 𝛼𝑠 (𝑀𝑍 ) = 0.1185 ± 0.0010.
232 7 the strong interaction

0.5
February 2007

α s(Q)
Deep inelastic scattering
0.4 e+e– annihilation
Hadron collisions
Heavy quarkonia
QCD calculation

0.3

0.2 F I G U R E 136 The measured
and the calculated variation
of the strong coupling with

Motion Mountain – The Adventure of Physics
energy, showing the
precision of the QCD
0.1 Lagrangian and the
α s ( M Z ) = 0.1185 ± 0.0010 asymptotic freedom of the
strong interaction
1 10 100 (© Siegfried Bethke, updated
Q [G eV ] from Ref. 193).

The energy dependence of the strong coupling can be calculated with the standard
renormalization procedures and is expected to be

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 186, Ref. 188

12π 1 (918 − 114𝑛𝑓 ) ln 𝐿 𝑄2
𝛼𝑠 (𝑄2 ) = (1 − + ...) where 𝐿 = ln (85)
33 − 2𝑛𝑓 𝐿 (33 − 2𝑛𝑓 )2 𝐿 Λ2 (𝑛𝑓 )

where 𝑛𝑓 is the number of quarks with mass below the energy scale 𝑄, thus a number
between 3 and 6. (The expression has been expanded to many additional terms with the
help of computer algebra.)
The third way to describe the strong coupling is thus the energy parameter Λ(𝑛𝑓 ).
Experiments yield Λ(3) =230(60) GeV, Λ(4) =180(50) GeV and Λ(5) =120(30) GeV.
The accelerator experiments that measure the coupling are extremely involved, and
hundreds of people across the world have worked for many years to gather the relevant
data. The comparison of QCD and experiment, shown in Figure 136, does not show any
contradiction between the two.
Figure 136 and expression (85) illustrate what is called asymptotic freedom: 𝛼𝑠 de-
creases at high energies. In other words, at high energies quarks are freed from the strong
interaction; they behave as free particles.* As a result of asymptotic freedom, in QCD, a
perturbation expansion can be used only at energies much larger than Λ. Historically,

* Asymptotic freedom was discovered in 1972 by Gerard ’t Hooft; since he had received the Nobel Prize
already, the 2004 Prize was then given to the next people who highlighted it: David Gross, David Politzer
and Frank Wilczek, who studied it extensively in 1973.
7 inside nuclei and nucleons 233

the discovery of asymptotic freedom was essential to establish QCD as a theory of the
strong interaction.
Asymptotic freedom can be understood qualitatively if the situation is compared to
QED. The electron coupling increases at small distances, because the screening due to the
virtual electron-positron pairs has less and less effect. In QCD, the effective colour coup-
ling also changes at small distances, due to the smaller number of virtual quark-antiquark
pairs. However, the gluon properties lead to the opposite effect, an antiscreening that is
even stronger: in total, the effective strong coupling decreases at small distances.

The sizes and masses of quarks
The size of quarks, like that of all elementary particles, is predicted to vanish by QCD, as in
all quantum field theory. So far, no experiment has found any effect due to a finite quark
Ref. 194 size. Measurements show that quarks are surely smaller than 10−19 m. No size conjecture
has been given by any hypothetical theory. Quarks are assumed point-like, or at most
Planck-sized, in all descriptions so far.

Motion Mountain – The Adventure of Physics
We noted in several places that a neutral compound of charged particles is always
less massive than its components. But if you look up the mass values for quarks in most
tables, the masses of 𝑢 and 𝑑 quarks are only of the order of a few MeV/𝑐2 , whereas the
proton’s mass is 938 MeV/c2 . What is the story here?
It turns out that the definition of the mass is more involved for quarks than for other
particles. Quarks are never found as free particles, but only in bound states. As a result,
the concept of quark mass depends on the calculation framework one is using.
Due to asymptotic freedom, quarks behave almost like free particles only at high ener-
gies. The mass of such a ‘free’ quark is called the current quark mass; for the light quarks

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
it is only a few MeV/c2 , as shown in Table 17.
At low energy, for example inside a proton, quarks are not free, but must carry along
a large amount of energy due to the confinement process. As a result, bound quarks
have a much larger effective, so-called constituent quark mass, which takes into account
this confinement energy. To give an idea of the values, take a proton; the indeterminacy
relation for a particle inside a sphere of radius 0.9 fm gives a momentum indeterminacy
of around 190 MeV/c. In three dimensions this gives an energy of √3 times that value, or
an effective, constituent quark mass of about 330 MeV/c2 . Three confined quarks are thus
heavier than a proton, whose mass is 938 MeV/c2 ; we can thus still say that a compound
proton is less massive than its constituents.
In short, the mass of the proton and the neutron is (almost exclusively) the kinetic
Ref. 195 energy of the quarks inside them, as their rest mass is almost negligible. As Frank Wilczek
says, some people put on weight even though they never eat anything heavy.
But also the small current quark mass values for the up, down, strange and charmed
quarks that appear in the QCD Lagrangian depend on the calculation framework that is
used. The values of Table 17 are those for a renormalization scale of 2 GeV. For half that
Ref. 186 energy, the mass values increase by 35 %. The heavy quark masses are those used in the
so-called 𝑀𝑆 scheme, a particular way to perform perturbation expansions.
234 7 the strong interaction

F I G U R E 137 An illustration of a prolate
(left) and an oblate (right) ellipsoidal shape
(© Sam Derbyshire).

The mass, shape and colour of protons
Ref. 195 Frank Wilczek mentions that one of the main results of QCD, the theory of strong inter-
actions, is to explain mass relations such as

Motion Mountain – The Adventure of Physics
𝑚proton ∼ e−𝑘/𝛼 𝑚Planck and 𝑘 = 11/2π , 𝛼unif = 1/25 . (86)

Here, the value of the coupling constant 𝛼unif is taken at the grand unifying energy, a
Page 268 factor of 1000 below the Planck energy. (See the section of grand unification below.) In
other words, a general understanding of masses of bound states of the strong interaction,
such as the proton, requires almost purely a knowledge of the unification energy and the
coupling constant at that energy. The approximate value 𝛼unif = 1/25 is an extrapolation
from the low energy value, using experimental data. The proportionality factor 𝑘 in ex-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
pression (86) is not easy to calculate. It is usually determined on computers using lattice
QCD.
But the mass is not the only property of the proton. Being a cloud of quarks and
gluons, it also has a shape. Surprisingly, it took a long time before people started to be-
come interested in this aspect. The proton, being made of two up quarks and one down
quark, resembles a ionized 𝐻2+ molecule, where one electron forms a cloud around two
protons. Obviously, the 𝐻2+ molecule is elongated, or prolate, as shown in Figure 137.
Is the proton prolate? There is no spectroscopically measurable asphericity – or quad-
rupole moment – of the proton. However, the proton has an intrinsic quadrupole mo-
ment. The quadrupole moments of the proton and of the neutron are predicted to be
Ref. 196 positive in all known calculation methods, implying an prolate shape. Recent measure-
ments at Jefferson Laboratories confirm this prediction. A prolate shape is predicted for
all 𝐽 = 1/2 baryons, in contrast to the oblate shape predicted for the 𝐽 = 3/2 baryons.
The spin 0 pseudoscalar mesons are predicted to be prolate, whereas the spin 1 vector
mesons are expected to be oblate.
The shape of any molecule will depend on whether other molecules surround it. Re-
cent research showed that similarly, both the size and the shape of the proton in nuclei
Ref. 197 is slightly variable; both seem to depend on the nucleus in which the proton is built-in.
Apart from shapes, molecules also have a colour. The colour of a molecule, like that
of any object, is due to the energy absorbed when it is irradiated. For example, the 𝐻2+
molecule can absorb certain light frequencies by changing to an excited state. Molecules
change mass when they absorb light; the excited state is heavier than the ground state.
7 inside nuclei and nucleons 235

N (I=1/2) Mass/(MeV/c2) Δ (I=3/2)
Experiment Calculation Calculation Experiment
H 3,11 (2420)
2400 F 37 (2390)
G 19 (2250) D 35 (2350)
H 19 (2220) H 39 (2300)
D 15 (2200)
G 17 (2190) 2200
P 11 (2100) S 31 (2150)
S 11 (2090) F 35 (2000)
D 13 (2080) F 37 (1950)
F 15 (2000) D 33 (1940)
2000 D 35 (1930)
F 17 (1990)
P 33 (1920)
P 13 (1900) P 31 (1910)
P 13 (1720) F 35 (1905)
P 11 (1710) 1800 S 31 (1900)
D 13 (1700) P 31 (1750)
F 15 (1680) D 33 (1700)
D 15 (1675)
S 11 (1650) S 31 (1620)
1600 P 33 (1600)
S 11 (1535)
D 13 (1520)

Motion Mountain – The Adventure of Physics
P 11 (1440)
1400

P 33 (1232)
1200

1000
P 11 (939)

F I G U R E 138 The mass spectrum of the excited states of the proton: experimental and calculated values
(from Ref. 186).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In the same way, protons and neutrons can be excited. In fact, their excited states have
Ref. 186 been studied in detail; a summary, also showing the limitation of the approach, is shown
in Figure 138. Many excitations can be explained as excited quarks states, but many more
are predicted. The calculated masses agree with observations to within 10 %. The quark
model and QCD thus structure and explain a large part of the baryon spectrum; but the
agreement is not yet perfect.
Obviously, in our everyday environment the energies necessary to excite nucleons do
not appear – in fact, they do not even appear inside the Sun – and these excited states
can be neglected. They only appear in particle accelerators and in cosmic rays. In a sense,
we can say that in our corner of the universe, the colour of protons usually is not visible.

Curiosities ab ou t the strong interaction
In a well-known analogy, QCD can be compared to superconductivity. Table 18 gives an
overview of the correspondence.
∗∗
The computer calculations necessary to extract particle data from the Lagrangian of
quantum chromodynamics are among the most complex calculations ever performed.
They beat weather forecasts, fluid simulations and the like by orders of magnitude.
236 7 the strong interaction

TA B L E 18 Correspondence between QCD and superconductivity.

QCD Superconductivity

Quark magnetic monopole
Colour force non-linearities Electron–lattice interaction
Chromoelectric flux tube magnetic flux tube
Gluon-gluon attraction electron–electron attraction
Glueballs Cooper pairs
Instability of bare vacuum instability of bare Fermi surface
Discrete centre symmetry continuous U(1) symmetry
High temperature breaks symmetry low temperature breaks symmetry

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 139 A three jet event
observed at the PETRA collider in
Hamburg in Germany. The event,
triggered by an electron-positron
collision, allowed detecting the
decay of a gluon and measuring
its spin (© DESY).

Nobody knows whether this will be necessary also in the future: the race for a simple
approximation method for finding solutions is still open.
∗∗
Even though gluons are massless, like photons and gravitons are, there is no colour ra-
diation in nature. Gluons carry colour and couple to themselves; as a result, free gluons
were predicted to directly decay into quark–antiquark pairs.
In 1979, the first clear decays of gluons have been observed at the PETRA particle col-
Ref. 191 lider in Hamburg. The occurrence of certain events, called gluon jets, are due to the de-
cay of high-energy gluons into narrow beams of particles. Gluon jets appear in coplanar
7 inside nuclei and nucleons 237

three-jet events. The observed rate and the other properties of these events confirmed
the predictions of QCD. Experiments at PETRA also determined the spin 𝑆 = 1 of the
gluon and the running of the strong coupling constant. The hero of those times was the
project manager Gustav-Adolf Voss, who completed the accelerator on budget and six
months ahead of schedule.
∗∗
Something similar to colour radiation, but still stranger, might have been found in 1997.
It might be that a scalar meson with a mass of 1.5 GeV/c2 is a glueball. This is a hypothet-
ical meson composed of gluons only. Numerical results from lattice gauge theory seem
Ref. 198 to confirm the possibility of a glueball in that mass range. The existence of glueballs is
hotly debated and still open.
∗∗
There is a growing consensus that most light scalar mesons below 1 GeV/c2 , are

Motion Mountain – The Adventure of Physics
Ref. 199 tetraquarks. In 2003, experiments provided also candidates for heavier tetraquarks,
namely the X(3872), Ds(2317) and Ds(2460). The coming years will show whether this
interpretation is correct.
∗∗
Do particles made of five quarks, so-called pentaquarks, exist? So far, they seem to exist
only in a few laboratories in Japan, whereas in other laboratories across the world they
Ref. 200 are not seen. Most researchers do not believe the results any more.
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Whenever we look at a periodic table of the elements, we look at a manifestation of the
strong interaction. The Lagrangian of the strong interaction describes the origin and
properties of the presently known 115 elements.
Nevertheless one central aspect of nuclei is determined by the electromagnetic in-
teraction. Why are there around 115 different elements? Because the electromagnetic
coupling constant 𝛼 is around 1/137.036. In more detail, the answer is the following. If
the charge of a nucleus were much higher than around 137, the electric field around nuc-
lei would lead to spontaneous electron–positron pair generation; the generated electron
would fall into the nucleus and transform one proton into a neutron, thus inhibiting a
larger proton number. The finite number of the elements is thus due to the electromag-
netic interaction.
∗∗
To know more about radioactivity, its effects, its dangers and what a government can do
about it, see the English and German language site of the Federal Office for Radiation
Protection at www.bfs.de.
∗∗
From the years 1990 onwards, it has regularly been claimed that extremely poor countries
Challenge 146 s are building nuclear weapons. Why is this highly unlikely?
238 7 the strong interaction

∗∗
Historically, nuclear reactions provided the first test of the relation 𝐸 = 𝑐2 𝛾𝑚. This was
achieved in 1932 by Cockcroft and Walton. They showed that by shooting protons into
lithium one gets the reaction
7
3 Li + 11 H → 84 Be → 42 He + 42 He + 17 MeV . (87)

The measured energy on the right is exactly the same value that is derived from the dif-
ferences in total mass of the nuclei on both sides.
∗∗
A large fraction of researchers say that QCD is defined by two parameters. Apart from the
coupling constant, they count also the strong CP parameter. Indeed, it might be that the
strong interaction violates CP invariance. This violation would be described by a second
term in the Lagrangian; its strength would be described by a second parameter, a phase

Motion Mountain – The Adventure of Physics
usually called 𝜃𝐶𝑃 . However, many high-precision experiments have been performed to
search for this effect, and no CP violation in the strong interaction has ever been detected.

A summary of QCD and its open issues
Quantum chromodynamics, the non-Abelian gauge theory based on the Lagrangian with
SU(3) symmetry, describes the properties of gluons and quarks, the properties of the
proton, the neutron and all other hadrons, the properties of atomic nuclei, the working
of the stars and the origin of the atoms inside us and around us. Without the strong
interaction, we would not have flesh and blood. And all these aspects of nature follow

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
from a single number, the strong coupling constant, and the SU(3) gauge symmetry.
The strong interaction acts only on quarks and gluons. It conserves particle type, col-
our, electric charge, weak charge, spin, as well as C, P and T parity.
QCD and experiment agree wherever comparisons have been made. QCD is a perfect
description of the strong interaction. The limitations of QCD are only conceptual. Like
in all of quantum field theory, also in the case of QCD the mathematical form of the
Lagrangian is almost uniquely defined by requiring renormalizability, Lorentz invari-
ance and gauge invariance – SU(3) in this case. We say ‘almost’, because the Lagrangian,
despite describing correctly all experiments, contains a few parameters that remain un-
explained:
— The number, 6, and the masses 𝑚𝑞 of the quarks are not explained by QCD.
— The coupling constant of the strong interaction 𝑔s , or equivalently, 𝛼𝑠 or Λ, is unex-
plained. QCD predicts its energy dependence, but not its absolute value.
— Experimentally, the strong interaction is found to be CP conserving. This is not obvi-
ous; the QCD Lagrangian assumes that any possible CP-violating term vanishes, even
though there exist CP-violating Lagrangian terms that are Lorentz-invariant, gauge-
invariant and renormalizable.
— The properties of space-time, in particular its Lorentz invariance, its continuity and
the number of its dimensions are assumed from the outset and are obviously all un-
explained in QCD.
7 inside nuclei and nucleons 239

— It is also not known how QCD has to be modified in strong gravity, thus in strongly
curved space-time.
We will explore ways to overcome these limits in the last part of our adventure. Before
we do that, we have a look at the other nuclear interaction observed in nature.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Chapter 8

T H E W E A K N U C L E A R I N T E R AC T ION
A N D T H E HA N DE DN E S S OF NAT U R E

T
he weirdest interaction in nature is the weak interaction. The weak interaction
ransforms elementary particles into each other, has radiation particles
hat have mass, violates parity and treats right and left differently. Fortunately,
we do not experience the weak interaction in our everyday life, as its properties violate

Motion Mountain – The Adventure of Physics
much of what we normally experience. This contrast makes the weak interaction the
most fascinating of the four interactions in nature.

Transformation of elementary particles
Radioactivity, in particular the so-called β decay, is a bizarre phenomenon. Experiments
in the 1910s showed that when β sources emit electrons, atoms are transformed from one
chemical element to another. For example, experiments such as those of Figure 140 show
that tritium, an isotope of hydrogen, decays into helium as

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
3
1H → 32 He + 𝑒− + 𝜈e . (88)

In fact, new elements appear in all β decays. In the 1930s it became clear that the trans-
formation process is due to a neutron in the nucleus changing into a proton (and more):

𝑛 → 𝑝 + 𝑒− + 𝜈 e . (89)

This reaction explains all β decays. In the 1960s, the quark model showed that β decay is
in fact due to a down quark changing to an up quark:

𝑑 → 𝑢 + 𝑒− + 𝜈 e . (90)

This reaction explains the transformation of a neutron – a 𝑢𝑑𝑑 state – into a proton –
a 𝑢𝑢𝑑 state. In short, matter particles can transform into each other. We note that this
transformation differs from what occurs in other nuclear processes. In fusion, fission or α
decay, even though nuclei change, every neutron and every proton retains its nature. In β
decay, elementary particles are not immutable. The dream of Democritus and Leucippus
about immutable basic building blocks is definitely not realized in nature.
Experiments show that quark transformations cannot be achieved with the help of
electromagnetic fields, nor with the help of gluon fields, nor with the help of gravita-
tion. There must be another type of radiation in nature, and thus another, fourth in-
8 and the handedness of nature 241

1.2

Count rate (arbitrary units)
1.0 neutron proton
0.8
u d u d
d u
0.6
W e
0.4
ν
0.2

0 time
0 5 10 15 20
Energy (keV)

F I G U R E 140 β decay (beta decay) in tritium: a modern, tritium-powered illuminated watch, the
measured continuous energy spectrum of the emitted electrons from tritium, and the process occurring
in the tritium nucleus (© Traser, Katrin collaboration).

Motion Mountain – The Adventure of Physics
teraction. Chemical transformations are also rare, otherwise we would not be running
around with a constant chemical composition. The fourth interaction is therefore weak.
Since the transformation processes were observed in the nucleus first, the interaction
was named the weak nuclear interaction.
In β decay, the weak nuclear interaction transforms quarks into each other. In fact,
the weak nuclear interaction can also transform leptons into each other, such as muons
into electrons. But where does the energy released in β decay go to? Measurements in
1911 showed that the energy spectrum of the emitted electron is continuous. This is il-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
lustrated in Figure 140. How can this be? In 1930, Wolfgang Pauli had the courage and
genius to explain this observation with a daring hypothesis: the energy of the decay is
split between the electron and a new, truly astonishing particle, the neutrino – more pre-
cisely, the electron anti-neutrino 𝜈ē . In order to agree with data, the neutrino must be
uncharged, cannot interact strongly, and must be of very low mass. As a result, neut-
rinos interact with ordinary matter only extremely rarely, and usually fly through the
Earth without being affected. This property makes their detection very difficult, but not
impossible; the first neutrino was finally detected in 1952. Later on, it was discovered that
there are three types of neutrinos, now called the electron neutrino, the muon neutrino
and the tau neutrino, each with its own antiparticle. For the summary of these experi-
Page 242 mental efforts, see Table 19.

The weakness of the weak nuclear interaction
From the observation of β decay, and helped by the quark model, physicists quickly con-
cluded that there must be an intermediate particle that carries the weak nuclear interac-
tion, similar to the photon that carries the electromagnetic interaction. This ‘weak radi-
ation’, in contrast to all other types of radiation, consists of massive particles.

⊳ There are two types of weak radiation particles: the neutral Z boson with
a mass of 91.2 GeV – that is the roughly mass of a silver atom – and the
electrically charged W boson with a mass of 80.4 GeV.
242 8 the weak nuclear interaction

TA B L E 19 The leptons: the three neutrinos and the three charged leptons (antiparticles have opposite
charge Q and parity P).

Neutrino Mass 𝑚 Spin 𝐽 Colour; C h a r g e 𝑄, L epton
a n d d e c ay pa r i t y p o s s i b l e i s o s p i n 𝐼, number
(see text) 𝑃 weak be - s t r a n g eness 𝑆, 𝐿,
h av i o u r charm 𝐶, beauty baryon
𝐵󸀠 , topness 𝑇 number
𝐵
1+
Electron < 2 eV/𝑐2 , 2
white; singlet, 0, 0, 0, 0, 0, 0 1, 0
neutrino 𝜈𝑒 oscillates doublet
1+
Muon < 2 eV/𝑐2 , 2
white; singlet, 0, 0, 0, 0, 0, 0 1, 0
neutrino 𝜈𝑒 oscillates doublet
1+
Tau < 2 eV/𝑐2 , 2
white; singlet, 0, 0, 0, 0, 0, 0 1, 0
neutrino 𝜈𝑒 oscillates doublet

Motion Mountain – The Adventure of Physics
1+
Electron 𝑒 2
white; singlet, −1, 0, 0, 0, 0, 0 1, 0
0.510 998 928(11) doublet
MeV/𝑐2 , stable
1+
Muon 𝜇 105.658 3715(35) 2
white; singlet, −1, 0, 0, 0, 0, 0 1, 0
MeV/𝑐2 , doublet
c. 99 % 𝑒𝜈𝑒 𝜈𝜇
1+
Tau 𝜏 1.776 82(16) 2
white; singlet, −1, 0, 0, 0, 0, 0 1, 0
GeV/𝑐2 , doublet
c. 17 % 𝜇𝜈𝜇 𝜈𝜏 ,
c. 18 % 𝑒𝜈𝑒 𝜈𝜏

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Together, the W and Z bosons are also called the weak vector bosons, or the weak inter-
mediate bosons.
The masses of the weak vector bosons are so large that free weak radiation exists only
for an extremely short time, about 0.1 ys; then the bosons decay. The large mass is the
reason that the weak interaction is extremely short range and thus extremely weak. In-
deed, any exchange of virtual carrier particles scales with the negative exponential of the
intermediate particle’s mass. A few additional properties are given in Table 20. In fact,
the weak interaction is so weak that neutrinos, particles which interact only weakly, have
a large probability to fly through the Sun without any interaction.
The existence of a massive charged intermediate vector boson, today called the W, was
already deduced and predicted in the 1940s; but theoretical physicists did not accept the
idea until the Dutch physicists Martin Veltman and Gerard ’t Hooft proved that it was
possible to have such a mass without having problems in the rest of the theory. For this
proof they later received the Nobel Prize in Physics – after experiments confirmed their
prediction.
The existence of an additional massive neutral intermediate vector boson, the Z boson,
was predicted only much after the W boson, by Salam, Weinberg and Glashow. Experi-
mentally, the Z boson was first observed as a virtual particle in 1973 at CERN in Geneva.
The discovery was made by looking, one by one, at over 700 000 photographs made at
8 and the handedness of nature 243

TA B L E 20 The intermediate vector bosons of the weak interaction (the Z boson is its own antiparticle;
the W boson has an antiparticle of opposite charge).

Boson Mass 𝑚 Spin 𝐽 C o l o u r ; C h a r g e 𝑄, L epton
w e a k b e - i s o s p i n 𝐼, number
h a v i o u r s t r a n g eness 𝑆, 𝐿,
charm 𝐶, beauty baryon
𝐵󸀠 , topness 𝑇 number
𝐵

Z boson 91.1876(21) 1 white; 0, 0, 0, 0, 0, 0 0, 0
GeV/𝑐2 ‘triplet’
W boson 80.398(25) 1 white; 1, 0, 0, 0, 0, 0 0, 0
GeV/𝑐2 ‘triplet’

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 141 The ﬁrst observation of a virtual Z boson: only neutral weak currents allow that a neutrino
collides with an electron in the bubble chamber and leaves again (© CERN).

the Gargamelle bubble chamber. Only a few interesting pictures were found; the most
famous one is shown in Figure 141.
In 1983, CERN groups produced and detected the first real W and Z bosons. This ex-
periment was a five-year effort by thousands of people working together. The results are
244 8 the weak nuclear interaction

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 142 Top: the SPS, the proton–antiproton accelerator and collider at CERN, with 7 km
circumference, that was used to make the ﬁrst observations of real W and Z bosons. Bottom: the
beautifully simple Z observation made with LEP, the successor machine (© CERN)
8 and the handedness of nature 245

F I G U R E 143 A measurement of the W boson mass at LEP (© CERN)

Motion Mountain – The Adventure of Physics
summarized in Table 20. The energetic manager of the project, Carlo Rubbia, whose bad
temper made his secretaries leave, on average, after three weeks, and the chief technolo-
gist, Simon van der Meer, received the 1984 Nobel Prize in Physics for the discovery. This
again confirmed the ‘law’ of nature that bosons are discovered in Europe and fermions
in America. The simplest data that show the Z and W bosons is shown in Figure 142 and
Figure 143; both results are deduced from the cross section of electron-positron collisions
at LEP, a decade after the original discovery.
In the same way that photons are emitted by accelerated electric charges, W and Z

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
bosons are emitted by accelerated weak charges. Due to the high mass of the W and Z
bosons, the required accelerations are very high, so that they are only found in certain
nuclear decays and in particle collisions. Nevertheless, the W and Z bosons are, apart
from their mass and their weak charge, similar to the photon in most other aspects. In
particular, the W and the Z are observed to be elementary. For example, the W gyromag-
Ref. 186 netic ratio has the value predicted for elementary particles.

Distinguishing left from right
Another weird characteristic of the weak interaction is the non-conservation of parity
P under spatial inversion. The weak interaction distinguishes between mirror systems;
this is in contrast to everyday life, to gravitation, to electromagnetism, and to the strong
interaction. The non-conservation of parity by the weak interaction had been predicted
by 1956 by Lee Tsung-Dao and Yang Chen Ning in order to explain the ability of 𝐾0
mesons to decay sometimes into 2 pions, which have even parity, and sometimes into 3
Ref. 202 pions, which have odd parity.
Lee and Yang suggested an experiment to Wu Chien-Shiung* The experiment she
performed with her team is shown schematically in Figure 144. A few months after the

* Wu Chien-Shiung (b. 1912 Shanghai, d. 1997 New York) was called ‘madame Wu’ by her colleagues. She
was a bright and driven physicist born in China. She worked also on nuclear weapons; later in life she was
president of the American Physical Society.
246 8 the weak nuclear interaction

Observed situation : Situation after spatial inversion P,
not observed:

ν spin S=1/2

most ν momenta most e

J=5 (Co), then 4 (Ni)
Mag- Mag-
netic nucleus netic nucleus
60Co,
field field
B then virtual W– spin S=1 B
60Ni

most el. momenta most ν

Motion Mountain – The Adventure of Physics
electron spin S=1/2

Magnetic field Magnetic field
and most electron and most electron
motion are motion would
antiparallel be parallel

F I G U R E 144 The measured behaviour of β decay, and its imagined, but unobserved behaviour under
spatial inversion P (corresponding to a mirror reﬂection plus subsequent rotation by π around an axis
perpendicular to the mirror plane).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
first meetings with Lee and Yang, Wu and her team found that in the β decay of cobalt
nuclei aligned along a magnetic field, the electrons are emitted mostly against the spin of
the nuclei. In the experiment with inversed parity, the electrons would be emitted along
the spin direction; however, this case is not observed. Parity is violated. This earned Lee
and Yang a Nobel Prize in 1957.
Parity is thus violated in the weak interaction. Parity violation does not only occur in
β decay. Parity violation has been found in muon decay and in every other weak process
studied so far. In particular, when two electrons collide, those collisions that are medi-
ated by the weak interaction behave differently in a mirror experiment. The number of
Ref. 203 experiments showing this increases from year to year. In 2004, two polarized beams of
electrons – one left-handed and one right-handed – were shot at a matter target and the
reflected electrons were counted. The difference was 0.175 parts per million – small, but
measurable. The experiment also confirmed the predicted weak charge of −0.046 of the
electron.
A beautiful consequence of parity violation is its influence on the colour of certain
Ref. 204 atoms. This prediction was made in 1974 by Bouchiat and Bouchiat. The weak interac-
tion is triggered by the weak charge of electrons and nuclei; therefore, electrons in atoms
do not exchange only virtual photons with the nucleus, but also virtual Z particles. The
chance for this latter process is extremely small, around 10−11 times smaller than for ex-
8 and the handedness of nature 247

Motion Mountain – The Adventure of Physics
F I G U R E 145 Wu Chien-Shiung (1912
–1997) at her parity-violation
experiment.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
change of virtual photons. But since the weak interaction is not parity conserving, this
process allows electron transitions which are impossible by purely electromagnetic ef-
fects. In 1984, measurements confirmed that certain optical transitions of caesium atoms
that are impossible via the electromagnetic interaction, are allowed when the weak inter-
Ref. 205 action is taken into account. Several groups have improved these results and have been
able to confirm the calculations based on the weak interaction properties, including the
Ref. 206 weak charge of the nucleus, to within a few per cent.
The weak interaction thus allows one to distinguish left from right. Nature contains
processes that differ from their mirror version. In short, particle physics has shown that
nature is (weakly) left-handed.
The left-handedness of nature is to be taken literally. All experiments confirmed two
Challenge 147 e central statements on the weak interaction that can be already guessed from Figure 144.

⊳ The weak interaction only couples to left-handed particles and to right-
handed antiparticles. Parity is maximally violated in the weak interaction.

⊳ All neutrinos observed so far are left-handed, and all antineutrinos are right-
handed.

This result can only hold if neutrino masses vanish or are negligibly small. These two
experimental results fix several aspects of the Lagrangian of the weak interaction.
248 8 the weak nuclear interaction

Motion Mountain – The Adventure of Physics
F I G U R E 146 A map of the intensity distribution of the 3(2) ⋅ 1025 antineutrinos between 0 and 11 MeV
radiated every second from the Earth. Around 99 % of the ﬂux is from natural sources and around 1 %
from civil and military nuclear processing plants and reactors. The map is from www.nga.mil, thus
cannot be completely trusted about the human sources.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Distinguishing particles and antiparticles, CP violation
In the weak interaction, the observation that only right-handed particles and left-handed
antiparticles are affected has an important consequence: it implies a violation of charge
Challenge 148 e conjugation parity C. Observations of muons into electrons shows this most clearly: anti-
muon decay differs from muon decay. The weak interaction distinguishes particles from
antiparticles.

⊳ Experiments show that C parity, like P parity, is maximally violated in the
weak interaction.

Also this effect has been confirmed in all subsequent observations ever performed on
the weak interaction. But that is not all. In 1964, a now famous observation was made by
Val Fitch and James Cronin in the decay of the neutral K mesons.

⊳ The weak interaction also violates the combination of parity inversion with
particle-antiparticle symmetry, the so-called CP invariance. In contrast to P
violation and C violation, which are maximal, CP violation is a tiny effect.

The experiment, shown in Figure 147 earned them the Nobel Prize in 1980. CP violation
8 and the handedness of nature 249

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net

F I G U R E 147 Top: the experimental set-up for measuring the behaviour of neutral K meson decay.
Bottom: the measured angular dependence; the middle graph shows a peak at the right side that
would not appear if CP symmetry would not be violated (© Brookhaven National Laboratory, Nobel
Foundation).
250 8 the weak nuclear interaction

The electroweak Feynman diagrams (without the Higgs boson)

q‘ W or Z or γ W Z or γ W Z or γ or Z

q‘ W W Z or γ or γ
l‘ Z or γ ν W W W

Motion Mountain – The Adventure of Physics
l‘ l W W

q’ and l’ indicate quark and lepton mixing

F I G U R E 148 The essence of the electroweak interaction Lagrangian.

has also been observed in neutral B mesons, in several different processes and reactions.
The search for other manifestations of CP violation, such as in non-vanishing electric

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
dipole moments of elementary particles, is an intense research field. The search is not
simple because CP violation is a small effect in an already very weak interaction; this
tends to makes experiments large and expensive.
Since the weak interaction violates CP invariance, it also violates motion (or time)
reversal T.

⊳ But like all gauge theories, the weak interaction does not violate the com-
bined CPT symmetry: it is CPT invariant.

If CPT would be violated, the masses, lifetimes and magnetic moments of particles and
antiparticles would differ. That is not observed.

Weak charge and mixings
All weak interaction processes can be described by the Feynman diagrams of Figure 148.
But a few remarks are necessary. First of all, the W and Z act only on left-handed fermion
and on right-handed anti-fermions. Secondly, the weak interaction conserves a so-called
weak charge 𝑇3 , also called weak isospin. The three quarks u, c and t, as well as the three
neutrinos, have weak isospin 𝑇3 = 1/2; the other three quarks and the charged leptons
have weak isospin 𝑇3 = −1/2. In an idealized, SU(2)-symmetric world, the three vec-
tor bosons 𝑊+ , 𝑊0 , 𝑊− would have weak isospin values 1, 0 and −1 and be massless.
However, a few aspects complicate the issue.
8 and the handedness of nature 251

First of all, it turns out that the quarks appearing in Figure 148 are not those of the
strong interaction: there is a slight difference, due to quark mixing. Secondly, also neut-
rinos mix. And thirdly, the vector bosons are massive and break the SU(2) symmetry of
the imagined idealized world; the Lie group SU(2) is not an exact symmetry of the weak
interaction, and the famous Higgs boson has mass. We now explore these aspects in this
order.
Surprisingly, the weak interaction eigenstates of the quarks are not the same as
the mass eigenstates. This discovery by Nicola Cabibbo is described by the so-called
Cabibbo–Kobayashi–Maskawa or CKM mixing matrix. The matrix is defined by

𝑑󸀠 𝑑
( 𝑠󸀠 ) = (𝑉𝑖𝑗) ( 𝑠 ) . (91)
𝑏󸀠 𝑏

where, by convention, the states of the +2/3 quarks (𝑢, 𝑐, 𝑡) are unmixed. In its standard

Motion Mountain – The Adventure of Physics
parametrization, the CKM matrix reads

𝑐12 𝑐13 𝑠12 𝑐13 𝑠13 e−𝑖𝛿13
𝑉 = (−𝑠12 𝑐23 − 𝑐12 𝑠23 𝑠13 e𝑖𝛿13 𝑐12 𝑐23 − 𝑠12 𝑠23 𝑠13 e𝑖𝛿13 𝑠23 𝑐13 ) (92)
𝑠12 𝑠23 − 𝑐12 𝑐23 𝑠13 e𝑖𝛿13 −𝑐12 𝑠23 − 𝑠12 𝑐23 𝑠13 e𝑖𝛿13 𝑐23 𝑐13

where 𝑐𝑖𝑗 = cos 𝜃𝑖𝑗 , 𝑠𝑖𝑗 = sin 𝜃𝑖𝑗 and 𝑖 and 𝑗 label the generation (1 ⩽ 𝑖, 𝑗 ⩽ 3). In the
limit 𝜃23 = 𝜃13 = 0, i.e., when only two generations mix, the only remaining parameter
is the angle 𝜃12 , called the Cabibbo angle, which was introduced when only the first two

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
generations of fermions were known. The phase 𝛿13 , lying between 0 and 2π, is different
from zero in nature, and expresses the fact that CP invariance is violated in the case of the
weak interactions. It appears in the third column and shows that CP violation is related
to the existence of (at least) three generations.
The CP violating phase 𝛿13 is usually expressed with the Jarlskog invariant, defined
2
as 𝐽 = sin 𝜃12 sin 𝜃13 sin 𝜃23 cos 𝜃12 cos 𝜃13 cos 𝜃23 sin 𝛿13 . This expression is independent
of the definition of the phase angles; it was discovered by Cecilia Jarlskog, an important
Ref. 207 Swedish particle physicist. Its measured value is 𝐽 = 2.96(20) ⋅ 10−5 .
The CKM mixing matrix is predicted to be unitary. The unitarity has been confirmed
Ref. 186 by all experiments so far. The 90 % confidence upper and lower limits for the magnitude
of the complex CKM matrix 𝑉 are given by

0.97428(15) 0.2253(7) 0.00347(16)
|𝑉| = ( 0.2252(7) 0.97345(16) 0.0410(11) ) . (93)
0.00862(26) 0.0403(11) 0.999152(45)

The values have been determined in dozens of experiments by thousands of physicists.
Also neutrinos mix, in the same way as the d, s and b quarks. The determination of the
matrix elements is not as complete as for the quark case. This is an intense research field.
Like for quarks, also for neutrinos the mass eigenstates and the flavour eigenstates differ.
There is a dedicated neutrino mixing matrix, called the Pontecorvo–Maki–Nakagawa–
252 8 the weak nuclear interaction

Sakata mixing matrix or PMNS mixing matrix, with 4 angles for massive neutrinos (it
would have 6 angles if neutrinos were massless). In 2012, the measured matrix values
were
0.82 0.55 −0.15 + 0.038𝑖
𝑃 = (−0.36 + 0.020𝑖 0.70 + 0.013𝑖 0.61 ) . (94)
0.44 + 0.026𝑖 −0.45 + 0.017𝑖 0.77

Many experiments are trying to measure these parameters with higher precision.

Symmetry breaking – and the lack of electroweak unification
The intermediate W and Z bosons are massive and their masses differ. Thus, the weak
interaction does not show a SU(2) symmetry. In addition, electromagnetic and weak
processes mix.
Beautiful research in the 1960s showed that the mixing of the electromagnetic and the
weak interactions can be described by an ‘electroweak’ coupling constant 𝑔 and a weak

Motion Mountain – The Adventure of Physics
Ref. 186 mixing angle 𝜃𝑊 . The mixing angle describes the strength of the breaking of the SU(2)
symmetry.
It needs to be stressed that in contrast to what is usually said and written, the weak
and the electromagnetic interactions do not unify. They have never been unified. Despite
the incessant use of the term ‘electroweak unification’ since several decades, the term is
wrong. The electromagnetic and the weak interactions are two independent interactions,
with two coupling constants, that mix. But they do not unify. Even though the Nobel
Prize committee used the term ‘unification’, the relevant Nobel Prize winners confirm
that the term is not correct.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
⊳ The electromagnetic and the weak interaction have not been unified. Their
mixing has been elucidated.

The usual electromagnetic coupling constant 𝑒 is related to the ‘electroweak’ coupling
𝑔 and the mixing angle 𝜃w by
𝑒 = 𝑔 sin 𝜃w , (95)

which at low four-momentum transfers is the fine structure constant with the value
1/137.036. The electroweak coupling constant 𝑔 also defines the historically defined Fermi
constant 𝐺F by
𝑔2 √2
𝐺F = 2
. (96)
8𝑀𝑊

The broken SU(2) symmetry implies that in the real world, in contrast to the ideal SU(2)
world, the intermediate vector bosons are

— the massless, neutral photon, given as 𝐴 = 𝐵 cos 𝜃𝑊 + 𝑊3 sin 𝜃𝑊 ;
— the massive neutral Z boson, given as 𝑍 = −𝐵 sin 𝜃𝑊 + 𝑊3 cos 𝜃𝑊 ;
— the massive charged W bosons, given as 𝑊± = (𝑊1 ∓ 𝑖𝑊2 )/√2 .

Together, the mixing of the electromagnetic and weak interactions as well as the breaking
8 and the handedness of nature 253

of the SU(2) symmetry imply that the electromagnetic coupling 𝑒, the weak coupling 𝑔
and the intermediate boson masses by the impressive relation

𝑚𝑊 2 𝑒 2
( ) +( ) =1. (97)
𝑚𝑍 𝑔

The relation is well verified by experiments.
The mixing of the electromagnetic and weak interactions also suggests the existence
of a scalar, elementary Higgs boson. This prediction, from the year 1963, was made by
Ref. 208 Peter Higgs and a number of other particles physicists, who borrowed ideas that Yoichiro
Nambu and, above all, Philip Anderson introduced in solid state physics. The Higgs bo-
son maintains the unitarity of longitudinal boson scattering at energies of a few TeV and
influences the mass of all other elementary particles. In 2012, the Higgs boson has finally
been observed in two large experiments at CERN.

Motion Mountain – The Adventure of Physics
The L agrangian of the weak and electromagnetic interactions
If we combine the observed properties of the weak interaction mentioned above, namely
its observed Feynman diagrams, its particle transforming ability, P and C violation, quark
mixing, neutrino mixing and symmetry breaking, we arrive at the full Lagrangian dens-
ity. It is given by:

𝑔𝑚𝑘 𝐻
LE&W = ∑𝑘 𝜓𝑘 (𝑖∂/ − 𝑚𝑘 − 2𝑚𝑊
)𝜓𝑘 } 1. fermion mass terms
𝜇
−𝑒 ∑𝑘 𝑞𝑘 𝜓𝑘 𝛾 𝜓𝑘 𝐴 𝜇 } 2. e.m. interaction

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑔
− ∑𝑘 𝜓𝑘 𝛾𝜇 (1 − 𝛾5 )(𝑇+ 𝑊𝜇+ + 𝑇− 𝑊𝜇− )𝜓𝑘 } 3. charged weak currents
2√2
𝑔
− 2 cos 𝜃 ∑𝑘 𝜓𝑘 𝛾𝜇 (𝑔𝑉𝑘 − 𝑔𝐴𝑘 𝛾5 )𝜓𝑘 𝑍𝜇 } 4. neutral weak currents
𝑊

− 14 𝐹𝜇𝜈 𝐹𝜇𝜈 } 5. electromagnetic field
− 12 𝑊𝜇𝜈+
𝑊−𝜇𝜈 − 14 𝑍𝜇𝜈 𝑍𝜇𝜈 } 6. weak W and Z fields
+𝑚2𝑊 𝑊+ 𝑊− + 12 𝑚2𝑍 𝑍2 } 7. W and Z mass terms
−𝑔𝑊𝑊𝐴 − 𝑔𝑊𝑊𝑍 } 8. cubic interaction
𝑔2
− 4 (𝑊4 + 𝑍4 + 𝑊2 𝐹2 + 𝑍2 𝐹2 ) } 9. quartic interaction
+ 12 (∂𝜇 𝐻)(∂𝜇 𝐻) − 12 𝑚2𝐻 𝐻2 } 10. Higgs boson mass
𝑔𝑚2 𝑔2 𝑚2
− 4𝑚 𝐻 𝐻3 − 32𝑚2𝐻 𝐻4 } 11. Higgs self-interaction
𝑊 𝑊
𝑔2
+(𝑔𝑚𝑊𝐻 + 4 𝐻2 )(𝑊𝜇+ 𝑊−𝜇 + 2 cos12 𝜃 𝑍𝜇 𝑍𝜇 ) } 12. Higgs–W and Z int.
w
(98)
The terms in the Lagrangian are easily associated to the Feynman diagrams of Figure 148:
1. this term describes the inertia of every object around us, yields the motion of fermi-
ons, and represents the kinetic energy of the quarks and leptons, as it appears in the
usual Dirac equation, modified by the so-called Yukawa coupling to the Higgs field
𝐻 and possibly by a Majorana term for the neutrinos (not shown);
254 8 the weak nuclear interaction

2. the second term describes the well-known interaction of matter and electromagnetic
radiation, and explains practically all material properties and colours observed in
daily life;
3. the term is the so-called charged weak current interaction, due to exchange of virtual
W bosons, that is responsible for the β decay and for the fact that the Sun is shining;
4. this term is the neutral weak current interaction, the ‘𝑉 − 𝐴 theory’ of George Sudar-
shan, that explains the elastic scattering of neutrinos in matter;
5. this term represents the kinetic energy of photons and yields the evolution of the
electromagnetic field in vacuum, thus the basic Maxwell equations;
6. this term represents the kinetic energy of the weak radiation field and gives the evolu-
tion of the intermediate W and Z bosons of the weak interaction;
7. this term is the kinetic energy of the vector bosons;
8. this term represents the triple vertex of the self-interaction of the vector boson;
9. this term represents the quadruple vertex of the self-interaction of the vector boson;
10. this term is the kinetic energy of Higgs boson;

Motion Mountain – The Adventure of Physics
11. this term is the self-interaction of the Higgs boson;
12. the last term is expected to represent the interaction of the vector bosons with the
Higgs boson that restore unitarity at high energies.
Let us look into the formal details. The quantities appearing in the Lagrangian are:
— The wave functions 𝜓𝑘 = (𝜈𝑘󸀠 𝑙𝑘− ) for leptons and (𝑢𝑘 𝑑󸀠𝑘 ) for quarks are the left-handed
fermion fields of the 𝑘-th fermion generation; every component is a spinor. The index
𝑘 = 1, 2, 3 numbers the generation: the value 1 corresponds to (u d 𝜈𝑒 𝑒− ), the second
generation is (c s 𝜈𝜇 𝜇− ) and the third (t b 𝜈𝜏 𝜏− ). The 𝜓𝑘 transform as doublets under
SU(2); the right handed fields are SU(2) singlets.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In the doublets, one has
𝑑󸀠𝑘 = ∑ 𝑉𝑘𝑙 𝑑𝑙 , (99)
𝑙

where 𝑉𝑘𝑙 is the Cabibbo–Kobayashi–Maskawa mixing matrix, 𝑑󸀠𝑘 are the quark fla-
vour eigenstates and 𝑑𝑘 are the quark mass eigenstates. A similar expression holds for
the mixing of the neutrinos:
𝜈𝑘󸀠 = ∑ 𝑃𝑘𝑙 𝜈𝑙 , (100)
𝑙

where 𝑃𝑘𝑙 is the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix, 𝜈𝑙󸀠 the neutrino
flavour eigenstates and 𝜈𝑙 the neutrino mass eigenstates.
— For radiation, 𝐴𝜇 and 𝐹𝜇𝜈 is the field of the massless vector boson of the electromag-
netic field, the photon γ.
𝑊𝜇± are the massive charged gauge vector bosons of the weak interaction; the cor-
responding particles, 𝑊+ and 𝑊− , are each other’s antiparticles.
𝑍𝜇 is the field of the massive neutral gauge vector boson of the weak interactions;
the neutral vector boson itself is usually called Z0 .
— 𝐻 is the field of the neutral scalar Higgs boson H0 , the only elementary scalar particle
in the standard model.
— Two charges appear, one for each interaction. The number 𝑞𝑘 is the well-known elec-
8 and the handedness of nature 255

tric charge of the particle 𝜓𝑘 in units of the positron charge. The number 𝑡3𝐿 (𝑘) is the
weak isospin, or weak charge, of fermion 𝑘, whose value is +1/2 for 𝑢𝑘 and 𝜈𝑘 and is
−1/2 for 𝑑𝑘 and 𝑙𝑘 . These two charges together define the so-called vector coupling

𝑔𝑉𝑘 = 𝑡3𝐿 (𝑘) − 2𝑞𝑘 sin2 𝜃𝑊 (101)

and the axial coupling
𝑔𝐴𝑘 = 𝑡3𝐿 (𝑘) . (102)

The combination 𝑔𝑉𝑘 − 𝑔𝐴𝑘 , or 𝑉 − 𝐴 for short, expresses the maximal violation of P
and C parity in the weak interaction.
— The operators 𝑇+ and 𝑇− are the weak isospin raising and lowering operators. Their
action on a field is given e.g. by 𝑇+ 𝑙𝑘− = 𝜈𝑘 and 𝑇− 𝑢𝑘 = 𝑑𝑘 .

We see that the Lagrangian indeed contains all the ideas developed above. The elec-

Motion Mountain – The Adventure of Physics
troweak Lagrangian is essentially unique: it could not have a different mathematical form,
because both the electromagnetic terms and the weak terms are fixed by the requirements
of Lorentz invariance, U(1) and broken SU(2) gauge invariance, permutation symmetry
and renormalizability.
The Lagrangian of the weak interaction has been checked and confirmed by thou-
sands of experiments. Many experiments have been designed specifically to probe it to
the highest precision possible. In all these cases, no contradictions between observation
Ref. 186 and theory has ever been found. Even though the last three terms of the Lagrangian are
not fully confirmed, this is – most probably – the exact Lagrangian of the weak interac-

Curiosities ab ou t the weak interaction
The weak interaction, with its breaking of parity and its elusive neutrino, exerts a deep
Ref. 209 fascination on all those who have explored it. Let us explore this fascination a bit more.
∗∗
The weak interaction is required to have an excess of matter over antimatter. Without the
parity violation of the weak interactions, there would be no matter at all in the universe,
because all matter and antimatter that appeared in the big bang would have annihilated.
The weak interaction prevents the symmetry between matter and antimatter, which is
required to have an excess of one over the other in the universe. In short, the parity
violation of the weak interaction is a necessary condition for our own existence.
∗∗
The weak interaction is also responsible for the heat produced inside the Earth. This
heat keeps the magma liquid. As a result, the weak interaction, despite its weakness, is
responsible for most earthquakes, tsunamis and volcanic eruptions.
∗∗
The Lagrangian of the weak interaction was clarified by Steven Weinberg, Sheldon
256 8 the weak nuclear interaction

Glashow and Abdus Salam. They received the 1979 Nobel Prize in physics for their work.
Abdus Salam (b. 1926 Santokdas, d. 1996 Oxford) was a physics genius, the greatest
Pakistani scientist by far, an example to many scientists across the world, the first muslim
science Nobel-Prize winner, and a deeply spiritual man. In his Nobel banquet speech he
explained: ‘This, in effect, is the faith of all physicists: the deeper we seek, the more is our
wonder excited, the more is the dazzlement for our gaze.’ Salam often connected his re-
search to the spiritual aspects of Islam. Once he was asked in Pakistani television why he
believed in unification of physics. He answered: ‘Because god is one!’ When the parlia-
ment of Pakistan, in one of the great injustices of the twentieth century, declared Ahmadi
Muslims to be non-Muslims and thus effectively started a religious persecution, Salam
left Pakistan and never returned. The religious persecution continues to this day: on his
tombstone in Pakistan, the word ‘muslim’ has been hammered away, and the internet
is full of offensive comments about him by other muslims, even on Wikipedia. Salam
was also an important science manager. With support of UNESCO, Salam founded the
International Centre for Theoretical Physics and the Third World Academy of Sciences,

Motion Mountain – The Adventure of Physics
both in Trieste, in Italy, and attracted there the best scientists from developing countries.
∗∗
β decay, due to the weak interaction, separates electrons and protons. Finally, in 2005,
people have proposed to use this effect to build long-life batteries that could be used in
satellites. Future will tell whether the proposals will be successful.
∗∗
16
Ref. 211 Every second around 10 neutrinos fly through our body. They have five sources:

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
— Solar neutrinos arrive on Earth at 6 ⋅ 1014 /m2 s, with an energy from 0 to 0.42 MeV;
they are due to the p-p reaction in the sun; a tiny is due to the 8 B reaction and has
energies up to 15 MeV.
— Atmospheric neutrinos are products of cosmic rays hitting the atmosphere, consist of
2/3 of muon neutrinos and one third of electron neutrinos, and have energies mainly
between 100 MeV and 5 GeV.
Page 184 — Earth neutrinos from the radioactivity that keeps the Earth warm form a flux of
6 ⋅ 1010 /m2 s.
— Fossil neutrinos from the big bang, with a temperature of 1.95 K are found in the
universe with a density of 300 cm−3 , corresponding to a flux of 1015 /m2 s.
— Man-made neutrinos are produced in nuclear reactors (at 4 MeV) and as neutrino
beams in accelerators, using pion and kaon decay. A standard nuclear plant produces
5 ⋅ 1020 neutrinos per second. Neutrino beams are produced, for example, at the CERN
in Geneva. They are routinely sent 700 km across the Earth to the Gran Sasso labor-
atory in central Italy, where they are detected. (In 2011, a famous measurement er-
ror led some people to believe, incorrectly, that these neutrinos travelled faster than
light.)
Neutrinos are mainly created in the atmosphere by cosmic radiation, but also com-
ing directly from the background radiation and from the centre of the Sun. Never-
theless, during our own life – around 3 thousand million seconds – we have only a
10 % chance that one of these neutrinos interacts with one of the 3 ⋅ 1027 atoms of
8 and the handedness of nature 257

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 149 A typical
underground
experiment to observe
neutrino oscillations:
the Sudbury Neutrino
Observatory in Canada
(© Sudbury Neutrino
Observatory)

our body. The reason is, as usual, that the weak interaction is felt only over distances
less than 10−17 m, about 1/100th of the diameter of a proton. The weak interaction is
indeed weak.
∗∗
Already in 1957, the great physicist Bruno Pontecorvo imagined that travelling neutri-
nos could spontaneously change into their own antiparticles. Today, it is known exper-
imentally that travelling neutrinos can change generation, and one speaks of neutrino
oscillations. Such experiments are carried out in large deep underground caves; the most
famous one is shown in Figure 149. In short, experiments show that the weak interaction
mixes neutrino types in the same way that it mixes quark types.
258 8 the weak nuclear interaction

∗∗
Only one type of particles interacts (almost) only weakly: neutrinos. Neutrinos carry no
electric charge, no colour charge and almost no gravitational charge (mass). To get an
impression of the weakness of the weak interaction, it is usually said that the probability
of a neutrino to be absorbed by a lead screen of the thickness of one light-year is less than
50 %. The universe is thus essentially empty for neutrinos. Is there room for bound states
of neutrinos circling masses? How large would such a bound state be? Can we imagine
bound states, which would be called neutrinium, of neutrinos and antineutrinos circling
each other? The answer depends on the mass of the neutrino. Bound states of massless
particles do not exist. They could and would decay into two free massless particles.*
Since neutrinos are massive, a neutrino–antineutrino bound state is possible in prin-
ciple. How large would it be? Does it have excited states? Can they ever be detected?
Challenge 149 ny These issues are still open.
The weak interaction is so weak that a neutrino–antineutrino annihilation – which is
only possible by producing a massive intermediate Z boson – has never been observed

Motion Mountain – The Adventure of Physics
up to this day.
∗∗
Exploring the mixing of the weak and the electromagnetic interaction led to the predic-
tion of the Higgs boson. The fascination of the Higgs boson is underlined by the fact that
it is the only fundamental particle that bears the name of a physicist. By the way, the pa-
Ref. 210 per by Peter Higgs on the boson named after him is only 79 lines long, and has only five
equations.
∗∗

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In the years 1993 and 1994 an intense marketing campaign was carried out across the
United States of America by numerous particle physicists. They sought funding for the
‘superconducting supercollider’, a particle accelerator with a circumference of 80 km.
This should have been the largest machine ever built, with a planned cost of more than
twelve thousand million dollars, aiming at finding the Higgs boson before the Europeans
would do so at a fraction of that cost. The central argument brought forward was the fol-
lowing: since the Higgs boson is the basis of particle masses, it was central to US science
to know about it first. Apart from the issue of the relevance of the conclusion, the worst
is that the premise is wrong.
Page 233 We have seen above that 99 % of the mass of protons, and thus of the universe, is due
to quark confinement; this part of mass appears even if the quarks are approximated as
massless. The Higgs boson is not responsible for the origin of mass itself; it just might
Ref. 195 shed some light on the issue. In particular, the Higgs boson does not allow calculating
or understanding the mass of any particle. The whole campaign was a classic case of
disinformation, and many people involved have shown their lack of honesty.** In the
end, the project was stopped, mainly for financial reasons.
* In particular, this is valid for photons bound by gravitation; this state is not possible.
** We should not be hypocrites. The supercollider lie is negligible when compared to other lies. The biggest
lie in the world is probably the one that states that to ensure its survival, the USA government need to spend
more on the military than all other countries in the world combined. This lie is, every single year, around 40
times as big as the once-only supercollider lie. Many other governments devote even larger percentages of
8 and the handedness of nature 259

“ ”
Difficile est saturam non scribere.*
Juvenal, Saturae 1, 30.

∗∗
There is no generally accepted name for the quantum field theory of the weak interaction.
The expression quantum asthenodynamics (QAD) – from the Greek word for ‘weak’ – has
not been universally adopted.
∗∗
Do ruminating cows move their jaws equally often in clockwise and anticlockwise direc-
tion? In 1927, the theoretical physicists Pascual Jordan and Ralph de Laer Kronig pub-
Ref. 212 lished a study showing that in Denmark the two directions are almost equally distributed.
The rumination direction of cows is thus not related to the weak interaction.
∗∗

Motion Mountain – The Adventure of Physics
Of course, the weak interaction is responsible for radioactive β decay, and thus for part
of the radiation background that leads to mutations and thus to biological evolution.
∗∗
Due to the large toll it placed on society, research in nuclear physics, has almost disap-
peared from the planet, like poliomyelitis has. Like poliomyelitis, nuclear research is kept
alive only in a few highly guarded laboratories around the world, mostly by questionable
figures, in order to build dangerous weapons. Only a small number of experiments car-
ried on by a few researchers are able to avoid this involvement and continue to advance

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the topic.
∗∗
Interesting aspects of nuclear physics appear when powerful lasers are used. In 1999, a
British team led by Ken Ledingham observed laser induced uranium fission in 238 U nuc-
lei. In the meantime, this has even be achieved with table-top lasers. The latest feat, in
2003, was the transmutation of 129 I to 128 I with a laser. This was achieved by focussing
Ref. 213 a 360 J laser pulse onto a gold foil; the ensuing plasma accelerates electrons to relativ-
istic speed, which hit the gold and produce high energy γ rays that can be used for the
transmutation.

A summary of the weak interaction
The weak interaction is described by a non-Abelian gauge theory based on a broken
SU(2) gauge group for weak processes. The weak interaction mixes with the unbroken
U(1) gauge group of the electrodynamic interaction. This description matches the ob-
served properties of β decay, of particle transformations, of neutrinos and their mixing,
of the massive intermediate W and Z bosons, of maximal parity violation, of the heat

their gross national product to their own version of this lie. As a result, the defence spending lie is directly
responsible for most of the poverty in all the countries that use it.
* ‘It is hard not to be satirical.’
260 8 the weak nuclear interaction

production inside the Earth, of several important reactions in the Sun and of the origin
of matter in the universe. Even though the weak interaction is weak, it is a bit everywhere.
The weak interaction is described by a Lagrangian. After a century of intense research,
the Lagrangian is known in all its details, including, since 2012, the Higgs boson. Theory
and experiment agree whenever comparisons have been made.
All remaining limitations of the gauge theory of the weak interaction are only con-
ceptual. Like in all of quantum field theory, also in the case of the weak interaction the
mathematical form of the Lagrangian is almost uniquely defined by requiring renormal-
izability, Lorentz invariance, and (broken) gauge invariance – SU(2) in this case. We say
Page 238 again ‘almost’, as we did for the case of the strong interaction, because the Lagrangian
of the weak and electromagnetic interactions contains a few parameters that remain un-
explained:
— The two coupling constants 𝑔 and 𝑔󸀠 of the weak and the electromagnetic interaction
are unexplained. (They define weak mixing angle 𝜃w = arctan(𝑔󸀠 /𝑔).)
— The mass 𝑀𝑍 = 91 GeV/𝑐2 of the neutral Z boson is unexplained.

Motion Mountain – The Adventure of Physics
— The number 𝑛 = 3 of generations is unexplained.
— The masses of the six leptons and the six quarks are unexplained.
— The four parameters of the Cabibbo–Kobayashi–Maskawa quark mixing matrix and
the six parameters of the neutrino mixing matrix are unexplained, including the re-
spective CP violating phases.
— The properties of space-time, in particular its Lorentz invariance, its continuity and
the number of its dimensions are assumed from the outset and are obviously all un-
explained.
— It is also not known how the weak interaction behaves in strong gravity, thus in

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
strongly curved space-time.
Before exploring how to overcome these limitations, we summarize all results found so
far in the so-called standard model of particle physics.
Chapter 9

T H E STA N DA R D MODE L OF PA RT IC L E
PH YSIC S – A S SE E N ON T E L EV I SION

T
he expression standard model of elementary particle physics stands for
he summary of all knowledge about the motion of quantum particles in nature.
he standard model can be explained in four tables: the table of the elementary
particles, the table of their properties, the table of possible Feynman diagrams and the

Motion Mountain – The Adventure of Physics
table of fundamental constants.
The following table lists the known elementary particles found in nature.

TA B L E 21 The elementary particles.

Radiation electromagnetic weak strong
γ 𝑊 +
, 𝑊 −
𝑔1 ... 𝑔8

𝑍0

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
photon weak bosons gluons

Radiation particles are intermediate vector bosons, thus
with spin 1. 𝑊− is the antiparticle of 𝑊+ ; the photons
and the 𝑍0 are their own antiparticles. Only the 𝑊± and
𝑍0 are massive.

Matter generation 1 generation 2 generation 3
Leptons 𝑒 𝜇 𝜏
𝜈𝑒 𝜈𝜇 𝜈𝜏

Quarks 𝑑 𝑠 𝑡
(each in three colours) 𝑢 𝑐 𝑏

Matter particles are fermions with spin 1/2; all have a cor-
responding antiparticle. Leptons mix among themselves;
so do quarks. All fermions are massive.

Vacuum state
Higgs boson 𝐻 Has spin 0 and a mass of 126 GeV.
262 9 the standard model – as seen on television

The table has not changed much since the mid-1970s, except for the Higgs boson,
which has been found in 2012. Assuming that the table is complete, it contains all con-
stituents that make up all matter and all radiation in nature. Thus the table lists all con-
stituents – the real ‘uncuttables’ or ‘atoms’, as the Greek called them – of material ob-
jects and beams of radiation. The elementary particles are the basis for materials science,
geology, astronomy, engineering, chemistry, biology, medicine, the neurosciences and
psychology. For this reason, the table regularly features in mass tabloids, on television
and on the internet.
The full list of elementary particles allows us to put together a full table of particle
properties, shown in Table 22. It lists all properties of the elementary particles. To save
space, colour and weak isospin are not mentioned explicitly. Also the decay modes of the
Ref. 186 unstable particles are not given in detail; they are found in the standard references.
The Table 22 on particle properties is fascinating. It allows us to give a complete char-
acterization of the intrinsic properties of any composed object or image. At the beginning
Vol. I, page 29 of our study of motion, we were looking for a complete list of the permanent, intrinsic

Motion Mountain – The Adventure of Physics
properties of moving entities. Now we have it.

TA B L E 22 Elementary particle properties.

Elementary radiation (bosons)
photon γ 0 (< 2 ⋅ 10−54 kg) stable 𝐼(𝐽𝑃𝐶 ) = 000000 0, 0
0, 1(1−− )
𝑊± 80.385(15) GeV/𝑐2 2.085(42) GeV 𝐽 = 1 ±100000 0, 0
67.60(27) % hadrons,
32.40(27) % 𝑙+ 𝜈
𝑍 91.1876(21) GeV/𝑐2 0.265(1) ys 𝐽=1 000000 0, 0
= 2.4952(23) GeV/𝑐2
69.91(6) % hadrons
10.0974(69) % 𝑙+ 𝑙−
gluon 0 stable 𝐼(𝐽𝑃 ) = 0(1− ) 000000 0, 0
Elementary matter (fermions): leptons
electron 𝑒 9.109 382 91(40) ⋅ > 13 ⋅ 1030 s 𝐽 = 12 −100 000 1, 0
−31 2
10 kg = 81.871 0506(36) pJ/𝑐
= 0.510 998 928(11) MeV/𝑐2 = 0.000 548 579 909 46(22) u
gyromagnetic ratio 𝜇𝑒 /𝜇B = −1.001 159 652 180 76(27)
electric dipole moment 𝑒 𝑑 =< 0.87 ⋅ 10−30 𝑒 m
9 the standard model – as seen on television 263

Particle Mass 𝑚 𝑎 Lifetime 𝜏 or Isospin 𝐼, Charge, Lepton
energy spin 𝐽, 𝑐 isospin, &
𝑏
width, main parity 𝑃, strange- baryon
decay modes charge ness, 𝑐 num-
parity 𝐶 charm, bers
beauty, 𝐿𝐵
topness:
𝑄𝐼𝑆𝐶𝐵𝑇
muon 𝜇 0.188 353 109(16) yg 2.196 9811(22) μs 𝐽 = 12 −100000 1, 0
−
99 % 𝑒 𝜈𝑒̄ 𝜈𝜇
= 105.658 3715(35) MeV/𝑐2 = 0.113 428 9267(29) u
gyromagnetic ratio 𝜇𝜇 /(𝑒ℏ/2𝑚𝜇 ) = −1.001 165 9209(6)
electric dipole moment 𝑑 = (−0.1 ± 0.9) ⋅ 10−21 𝑒 m
1
tau 𝜏 1.776 82(16) GeV/𝑐2 290.6(1.0) fs 𝐽= 2
−100000 1, 0

Motion Mountain – The Adventure of Physics
1
el. neutrino < 2 eV/𝑐2 𝐽= 2
1, 0
𝜈e
1
muon < 2 eV/𝑐2 𝐽= 2
1, 0
neutrino 𝜈𝜇
1
tau neutrino < 2 eV/𝑐2 𝐽= 2
1, 0
𝜈𝜏
Elementary matter (fermions): quarks 𝑓
+
up 𝑢 1.8 to 3.0 MeV/𝑐2 see proton 𝐼(𝐽𝑃 ) = 12 ( 21 ) + 23 + 12 0000 0, 13

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
+
down 𝑑 4.5 to 5.5 MeV/𝑐2 see proton 𝐼(𝐽𝑃 ) = 12 ( 21 ) − 13 − 12 0000 0, 13
+
strange 𝑠 95(5) MeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) − 13 0−1000 0, 13
+
charm 𝑐 1.275(25) GeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) + 23 00+100 0, 13
+
bottom 𝑏 4.18(17) GeV/𝑐2 𝜏 = 1.33(11) ps 𝐼(𝐽𝑃 ) = 0( 12 ) − 13 000−10 0, 13
+
top 𝑡 173.5(1.4) GeV/𝑐2 𝐼(𝐽𝑃 ) = 0( 12 ) + 23 0000+1 0, 13
Elementary boson
Higgs H 126(1) GeV/𝑐2 not measured 𝐽=0

Notes:
𝑎. See also the table of SI prefixes on page 326. About the eV/𝑐2 mass unit, see page 330.
𝑏. The energy width Γ of a particle is related to its lifetime 𝜏 by the indeterminacy relation Γ𝜏 = ℏ.
There is a difference between the half-life 𝑡1/2 and the lifetime 𝜏 of a particle: they are related by
𝑡1/2 = 𝜏 ln 2, where ln 2 ≈ 0.693 147 18; the half-life is thus shorter than the lifetime. The unified
atomic mass unit u is defined as 1/12 of the mass of a carbon 12 atom at rest and in its ground
1
state. One has 1 u = 12 𝑚(12 C) = 1.660 5402(10) yg.
𝑐. To keep the table short, the header does not explicitly mention colour, the charge of the strong
interactions. This has to be added to the list of basic object properties. Quantum numbers con-
taining the word ‘parity’ are multiplicative; all others are additive. Time parity 𝑇 (not to be con-
fused with topness 𝑇), better called motion inversion parity, is equal to CP. The isospin 𝐼 (or 𝐼Z ) is
defined only for up and down quarks and their composites, such as the proton and the neutron.
In the literature one also sees references to the so-called 𝐺-parity, defined as 𝐺 = (−1)𝐼𝐶 .
264 9 the standard model – as seen on television

The table also does not mention the weak charge of the particles. The details on weak charge
𝑔, or, more precisely, on the weak isospin, a quantum number assigned to all left-handed fer-
mions (and right-handed anti-fermions), but to no right-handed fermion (and no left-handed
Page 245 antifermion), are given in the section on the weak interactions.
𝑑. ‘Beauty’ is now commonly called bottomness; similarly, ‘truth’ is now commonly called top-
ness. The signs of the quantum numbers 𝑆, 𝐼, 𝐶, 𝐵, 𝑇 can be defined in different ways. In the
standard assignment shown here, the sign of each of the non-vanishing quantum numbers is
given by the sign of the charge of the corresponding quark.
Ref. 214, Ref. 215 𝑒. The electron radius is observed to be less than 10−22 m. It is possible to store single electrons
in traps for many months.
𝑓. See page 233 for the precise definition and meaning of the quark masses.

The other aim that we formulated at the beginning of our adventure was to find the
complete list of all state properties. This aim is also achieved, namely by the wave function
and the field values due to the various bosons. Were it not for the possibility of space-time
curvature, we would be at the end of our exploration.

Motion Mountain – The Adventure of Physics
The main ingredient of the standard model are the Lagrangians of the electromag-
netic, the weak and the strong interactions. The combination of the Lagrangians, based
on the U(1), SU(3) and broken SU(2) gauge groups, is possible only in one specific way.
The Lagrangian can be summarized by the Feynman diagram of Figure 150.
To complete the standard model, we need the coupling constants of the three gauge
interactions, the masses of all the particles, and the values of the mixing among quarks
and among leptons. Together with all those constants of nature that define the SI system
and the number of space-time dimensions, the following table therefore completes the
standard model.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
TA B L E 23 Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Constants that define the SI measurement units
Vacuum speed of light𝑐 𝑐 299 792 458 m/s 0
𝑐 −7
Vacuum permeability 𝜇0 4π ⋅ 10 H/m 0
= 1.256 637 061 435 ... μH/m0
Vacuum permittivity𝑐 𝜀0 = 1/𝜇0 𝑐2 8.854 187 817 620 ... pF/m 0
Original Planck constant ℎ 6.626 069 57(52) ⋅ 10−34 Js 4.4 ⋅ 10−8
Reduced Planck constant, ℏ 1.054 571 726(47) ⋅ 10−34 Js 4.4 ⋅ 10−8
quantum of action
Positron charge 𝑒 0.160 217 656 5(35) aC 2.2 ⋅ 10−8
Boltzmann constant 𝑘 1.380 6488(13) ⋅ 10 J/K 9.1 ⋅ 10−7
−23

Gravitational constant 𝐺 6.673 84(80) ⋅ 10−11 Nm2 /kg2 1.2 ⋅ 10−4
4
Gravitational coupling constant𝜅 = 8π𝐺/𝑐 2.076 50(25) ⋅ 10−43 s2 /kg m 1.2 ⋅ 10−4
Fundamental constants (of unknown origin)
Number of space-time dimensions 3+1 0𝑏
2
Fine-structure constant𝑑 or 𝛼 = 4π𝜀𝑒 ℏ𝑐 1/137.035 999 074(44) 3.2 ⋅ 10−10
0

e.m. coupling constant = 𝑔em (𝑚2e 𝑐2 ) = 0.007 297 352 5698(24) 3.2 ⋅ 10−10
9 the standard model – as seen on television 265

TA B L E 23 (Continued) Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Fermi coupling constant𝑑 or 𝐺F /(ℏ𝑐)3 1.166 364(5) ⋅ 10−5 GeV−2 4.3 ⋅ 10−6
2
weak coupling constant 𝛼w (𝑀Z ) = 𝑔w /4π 1/30.1(3) 1 ⋅ 10−2
Weak mixing angle sin2 𝜃W (𝑀𝑆) 0.231 24(24) 1.0 ⋅ 10−3
sin2 𝜃W (on shell)0.2224(19) 8.7 ⋅ 10−3
= 1 − (𝑚W /𝑚Z )2
Strong coupling constant𝑑 𝛼s (𝑀Z ) = 𝑔s2 /4π 0.118(3) 25 ⋅ 10−3
0.97428(15) 0.2253(7) 0.00347(16)
CKM quark mixing matrix |𝑉| ( 0.2252(7) 0.97345(16) 0.0410(11) )
0.00862(26) 0.0403(11) 0.999152(45)
Jarlskog invariant 𝐽 2.96(20) ⋅ 10−5
0.82 0.55 −0.15 + 0.038𝑖

Motion Mountain – The Adventure of Physics
PMNS neutrino mixing m. 𝑃 (−0.36 + 0.020𝑖 0.70 + 0.013𝑖 0.61 )
0.44 + 0.026𝑖 −0.45 + 0.017𝑖 0.77
Particle masses: see previous table

𝑎. Uncertainty: standard deviation of measurement errors.
𝑏. Only measured from to 10−19 m to 1026 m.
𝑐. Defining constant.
𝑑. All coupling constants depend on the 4-momentum transfer, as explained in the section on
Page 131 renormalization. Fine-structure constant is the traditional name for the electromagnetic coup-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ling constant 𝑔em in the case of a 4-momentum transfer of 𝑄2 = 𝑚2e 𝑐2 , which is the smallest
one possible. At higher momentum transfers it has larger values, e.g., 𝑔em (𝑄2 = 𝑀W
2 2
𝑐 ) ≈ 1/128.
In contrast, the strong coupling constant has lower values at higher momentum transfers; e.g.,
𝛼s (34 GeV) = 0.14(2).

In short, with the three tables and the figure, the standard model describes every ob-
servation ever made in flat space-time. In particular, the standard model includes a min-
imum action, a maximum speed, electric charge quantization and the least action prin-
ciple.

Summary and open questions
The standard model of particle physics clearly distinguishes elementary from composed
particles. The standard model provides the full list of properties that characterizes a
particle – and thus any moving object and image. These properties are: mass, spin, charge,
colour, weak isospin, parity, charge parity, isospin, strangeness, charm, topness, beauty,
lepton number and baryon number.
The standard model describes electromagnetic and nuclear interactions as as ex-
changes of virtual radiation particles. In particular, the standard model describes the
three types of radiation that are observed in nature with full precision, using gauge
groups. The standard model is based on quantization and conservation of electric charge,
weak charge and colour, as well as on a smallest action value ℏ and a maximum energy
speed 𝑐. As a result, the standard model describes the structure of the atoms, their form-
266 9 the standard model – as seen on television

QED, describing the electromagnetic interaction
charged photon
particle

charged particle
QCD, describing the strong nuclear interaction
quark gluon gluon gluon gluon gluon

Motion Mountain – The Adventure of Physics
quark gluon gluon gluon

QAD, describing the weak nuclear interaction
q’ and l’ indicate quark and lepton mixing.
q‘ W or Z W Z or γ W Z or γ or Z

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
q‘ W W Z or γ or γ
l‘ Z ν W W W

l‘ l W W
Higgs couplings
fermion
H H H H H H H
or boson

fermion or boson W or Z W or Z H H H

F I G U R E 150 The Feynman diagrams of the standard model.
9 the standard model – as seen on television 267

ation in the history of the universe, the properties of matter and the mechanisms of life.
Despite the prospect of fame and riches, not one deviation between experiment and the
standard model has been found.
In short, the standard model realizes the dream of Leucippus and Democritus, plus a
bit more: we know the bricks that compose all of matter and radiation, and in addition we
know how they move, interact and transform, in flat space-time, with perfect accuracy.
Despite this perfect accuracy, we also know what we still do not know:
— We do not know the origin of the coupling constants.
— We do not know why positrons and protons have the same charge.
— We do not know the origin of the masses of the particles.
— We do not know the origin of the mixing and CP violation parameters.
— We do not know the origin of the gauge groups.
— We do not know the origin of the three particle generations.
— We do not know whether the particle concept survives at high energy.
— We do not know what happens in curved space-time.

Motion Mountain – The Adventure of Physics
To study these issues, the simplest way is to explore nature at particle energies that are
as high as possible. There are two methods: building large experiments and exploring
hypothetical models. Both are useful.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
C h a p t e r 10

DR E A M S OF U N I F IC AT ION

“ ”
Materie ist geronnenes Licht.**
Albertus Magnus

I
s there a common origin to the three particle interactions? We have seen

Motion Mountain – The Adventure of Physics
n the preceding chapters that the Lagrangian densities of the three gauge
nteractions are determined almost uniquely by two types of requirements: to possess
a certain gauge symmetry, and to be consistent with space-time, through Lorentz invari-
ance and renormalizability. The search for unification of the interactions thus seems to
Challenge 150 s require the identification of the one, unified symmetry of nature. (Do you agree?)
Between 1970 and 2015, several conjectures have fuelled the hope to achieve unific-
ation through higher symmetry. The most popular were grand unification, supersym-
metry, conformal field theory, coupling constant duality and mathematical quantum field
theory. We give a short summary of these efforts; we start with the first candidate, which
is conceptually the simplest.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Grand unification
At all measured energies up to the year 2016, thus below about 3 TeV, there are no con-
tradictions between the Lagrangian of the standard model and observation. On the other
hand, the Lagrangian itself can be conjectured to be a low energy approximation to the
unified theory. It should thus be possible – attention, this a belief – to find a unifying
symmetry that contains the symmetries of the electroweak and strong interactions as
subgroups. In this way, the three gauge interactions would be different aspects of a single,
‘unified’ interaction. We can then examine the physical properties that follow from this
unifying symmetry and compare them with observation. This approach, called grand
unification, attempts the unified description of all types of matter. All known elementary
particles are seen as fields which appear in a Lagrangian determined by a single gauge
symmetry group.
Like for each gauge theory described so far, also the grand unified Lagrangian is fixed
by the symmetry group, the representation assignments for each particle, and the cor-
responding coupling constant. A general search for the grand symmetry group starts
Ref. 216 with all those (semisimple) Lie groups which contain 𝑈(1) × 𝑆𝑈(2) × 𝑆𝑈(3). The smallest
groups with these properties are SU(5), SO(10) and E(6); they are defined in Appendix C.

** ‘Matter is coagulated light.’ Albertus Magnus (b. c. 1192 Lauingen, d. 1280 Cologne) was the most im-
portant thinker of his time.
10 dreams of unification 269

For each of these candidate groups, the predicted consequences of the model can be stud-
Ref. 217 ied and compared with experiment.

C omparing predictions and data
Grand unification models – also incorrectly called GUTs or grand unified theories – make
several predictions that can be matched with experiment. First of all, any grand unified
model predicts relations between the quantum numbers of quarks and those of leptons.
In particular, grand unification explains why the electron charge is exactly the opposite
of the proton charge.
Grand unification models predict a value for the weak mixing angle 𝜃w ; this angle is
Ref. 216 not fixed by the standard model. The most frequently predicted value,

sin2 𝜃w,th = 0.2 (103)

Motion Mountain – The Adventure of Physics
is close to the measured value of

sin2 𝜃w,ex = 0.231(1) , (104)

which is not a good match, but might be correct, in view of the approximations in the
prediction.
All grand unified models predict the existence of magnetic monopoles, as was shown
Ref. 218 by Gerard ’t Hooft. However, despite extensive searches, no such particles have been
found yet. Monopoles are important even if there is only one of them in the whole uni-
Ref. 219 verse: the existence of a single monopole would imply that electric charge is quantized.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
If monopoles were found, grand unification would explain why electric charge appears
in multiples of a smallest unit.
Grand unification predicts the existence of heavy intermediate vector bosons, called
X bosons. Interactions involving these bosons do not conserve baryon or lepton number,
but only the difference 𝐵 − 𝐿 between baryon and lepton number. To be consistent with
data, the X bosons must have a mass of the order of 1016 GeV. However, this mass is and
always will be outside the range of experiments, so that the prediction cannot be tested
directly.
Most spectacularly, the X bosons of grand unification imply that the proton decays.
This prediction was first made by Pati and Abdus Salam in 1974. If protons decay, means
that neither coal nor diamond* – nor any other material – would be for ever. Depending
on the precise symmetry group, grand unification predicts that protons decay into pions,
electrons, kaons or other particles. Obviously, we know ‘in our bones’ that the proton
lifetime is rather high, otherwise we would die of leukaemia; in other words, the low
level of cancer in the world already implies that the lifetime of the proton is larger than
about 1016 years.
Detailed calculations for the proton lifetime 𝜏𝑝 using the gauge group SU(5) yield the

* As is well known, diamond is not stable, but metastable; thus diamonds are not for ever, but coal might
be, as long as protons do not decay.
270 10 dreams of unification

Ref. 216 expression
1 𝑀X4
𝜏p ≈ 2 5
≈ 1031±1 a (105)
𝛼𝐺 (𝑀X ) 𝑀p

where the uncertainty is due to the uncertainty of the mass 𝑀X of the gauge bosons
involved and to the exact decay mechanism. Several large experiments have tried and
are still trying to measure this lifetime. So far, the result is simple but clear. Not a single
Ref. 220 proton decay has ever been observed. The data can be summarized by

𝜏(p → 𝑒+ π0 ) > 5 ⋅ 1033 a
𝜏(p → 𝐾+ 𝜈)̄ > 1.6 ⋅ 1033 a
𝜏(n → 𝑒+ π− ) > 5 ⋅ 1033 a
𝜏(n → 𝐾0 𝜈)̄ > 1.7 ⋅ 1032 a (106)

Motion Mountain – The Adventure of Physics
These values are higher than the prediction by SU(5) – and SO(10) – models. For other
gauge group candidates proton decay measurements require more time.

The state of grand unification
To settle the issue of grand unification definitively, one last prediction of grand unific-
ation remains to be checked: the unification of the coupling constants. Most estimates
of the grand unification energy are near the Planck energy, the energy at which gravit-
ation starts to play a role even between elementary particles. As grand unification does

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
not take gravity into account, for a long time there was a doubt whether something was
lacking in the approach. This doubt changed into certainty when the precision measure-
Ref. 221 ments of the coupling constants became available, in 1991, and were put into the diagram
of Figure 151. The GUT prediction of the way the constants evolve with energy implies
that the three constants do not meet at one energy. Simple grand unification by SU(5),
SU(10) or E6 is thus ruled out by experiment.
This state of affairs is changed if supersymmetry is taken into account. Supersymmetry
is a conjecture on the way to take into account low-energy effects of gravitation in the
Page 271 particle world. Supersymmetry conjectures new elementary particles that change the
curves at intermediate energies, so that they all meet at a grand unification energy of
about 1016 GeV. (The line thicknesses in Figure 151 represent the experimental errors.)
The inclusion of supersymmetry also puts the proton lifetime prediction back to a value
higher (but not by much) than the present experimental bound and predicts the correct
value of the mixing angle. With supersymmetry, one can thus retain the advantages of
grand unification (charge quantization, one coupling constant) without being in contra-
diction with experiments.
In summary, pure grand unification is in contradiction with experiments. This is not
a surprise, as its goal, to unify the description of matter, cannot been achieved in this
way. Indeed, the unifying gauge group must be introduced, i.e., added, at the very begin-
ning. Adding the group is necessary because grand unification cannot deduce the gauge
group from a general principle. Neither does pure grand unification tell us completely
10 dreams of unification 271

1/α i 1/α i
60 60
1/α1 SM 1/α1 MSSM
50 50

40 40
1/α2
1/α2
30 30

20 20
1/α3
1/α3
10 10

0 0
0 5 10 15 0 5 10 15
10 10 10 10 Q/GeV 10 10 10 10 Q/GeV

Motion Mountain – The Adventure of Physics
F I G U R E 151 The behaviour of the three coupling constants with energy, for simple grand uniﬁcation
(left) and for the minimal supersymmetric model (right); the graph shows the constants
𝛼1 = 53 𝛼em / cos2 𝜃W for the electromagnetic interaction (the factor 5/3 appears in GUTs),
𝛼2 = 𝛼em / sin2 𝜃W for the weak interaction and the strong coupling constant 𝛼3 = 𝛼s (© W. de Boer).

which elementary particles exist in nature. In other words, grand unification only shifts
the open questions of the standard model to the next level, while keeping most of the
open questions unanswered. The name ‘grand unification’ was wrong from the begin-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ning. We definitively need to continue our adventure.

Searching for higher symmetries
Since we want to reach the top of Motion Mountain, we go on. We have seen in the
preceding sections that the main ingredients of the Lagrangian that describes motion are
the symmetry properties. We recall that a Lagrangian is just the mathematical name for
the concept that measures change. The discovery of the correct symmetry, together with
mathematical consistency, usually restricts the possible choices for a Lagrangian down
to a limited number, and to one in the best case. This Lagrangian then allows making
experimental predictions.
The history of particle physics from 1920 to 1965 has shown that progress was always
coupled to the discovery of larger symmetries, in the sense that the newly discovered
symmetries always included the old ones as a subgroup. Therefore, in the twentieth cen-
tury, researchers searched for the largest possible symmetry that is consistent with exper-
iments on the one hand and with gauge theories on the other hand. Since grand unific-
ation failed, a better approach is to search directly for a symmetry that includes gravity.

Supersymmetry
In the search for possible symmetries of a Lagrangian describing a gauge theory and
gravity, one way to proceed is to find general mathematical theorems which restrict the
Ref. 222 symmetries that a Lagrangian can possibly have.
272 10 dreams of unification

Ref. 223 A well-known theorem by Coleman and Mandula states that if the symmetry trans-
formations transform fermions into fermions and bosons into bosons, no quantities
other than the following can be conserved:
the energy momentum tensor 𝑇𝜇𝜈 , a consequence of the external Poincaré space-
time symmetry, and
the internal quantum numbers, all scalars, associated with each gauge group gener-
ator – such as electric charge, colour, etc. – and consequences of the internal symmetries
of the Lagrangian.
But, and here comes a way out, if transformations that mix fermions and bosons are con-
sidered, new conserved quantities become possible. This family of symmetries includes
gravity and came to be known as supersymmetry. Its conserved quantities are not scalars
but spinors. Therefore, classical supersymmetry does not exist; it is a purely quantum-
mechanical symmetry. The study of supersymmetry has been a vast research field. For
example, supersymmetry generalizes gauge theory to super-gauge theory. The possible

Motion Mountain – The Adventure of Physics
super-gauge groups have been completely classified.
Supersymmetry can be extended to incorporate gravitation by changing it into a local
gauge theory; in that case one speaks of supergravity. Supergravity is based on the idea
that coordinates can be fermionic as well as bosonic. Supergravity thus makes specific
statements on the behaviour of space-time at small distances. Supergravity predicts 𝑁
additional conserved, spinorial charges. The number 𝑁 lies between 1 and 8; each value
leads to a different candidate Lagrangian. The simplest case is called 𝑁 = 1 supergravity.
In short, supersymmetry is a conjecture to unify matter and radiation at low energies.
Many researchers conjectured that supersymmetry, and in particular 𝑁 = 1 supergravity,
might be an approximation to reality.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Supersymmetric models have around 100 fundamental constants, in comparison to
around 25 for the standard model of particle physics. The precise experimental predic-
tions depend on the values of these constants. Nevertheless, a number of general predic-
tions are possible and can be tested by experiment.
— Supersymmetry predicts partners to the usual elementary particles, called sparticles.
Fermions are predicted to have boson partners, and vice versa. For example, super-
symmetry predicts a photino as fermionic partner of the photon, gluinos as partner of
the gluons, a selectron as partner of the electron, etc. However, none of these particles
have been observed yet.
— Supersymmetry allows for the unification of the coupling constants in a way compat-
Page 270 ible with the data, as shown already above.
— Supersymmetry slows down the proton decay rates predicted by grand unified the-
ories. The slowed-down rates are compatible with observation.
— Supersymmetry predicts electric dipole moments for the neutron and other element-
ary particles. The largest predicted values for the neutron, 10−30 𝑒 m, are in contra-
diction with observations; the smallest predictions have not yet been reached by ex-
periment. In comparison, the values expected from the standard model are at most
10−33 𝑒 m. This is a vibrant experimental research field that can save tax payers from
financing an additional large particle accelerator.
However, up to the time of this writing, the year 2016, there was no experimental evid-
10 dreams of unification 273

ence for supersymmetry. In particular, the Large Hadron Collider at CERN in Geneva
has not found any hint of supersymmetry. In fact, experiments excluded almost all su-
persymmetric particle models proposed in the past.
Is supersymmetry an ingredient of the unified theory? The safe answer is: this is un-
clear. The optimistic answer is: there is still a small chance that supersymmetry can hold
in nature. The pessimistic answer is: supersymmetry is a belief system contradicting ob-
servations and made up to correct the failings of grand unified theories. The last volume
of this adventure will tell which answer is correct.

Other at tempts
If supersymmetry is not successful, it might be that even higher symmetries are required
for unification. Therefore, researchers have explored quantum groups, non-commutative
space-time, conformal symmetry, topological quantum field theory, and other abstract
symmetries. None of these approaches led to useful results; neither experimental pre-
dictions nor progress towards unification. But two further approaches deserve special

Motion Mountain – The Adventure of Physics
mention: duality symmetries and extensions to renormalization.

D ualities – the most incredible symmetries of nature
An important discovery of mathematical physics took place in 1977, when Claus Mon-
tonen and David Olive proved that the standard concept of symmetry could be expanded
dramatically in a different and new way.
The standard class of symmetry transformations, which turns out to be only the first
class, acts on fields. This class encompasses gauge symmetries, space-time symmetries,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
motion reversal, parities, flavour symmetries and supersymmetry.
The second, new class is quite different. If we take the coupling constants of nature,
we can imagine that they are members of a continuous space of all possible coupling
constants, called the parameter space.* Montonen and Olive showed that there are trans-
formations in parameter space that leave nature invariant. These transformations thus
form a new, second class of symmetries of nature.
In fact, we already encountered a member of this class: renormalization symmetry.
But Olive and Montonen expanded the symmetry class considerably by the discovery of
Vol. III, page 91 electromagnetic duality. Electromagnetic duality is a discrete symmetry exchanging

4πℏ𝑐
𝑒↔ (107)
𝑒
where the right hand side turns out to be the unit of magnetic charge. Electro-magnetic
duality thus relates the electric charge 𝑒 and the magnetic charge 𝑚

𝑄el = 𝑚𝑒 and 𝑄mag = 𝑛𝑔 = 2πℏ𝑐/𝑒 (108)

* The space of solutions for all value of the parameters is called the moduli space.
274 10 dreams of unification

and puts them on equal footing. In other words, the transformation exchanges

1 1
𝛼↔ or ↔ 137.04 , (109)
𝛼 137.04
and thus exchanges weak and strong coupling. In other words, electromagnetic duality
relates a regime where particles make sense (the low coupling regime) with one where
particles do not make sense (the strong coupling regime). It is the most mind-boggling
symmetry ever conceived.
Dualities are among the deepest connections of physics. They contain ℏ and are
thus intrinsically quantum. They do not exist in classical physics and thus confirm that
quantum theory is more fundamental than classical physics. More clearly stated, dual-
ities are intrinsically non-classical symmetries. Dualities confirm that quantum theory
stands on its own.
If one wants to understand the values of unexplained parameters such as coupling

Motion Mountain – The Adventure of Physics
constants, an obvious thing to do is to study all possible symmetries in parameter space,
thus all possible symmetries of the second class, possibly combining them with those
of the first symmetry class. Indeed, the combination of duality with supersymmetry is
Vol. VI, page 141 studied in superstring theory.
These investigations showed that there are several types of dualities, which all are non-
Ref. 224 perturbative symmetries:
— S duality, the generalization of electromagnetic duality for all interactions;
— T duality, also called space-time duality, a mapping between small and large lengths
2
and times following 𝑙 ↔ 𝑙Pl /𝑙;*
— infrared dualities.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Despite the fascination of the idea, research into dualities has not led to any experimental
prediction. However, the results highlighted a different way to approach quantum field
theory. Dualities play an important role in superstring theory, which we will explore later
Vol. VI, page 141 on.

C ollective aspects of quantum field theory
For many decades, mathematicians asked physicists: What is the essence of quantum
field theory? Despite intensive research, this question has yet to be answered precisely.
Half of the answer is given by the usual definition found in physics textbooks: QFT
is the most general known way to describe quantum mechanically continuous systems
with a finite number of types of quanta but with an infinite number of degrees of freedom.
For example, this definition implies that the Lagrangian must be relativistically invariant
and must be described by a gauge theory. However, this half of the answer is already
Vol. VI, page 40 sufficient to spell trouble. We will show in the next part of our ascent that space and time
have a minimal distance scale and that nature does not have infinite numbers of degrees
of freedom. In other words, quantum field theory is an effective theory; this is the modern

* Space-time duality, the transformation between large and small sizes, leads one to ask whether there is
Vol. IV, page 112 an inside and an outside to particles. We encountered this question already in our study of gloves. We will
Page 292 encounter the issue again below, when we explore eversion and inversion. The issue will be fully clarified
only in the last volume of our adventure.
10 dreams of unification 275

way to say that it is approximate, or more bluntly, that it is wrong. But let us put these
issues aside for the time being.
The second, still partly unknown half of the answer would specify which (mathemat-
ical) conditions a physical system, i.e., a Lagrangian, actually needs to realize in order to
become a quantum field theory. Despite the work of many mathematicians, no complete
list of conditions is known yet. But it is known that the list includes at least two condi-
tions. First of all, a quantum field theory must be renormalizable. Secondly, a quantum
field theory must be asymptotically free; in other words, the coupling must go to zero
when the energy goes to infinity. This condition ensures that interactions are defined
properly. Only a subset of renormalizable Lagrangians obey this condition.
In four dimensions, the only known quantum field theories with these two proper-
ties are the non-Abelian gauge theories. These Lagrangians have several general aspects
which are not directly evident when we arrive at them through the usual way, i.e., by
generalizing naive wave quantum mechanics. This standard approach, the historical one,
emphasizes the perturbative aspects: we think of elementary fermions as field quanta and

Motion Mountain – The Adventure of Physics
of interactions as exchanges of virtual bosons, to various orders of perturbation.
On the other hand, all field theory Lagrangians also show two other configurations,
apart from particles, which play an important role. These mathematical solutions appear
when a non-perturbative point of view is taken; they are collective configurations.
— Quantum field theories show solutions which are static and of finite energy, created
by non-local field combinations, called solitons. In quantum field theories, solitons
are usually magnetic monopoles and dyons; also the famous skyrmions are solitons.
In this approach to quantum field theory, it is assumed that the actual equations of
Vol. I, page 316 nature are non-linear at high energy. Like in liquids, one then expects stable, local-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ized and propagating solutions, the solitons. These solitons could be related to the
observed particles.
— Quantum field theories show self-dual or anti-self dual solutions, called instantons.
Instantons play a role in QCD, and could also play a role in the fundamental Lag-
rangian of nature.
— Quantum field theory defines particles and interactions using perturbation expan-
sions. Do particles exist non-perturbatively? When does the perturbation expansion
break down? What happens in this case? Despite these pressing issues, no answer has
ever been found.
Ref. 225 All these fascinating topics have been explored in great detail by mathematical physicists.
This research has deepened the understanding of gauge theories. However, none of the
available results has yet helped to approach unification.

Curiosities ab ou t unification
From the 1970s onwards, it became popular to draw graphs such as the one of Figure 152.
They are found in many books. This approach towards the final theory of motion was
inspired by the experimental success of electroweak unification and to the success among
theoreticians of the idea of grand unification.
Unfortunately, grand unification contradicts experiment. In fact, as explained above,
Page 252 not even the electromagnetic and the weak interactions have been unified. (It took about
a decade to brainwash people into believing the contrary; this was achieved by intro-
276 10 dreams of unification

false beliefs, correct
common from descriptions
c. 1975 to c. 2015

Motion Mountain – The Adventure of Physics
F I G U R E 152 Blue and black: a typical graph on uniﬁcation, as found for years in publications on the
topic (© CERN). Red and green: its correct and incorrect parts.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ducing the incorrect term ‘electroweak unification’ instead of a correct term akin to
‘electroweak compatibility’ or ‘electroweak mixing’.) In other words, the larger part of
Figure 152 is not correct. Unfortunately, the graph and the story behind it has led most
researchers along the wrong path for several decades.
∗∗
Over the years, the length of this chapter became shorter and shorter. This was due to
the large number of unification attempts that were found to be in contradiction with
experiment. It is hard to describe the vast amount of effort that has been invested, usually
in vain, in the quest for unification of the description of motion.
∗∗
For a professional and up-to-date introduction into modern particle research, see the
summer student lectures that are given at CERN every year. They can be found at cdsweb.
cern.ch/collection/SummerStudentLectures?ln=en.

A summary on unification, mathematics and higher symmetries
The decades of theoretical research since the 1970s have shown:

⊳ Mathematical physics is not the way to search for unification of the de-
10 dreams of unification 277

scription of motion.

All the searches for unification that were guided by mathematical ideas – by mathematical
theorems or by mathematical generalizations – have failed. Mathematics is not helpful in
this quest. The standard model of particle physics and general relativity remain separate.
In addition, the research effort has led to a much more concrete result:

⊳ The search for a higher symmetry in nature has failed.

Despite thousands of extremely smart people exploring many possible higher symmet-
ries of nature, their efforts have not been successful.
Symmetry considerations are not helpful in the search for unification. Symmetry sim-
plifies the description of nature; but symmetry does not specify the description. It seems
that researchers have fallen into the trap of music theory. Anybody who has learned to

Motion Mountain – The Adventure of Physics
play an instrument has heard the statement that ‘mathematics is at the basis of music’.
Of course, this is nonsense; emotions are at the basis of music. But the incorrect state-
ment about mathematics lurks in the head of every musician. Looking back to research
in the twentieth century, it seems that the same has happened to researchers in the field
of unification. From these failures we conclude:

⊳ Unification requires to extract an underlying physical principle.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
On the other hand, in the twentieth century, researchers have failed to find such a prin-
ciple. This failure leads to two questions. In the quest for unification, is there really an
alternative to the search for higher symmetry? And did researchers rely on implicit, in-
correct assumptions about the structure of particles or of space-time? Before we explore
these fascinating issues, we take a break to inspire us.
C h a p t e r 11

BAC T E R IA , F L I E S A N D K NOT S

“
La première et la plus belle qualité de la nature est le
mouvement qui l’agite sans cesse ; mais ce
mouvement n’est qu’une suite perpétuelle de crimes ;

”
ce n’est que par des crimes qu’elle le conserve.
Donatien de Sade, Justine, ou les malheurs de la

Motion Mountain – The Adventure of Physics
vertu.**

W
obbly entities, in particular jellyfish or amoebas, open up a fresh vision of the
orld of motion, if we allow being led by the curiosity to study them in detail.
e have missed many delightful insights by leaving them aside up to now. In fact,
wobbly entities yield surprising connections between shape change and motion that will
be of great use in the last part of our mountain ascent. Instead of continuing to look at the
smaller and smaller, we now take a second look at everyday motion and its mathematical
description.
To enjoy this chapter, we change a dear habit. So far, we always described any general

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
example of motion as composed of the motion of point particles. This worked well in
classical physics, in general relativity and in quantum theory; we based the approach on
the silent assumption that during motion, each point of a complex system can be followed
separately. We will soon discover that this assumption is not realized at smallest scales.
Therefore the most useful description of motion of extended bodies uses methods that
do not require that body parts be followed piece by piece. We explore these methods in
this chapter; doing so is a lot of fun in its own right.
Vol. VI, page 117 If we describe elementary particles as extended entities – as we soon will have to –
a particle moving through space is similar to a dolphin swimming through water, or to
a bee flying through air, or to a vortex advancing in a liquid. Therefore we explore how
this happens.

Bumblebees and other miniature flying systems
If a butterfly passes by during our mountain ascent, we can stop a moment to appreciate a
simple fact: a butterfly flies, and it is rather small. If we leave some cut fruit in the kitchen
until it rots, we observe the even smaller fruit flies (Drosophila melanogaster), just about

** ‘The primary and most beautiful of nature’s qualities is motion, which agitates her at all times; but this
motion is simply a perpetual consequence of crimes; she conserves it by means of crimes only.’ Donatien Al-
phonse François de Sade (b. 1740 Paris, d. 1814 Charenton-Saint-Maurice) is the intense writer from whom
the term ‘sadism’ was deduced.
11 bacteria, flies and knots 279

F I G U R E 153 A ﬂying fruit ﬂy, tethered to a force-measuring
microelectromechanical system (© Bradley Nelson).

Motion Mountain – The Adventure of Physics
F I G U R E 154 Vortices around a butterﬂy wing (© Robert Srygley/Adrian Thomas).

two millimetres in size. Figure 153 shows a fruit fly in flight. If you have ever tried to build

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
small model aeroplanes, or if you even only compare these insects to paper aeroplanes
– possibly the smallest man-made flying thing you might have seen – you start to get a
feeling for how well evolution has optimized flying insects.
Compared to paper planes, insects also have engines, flapping wings, sensors, naviga-
tion systems, gyroscopic stabilizers, landing gear and of course all the features due to life,
reproduction and metabolism, built into an incredibly small volume. Evolution really is
an excellent engineering team. The most incredible flyers, such as the common house fly
(Musca domestica), can change flying direction in only 30 ms, using the stabilizers that
nature has given them by reshaping the original second pair of wings. Human engineers
are getting more and more interested in the technical solutions evolution has developed;
Ref. 226 many engineers are trying to achieve similar miniaturization. The topic of miniature fly-
ing systems is extremely vast, so that we will pick out only a few examples.
How does a bumblebee (Bombus terrestris) fly? The lift 𝑚𝑔 generated by a fixed wing
Vol. I, page 363 (as explained before) follows the empirical relation

𝑚𝑔 = 𝑓 𝐴 𝑣2 𝜌 (110)

where 𝐴 is the surface of the wing, 𝑣 is the speed of the wing in the fluid of density 𝜌.
The factor 𝑓 is a pure number, usually with a value between 0.2 and 0.4, that depends
Ref. 227 on the angle of the wing and its shape; here we use the average value 0.3. For a Boeing
Vol. I, page 38 747, the surface is 511 m2 , the top speed at sea level is 250 m/s; at an altitude of 12 km the
280 11 bacteria, flies and knots

density of air is only a quarter of that on the ground, thus only 0.31 kg/m3 . We deduce
Challenge 151 e (correctly) that a Boeing 747 has a mass of about 300 ton. For bumblebees with a speed
of 3 m/s and a wing surface of 1 cm2 , we get a lifted mass of about 35 mg, far less than the
weight of the bee, namely about 1 g. The mismatch is even larger for fruit flies. In other
words, an insect cannot fly if it keeps its wings fixed. It could not fly with fixed wings
even if it had tiny propellers attached to them!
Due to the limitations of fixed wings at small dimensions, insects and small birds
must move their wings, in contrast to aeroplanes. They must do so not only to take off
or to gain height, but also to simply remain airborne in horizontal flight. In contrast,
aeroplanes generate enough lift with fixed wings. Indeed, if you look at flying animals,
such as the ones shown in Figure 155, you note that the larger they are, the less they need
to move their wings (at cruising speed).
Can you deduce from equation (110) that birds or insects can fly but people cannot?
Challenge 152 s Conversely, the formula also (partly) explains why human-powered aeroplanes must be
so large.*

Motion Mountain – The Adventure of Physics
But how do insects, small birds, flying fish or bats have to move their wings in order to
fly? This is a tricky question and the answer has been uncovered only recently. The main
Ref. 228 point is that insect wings move in a way to produce eddies at the front edge which in
Ref. 229 turn thrust the insect upwards. Aerodynamic studies of butterflies – shown in Figure 154
– and studies of enlarged insect models moving in oil instead of in air are exploring the
precise way insects make use of vortices. At the same time, more and more ‘mechanical
birds’ and ‘model aeroplanes’ that use flapping wings for their propulsion are being built
around the world. The field is literally in full swing.** Researchers are especially inter-
ested in understanding how vortices allow change of flight direction at the small dimen-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
sions typical for insects. Another aim is to reduce the size of flying machines.
The expression (110) for the lift of fixed wings also shows what is necessary for safe
take-off and landing. The lift of all wings decreases for smaller speeds. Thus both animals
and aeroplanes increase their wing surface in these occasions. Many birds also vigorously
increase the flapping of wings in these situations. But even strongly flapping, enlarged
wings often are insufficient for take-off. Many flying animals, such as swallows, therefore
avoid landing completely. For flying animals which do take off from the ground, nature
most commonly makes them hit the wings against each other, over their back, so that
when the wings separate again, the low pressure between them provides the first lift.
This method is used by insects and many birds, including pheasants. As bird watchers

* Another part of the explanation requires some aerodynamics, which we will not study here. Aerodynamics
shows that the power consumption, and thus the resistance of a wing with given mass and given cruise
speed, is inversely proportional to the square of the wingspan. Large wingspans with long slender wings are
thus of advantage in (subsonic) flying, especially when energy is scarce.
One issue is mentioned here only in passing: why does an aircraft fly? The correct general answer is:
because it deflects air downwards. How does an aeroplane achieve this? It can do so with the help of a tilted
plank, a rotor, flapping wings, or a fixed wing. And when does a fixed wing deflect air downwards? First of
all, the wing has to be tilted with respect to the air flow; in addition, the specific cross section of the wing
can increase the downward flow. The relation between wing shape and downward flow is a central topic of
applied aerodynamics.
** The website www.aniprop.de presents a typical research approach and the sites ovirc.free.fr and www.
ornithopter.org give introductions into the way to build such systems for hobbyists.
11 bacteria, flies and knots 281

F I G U R E 155 Examples of the three larger wing types in nature, all optimized for rapid ﬂows: turkey
vulture (Cathartes aura), ruby-throated hummingbird (Archilochus colubris) and a dragonﬂy (© S.L.
Brown, Pennsylvania Game Commission/Joe Kosack and nobodythere).

know, pheasants make a loud ‘clap’ when they take off. The clap is due to the low pressure
region thus created.
Both wing use and wing construction depend on size. In fact, there are four types of
wings in nature.

Motion Mountain – The Adventure of Physics
1. First of all, all large flying objects, such aeroplanes and large birds, fly using fixed
wings, except during take-off and landing. This wing type is shown on the left-hand
side of Figure 155.
2. Second, common size birds use flapping wings. (Hummingbirds can have over 50
wing beats per second.) These first two types of wings have a thickness of about 10 to
15 % of the wing depth. This wing type is shown in the centre of Figure 155.
3. At smaller dimensions, a third wing type appears, the membrane wing. It is found
in dragonflies and most everyday insects. At these scales, at Reynolds numbers of
around 1000 and below, thin membrane wings are the most efficient. The Reynolds

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
number measures the ratio between inertial and viscous effects in a fluid. It is defined
as
𝑙𝑣𝜌
Re = (111)
𝜂

where 𝑙 is a typical length of the system, 𝑣 the speed, 𝜌 the density and 𝜂 the dynamic
viscosity of the fluid.* A Reynolds number much larger than one is typical for rapid air
flow and fast moving water. In fact, the value of the Reynolds number distinguishes
a ‘rapid’ or ‘turbulent’ flow on the one hand, and a ‘slow’, ‘laminar’ or ‘viscous’
flow on the other. An example of membrane wing is shown on the right-hand side of
Figure 155. All the first three wing types are designed for turbulent flows.
* The viscosity is the resistance to flow a fluid poses. It is defined by the force 𝐹 necessary to move a layer of
surface 𝐴 with respect to a second, parallel one at distance 𝑑; in short, the (coefficient of) dynamic viscosity
is defined as 𝜂 = 𝑑 𝐹/𝐴 𝑣. The unit is 1 kg/s m or 1 Pa s or 1 N s/m2 , once also called 10 P or 10 poise. In
other words, given a horizontal tube, the viscosity determines how strong the pump needs to be to pump
the fluid through the tube at a given speed. The viscosity of air 20°C is 1.8 × 10−5 kg/s m or 18 μPa s and
Challenge 153 ny increases with temperature. In contrast, the viscosity of liquids decreases with temperature. (Why?) The
viscosity of water at 0°C is 1.8 mPa s, at 20°C it is 1.0 mPa s (or 1 cP), and at 40°C is 0.66 mPa s. Hydrogen
has a viscosity smaller than 10 μPa s, whereas honey has 25 Pa s and pitch 30 MPa s.
Physicists also use a quantity 𝜈 called the kinematic viscosity. It is defined with the help of the mass
density of the fluid as 𝜈 = 𝜂/𝜌 and is measured in m2 /s, once called 104 stokes. The kinematic viscosity of
water at 20°C is 1 mm2 /s (or 1 cSt). One of the smallest values is that of acetone, with 0.3 mm2 /s; a larger
one is glycerine, with 2000 mm2 /s. Gases range between 3 mm2 /s and 100 mm2 /s.
282 11 bacteria, flies and knots

F I G U R E 156 The wings of a few types
of insects smaller than 1 mm (thrips,
Encarsia, Anagrus, Dicomorpha)
(HortNET).

Motion Mountain – The Adventure of Physics
4. The fourth type of wings is found at the smallest possible dimensions, for insects
smaller than one millimetre; their wings are not membranes at all, but are optimized
for viscous air flow. Typical are the cases of thrips and of parasitic wasps, which can
be as small as 0.3 mm. All these small insects have wings which consist of a central
stalk surrounded by hair. In fact, Figure 156 shows that some species of thrips have
wings which look like miniature toilet brushes.
5. At even smaller dimensions, corresponding to Reynolds number below 10, nature

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
does not use wings any more, though it still makes use of air transport. In principle,
at the smallest Reynolds numbers gravity plays no role any more, and the process of
flying merges with that of swimming. However, air currents are too strong compared
with the speeds that such a tiny system could realize. No active navigation is then
possible any more. At these small dimensions, which are important for the transport
through air of spores and pollen, nature uses the air currents for passive transport,
making use of special, but fixed shapes.
We summarize: active flying is only possible through shape change. Only two types of
shape changes are possible for active flying: that of wings and that of propellers (including
Ref. 230 turbines). Engineers are studying with intensity how these shape changes have to take
Ref. 231 place in order to make flying most effective. Interestingly, a similar challenge is posed by
swimming.

Swimming
Swimming is a fascinating phenomenon. The Greeks argued that the ability of fish to
swim is a proof that water is made of atoms. If atoms would not exist, a fish could not
advance through it. Indeed, swimming is an activity that shows that matter cannot be
continuous. Studying swimming can thus be quite enlightening. But how exactly do fish
swim?
Whenever dolphins, jellyfish, submarines or humans swim, they take water with their
fins, body, propellers, hands or feet and push it backwards. Due to momentum conser-
11 bacteria, flies and knots 283

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 157 A selection of animals using undulatory swimming and the main variables that describe it.
Ref. 232 (© L. Mahadevan/Macmillan from )

vation they then move forward.*
Most fish and aquatic mammals swim by bending their bodies and alternating
between two extreme deformations. An overview of animals that use this type of swim-
Ref. 232 ming – experts call it undulatory gait – is given in Figure 157. For all such swimming,
the Reynolds number obeys Re = 𝑣𝐿/𝜈 >> 1; here 𝑣 is the swimming speed, 𝐿 the
body length and 𝜈 the kinematic viscocity. The wide range of Reynolds number values
observed for swimming living beings is shown in the graph. The swimming motion is
best described by the so-called swimming number Sw = 𝜔𝐴𝐿/𝜈, where 𝜔 and 𝐴 are the
circular beat frequency and amplitude. The next graph, Figure 158, shows that for tur-

Vol. I, page 89 * Fish could use propellers, as the arguments against wheels we collected at the beginning of our walk do
not apply for swimming. But propellers with blood supply would be a weak point in the construction, and
thus make fish vulnerable. Therefore, nature has not developed fish with propellers.
284 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 158 Undulatory swimming follows simple scaling rules for organisms that range from a few
mm to 30 m in length. The Reynolds number Re describes the ratio between inertial effects (the water
thrown behind) and the disspiative effects (the friction of the water). The swimming number Sw
Ref. 232 describes the body kinematics. (© L. Mahadevan/Macmillan from )

bulent swimming, the swimming speed 𝑣 almost exclusively depends on the amplitude
Challenge 154 e and frequency of the undulatory motion; for smaller organisms that swim in the laminar
regime, the swimming speed also depends on the length of the organism.
In short, fish, dolphins, submarines and people swim in the same way that fireworks
or rockets fly: by throwing matter behind them, through lift. Lift-based propulsion is the
Ref. 234 main type of macroscopic swimming. Do all organisms swim in this way? No. Several
organisms swim by expelling water jets, for example cephalopods such as squids. And
above all, small organisms advancing through the molecules of a liquid use a completely
different, microscopic way of swimming.
Small organisms such as bacteria do not have the capacity to propel or accelerate water
against their surroundings. Indeed, the water remains attached around a microorganism
without ever moving away from it. Physically speaking, in these cases of swimming the
kinetic energy of the water is negligible. In order to swim, unicellular beings thus need to
11 bacteria, flies and knots 285

F I G U R E 159 A swimming scallop
(here from the genus Chlamys)
(© Dave Colwell).

use other effects. In fact, their only possibility is to change their body shape in controlled

Motion Mountain – The Adventure of Physics
ways. Seen from far away, the swimming of microorganisms thus resembles the motion
of particles through vacuum: like microorganisms, also particles have nothing to throw
behind* them.
A good way to distinguish macroscopic from microscopic swimming is provided by
scallops. Scallops are molluscs up to a few cm in size; an example is shown in Figure 159.
Scallops have a double shell connected by a hinge that they can open and close. If they
close it rapidly, water is expelled and the mollusc is accelerated; the scallop then can
glide for a while through the water. Then the scallop opens the shell again, this time
slowly, and repeats the feat. When swimming, the larger scallops look like clockwork

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
false teeth. Scallops thus use a macroscopic swimming process.
If we reduce the size of the scallop by a thousand times to the size of a single cell we
get a simple result: such a tiny scallop cannot swim. The lack of scalability of swimming
methods is due to the changing ratio between inertial and dissipative effects at different
Vol. I, page 429 scales. This ratio is measured by the Reynolds number 𝑅𝑒. For the scallop the Reynolds
number, is defined as 𝑅𝑒 = 𝑣𝐿/𝜂, where 𝑣 is the relative speed between swimmer and
water, 𝐿 the length of the swimmer and 𝜂 is the kinematic viscosity of the water, around
1 mm2 /s.
For the scallop, the Reynolds number is about 100, which shows that when it swims,
inertial effects are much more important than dissipative, viscous effects. For a bacterium
the Reynolds number is much smaller than 1, so that inertial effects effectively play no
role. There is no way to accelerate water away from a bacterial-sized scallop, and thus
no way to glide. Bacteria cannot swim like scallops or people do; bacteria cannot throw
water behind them. And this is not the only problem microorganism face when they
want to swim.
A well-known mathematical theorem states that no cell-sized being can move if the
shape change is the same in the two halves of the motion, i.e., when opening and closing
Ref. 233 are just the inverse of each other. Such a shape change would simply make it move back
and forward. Another mathematical theorem, the so-called scallop theorem, that states

* There is an exception: gliding bacteria move by secreting slime, even though it is still not fully clear why
this leads to motion.
286 11 bacteria, flies and knots

that no microscopic system can swim if it uses movable parts with only one degree of
freedom. Thus it is impossible to move, at cell dimensions, using the method that the
scallop uses on centimetre scale.
In order to swim, microorganisms thus need to use a more evolved, two-dimensional
motion of their shape. Indeed, biologists found that all microorganisms use one of the
following four swimming styles:
1. Microorganisms of compact shape of diameter between 20 μm and about 20 mm, use
cilia. Cilia are hundreds of little hairs on the surface of the organism. Some organisms
have cilia across their full surface, other only on part of it. These organisms move
the cilia in waves wandering around their surface, and these surface waves make the
body advance through the fluid. All children watch with wonder Paramecium, the
unicellular animal they find under the microscope when they explore the water in
which some grass has been left for a few hours. Paramecium, which is between 100 μm
and 300 μm in size, as well as many plankton species* use cilia for its motion. The cilia
and their motion are clearly visible in the microscope. A similar swimming method

Motion Mountain – The Adventure of Physics
is even used by some large animals; you might have seen similar waves on the borders
of certain ink fish; even the motion of the manta (partially) belongs into this class.
Ref. 235 Ciliate motion is an efficient way to change the shape of a body making use of two
dimensions and thus avoiding the scallop theorem.
2. Sperm and eukaryote microorganisms whose sizes are in the range between 1 μm
and 50 μm swim using an (eukaryote) flagellum.** Flagella, Latin for ‘small whips’,
work like flexible oars. Even though their motion sometimes appears to be just an
oscillation, flagella get a kick only during one half of their motion, e.g. at every swing
to the left. Flagella are indeed used by the cells like miniature oars. Some cells even

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
twist their flagellum in a similar way that people rotate an arm. Some microorgan-
isms, such as Chlamydomonas, even have two flagella which move in the same way
Ref. 237 as people move their legs when they perform the breast stroke. Most cells can also
change the sense in which the flagellum is kicked, thus allowing them to move either
Ref. 238 forward or backward. Through their twisted oar motion, bacterial flagella avoid re-
tracing the same path when going back and forward. As a result, the bacteria avoid
the scallop theorem and manage to swim despite their small dimensions. The flex-
ible oar motion they use is an example of a non-adiabatic mechanism; an important
fraction of the energy is dissipated.
3. The smallest swimming organisms, bacteria with sizes between 0.2 μm and 5 μm,
Ref. 239 swim using bacterial flagella. These flagella, also called prokaryote flagella, are dif-
ferent from the ones just mentioned. Bacterial flagella move like turning corkscrews.
They are used by the famous Escherichia coli bacterium and by all bacteria of the
genus Salmonella. This type of motion is one of the prominent exceptions to the
non-existence of wheels in nature; we mentioned it in the beginning of our walk.
Vol. I, page 89 Corkscrew motion is an example of an adiabatic mechanism.
A Coli bacterium typically has a handful of flagella, each about 30 nm thick and of
corkscrew shape, with up to six turns; the turns have a ‘wavelength’ of 2.3 μm. Each

* See the www.liv.ac.uk/ciliate website for an overview.
Ref. 236 ** The largest sperm, of 5.8 cm length, are produced by the 1.5 mm sized Drosophila bifurca fly, a relative of
the famous Drosophila melanogaster.
11 bacteria, flies and knots 287

flagellum is turned by a sophisticated rotation motor built into the cell, which the cell
can control both in rotation direction and in angular velocity. For Coli bacteria, the
Ref. 240 range is between 0 and about 300 Hz.
A turning flagellum does not propel a bacterium like a propeller; as mentioned, the
velocities involved are much too small, the Reynolds number being only about 10−4 .
At these dimensions and velocities, the effect is better described by a corkscrew turn-
ing in honey or in cork: a turning corkscrew produces a motion against the material
around it, in the direction of the corkscrew axis. The flagellum moves the bacterium
in the same way that a corkscrew moves the turning hand with respect to the cork.
4. One group of bacteria, the spirochaetes, move as a whole like a cork-screw through
water. An example is Rhodospirillum rubrum, whose motion can be followed
in the video on www.microbiologybytes.com/video/motility.com. These bacteria
have an internal motor round an axial filament, that changes the cell shape in a
non-symmetrical fashion and yield cork-screw motion. A different bacterium is
Ref. 241 Spiroplasma, a helical bacterium – but not a spirochaete – that changes the cell shape,

Motion Mountain – The Adventure of Physics
again in a non-symmetrical fashion, by propagating kink pairs along its body surface.
Various other microorganisms move by changing their body shape.
To test your intuition, you may try the following puzzle: is microscopic swimming pos-
Challenge 155 s sible in two spatial dimensions? In four?
By the way, still smaller bacteria do not swim at all. Indeed, each bacterium faces a
minimum swimming speed requirement: it must outpace diffusion in the liquid it lives
Ref. 242 in. Slow swimming capability makes no sense; numerous microorganisms therefore do
not manage or do not swim at all. Some microorganisms are specialized to move along
liquid–air interfaces. In fact, there are many types of interfacial swimming, including

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
macroscopic types, but we do not cover them here. Other microorganisms attach them-
selves to solid bodies they find in the liquid. Some of them are able to move along these
solids. The amoeba is an example for a microorganism moving in this way. Also the smal-
lest active motion mechanisms known, namely the motion of molecules in muscles and
Page 21 in cell membranes, work this way.
Let us summarize these observations. All known active motion, or self-propulsion,
(in flat space) takes place in fluids – be it air or liquids. All active motion requires shape
change. Macroscopic swimming works by accelerating the fluid in the direction opposite
to the direction of motion. Microscopic swimming works through smart shape change
that makes the swimmer advance through the fluid. In order that shape change leads
to motion, the environment, e.g. the fluid, must itself consist of moving components
always pushing onto the swimming entity. The motion of the swimming entity can then
be deduced from the particular shape change it performs. The mathematics of swimming
through shape change is fascinating; it deserves to be explored.

Rotation, falling cats and the theory of shape change
At small dimensions, flying and swimming takes place through shape change. In the
last decades, the description of shape change has changed from a fashionable piece of
research to a topic whose results are both appealing and useful. There are many studies,
both experimental and theoretical, about the exact way small systems move in water and
288 11 bacteria, flies and knots

F I G U R E 160 Cats can turn
themselves, even with no initial
angular momentum
(photographs by Etienne-Jules
Marey, 1894).

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 161 Humans can turn themselves in mid air like
cats: see the second, lateral rotation of Artem Silchenko, at
the 2006 cliff diving world championship (© World High
Diving Federation).

air, about the achievable and achieved efficiency, and much more.
It is not a surprise that organized shape change can lead to translational motion.
Amoebas, earthworms, caterpillars, snakes, and even human themselves move though
shape change.
But shape change can also lead to a rotation of a body. In this case, the ideas are not
restricted to microscopic systems, but apply at all scales. In particular, the theory of shape
change is useful in explaining how falling cats manage to fall always on their feet. Cats
are not born with this ability; they have to learn it. But the feat has fascinated people for
11 bacteria, flies and knots 289

𝑡1 𝑡2 𝑡3 𝑡4 𝑡5

𝑎 centre
of mass
𝜃

F I G U R E 162 The square cat: in free space, or also on perfect ice, a deformable body in the shape of a
parallelogram made of four masses and rods that is able to change the body angle 𝜃 and two rod
lengths 𝑎 is able to rotate itself around the centre of mass without outside help. One mass and the
length-changing rods are coloured to illustrate the motion.

Motion Mountain – The Adventure of Physics
centuries, as shown in the old photograph given in Figure 160. In fact, cats confirm in
Vol. I, page 120 three dimensions what we already knew for two dimensions:

⊳ A deformable body can change its own orientation in space without outside
help.

This is in strong contrast to translation, for which outside help is always needed.
Archimedes famously said: Give me a place to stand, and I’ll move the Earth. But to

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
rotate the Earth, a place to stand is not needed!
Not only cats, also humans can perform the feat: simply observe the second, lateral
rotation of the diver in Figure 161. Cosmonauts in space stations and passengers of para-
bolic ‘zero-gravity’ flights regularly do the same, as do many artificial satellites sent into
space.
In the 1980s, the work by Berry, Wilczek, Zee and ShapereBerry, MichaelWilczek,
FrankZee, AnthonyShapere, Alfred showed that all motion due to shape change is de-
Ref. 243 scribed by a gauge theory. The equivalence between the two situations is detailed in
Table 24. A simple and beautiful example for these ideas has been given by Putterman
Ref. 244 and Raz and is illustrated in Figure 162. Imagine four spheres on perfect ice, all of the
same mass and size, connected by four rods forming a parallelogram. Now imagine that
this parallelogram, using some built-in motors, can change length along one side, called
𝑎, and that it can also change the angle 𝜃 between the sides. Putterman and Raz call this
the square cat. The figure shows that the square cat can change its own orientation on
the ice while, obviously, keeping its centre of mass at rest. The figure also shows that the
change of orientation only works because the two motions that the cat can perform, the
stretching and the angle change, do not commute. The order in which these deformations
occur is essential for achieving the desired rotation.
The rotation of the square cat occurs in strokes; large rotations are achieved by repeat-
ing strokes, similar to the situation of swimmers. If the square cat would be swimming
in a liquid, the cat could thus rotate itself – though it could not advance.
When the cat rotates itself, each stroke results in a rotation angle that is independent
290 11 bacteria, flies and knots

TA B L E 24 The correspondence between shape change and gauge theory.

Concept Shape change G au ge t he ory

System deformable body matter–field combination
Gauge freedom freedom of description of body freedom to define vector potential
orientation and position
Gauge-dependent shape’s angular orientation and vector potential, phase
quantity position
orientation and position change vector potential and phase change
along an open path along open path
Gauge changes angular orientation and changes vector potential
transformation position
Gauge-independent orientation and position after full phase difference on closed path,
quantities stroke integral of vector potential along a
closed path

Motion Mountain – The Adventure of Physics
deformations field strengths
Gauge group e.g. possible rotations SO(3) or U(1), SU(2), SU(3)
motions E(3)

of the speed of the stroke. The same experience can be made when rotating oneself on an

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
office chair by rotating the arm above the head: the chair rotation angle after arm turn is
independent of the arm speed. Stroke motion leads to a puzzle: what is the largest angle
Challenge 156 d that a cat can turn in one stroke?
Rotation in strokes has a number of important implications. First of all, the number
of strokes is a quantity that all observers agree upon: it is observer-invariant. Secondly,
the orientation change after a complete stroke is also observer-invariant. Thirdly, the ori-
entation change for incomplete strokes is observer-dependent: it depends on the way that
orientation is defined. For example, if orientation is defined by the direction of the body
diagonal through the black mass (see Figure 162), it changes in a certain way during a
stroke. If the orientation is defined by the direction of the fixed bar attached to the black
mass, it changes in a different way during a stroke. Only when a full stroke is completed
do the two values coincide. Mathematicians say that the choice of the definition and
thus the value of the orientation is gauge-dependent, but that the value of the orientation
change at a full stroke is gauge-invariant.
In summary, the square cat shows three interesting points. First, already rather simple
deformable bodies can change their orientation in space. Secondly, the orientation of a
deformable body can only change if the deformations it can perform are non-commuting.
Thirdly, such deformable bodies are described by gauge theories: certain aspects of the
bodies are gauge-invariant, others are gauge-dependent. This summary leads to a ques-
tion: Can we use these ideas to increase our understanding of the gauge theories of the
electromagnetic, weak and strong interaction? Shapere and Wilczek say no. We will ex-
plore this issue in the next volume. In fact, shape change bears even more surprises.
11 bacteria, flies and knots 291

𝑧

𝑦
𝜑
𝜓 𝜃
𝜑
Equator

𝜃
𝑥
F I G U R E 163 Swimming on a curved surface using two discs.

Motion Mountain – The Adventure of Physics
Swimming in curved space
In flat space it is not possible to produce translation through shape change. Only orient-
ation changes are possible. Surprisingly, if space is curved, motion does become possible.
Ref. 245 A simple example was published in 2003 by Jack Wisdom. He found that cyclic changes
in the shape of a body can lead to net translation, a rotation of the body, or both.
Indeed, we know from Galilean physics that on a frictionless surface we cannot move,
but that we can change orientation. This is true only for a flat surface. On a curved sur-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
face, we can use the ability to turn and translate it into motion.
Take two massive discs that lie on the surface of a frictionless, spherical planet, as
shown in Figure 163. Consider the following four steps: 1. the disc separation 𝜑 is in-
creased by the angle Δ𝜑, 2. the discs are rotated oppositely about their centres by the
angle Δ𝜃, 3. their separation is decreased by −Δ𝜑, and 4. they are rotated back by −Δ𝜃.
Due to the conservation of angular momentum, the two-disc system changes its longit-
Challenge 157 ny ude Δ𝜓 as
1
Δ𝜓 = 𝛾2 Δ𝜃Δ𝜑 , (112)
2
where 𝛾 is the angular radius of the discs. This cycle can be repeated over and over. The
cycle allows a body, located on the surface of the Earth, to swim along the surface. Un-
fortunately, for a body of size of one metre, the motion for each swimming cycle is only
around 10−27 m.
Wisdom showed that the same procedure also works in curved space, thus in the
presence of gravitation. The mechanism thus allows a falling body to swim away from
the path of free fall. Unfortunately, the achievable distances for everyday objects are neg-
ligibly small. Nevertheless, the effect exists.
In other words, there is a way to swim through curved space that looks similar to
swimming at low Reynolds numbers, where swimming results of simple shape change.
Does this tell us something about fundamental descriptions of motion? The last part of
our ascent will tell.
292 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
F I G U R E 164 A way to
turn a sphere inside
out, with intermediate
steps ordered
clockwise (© John
Sullivan).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Turning a sphere inside ou t

“
A text should be like a lady’s dress; long enough
to cover the subject, yet short enough to keep it

”
interesting.
Anonymous

Exploring the theme of motion of wobbly entities, a famous example cannot be avoided.
Ref. 246 In 1957, the mathematician Stephen Smale proved that a sphere can be turned inside out.
The discovery brought him the Fields medal in 1966, the highest prize for discoveries in
mathematics. Mathematicians call his discovery the eversion of the sphere.
To understand the result, we need to describe more clearly the rules of mathematical
eversion. First of all, it is assumed that the sphere is made of a thin membrane which has
the ability to stretch and bend without limits. Secondly, the membrane is assumed to be
able to intersect itself. Of course, such a ghostly material does not exist in everyday life;
but in mathematics, it can be imagined. A third rule requires that the membrane must
be deformed in such a way that the membrane is not punctured, ripped nor creased;
in short, everything must happen smoothly (or differentiably, as mathematicians like to
say).
Even though Smale proved that eversion is possible, the first way to actually perform
Ref. 247, Ref. 248 it was discovered by the blind topologist Bernard Morin in 1961, based on ideas of Arnold
Shapiro. After him, several additional methods have been discovered.
11 bacteria, flies and knots 293

Ref. 249 Several computer videos of sphere eversions are now available.* The most famous ones
are Outside in, which shows an eversion due to William P. Thurston, and The Optiverse,
which shows the most efficient method known so far, discovered by a team led by John
Sullivan and shown in Figure 164.
Why is sphere eversion of interest to physicists? If elementary particles were extended
and at the same time were of spherical shape, eversion might be a particle symmetry. To
see why, we summarize the effects of eversion on the whole surrounding space, not only
on the sphere itself. The final effect of eversion is the transformation

(𝑥, 𝑦, −𝑧) 𝑅2
(𝑥, 𝑦, 𝑧) → (113)
𝑟2
where 𝑅 is the radius of the sphere and 𝑟 is the length of the coordinate vector (𝑥, 𝑦, 𝑧),
thus 𝑟 = √𝑥2 + 𝑦2 + 𝑧2 . Due to the minus sign in the 𝑧-coordinate, eversion differs from
inversion, but not by too much. As we will find out in the last part of our adventure, a

Motion Mountain – The Adventure of Physics
transformation similar to eversion, space-time duality, is a fundamental symmetry of
Vol. VI, page 114 nature.

Clouds
Clouds are another important class of wobbly objects. The lack of a definite boundary
makes them even more fascinating than amoebas, bacteria or falling cats. We can observe
the varieties of clouds from any aeroplane.
Vol. III, page 218 The common cumulus or cumulonimbus in the sky, like all the other meteorological
clouds, are vapour and water droplet clouds. Galaxies are clouds of stars. Stars are clouds

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of plasma. The atmosphere is a gas cloud. Atoms are clouds of electrons. Nuclei are clouds
of protons and neutrons, which in turn are clouds of quarks. Comparing different cloud
types is illuminating and fun.
Clouds of all types can be described by a shape and a size, even though in theory they
have no bound. An effective shape and size can be defined by that region in which the
cloud density is only, say, 1 % of the maximum density; slightly different procedures can
also be used. All clouds are described by probability densities of the components making
up the cloud. All clouds show conservation of the number of their constituents.
Whenever we see a cloud, we can ask why it does not collapse. Every cloud is an
Vol. I, page 257 aggregate; all aggregates are kept from collapse in only three ways: through rotation,
through pressure, or through the Pauli principle, i.e., the quantum of action. For example,
galaxies are kept from collapsing by rotation. Most stars, the atmosphere and rain clouds
are kept from collapsing by gas pressure. Neutron stars, the Earth, atomic nuclei, protons
or the electron clouds of atoms are kept apart by the quantum of action.
A rain cloud is a method to keep several thousand tons of water suspended in the air.
Can you explain what keeps it afloat, and what else keeps it from continuously diffusing

* Summaries of the videos can be seen at the website www.geom.umn.edu/docs/outreach/oi, which also
has a good pedagogical introduction. Another simple eversion and explanation is given by Erik de Neve
on his website www.usefuldreams.org/sphereev.htm. It is even possible to run the eversion film software
at home; see the website www.cslub.uwaterloo.ca/~mjmcguff/eversion. Figure 164 is from the website new.
math.uiuc.edu/optiverse.
294 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
F I G U R E 165 A vortex in nature: a waterspout (© Zé Nogueira).

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Challenge 158 s into a thinner and thinner structure?
Two rain clouds can merge. So can two atomic electron clouds. So can galaxies. But
only atomic clouds are able to cross each other. We remember that a normal atom can be
Vol. IV, page 186 inside a Rydberg atom and leave it again without change. In contrast, rain clouds, stars,
galaxies or other macroscopic clouds cannot cross each other. When their paths cross,
they can only merge or be ripped into pieces. Due to this lack of crossing ability, only
microscopic clouds can be counted. In the macroscopic cases, there is no real way to
define a ‘single’ cloud in an accurate way. If we aim for full precision, we are unable to
claim that there is more than one rain cloud, as there is no clear-cut boundary between
them. Electronic clouds are different. True, in a piece of solid matter we can argue that
there is only a single electronic cloud throughout the object; however, when the object
is divided, the cloud is divided in a way that makes the original atomic clouds reappear.
We thus can speak of ‘single’ electronic clouds.
If one wants to be strict, galaxies, stars and rain clouds can be seen as made of localized
particles. Their cloudiness is only apparent. Could the same be true for electron clouds?
And what about space itself? Let us explore some aspects of these questions.

Vortices and the S chrödinger equation
Fluid dynamics is a topic with many interesting aspects. Take the vortex that can be ob-
served in any deep, emptying bath tub: it is an extended, one-dimensional ‘object’, it is
11 bacteria, flies and knots 295

𝑤 𝑣

𝑒

𝑛

F I G U R E 166 The mutually perpendicular tangent 𝑒,
normal 𝑛, torsion 𝑤 and velocity 𝑣 of a vortex in a
rotating ﬂuid.

deformable, and it is observed to wriggle around. Larger vortices appear as tornadoes
on Earth and on other planets, as waterspouts, and at the ends of wings or propellers of

Motion Mountain – The Adventure of Physics
Page 105 all kinds. Smaller, quantized vortices appear in superfluids. An example is shown in Fig-
ure 165; also the spectacular fire whirls and fire tornados observed every now and then
are vortices.
Vortices, also called vortex tubes or vortex filaments, are thus wobbly entities. Now,
Ref. 250 a beautiful result from the 1960s states that a vortex filament in a rotating liquid is de-
scribed by the one-dimensional Schrödinger equation. Let us see how this is possible.
Any deformable linear vortex, as illustrated in Figure 166, is described by a continuous
set of position vectors 𝑟(𝑡, 𝑠) that depend on time 𝑡 and on a single parameter 𝑠. The
parameter 𝑠 specifies the relative position along the vortex. At each point on the vortex,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
there is a unit tangent vector 𝑒(𝑡, 𝑠), a unit normal curvature vector 𝑛(𝑡, 𝑠) and a unit
torsion vector 𝑤(𝑡, 𝑠). The three vectors, shown in Figure 166, are defined as usual as

∂𝑟
𝑒= ,
∂𝑠
∂𝑒
𝜅𝑛 = ,
∂𝑠
∂(𝑒 × 𝑛)
𝜏𝑤 = − , (114)
∂𝑠
where 𝜅 specifies the value of the curvature and 𝜏 specifies the value of the torsion. In
general, both numbers depend on time and on the position along the line.
In the simplest possible case the rotating environment induces a local velocity 𝑣 for
the vortex that is proportional to the curvature 𝜅, perpendicular to the tangent vector 𝑒
and perpendicular to the normal curvature vector 𝑛:

𝑣 = 𝜂𝜅(𝑒 × 𝑛) , (115)

Ref. 250 where 𝜂 is the so-called coefficient of local self-induction that describes the coupling
between the liquid and the vortex motion. This is the evolution equation of the vortex.
We now assume that the vortex is deformed only slightly from the straight configura-
tion. Technically, we are thus in the linear regime. For such a linear vortex, directed along
296 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
F I G U R E 167 Motion of a vortex: the fundamental helical solution and a moving helical ‘wave packet’.

the 𝑥-axis, we can write
𝑟 = (𝑥, 𝑦(𝑥, 𝑡), 𝑧(𝑥, 𝑡)) . (116)

Slight deformations imply ∂𝑠 ≈ ∂𝑥 and therefore

∂𝑦 ∂𝑧
𝑒 = (1, , ) ≈ (1, 0, 0) ,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∂𝑥 ∂𝑥
∂2 𝑦 ∂2 𝑧
𝜅𝑛 ≈ (0, 2 , 2 ) , and
∂𝑥 ∂𝑥
∂𝑦 ∂𝑧
𝑣 = (0, , ) . (117)
∂𝑡 ∂𝑡
We can thus rewrite the evolution equation (115) as

∂𝑦 ∂𝑧 ∂2 𝑧 ∂2 𝑦
(0, , ) = 𝜂 (0, − 2 , 2 ) . (118)
∂𝑡 ∂𝑡 ∂𝑥 ∂𝑥
This equation is well known; if we drop the first coordinate and introduce complex num-
bers by setting Φ = 𝑦 + 𝑖𝑧, we can rewrite it as

∂Φ ∂2 Φ
= 𝑖𝜂 2 . (119)
∂𝑡 ∂𝑥
This is the one-dimensional Schrödinger equation for the evolution of a free wave func-
Ref. 251 tion! The complex function Φ specifies the transverse deformation of the vortex. In other
words, we can say that the Schrödinger equation in one dimension describes the evolu-
tion of the deformation for an almost linear vortex in a rotating liquid. We note that there
is no constant ℏ in the equation, as we are exploring a classical system.
11 bacteria, flies and knots 297

Schrödinger’s equation is linear in Φ. Therefore the fundamental solution is

Φ(𝑥, 𝑦, 𝑧, 𝑡) = 𝑎 e𝑖(𝜏𝑥−𝜔𝑡) with 𝜔 = 𝜂𝜏2 and 𝜅 = 𝑎𝜏2 . (120)

The amplitude 𝑎 and the wavelength or pitch 𝑏 = 1/𝜏 can be freely chosen, as long as
the approximation of small deviation is fulfilled; this condition translates as 𝑎 ≪ 𝑏.* In
the present interpretation, the fundamental solution corresponds to a vortex line that is
deformed into a helix, as shown in Figure 167. The angular speed 𝜔 is the rotation speed
around the axis of the helix.
Challenge 159 ny A helix moves along the axis with a speed given by

𝑣helix along axis = 2𝜂𝜏 . (121)

In other words, for extended entities following evolution equation (115), rotation and
translation are coupled.** The momentum 𝑝 can be defined using ∂Φ/∂𝑥, leading to

Motion Mountain – The Adventure of Physics
1
𝑝=𝜏= . (122)
𝑏
Momentum is thus inversely proportional to the helix wavelength or pitch, as expected.
The energy 𝐸 is defined using ∂Φ/∂𝑡, leading to
𝜂
𝐸 = 𝜂𝜏2 = . (123)
𝑏2

𝑝2 1
𝐸= where 𝜇 = . (124)
2𝜇 2𝜂

In other words, a vortex with a coefficient 𝜂 – describing the coupling between environ-
Page 295 ment and vortex – is thus described by a number 𝜇 that behaves like an effective mass. We
can also define the (real) quantity |Φ| = 𝑎; it describes the amplitude of the deformation.
In the Schrödinger equation (119), the second derivative implies that the deforma-
tion ‘wave packet’ has tendency to spread out over space. Can you confirm that the
wavelength–frequency relation for a vortex wave group leads to something like the in-
Challenge 161 ny determinacy relation (however, without a ℏ appearing explicitly)?
In summary, the complex amplitude Φ for a linear vortex in a rotating liquid behaves
like the one-dimensional wave function of a non-relativistic free particle. In addition,
we found a suggestion for the reason that complex numbers appear in the Schrödinger
equation of quantum theory: they could be due to the intrinsic rotation of an underlying
Vol. VI, page 174 substrate. Is this suggestion correct? We will find out in the last part of our adventure.

* The curvature is given by 𝜅 = 𝑎/𝑏2 , the torsion by 𝜏 = 1/𝑏. Instead of 𝑎 ≪ 𝑏 one can thus also write 𝜅 ≪ 𝜏.
Challenge 160 ny ** A wave packet moves along the axis with a speed given by 𝑣packet = 2𝜂𝜏0 , where 𝜏0 is the torsion of the
helix of central wavelength.
298 11 bacteria, flies and knots

B
A

B
screw edge effective size
dislocation dislocation Burgers vector

F I G U R E 168 The two pure dislocation types, edge and screw dislocations, seen from the outside of a

Motion Mountain – The Adventure of Physics
cubic crystal (left) and the mixed dislocation – a quarter of a dislocation loop – joining them in a
horizontal section of the same crystal (right) (© Ulrich Kolberg).

Fluid space-time
General relativity shows that space can move and oscillate: space is a wobbly entity. Is
space more similar to clouds, to fluids, or to solids?
An intriguing approach to space-time as a fluid was published in 1995 by Ted
Ref. 252 Jacobson. He explored what happens if space-time, instead of assumed to be continuous,
is assumed to be the statistical average of numerous components moving in a disordered

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
fashion.
The standard description of general relativity describes space-time as an entity similar
Vol. II, page 144 to a flexible mattress. Jacobson studied what happens if the mattress is assumed to be
made of a fluid. A fluid is a collection of (undefined) components moving randomly and
described by a temperature varying from place to place.
Page 129 Jacobson started from the Fulling–Davies–Unruh effect and assumed that the local
fluid temperature is given by a multiple of the local gravitational acceleration. He also
used the proportionality – correct on horizons – between area and entropy. Since the
energy flowing through a horizon can be called heat, one can thus translate the expres-
sion 𝛿𝑄 = 𝑇𝛿𝑆 into the expression 𝛿𝐸 = 𝑎𝛿𝐴(𝑐2 /4𝐺), which describes the behaviour
Vol. VI, page 33 of space-time at horizons. As we have seen, this expression is fully equivalent to general
relativity.
In other words, imagining space-time as a fluid is a powerful analogy that allows de-
ducing general relativity. Does this mean that space-time actually is similar to a fluid?
So far, the analogy is not sufficient to answer the question and we have to wait for the
last part of our adventure to settle it. In fact, just to confuse us a bit more, there is an old
argument for the opposite statement.

Dislo cations and solid space-time
General relativity tells us that space behaves like a deformable mattress; space thus be-
haves like a solid. There is a second argument that underlines this point and that exerts a
11 bacteria, flies and knots 299

continuing fascination. This argument is connected to a famous property of the motion
of dislocations.
Dislocations are one-dimensional construction faults in crystals, as shown in Fig-
ure 168. A general dislocation is a mixture of the two pure dislocation types: edge dis-
locations and screw dislocations. Both are shown in Figure 168.
If one explores how the atoms involved in dislocations can rearrange themselves, one
Challenge 162 e finds that edge dislocations can only move perpendicularly to the added plane. In con-
trast, screw dislocations can move in all directions.* An important case of general, mixed
dislocations, i.e., of mixtures of edge and screw dislocations, are closed dislocation rings.
On such a dislocation ring, the degree of mixture changes continuously from place to
place.
Any dislocation is described by its strength and by its effective size; they are shown,
respectively, in red and blue in Figure 168. The strength of a dislocation is measured by
the so-called Burgers vector; it measures the misfits of the crystal around the dislocation.
More precisely, the Burgers vector specifies by how much a section of perfect crystal

Motion Mountain – The Adventure of Physics
needs to be displaced, after it has been cut open, to produce the dislocation. Obviously,
the strength of a dislocation is quantized in multiples of a minimal Burgers vector. In
fact, dislocations with large Burgers vectors can be seen as composed of dislocations of
minimal Burgers vector, so that one usually studies only the latter.
The size or width of a dislocation is measured by an effective width 𝑤. Also the width
is a multiple of the lattice vector. The width measures the size of the deformed region of
the crystal around the dislocation. Obviously, the size of the dislocation depends on the
elastic properties of the crystal, can take continuous values and is direction-dependent.
The width is thus related to the energy content of a dislocation.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
A general dislocation can move, though only in directions which are both perpendic-
ular to its own orientation and to its Burgers vector. Screw dislocations are simpler: they
can move in any direction. Now, the motion of screw dislocations has a peculiar prop-
erty. We call 𝑐 the speed of sound in a pure (say, cubic) crystal. As Frenkel and Kontorowa
Ref. 253 found in 1938, when a screw dislocation moves with velocity 𝑣, its width 𝑤 changes as

𝑤0
𝑤= . (125)
√1 − 𝑣2 /𝑐2

In addition, the energy of the moving dislocation obeys

𝐸0
𝐸= . (126)
√1 − 𝑣2 /𝑐2

A screw dislocation thus cannot move faster than the speed of sound 𝑐 in a crystal and its
width shows a speed-dependent contraction. (Edge dislocations have similar, but more
complex behaviour.) The motion of screw dislocations in solids is thus described by the
same effects and formulae that describe the motion of bodies in special relativity; the

* See the uet.edu.pk/dmems/edge_dislocation.htm, uet.edu.pk/dmems/screw_dislocation.htm and uet.edu.
pk/dmems/mixed_dislocation.htm web pages to watch a moving dislocation.
300 11 bacteria, flies and knots

contracted state stretched state
(high entropy) (low entropy)

molecule

cross-link

Motion Mountain – The Adventure of Physics
F I G U R E 169 An illustration of the relation between polymer conﬁgurations and elasticity. The
molecules in the stretched situation have fewer possible shape conﬁgurations and thus lower entropy;
therefore, the material tends back to the contracted situation.

speed of sound is the limit speed for dislocations in the same way that the speed of light
is the limit speed for objects.
Does this mean that elementary particles are dislocations of space or even of space-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
time, maybe even dislocation rings? The speculation is appealing, even though it sup-
poses that space-time is a solid crystal, and thus contradicts the model of space or space-
Vol. VI, page 76 time as a fluid. Worse, we will soon encounter other reasons to reject modelling space-
Challenge 163 s time as a lattice; maybe you can find a few arguments already by yourself. Still, expres-
sions (125) and (126) for dislocations continue to fascinate.
At this point, we are confused. Space-time seems to be solid and liquid at the same
time. Despite this contrast, the discussion somehow gives the impression that there is
something waiting to be discovered. But what? We will find out in the last part of our
adventure.

Polymers
Ref. 254 The study of polymers is both economically important and theoretically fascinating.
Polymers are materials built of long and flexible macromolecules that are sequences of
many (‘poly’ in Greek) similar monomers. These macromolecules are thus wobbly en-
tities.
Polymers form solids, like rubber or plexiglas, melts, like those used to cure teeth, and
many kinds of solutions, like glues, paints, eggs, or people. Polymer gases are of lesser
importance.
All the material properties of polymers, such as their elasticity, their viscosity,
their electric conductivity or their unsharp melting point, depend on the number of
monomers and the topology of their constituent molecules. In many cases, this depend-
11 bacteria, flies and knots 301

ence can be calculated. Let us explore an example.
If 𝐿 is the contour length of a free, ideal, unbranched polymer molecule, the average
end-to-end distance 𝑅 is proportional to the square root of the length 𝐿:

𝑅 = √𝐿𝑙 ∼ √𝐿 or 𝑅 ∼ √𝑁 (127)

where 𝑁 is the number of monomers and 𝑙 is an effective monomer length describing the
scale at which the polymer molecule is effectively stiff. 𝑅 is usually much smaller than 𝐿;
this means that free, ideal polymer molecules are usually in a coiled state.
Obviously, the end-to-end distance 𝑅 varies from molecule to molecule, and follows
a Gaussian distribution for the probability 𝑃 of a end-to-end distance 𝑅:

−3𝑅2
𝑃(𝑅) ∼ e 2𝑁𝑙2 . (128)

The average end-to-end distance mentioned above is the root-mean-square of this distri-

Motion Mountain – The Adventure of Physics
bution. Non-ideal polymers are polymers which have, like non-ideal gases, interactions
with neighbouring molecules or with solvents. In practice, polymers follow the ideal be-
haviour quite rarely: polymers are ideal only in certain solvents and in melts.
If a polymer is stretched, the molecules must rearrange. This changes their entropy
and produces an elastic force 𝑓 that tries to inhibit the stretching. For an ideal polymer,
the force is not due to molecular interactions, but is entropic in nature. Therefore the
force can be deduced from the free energy

𝐹 ∼ −𝑇 ln 𝑃(𝑅) (129)

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of the polymer: the force is then simply given as 𝑓 = ∂𝐹(𝑅)/∂𝑅. For an ideal polymer,
using its probability distribution, the force turns out to be proportional to the stretched
Challenge 164 e length. Thus the spring constant 𝑘 can be introduced, given by

𝑓 3𝑇
𝑘= = . (130)
𝑅 𝐿𝑙
We thus deduced a material property, the spring constant 𝑘, from the simple idea that
polymers are made of long, flexible molecules. The proportionality to temperature 𝑇 is
a result of the entropic nature of the force; the dependence on 𝐿 shows that longer mo-
lecules are more easy to stretch. For a real, non-ideal polymer, the calculation is more
complex, but the procedure is the same. Indeed, this is the mechanism at the basis of the
elasticity of rubber.
Using the free energy of polymer conformations, we can calculate the material proper-
ties of macromolecules in many other situations, such as their reaction to compression,
their volume change in the melt, their interactions in solutions, the effect of branched
molecules, etc. This is a vast field of knowledge on its own, which we do not pursue here.
Modern research topics include the study of knotted polymers and the study of polymer
mixtures. Extensive computer calculations and experiments are regularly compared.
Do polymers have some relation to the structure of physical space? The issue is open.
302 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 170 The knot diagrams for the simplest prime knots (© Robert Scharein).

It is sure, however, that polymers are often knotted and linked.

Knots and links

“
Don’t touch this, or I shall tie your fingers into

”
knots!
(Nasty, but surprisingly efficient child
education technique.)

Knots and their generalization are central to the study of wobbly object motion. A (math-
ematical) knot is a closed piece of rubber string, i.e., a string whose ends have been glued
together, which cannot be deformed into a circle or a simple loop. The simple loop is also
called the trivial knot.
Knots are of importance in the context of this chapter as they visualize the limitations
of the motion of wobbly entities. In addition, we will discover other reasons to study
Ref. 255 knots later on. In this section, we just have a bit of fun.*

* Beautiful illustrations and detailed information about knots can be found on the Knot Atlas website at
katlas.math.toronto.edu and at the KnotPlot website at www.knotplot.com.
11 bacteria, flies and knots 303

In 1949, Schubert proved that every knot can be decomposed in a unique way as sum
Ref. 256 of prime knots. Knots thus behave similarly to integers.
If prime knots are ordered by their crossing numbers, as shown in Figure 170, the
trivial knot (01 ) is followed by the trefoil knot (31 ) and by the figure-eight knot (41 ). The
figure only shows prime knots, i.e., knots that cannot be decomposed into two knots that
are connected by two parallel strands. In addition, the figure only shows one of the often
possible two mirror versions.
Together with the search for invariants, the tabulation of knots – a result of their clas-
sification – is a modern mathematical sport. Flat knot diagrams are usually ordered by
the minimal number of crossings as done in Figure 170. There is 1 knot with zero, 1 with
Ref. 256 three and 1 with four crossings (not counting mirror knots); there are 2 knots with five
and 3 with six crossings, 7 knots with seven, 21 knots with eight, 41 with nine, 165 with
ten, 552 with eleven, 2176 with twelve, 9988 with thirteen, 46 972 with fourteen, 253 293
with fifteen and 1 388 705 knots with sixteen crossings.
The mirror image of a knot usually, but not always, is different from the original. If

Motion Mountain – The Adventure of Physics
you want a challenge, try to show that the trefoil knot, the knot with three crossings, is
different from its mirror image. The first mathematical proof was by Max Dehn in 1914.
Antiknots do not exist. An antiknot would be a knot on a rope that cancels out the
corresponding knot when the two are made to meet along the rope. It is easy to prove
that this is impossible. We take an infinite sequence of knots and antiknots on a string,
Ref. 257 𝐾 − 𝐾 + 𝐾 − 𝐾 + 𝐾 − 𝐾.... On the one hand, we could make them disappear in this way
𝐾−𝐾+𝐾−𝐾+𝐾−𝐾... = (𝐾−𝐾)+(𝐾−𝐾)+(𝐾−𝐾)... = 0. On the other hand, we could do
the same thing using 𝐾−𝐾+𝐾−𝐾+𝐾−𝐾... = 𝐾+(−𝐾+𝐾)+(−𝐾+𝐾)+(−𝐾+𝐾)... = 𝐾.
The only knot 𝐾 with an antiknot is thus the unknot 𝐾 = 0.*

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
How do we describe such a knot through the telephone? Mathematicians have spent
a lot of time to figure out smart ways to achieve it. The obvious way is to flatten the
knot onto a plane and to list the position and the type (below or above) of the crossings.
(See Figure 171.) But what is the simplest way to describe knots by the telephone? The
task is not completely finished, but the end is in sight. Mathematicians do not talk about
‘telephone messages’, they talk about knot invariants, i.e., about quantities that do not
depend on the precise shape of the knot. At present, the best description of knots use
polynomial invariants. Most of them are based on a discovery by Vaughan Jones in 1984.
However, though the Jones polynomial allows us to uniquely describe most simple knots,
it fails to do so for more complex ones. But the Jones polynomial finally allowed mathem-
aticians to prove that a diagram which is alternating and eliminates nugatory crossings
(i.e., if it is ‘reduced’) is indeed one which has minimal number of crossings. The poly-
nomial also allows showing that any two reduced alternating diagrams are related by a
sequence of flypes.
In short, the simplest way to describe a knot through the telephone is to give its Kauff-
man polynomial, together with a few other polynomials.
Since knots are stable in time, a knotted line in three dimensions is equivalent to a
knotted surface in space-time. When thinking in higher dimensions, we need to be care-
ful. Every knot (or knotted line) can be untied in four or more dimensions. However,

* This proof does not work when performed with numbers; we would be able to deduce 1 = 0 by setting
Challenge 165 s K=1. Why is this proof valid with knots but not with numbers?
304 11 bacteria, flies and knots

Reidemeister Reidemeister Reidemeister
right-hand left-hand move I move II move III
crossing +1 crossing -1 (untwist) (unpoke) (slide)

the flype
R
a nugatory crossing R
F I G U R E 171 Crossing types F I G U R E 172 The Reidemeister moves and the ﬂype.
in knots.

Motion Mountain – The Adventure of Physics
F I G U R E 173 A tight open overhand knot and a tight open ﬁgure-eight knot (© Piotr Pieranski)

there is no surface embedded in four dimensions which has as 𝑡 = 0 slice a knot, and as
𝑡 = 1 slice the circle. Such a surface embedding needs at least five dimensions.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In higher dimensions, knots are thus possible only if n-spheres are tied instead of
circles; for example, as just said, 2-spheres can be tied into knots in 4 dimensions, 3-
spheres in 5 dimensions and so forth.

The hardest open problems that you can tell your grandmother
Even though mathematicians have achieved good progress in the classification of knots,
surprisingly, they know next to nothing about the shapes of knots. Here are a few prob-
lems that are still open today:
— This is the simplest unsolved knot problem: Imagine an ideally wobbly rope, that is,
a rope that has the same radius everywhere, but whose curvature can be changed as
one prefers. Tie a trefoil knot into the rope. By how much do the ends of the rope get
Challenge 166 r nearer? In 2006, there are only numerical estimates for the answer: about 10.1 radi-
uses. There is no formula yielding the number 10.1. Alternatively, solve the following
problem: what is the rope length of a closed trefoil knot? Also in this case, only nu-
merical values are known – about 16.33 radiuses – but no exact formula. The same is
valid for any other knot, of course.

— For mathematical knots, i.e., closed knots, the problem is equally unsolved. For ex-
ample: the ropelength of the tight trefoil knot is known to be around 16.33 diameters,
Ref. 259 and that of the figure-eight knot about 21.04 diameters. For beautiful visualizations of
11 bacteria, flies and knots 305

F I G U R E 174 The ropelength problem for the simple clasp, and the candidate conﬁguration that
probably minimizes ropelength, leaving a gap between the two ropes (© Jason Cantarella).

Motion Mountain – The Adventure of Physics
the tightening process, see the animations on the website www.jasoncantarella.com/
movs. But what is the formula giving the ropelength values? Nobody knows, because
the precise shape of the trefoil knot – or of any other knot – is unknown. Lou Kauff-
man has a simple comment for the situation: ‘It is a scandal of mathematics!’
— Mathematicians also study more general structures than knots. Links are the gener-
alization of knots to several closed strands. Braids and long links are the generaliza-
tion of links to open strands. Now comes the next surprise, illustrated in Figure 174.
Even for two ropes that form a simple clasp, i.e., two linked letters ‘U’, the ropelength

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Ref. 260 problem is unsolved – and there is not even a knot involved! In fact, in 2004, Jason
Cantarella and his colleagues have presented a candidate for the shape that minimizes
ropelength. Astonishingly, the candidate configuration leaves a small gap between the
two ropes, as shown in Figure 174.

In short, the shape of knots is a research topic that has barely taken off. Therefore we
have to leave these questions for a future occasion.

Curiosities and fun challenges on knots and wobbly entities
Knots appear rarely in nature. For example, tree branches or roots do not seem to grow
Challenge 167 r many knots during the lifetime of a plant. How do plants avoid this? In other words, why
are there no knotted bananas or knotted flower stems in nature?
Recent research has also explored how octopusses avoid knots in their arms. It was
found that the arms secrete a chemical substance that prevents arms or parts of the arms
from sticking together.
∗∗
Not only knot, also links can be classified. The simplest links, i.e., the links for which the
simplest configuration has the smallest number of crossings, are shown in Figure 175.
∗∗
306 11 bacteria, flies and knots

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 176 A hagﬁsh tied into a knot F I G U R E 177 How apparent
(© Christine Ortlepp). order for long rope coils (left)
changes over time when shaking
the container (right) (© 2007
PNAS).

The physics of human swimming is fascinating. To learn the details on how to move in
order to swim as rapidly as possible, explore the wonderful website coachsci.sdsu.edu/
swim by Brent Rushall. He tells how to move the arms, the trunk and the legs and he
shows how champions perform these movements. Rushall also tells about the bizarre
theories that are aired in the field of swimming, such as the mistaken idea that lift plays
a role in human swimming.
∗∗
A famous type of eel, the knot fish Myxine glutinosa, also called hagfish or slime eel, is
Ref. 262 able to make a knot in his body and move this knot from head to tail. Figure 176 shows
11 bacteria, flies and knots 307

an example. The hagfish uses this motion to cover its body with a slime that prevents
predators from grabbing it; it also uses this motion to escape the grip of predators, to get
rid of the slime after the danger is over, and to push against a prey it is biting in order to
extract a piece of meat. All studied knot fish form only left handed trefoil knots, by the
way; this is another example of chirality in nature.
∗∗
Proteins, the molecules that make up many cell structures, are chains of aminoacids.
Ref. 261 It seems that very few proteins are knotted, and that most of these form trefoil knots.
However, a figure-eight knotted protein has been discovered in 2000 by William Taylor.
∗∗
One of the most incredible discoveries of recent years is related to knots in DNA mo-
lecules. The DNA molecules inside cell nuclei can be hundreds of millions of base pairs
long; they regularly need to be packed and unpacked. When this is done, often the same

Motion Mountain – The Adventure of Physics
happens as when a long piece of rope or a long cable is taken out of a closet.
It is well known that you can roll up a rope and put it into a closet in such a way
that it looks orderly stored, but when it is pulled out at one end, a large number of knots
Ref. 263 is suddenly found. In 2007, this effects was finally explored in detail. Strings of a few
metres in length were put into square boxes and shaken, in order to speed up the effect.
The result, shown partly in Figure 177, was astonishing: almost every imaginable knot –
up to a certain complexity that depends on the length and flexibility of the string – was
formed in this way.
To make a long story short, the tangling also happens to nature when it unpacks DNA

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
in cell nuclei. Life requires that DNA molecules move inside the cell nucleus without
hindrance. So what does nature do? Nature takes a simpler approach: when there are
unwanted crossings, it cuts the DNA, moves it over and puts the ends together again. In
cell nuclei, there are special enzymes, the so-called topoisomerases, which perform this
process. The details of this fascinating process are still object of modern research.
∗∗
The great mathematician Carl-Friedrich Gauß – often written as ‘Gauss’ in English –
was the first person to ask what happens when an electrical current 𝐼 flows along a wire
Ref. 264 𝐴 that is linked with a wire 𝐵. He discovered a beautiful result by calculating the effect of
the magnetic field of one wire onto the other:

1 1 (𝑥 − 𝑥𝐵 )
∫ d𝑥𝐴 ⋅ 𝐵𝐵 = ∫ d𝑥𝐴 ⋅ ∫ d𝑥𝐵 × 𝐴 =𝑛, (131)
4π𝐼 𝐴 4π 𝐴 𝐵 |𝑥𝐴 − 𝑥𝐵 |3

where the integrals are performed along the wires. Gauss found that the number 𝑛 does
not depend on the precise shape of the wires, but only on the way they are linked. De-
forming the wires does not change the resulting number 𝑛. Mathematicians call such a
number a topological invariant. In short, Gauss discovered a physical method to calculate
a mathematical invariant for links; the research race to do the same for other invariants,
and in particular for knots and braids, is still going on today.
In the 1980s, Edward Witten was able to generalize this approach to include the nuc-
308 11 bacteria, flies and knots

F I G U R E 178 A large raindrop falling F I G U R E 179 Is this possible?
downwards.

lear interactions, and to define more elaborate knot invariants, a discovery that brought
him the Fields medal.
∗∗

Motion Mountain – The Adventure of Physics
If we move along a knot and count the crossings where we stay above and subtract the
number of crossings where we pass below, we get a number called the writhe of the knot.
It is not an invariant, but usually a tool in building them. Indeed, the writhe is not ne-
cessarily invariant under one of the three Reidemeister moves. Can you see which one,
Challenge 168 e using Figure 172? However, the writhe is invariant under flypes.
∗∗
Modern knot research is still a topic with many open questions. A recent discovery is the
Ref. 265 quasi-quantization of three-dimensional writhe in tight knots. Many discoveries are still

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
expected in the domain of geometric knot theory.
∗∗
Challenge 169 e There are two ways to tie your shoes. Can you find them?
∗∗
Challenge 170 s What is the shape of raindrops? Try to picture it. However, use your reason, not your
prejudice! By the way, it turns out that there is a maximum size for raindrops, with a
Ref. 266 value of about 4 mm. The shape of such a large raindrop is shown in Figure 178. Can you
imagine where the limit comes from?
For comparison, the drops in clouds, fog or mist are in the range of 1 to 100 μm, with
Vol. III, page 166 a peak at 10 to 15 μm. In situations where all droplets are of similar size and where light
Vol. III, page 131 is scattered only once by the droplets, one can observe coronae, glories or fogbows.
∗∗
Challenge 171 s What is the entity shown in Figure 179 – a knot, a braid or a link?
∗∗
Challenge 172 d Can you find a way to classify tie knots?
∗∗
Challenge 173 s Are you able to find a way to classify the way shoe laces can be threaded?
11 bacteria, flies and knots 309

F I G U R E 180 A ﬂying snake, Chrysopelea paradisii, performing the feat that gave it its name (QuickTime
ﬁlm © Jake Socha).

Motion Mountain – The Adventure of Physics
∗∗
A striking example of how wobbly entities can behave is given in Figure 180. There is
indeed a family of snakes that like to jump off a tree and sail through the air to a neigh-
bouring tree. Both the jump and the sailing technique have been studied in recent years.
The website www.flyingsnake.org by Jake Socha provides additional films. His fascinat-
Ref. 267 ing publications tell more about these intriguing reptiles.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
Vol. I, page 327 When a plane moves at supersonic speed through humid air, sometimes a conical cloud
forms and moves with the plane. How does this cloud differ from the ones studied above?
Challenge 174 e

∗∗
Challenge 175 ny One of the toughest challenges about clouds: is it possible to make rain on demand? So
far, there are almost no positive results. Inventing a method, possibly based on hygro-
scopic salt injection or with the help of lasers, will be a great help to mankind.
∗∗
Do knots have a relation to elementary particles? The question is about 150 years old. It
was first investigated by William Thomson-Kelvin and Peter Tait in the late nineteenth
century. So far, no proof of a relation has been found. Knots might be of importance at
Planck scales, the smallest dimensions possible in nature. We will explore how knots and
the structure of elementary particles might be related in the last volume of this adventure.

Summary on wobbly objects
We can sum up the possible motions of extended systems in a few key themes. In earl-
Vol. I, page 316 ier chapters we studied waves, solitons and interpenetration. These observations are de-
scribed by wave equations. In this chapter we explored the way to move through shape
310 11 bacteria, flies and knots

change, explored eversion, studied vortices, fluids, polymers, knots and their rearrange-
ment, and explored the motion of dislocations in solids. We found that shape change
is described by gauge theory, eversion is described by space-duality, vortices follow the
Schrödinger equation, fluids and polymers resemble general relativity and black holes,
knot shapes are hard to calculate and dislocations behave relativistically.
The motion of wobbly objects is a neglected topic in textbooks on motion. Research
is progressing at full speed; it is expected that many beautiful analogies with traditional
physics will be discovered in the near future. For example, in this chapter we have not ex-
plored any possible analogy for the motion of light. Similarly, including quantum theory
Challenge 176 r into the description of wobbly bodies’ motion remains a fascinating issue for anybody
aiming to publish in a new field.
In summary, we found that wobbly entities can reproduce most fields of modern phys-
ics. Are there wobbly entities that reproduce all of modern physics? We will explore the
question in the last volume.

QUA N T UM PH YSIC S I N A N U T SH E L L
– AG A I N

C
ompared to classical physics, quantum theory is definitely more
omplex. The basic idea however, is simple: in nature there is a smallest
hange, or a smallest action, with the value ℏ = 1.1 ⋅ 10−34 Js. The smallest ac-
tion value leads to all the strange observations made in the microscopic domain, such

Motion Mountain – The Adventure of Physics
as wave behaviour of matter, indeterminacy relations, decoherence, randomness in
measurements, indistinguishability, quantization of angular momentum, tunnelling,
pair creation, decay, particle reactions and virtual particle exchange.

Q uantum field theory in a few sentences

“ ”
Deorum offensae diis curae.
Voltaire, Traité sur la tolérance.

All of quantum theory can be resumed in a few sentences.

The existence of a smallest action in nature directly leads to the main lesson we learned
about motion in the quantum part of our adventure:

⊳ If something moves, it is made of quantons, or quantum particles.
⊳ There are elementary quantum particles.

These statements apply to everything, thus to all objects and to all images, i.e., to mat-
ter and to radiation. Moving stuff is made of quantons. Stones, water waves, light, sound
waves, earthquakes, tooth paste and everything else we can interact with is made of mov-
ing quantum particles. Experiments show:

⊳ All intrinsic object properties observed in nature – such as electric charge,
weak charge, colour charge, spin, parity, lepton number, etc., with the only
exception of mass – appear as integer numbers of a smallest unit; in com-
posed systems they either add or multiply.
312 12 quantum physics in a nutshell – again

⊳ An elementary quantum particle or elementary quanton is a countable
Page 261 entity, smaller than its own Compton wavelength, described by energy–
momentum, mass, spin, C, P and T parity, electric charge, colour, weak
isospin, isospin, strangeness, charm, topness, beauty, lepton number and ba-
ryon number.

All moving entities are made of elementary quantum particles. To see how deep this
result is, you can apply it to all those moving entities for which it is usually forgotten,
such as ghosts, spirits, angels, nymphs, daemons, devils, gods, goddesses and souls. You
Challenge 177 e can check yourself what happens when their particle nature is taken into account.
Quantum particles are never at rest, cannot be localized, move probabilistically, be-
have as particles or as waves, interfere, can be polarized, can tunnel, are indistinguishable,
have antiparticles, interact locally, define length and time scales and they limit measure-
ment precision.

Motion Mountain – The Adventure of Physics
Quantum particles come in two types:

⊳ Matter is composed of fermions: quarks and leptons. There are 6 quarks
that make up nuclei, and 6 leptons – 3 charged leptons, including the elec-
tron, and 3 uncharged neutrinos. Elementary fermions have spin 1/2 and
obey the Pauli exclusion principle.
⊳ Radiation is due to the three gauge interactions and is composed of bosons:
photons, the weak vector bosons and the 8 gluons. These elementary bosons
all have spin 1.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
In our adventure, we wanted to know what matter and interactions are. Now we know:
they are due to elementary quantum particles. The exploration of motion inside matter,
including particle reactions and virtual particle exchange, showed us that matter is made
of a finite number of elementary quantum particles. Experiments show:

⊳ In flat space, elementary particles interact in one of three ways: there is
the electromagnetic interaction, the strong nuclear interaction and the weak
nuclear interaction.
⊳ The three interactions are exchanges of virtual bosons.
⊳ The three interactions are described by the gauge symmetries U(1), SU(3)
and a broken, i.e., approximate SU(2) symmetry.

The three gauge symmetries fix the Lagrangian of every physical system in flat space-
time. The most simple description of the Lagrangians is with the help of Feynman dia-
grams and the gauge groups.

⊳ In all interactions, energy, momentum, angular momentum, electric
charge, colour charge, CPT parity, lepton number and baryon number are
conserved.

The list of conserved quantities implies:
12 quantum physics in a nutshell – again 313

⊳ Quantum field theory is the part of quantum physics that includes the
description of particle transformations.

The possibility of particle transformations – including particle reactions, particle emis-
sion and particle absorption – results from the existence of a minimum action and of a
Vol. IV, page 31 maximum speed in nature. Emission of light, radioactivity, the burning of the Sun and
the history of the composite matter we are made of are due to particle transformations.
Due to the possibility of particle transformations, quantum field theory introduces a
limit for the localization of particles. In fact, any object of mass 𝑚 can be localized only
within intervals of the Compton wavelength

ℎ 2πℏ
𝜆C = = , (132)
𝑚𝑐 𝑚𝑐
where 𝑐 is the speed of light. At the latest at this distance we have to abandon the clas-

Motion Mountain – The Adventure of Physics
sical description and use quantum field theory. If we approach the Compton wavelength,
particle transformations become so important that classical physics and even simple
quantum theory are not sufficient.
Quantum electrodynamics is the quantum field description of electromagnetism. It
includes and explains all particle transformations that involve photons. The Lagrangian
of QED is determined by the electromagnetic gauge group U(1), the requirements of
space-time (Poincaré) symmetry, permutation symmetry and renormalizability. The lat-
ter requirement follows from the continuity of space-time. Through the effects of virtual
particles, QED describes electromagnetic decay, lamps, lasers, pair creation, Unruh radi-
ation for accelerating observers, vacuum energy and the Casimir effect, i.e., the attrac-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tion of neutral conducting bodies. Particle transformations due to quantum electrody-
namics also introduce corrections to classical electrodynamics; among others, particle
transformations produce small departures from the superposition principle for electro-
magnetic fields, including the possibility of photon-photon scattering.
The theory of weak nuclear interaction describes parity violation, quark mixings, neut-
rino mixings, massive vector bosons and the Higgs field for the breaking of the weak
SU(2) gauge symmetry. The weak interaction explains a large part of radioactivity, in-
cluding the heat production inside the Earth, and describes processes that make the Sun
shine.
Quantum chromodynamics, the field theory of the strong nuclear interaction, de-
scribes all particle transformations that involve gluons. At fundamental scales, the strong
interaction is mediated by eight elementary gluons. At larger, femtometre scales, the
strong interaction effectively acts through the exchange of spin 0 pions, is strongly at-
tractive, and leads to the formation of atomic nuclei. The strong interaction determines
nuclear fusion and fission. Quantum chromodynamics, or QCD, explains the masses of
Page 200 mesons and baryons through their description as bound quark states.
By including particle transformations, quantum field theory provides a common basis
of concepts and descriptions to materials science, nuclear physics, chemistry, biology,
medicine and to most of astronomy. For example, the same concepts allow us to answer
questions such as why water is liquid at room temperature, why copper is red, why the
rainbow is coloured, why the Sun and the stars continue to shine, why there are about
314 12 quantum physics in a nutshell – again

110 elements, where a tree takes the material to make its wood and why we are able to
move our right hand at our own will. Quantum theory explains the origin of material
properties and the origin of the properties of life.
Quantum field theory describes all material properties, be they mechanical, optical,
electric or magnetic. It describes all waves that occur in materials, such as sound and
phonons, magnetic waves and magnons, light, plasmons, and all localized excitations.
Quantum field theory also describes collective effects in matter, such as superconduct-
ivity, semiconductor effects and superfluidity. Finally, quantum field theory describes all
interactions between matter and radiation, from colour to antimatter creation.
Quantum field theory also clarifies that the particle description of nature, including
the conservation of particle number – defined as the difference between particles and
antiparticles – follows from the possibility to describe interactions perturbatively. A per-
turbative description of nature is possible only at low energies. At extremely high ener-
gies, higher than those observed in experiments, the situation is expected to change and
non-perturbative effects should come into play. These situations will be explored in the

Motion Mountain – The Adventure of Physics
next volume.

Achievements in precision
Classical physics is unable to predict any property of matter. Quantum field theory pre-
dicts all properties of matter, and to the full number of digits – sometimes thirteen –
that can be measured today. The precision is usually not limited by the inaccuracy of the-
ory, it is limited by the measurement accuracy. In other words, the agreement between
quantum field theory and experiment is only limited by the amount of money one is
willing to spend. Table 25 shows some predictions of classical physics and of quantum

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
field theory. The predictions are deduced from the properties of nature collected in the
millennium list, which is given in the next section.

TA B L E 25 Selected comparisons between classical physics, quantum theory and experiment.

O b s e r va b l e Clas - Prediction of Measure - Cost
sical qua ntu m ment esti-
predic - theory𝑎 m at e𝑏
tion
Simple motion of bodies
Indeterminacy 0 Δ𝑥Δ𝑝 ⩾ ℏ/2 (1 ± 10−2 ) ℏ/2 10 k€
Matter wavelength none 𝜆𝑝 = 2πℏ (1 ± 10−2 ) ℏ 10 k€
Tunnelling rate in α decay 0 1/𝜏 is finite (1 ± 10−2 ) 𝜏 5 k€
Compton wavelength none 𝜆 c = ℎ/𝑚e 𝑐 (1 ± 10−3 ) 𝜆 20 k€
Pair creation rate 0 𝜎𝐸 agrees 100 k€
Radiative decay time in none 𝜏 ∼ 1/𝑛3 (1 ± 10−2 ) 5 k€
hydrogen
Smallest angular 0 ℏ/2 (1 ± 10−6 ) ℏ/2 10 k€
momentum
Casimir effect/pressure 0 𝑝 = (π2 ℏ𝑐)/(240𝑟4 ) (1 ± 10−3 ) 30 k€
12 quantum physics in a nutshell – again 315

TA B L E 25 (Continued) Selected comparisons between classical physics, quantum theory and
experiment.

O b s e r va b l e Clas - Prediction of Measure - Cost
sical qua ntu m ment esti-
predic - theory𝑎 m at e𝑏
tion

Colours of objects
Spectrum of hot objects diverges 𝜆 max = ℎ𝑐/(4.956 𝑘𝑇) (1 ± 10−4 ) Δ𝜆 10 k€
Lamb shift none Δ𝜆 = 1057.86(1) MHz (1 ± 10−6 ) Δ𝜆 50 k€
Rydberg constant none 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ (1 ± 10−9 ) 𝑅∞ 50 k€
Stefan–Boltzmann none 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 (1 ± 3 ⋅ 10−8 ) 𝜎 20 k€
constant
Wien’s displacement none 𝑏 = 𝜆 max 𝑇 (1 ± 10−5 ) 𝑏 20 k€
constant

Motion Mountain – The Adventure of Physics
Refractive index of water none 1.34 a few % 1 k€
Photon-photon scattering 0 from QED: finite agrees 50 M€
Laser radiation exists no yes agrees 10€
Particle and interaction properties
Electron gyromagnetic 1 or 2 2.002 319 304 3(1) 2.002 319 304 30 M€
ratio 3737(82)
Z boson mass none 𝑚2𝑍 = 𝑚2𝑊 (1 + sin 𝜃𝑊
2
) (1 ± 10−3 ) 𝑚𝑍 100 M€
Proton mass none (1 ± 5 %) 𝑚p 𝑚p =1.67 yg 1 M€

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Proton lifetime ≈ 1 μs ∞ > 1035 a 100 M€
Chemical reaction rate 0 from QED correct within 2 k€
errors
Composite matter properties
Atom lifetime ≈ 1 μs ∞ > 1020 a 1€
Molecular size none from QED within 10−3 20 k€
Von Klitzing constant ∞ ℎ/𝑒2 = 𝜇0 𝑐/2𝛼 (1 ± 10−7 ) ℎ/𝑒2 1 M€
AC Josephson constant 0 2𝑒/ℎ (1 ± 10−6 ) 2𝑒/ℎ 5 M€
Heat capacity of metals at 25 J/K 0 < 10−3 J/K 10 k€
0K
Heat capacity of diatomic 25 J/K 0 < 10−3 J/K 10 k€
gas at 0 K
Water density none 1000.00 kg/m3 at 4°C agrees 10 k€
Minimum electr. 0 𝐺 = 2𝑒2 /ℏ G(1 ± 10−3 ) 3 k€
conductivity
Ferromagnetism none exists exists 2€
Superfluidity none exists exists 200 k€
Bose–Einsein none exists exists 2 M€
condensation
316 12 quantum physics in a nutshell – again

TA B L E 25 (Continued) Selected comparisons between classical physics, quantum theory and
experiment.

O b s e r va b l e Clas - Prediction of Measure - Cost
sical qua ntu m ment esti-
predic - theory𝑎 m at e𝑏
tion
Superconductivity none exists exists 100 k€
(metal)
Superconductivity (high none none yet exists 100 k€
T)

𝑎. All these predictions are calculated from the fundamental quantities given in the millennium
list.
𝑏. Sometimes the cost for the calculation of the prediction is higher than that of the experimental

Motion Mountain – The Adventure of Physics
Challenge 178 s observation. (Can you spot the examples?) The sum of the two is given.

We notice that the values predicted by quantum theory do not differ from the measured
ones. In contrast, classical physics does not allow us to calculate any of the observed
values. This shows the progress that quantum physics has brought in the description of
nature.
In short, in the microscopic domain quantum theory is in perfect correspondence
with nature; despite prospects of fame and riches, despite the largest number of research-
ers ever, no contradiction with observation has been found yet. But despite this impress-
ive agreement, there still are unexplained observations; they form the so-called millen-

What is unexplained by quantum theory and general relativity?
The material gathered in this quantum part of our mountain ascent, together with the
Vol. II, page 287 earlier summary of general relativity, allows us to describe all observed phenomena con-
nected to motion. For the first time, there are no known differences between theory and
practice.
Despite the precision of the description of nature, some things are missing. Whenever
we ask ‘why?’ about an observation and continue doing so after each answer, we arrive
at one of the unexplained properties of nature listed in Table 26. The table lists all issues
about fundamental motion that were unexplained in the year 2000, so that we can call it
the millennium list of open problems.

TA B L E 26 The millennium list: everything the standard model and general relativity cannot explain;
thus, also the list of the only experimental data available to test the ﬁnal, uniﬁed description of motion.

O b s e r va b l e P r o p e r t y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0

Local quantities unexplained by the standard model: particle properties
𝛼 = 1/137.036(1) the low energy value of the electromagnetic coupling or fine structure con-
stant
𝛼w or 𝜃w the low energy value of the weak coupling constant or the value of the weak
mixing angle
12 quantum physics in a nutshell – again 317

TA B L E 26 (Continued) Everything the standard model and general relativity cannot explain.

O b s e r va b l e P r o p e r t y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0

𝛼s the value of the strong coupling constant at one specific energy value
𝑚q the values of the 6 quark masses
𝑚l the values of 6 lepton masses
𝑚W the value of the mass of the 𝑊 vector boson
𝑚H the value of the mass of the scalar Higgs boson
𝜃12 , 𝜃13 , 𝜃23 the value of the three quark mixing angles
𝛿 the value of the CP violating phase for quarks
𝜈 𝜈 𝜈
𝜃12 , 𝜃13 , 𝜃23 the value of the three neutrino mixing angles
𝛿𝜈 , 𝛼1 , 𝛼2 the value of the three CP violating phases for neutrinos
3⋅4 the number of fermion generations and of particles in each generation
J, P, C, etc. the origin of all quantum numbers of each fermion and each boson

Motion Mountain – The Adventure of Physics
Concepts unexplained by the standard model
𝑐, ℏ, 𝑘 the origin of the invariant Planck units of quantum field theory
3+1 the number of dimensions of physical space and time
SO(3,1) the origin of Poincaré symmetry, i.e., of spin, position, energy, momentum
Ψ the origin and nature of wave functions
𝑆(𝑛) the origin of particle identity, i.e., of permutation symmetry
Gauge symmetry the origin of the gauge groups, in particular:
U(1) the origin of the electromagnetic gauge group, i.e., of the quantization of elec-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tric charge, of the vanishing of magnetic charge, and of minimal coupling
SU(2) the origin of weak interaction gauge group, its breaking and P violation
SU(3) the origin of strong interaction gauge group and its CP conservation
Renorm. group the origin of renormalization properties
𝛿𝑊 = 0 the origin of the least action principle in quantum theory
𝑊 = ∫𝐿 SM d𝑡 the origin of the Lagrangian of the standard model of particle physics
Global quantities unexplained by general relativity and cosmology
0 the observed flatness, i.e., vanishing curvature, of the universe
1.2(1) ⋅ 1026 m the distance of the horizon, i.e., the ‘size’ of the universe (if it makes sense)
𝜌de = Λ𝑐4 /(8π𝐺) the value and nature of the observed vacuum energy density, dark energy or
≈ 0.5 nJ/m3 cosmological constant
(5 ± 4) ⋅ 1079 the number of baryons in the universe (if it makes sense), i.e., the average
visible matter density in the universe
𝜌dm the density and nature of dark matter
𝑓0 (1, ..., c. 1090 ) the initial conditions for c. 1090 particle fields in the universe (if or as long as
they make sense), including the homogeneity and isotropy of matter distri-
bution, and the density fluctuations at the origin of galaxies
Concepts unexplained by general relativity and cosmology
𝑐, 𝐺 the origin of the invariant Planck units of general relativity
318 12 quantum physics in a nutshell – again

TA B L E 26 (Continued) Everything the standard model and general relativity cannot explain.

O b s e r va b l e P r o p e r t y u n e x p l a i n e d s i n c e t h e y e a r 2 0 0 0

R × S3 the observed topology of the universe
𝐺𝜇𝜈 the origin and nature of curvature, the metric and horizons
𝛿𝑊 = 0 the origin of the least action principle in general relativity
𝑊 = ∫𝐿 GR d𝑡 the origin of the Lagrangian of general relativity

The millennium list has several notable aspects. First of all, neither quantum mechanics
nor general relativity explain any property unexplained in the other field. The two theor-
ies do not help each other; the unexplained parts of both fields simply add up. Secondly,
both in quantum theory and in general relativity, motion still remains the change of pos-
ition with time. In short, so far, we did not achieve our goal: we still do not understand
motion! We are able to describe motion with full precision, but we still do not know what
it is. Our basic questions remain: What are time and space? What is mass? What is charge

Motion Mountain – The Adventure of Physics
and what are the other properties of objects? What are fields? Why are all the electrons
the same?
Page 316 We also note that the millennium list of open questions, Table 26, contains extremely
different concepts. This means that at this point of our walk there is a lot we do not
understand. Finding the answers will require effort.
On the other hand, the millennium list of unexplained properties of nature is also
short. The description of nature that our adventure has produced so far is concise and
precise. No discrepancies from experiments are known. In other words, we have a good
description of motion in practice. Going further is unnecessary if we only want to im-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
prove measurement precision. Simplifying the above list is mainly important from the
conceptual point of view. For this reason, the study of physics at university often stops
at this point. However, as the millennium list shows, even though we have no known
discrepancies with experiments, we are not at the top of Motion Mountain.

The physics cube
Page 8 Another review of the progress and of the open issues of physics, already given in the
introduction, is shown in Figure 181: the physics cube. From the lowest corner of the
cube, representing Galilean physics and related topics from everyday life, three edges –
labelled 𝑐, 𝐺 and ℏ, 𝑒, 𝑘 – lead to classical gravity, special relativity and quantum theory.
Each constant implies a limit to motion; in the corresponding theory, this one limit is
taken into account, thus improving the precision of the description. From these second
level theories, similar edges lead upwards to general relativity, quantum field theory and
quantum theory with gravity. Each of these third level theories takes into account two of
the limits and thus improves precision even more.* The present volume completes the
third level of precision. We stress that each theory in the second and third level is exact,
* Of course, Figure 181 gives a simplified view of the history of physics. A more precise diagram might
use different arrows for ℏ (with 𝑘) and 𝑒, making the figure a four-dimensional cube. However, not all of
Challenge 179 e its corners would have dedicated theories (can you confirm this?). Also the weak and the strong coupling
constants might have to be added. The diagram would be far less appealing. And most of all, the conclusions
mentioned in the text would not change.
12 quantum physics in a nutshell – again 319

PHYSICS: Final, unified theory of motion Final level
Describing motion with 2020? of precision
precision, i.e., with the
principle of least action.

ℏ, 𝑒, 𝑘 𝑐 𝐺
General Quantum Quantum Third level
relativity theory with field theory
1915 gravity 1926-1950
c. 1950

𝑐 ℏ, 𝑒, 𝑘 𝐺 ℏ, 𝑒, 𝑘 𝐺 𝑐

Motion Mountain – The Adventure of Physics
Classical Special relativity Quantum theory Second level
gravitation 1905 1900-1923
c. 1680

𝐺 𝑐 ℏ, 𝑒, 𝑘

An arrow indicates an Galilean physics, First level
increase in precision by heat, electricity of precision

F I G U R E 181 A simpliﬁed history of the description of motion in physics, by giving the limits to motion
included in each description. The arrows show which constant of nature needs to be added and taken
into account to reach the next level of precision. (The electric charge 𝑒 is taken to represent all three
discrete gauge charges.)

though only in its domain. And even though the limits of each domain are obvious, no
differences between experiment and theory are known.
From the third level theories, the edges lead to the last missing corner: the (unified)
theory of motion that takes into account all limits of nature. Only this theory is a com-
plete and unified description of nature. Since we already know all limits to motion, in
order to arrive at the last level, we do not need new experiments. We do not need new
knowledge. We only have to advance, in the right direction, with careful thinking. And
we can start from three different points. This is the topic of the last volume of our adven-
ture.
320 12 quantum physics in a nutshell – again

The intense emotions due to quantum field theory and general
relativity
It is sometimes deemed chic to pretend that the adventure is over at the stage we have
just reached,* the third level of Figure 181. The reasoning given is as follows. If we change
Page 316 the values of the unexplained constants in the millennium list of Table 26 only ever so
Ref. 270 slightly, nature would look completely different from what it does. Indeed, these con-
sequences have been studied in great detail; an overview of the connections is given in
the following table.

TA B L E 27 A selection of the consequences of changing the properties of nature.

O b s e r va b l e Change R e s u lt

Local quantities, from quantum theory
𝛼em smaller: Only short lived, smaller and hotter stars; no Sun.

Motion Mountain – The Adventure of Physics
larger: Darker Sun, animals die of electromagnetic radiation,
too much proton decay, no planets, no stellar
explosions, no star formation, no galaxy formation.
+60 %: Quarks decay into leptons.
+200 %: Proton-proton repulsion makes nuclei impossible.
𝛼w −50 %: Carbon nucleus unstable.
very weak: No hydrogen, no p-p cycle in stars, no C-N-O cycle.
+2 %: No protons from quarks.
𝐺𝐹 𝑚2𝑒 ≉ Either no or only helium in the universe.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
√𝐺𝑚2𝑒 :
much larger: No stellar explosions, faster stellar burning.
𝛼s −9 %: No deuteron, stars far less bright.
−1 %: No C resonance, no life.
+3.4 %: Diproton stable, faster star burning.
much larger: Carbon unstable, heavy nuclei unstable, widespread
leukaemia.
n-p mass larger: Neutron decays in proton inside nuclei; no elements.
difference
smaller: Free neutron not unstable, all protons into neutrons
during big bang; no elements.
smaller than Protons would capture electrons, no hydrogen atoms,
𝑚𝑒 : star life much shorter.
𝑚l changes:
* Actually this attitude is not new. Only the arguments have changed. Maybe the greatest physicist ever,
Ref. 269 James Clerk Maxwell, already fought against this attitude over a hundred years ago: ‘The opinion seems to
have got abroad that, in a few years, all great physical constants will have been approximately estimated, and
that the only occupation which will be left to men of science will be to carry these measurements to another
place of decimals. [...] The history of science shows that even during that phase of her progress in which
she devotes herself to improving the accuracy of the numerical measurement of quantities with which she
has long been familiar, she is preparing the materials for the subjugation of new regions, which would have
remained unknown if she had been contented with the rough methods of her early pioneers.’
12 quantum physics in a nutshell – again 321

TA B L E 27 (Continued) A selection of the consequences of changing the properties of nature.

O b s e r va b l e Change R e s u lt

e-p mass ratio much No molecules.
different:
much smaller: No solids.
3 generations 6-8: Only helium in nature.
>8: No asymptotic freedom and confinement.
Global quantities, from general relativity
horizon size much smaller: No people.
baryon number very different: No smoothness .
much higher: No solar system.
Initial condition changes:
Moon mass smaller: Small Earth magnetic field; too much cosmic radiation;

Motion Mountain – The Adventure of Physics
widespread child skin cancer.
Moon mass larger: Large Earth magnetic field; too little cosmic radiation;
no evolution into humans.
Sun’s mass smaller: Too cold for the evolution of life.
Sun’s mass larger: Sun too short lived for the evolution of life.
Jupiter mass smaller: Too many comet impacts on Earth; extinction of
animal life.
Jupiter mass larger: Too little comet impacts on Earth; no Moon; no
dinosaur extinction.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Oort cloud smaller: No comets; no irregular asteroids; no Moon; still
object number dinosaurs.
galaxy centre smaller: Irregular planet motion; supernova dangers.
distance
initial cosmic +0.1 %: 1000 times faster universe expansion.
speed
−0.0001 %: Universe recollapses after 10 000 years.
vacuum energy change by No flatness.
density 10−55 :
3 + 1 dimensions different: No atoms, no planetary systems.
Local structures, from quantum theory
permutation none: No matter.
symmetry
Lorentz symmetry none: No communication possible.
U(1) different: No Huygens principle, no way to see anything.
SU(2) different: No radioactivity, no Sun, no life.
SU(3) different: No stable quarks and nuclei.
Global structures, from general relativity
322 12 quantum physics in a nutshell – again

TA B L E 27 (Continued) A selection of the consequences of changing the properties of nature.

O b s e r va b l e Change R e s u lt

topology other: Unknown; possibly correlated γ ray bursts or star
images at the antipodes.

Note. Some researchers speculate that the whole of Table 27 can be condensed into a single sen-
Ref. 271 tence: if any parameter in nature is changed, the universe would either have too many or too
Challenge 180 r few black holes. However, the proof of this condensed summary is not complete yet. But it is a
beautiful hypothesis.

The effects of changing nature that are listed in Table 27 lead us to a profound ex-
perience: even the tiniest changes in the properties of nature are incompatible with our
existence. What does this experience mean? Answering this question too rapidly is dan-
gerous. Many have fallen into one of several traps:

Motion Mountain – The Adventure of Physics
— The first trap is to deduce, incorrectly, that the unexplained numbers and other prop-
erties from the millennium list do not need to or even cannot be explained, i.e., de-
duced from more general principles.
— The second trap is to deduce, incorrectly, that the universe has been created or de-
signed.
— The third trap is to deduce, incorrectly, that the universe is designed for people.
— The fourth trap is to deduce, incorrectly, that the universe is one of many.
All these traps are irrational and incorrect beliefs. All these beliefs have in common that

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
they have no factual basis, that they discourage further search and that they sell many
books.
The first trap is due to a combination of pessimism and envy; it is a type of wish-
ful thinking. But wishful thinking has no place in the study of motion. The second trap
Vol. VI, page 374 works because many physicists incorrectly speak of fine tuning in nature. Many research-
ers succumb to the belief in ‘creation’ and are unable to steer clear from the logical errors
Vol. III, page 330 contained in it. We discussed them earlier on. The third trap, is often, again incorrectly,
called the anthropic principle. The name is a mistake, because we saw that the anthropic
Vol. III, page 337 principle is indistinguishable both from the simian principle and from the simple request
that statements be based on observations. Around 2000, the third trap has even become
fashionable among frustrated particle theorists. The fourth trap, the belief in multiple
universes, is a minority view, but sells many books. Most people that hold this view are
found in institutions. And that is indeed where they belong.
Stopping our adventure, our mountain ascent, with an incorrect belief at the present
stage is not different from doing so directly at the beginning. Such a choice has been
taken in various societies that lacked the passion for rational investigation, and still is
taken in circles that discourage the use of reason among their members. Looking for
beliefs instead of looking for answers means to give up our ascent while pretending to
have reached the top. Every such case is a tragedy, sometimes a small one, sometimes a
larger one.
In fact, Table 27 purveys only one message: all evidence implies that we are only a tiny
part of the universe, but that we are linked with all other aspects of it. Due to our small
12 quantum physics in a nutshell – again 323

size and due to all the connections with our environment, any imagined tiny change
would make us disappear, like a water droplet is swept away by large wave. Our walk has
repeatedly reminded us of this smallness and dependence, and overwhelmingly does so
again at this point.
In our adventure, accepting the powerful message of Table 27 is one of the most awe-
inspiring, touching and motivating moments. It shows clearly how vast the universe is. It
also shows how much we are dependent on many different and distant aspects of nature.
Having faced this powerful experience, everybody has to make up his or her own mind
Challenge 181 s on whether to proceed with the adventure or not. Of course, there is no obligation to do
so.

What awaits us?
Assuming that you have decided to continue the adventure, it is natural to ask what
awaits you. The shortness of the millennium list of unexplained aspects of nature, given
in Table 26, means that no additional experimental data are available as check of the final

Motion Mountain – The Adventure of Physics
Page 316
description of nature. Everything we need to arrive at the final description of motion will
be deduced from the experimental data given in the millennium list, and from nothing
else. In other words, future experiments will not help us – except if they change some-
thing in the millennium list. Accelerator experiments might do this with the particle list
or astronomical experiments with the topology issue. Fantasy provides no limits; fortu-
nately, nature does.
The lack of new experimental data means that to continue the walk is a conceptual
adventure only. Nevertheless, storms rage near the top of Motion Mountain. We have to
walk keeping our eyes open, without any other guidance except our reason. This is not an

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
adventure of action, but an adventure of the mind. And it is a fascinating one, as we shall
soon find out. To provide an impression of what awaits us, we rephrase the remaining
issues in five simple challenges.
1 – What determines colours? In other words, what relations of nature fix the famous
fine structure constant? Like the hero of Douglas Adams’ books, physicists know the
answer to the greatest of questions: it is 137.036. But they do not know the question.
2 – What fixes the contents of a teapot? It is given by its size to the third power. But
why are there only three dimensions? Why is the tea content limited in this way?
3 – Was Democritus right? Our adventure has confirmed his statement up to this
point: nature is indeed well described by the concepts of particle and of vacuum. At
large scales, relativity has added a horizon, and at small scales, quantum field theory
added vacuum energy and pair creation. Nevertheless, both theories assume the exist-
ence of particles and the existence of space-time, and neither predicts them. Even worse,
both theories completely fail to predict the existence of any of the properties either of
space-time – such as its dimensionality – or of particles – such as their masses and other
quantum numbers. A lot is missing.
4 – Was Democritus wrong? It is often said that the standard model has only about
twenty unknown parameters; this common mistake negates about 1093 initial conditions!
To get an idea of the problem, we simply estimate the number 𝑁 of possible states of all
324 12 quantum physics in a nutshell – again

particles in the universe by
𝑁=𝑛𝑣𝑑𝑝𝑓 (133)

where 𝑛 is the number of particles, 𝑣 is the number of variables (position, momentum,
spin), 𝑑 is the number of different values each of them can take (limited by the maximum
of 61 decimal digits), 𝑝 is the number of visible space-time points (about 10183 ) and 𝑓
is a factor expressing how many of all these initial conditions are actually independent
of each other. We thus get the following number of possible states of all particles in the
universe:
𝑁 = 1092 ⋅ 8 ⋅ 1061 ⋅ 10183 ⋅ 𝑓 = 10336 ⋅ 𝑓 (134)

from which the 1093 initial conditions have to be explained. But no explanation is known.
Worse, there is also the additional problem that we know nothing whatsoever about 𝑓.
Its value could be 0, if all data were interdependent, or 1, if none were. Even worse, above
Vol. IV, page 169 we noted that initial conditions cannot be defined for the universe at all; thus 𝑓 should

Motion Mountain – The Adventure of Physics
be undefined and not be a number at all! Whatever the case, we need to understand how
all the visible particles acquire their present 1093 states.
Vol. I, page 438 5 – Were our efforts up to this point in vain? Quite at the beginning of our walk we
noted that in classical physics, space and time are defined using matter, whereas matter
is defined using space-time. Hundred years of general relativity and of quantum theory,
including dozens of geniuses, have not solved this oldest paradox of all. The issue is still
Challenge 182 e open at this point of our walk, as you might want to check by yourself.
The answers to these five challenges define the goal of our adventure: the top of Motion
Mountain. Answering the five challenges means to know everything about motion. It

⊳ What is motion?

In short, our quest for the unravelling of the essence of motion gets really interesting
from this point onwards!

“
That is why Leucippus and Democritus, who
say that the atoms move always in the void and
the unlimited, must say what movement is, and

”
in what their natural motion consists.
Ref. 272 Aristotle, Treaty of the Heaven
Appendix A

U N I T S , M E A SU R E M E N T S A N D
C ON STA N T S

M
easurements are comparisons with standards. Standards are based on units.
any different systems of units have been used throughout the world.
ost of these standards confer power to the organization in charge of them.
Such power can be misused; this is the case today, for example in the computer in-

Motion Mountain – The Adventure of Physics
dustry, and was so in the distant past. The solution is the same in both cases: organize
an independent and global standard. For measurement units, this happened in the
eighteenth century: in order to avoid misuse by authoritarian institutions, to eliminate
problems with differing, changing and irreproducible standards, and – this is not a joke
– to simplify tax collection and to make it more just, a group of scientists, politicians
and economists agreed on a set of units. It is called the Système International d’Unités,
abbreviated SI, and is defined by an international treaty, the ‘Convention du Mètre’.
The units are maintained by an international organization, the ‘Conférence Générale
des Poids et Mesures’, and its daughter organizations, the ‘Commission Internationale
des Poids et Mesures’ and the ‘Bureau International des Poids et Mesures’ (BIPM). All

SI units
All SI units are built from seven base units. Their simplest definitions, translated from
French into English, are the following ones, together with the dates of their formulation
and a few comments:
‘The second is the duration of 9 192 631 770 periods of the radiation corresponding
to the transition between the two hyperfine levels of the ground state of the caesium 133
atom.’ (1967) The 2019 definition is equivalent, but much less clear.*
‘The metre is the length of the path travelled by light in vacuum during a time inter-
val of 1/299 792 458 of a second.’ (1983) The 2019 definition is equivalent, but much less
clear.*
‘The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed
numerical value of the Planck constant h to be 6.626 070 15 ⋅ 10−34 when expressed in the
unit J ⋅ s, which is equal to kg ⋅ m2 ⋅ s−1 .’ (2019)*
‘The ampere, symbol A, is the SI unit of electric current. It is defined by taking the
fixed numerical value of the elementary charge e to be 1.602 176 634 ⋅ 10−19 when ex-
pressed in the unit C, which is equal to A ⋅ s.’ (2019)* This definition is equivalent to:
One ampere is 6.241 509 074... ⋅ 1018 elementary charges per second.
‘The kelvin, symbol K, is the SI unit of thermodynamic temperature. It is defined by
326 a units, measurements and constants

taking the fixed numerical value of the Boltzmann constant 𝑘 to be 1.380649 ⋅10−23 when
expressed in the unit J ⋅ K−1 .’ (2019)*
‘The mole, symbol mol, is the SI unit of amount of substance. One mole contains
exactly 6.02214076 ⋅ 1023 elementary entities.’ (2019)*
‘The candela is the luminous intensity, in a given direction, of a source that emits
monochromatic radiation of frequency 540 ⋅ 1012 hertz and has a radiant intensity in
that direction of (1/683) watt per steradian.’ (1979) The 2019 definition is equivalent, but
much less clear.*
We note that both time and length units are defined as certain properties of a standard
example of motion, namely light. In other words, also the Conférence Générale des Poids
et Mesures makes the point that the observation of motion is a prerequisite for the defin-
ition and construction of time and space. Motion is the fundament of every observation
and of all measurement. By the way, the use of light in the definitions had been proposed
already in 1827 by Jacques Babinet.**

Motion Mountain – The Adventure of Physics
From these basic units, all other units are defined by multiplication and division. Thus,
all SI units have the following properties:
SI units form a system with state-of-the-art precision: all units are defined with a pre-
cision that is higher than the precision of commonly used measurements. Moreover, the
precision of the definitions is regularly being improved. The present relative uncertainty
of the definition of the second is around 10−14 , for the metre about 10−10 , for the kilo-
gram about 10−9 , for the ampere 10−7 , for the mole less than 10−6 , for the kelvin 10−6 and
for the candela 10−3 .
SI units form an absolute system: all units are defined in such a way that they can

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
be reproduced in every suitably equipped laboratory, independently, and with high pre-
cision. This avoids as much as possible any error or misuse by the standard-setting or-
ganization. In fact, the SI units are as now as near as possible to Planck’s natural units,
which are presented below. In practice, the SI is now an international standard defining
the numerical values of the seven constants Δ𝜈Cs , 𝑐, ℏ, 𝑒, 𝑘, 𝑁A and 𝐾cd . After over 200
years of discussions, the CGPM has little left to do.
SI units form a practical system: the base units are quantities of everyday magnitude.
Frequently used units have standard names and abbreviations. The complete list includes
the seven base units just given, the supplementary units, the derived units and the ad-
mitted units.
The supplementary SI units are two: the unit for (plane) angle, defined as the ratio
of arc length to radius, is the radian (rad). For solid angle, defined as the ratio of the
subtended area to the square of the radius, the unit is the steradian (sr).
The derived units with special names, in their official English spelling, i.e., without
capital letters and accents, are:

* The symbols of the seven units are s, m, kg, A, K, mol and cd. The full offical definitions are found at
Ref. 274 www.bipm.org. For more details about the levels of the caesium atom, consult a book on atomic physics.
The Celsius scale of temperature 𝜃 is defined as: 𝜃/°C = 𝑇/K − 273.15; note the small difference with the
number appearing in the definition of the kelvin. In the definition of the candela, the frequency of the light
corresponds to 555.5 nm, i.e., green colour, around the wavelength to which the eye is most sensitive.
** Jacques Babinet (1794–1874), French physicist who published important work in optics.
a units, measurements and constants 327

Name A bbre v iat i o n Name A b b r e v i at i o n

hertz Hz = 1/s newton N = kg m/s2
pascal Pa = N/m2 = kg/m s2 joule J = Nm = kg m2 /s2
watt W = kg m2 /s3 coulomb C = As
volt V = kg m2 /As3 farad F = As/V = A2 s4 /kg m2
ohm Ω = V/A = kg m2 /A2 s3 siemens S = 1/Ω
weber Wb = Vs = kg m2 /As2 tesla T = Wb/m2 = kg/As2 = kg/Cs
henry H = Vs/A = kg m2 /A2 s2 degree Celsius °C (see definition of kelvin)
lumen lm = cd sr lux lx = lm/m2 = cd sr/m2
becquerel Bq = 1/s gray Gy = J/kg = m2 /s2
sievert Sv = J/kg = m2 /s2 katal kat = mol/s

The admitted non-SI units are minute, hour, day (for time), degree 1° = π/180 rad,
minute 1 󸀠 = π/10 800 rad, second 1 󸀠󸀠 = π/648 000 rad (for angles), litre, and tonne. All

Motion Mountain – The Adventure of Physics
other units are to be avoided.
All SI units are made more practical by the introduction of standard names and ab-
breviations for the powers of ten, the so-called prefixes:*

Power Name Power Name Power Name Power Name
1 −1 18
10 deca da 10 deci d 10 Exa E 10−18 atto a
102 hecto h 10−2 centi c 1021 Zetta Z 10−21 zepto z
103 kilo k 10−3 milli m 1024 Yotta Y 10−24 yocto y

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
106 Mega M 10−6 micro μ unofficial: Ref. 275
109 Giga G 10−9 nano n 1027 Xenta X 10−27 xenno x
1012 Tera T 10−12 pico p 1030 Wekta W 10−30 weko w
1015 Peta P 10−15 femto f 1033 Vendekta V 10−33 vendeko v
1036 Udekta U 10−36 udeko u

SI units form a complete system: they cover in a systematic way the full set of ob-
servables of physics. Moreover, they fix the units of measurement for all other sciences
as well.
SI units form a universal system: they can be used in trade, in industry, in commerce,

* Some of these names are invented (yocto to sound similar to Latin octo ‘eight’, zepto to sound similar
to Latin septem, yotta and zetta to resemble them, exa and peta to sound like the Greek words ἑξάκις and
πεντάκις for ‘six times’ and ‘five times’, the unofficial ones to sound similar to the Greek words for nine,
ten, eleven and twelve); some are from Danish/Norwegian (atto from atten ‘eighteen’, femto from femten
‘fifteen’); some are from Latin (from mille ‘thousand’, from centum ‘hundred’, from decem ‘ten’, from
nanus ‘dwarf’); some are from Italian (from piccolo ‘small’); some are Greek (micro is from μικρός ‘small’,
deca/deka from δέκα ‘ten’, hecto from ἑκατόν ‘hundred’, kilo from χίλιοι ‘thousand’, mega from μέγας
‘large’, giga from γίγας ‘giant’, tera from τέρας ‘monster’).
Translate: I was caught in such a traffic jam that I needed a microcentury for a picoparsec and that my
Challenge 183 e car’s fuel consumption was two tenths of a square millimetre.
328 a units, measurements and constants

at home, in education and in research. They could even be used by extraterrestrial civil-
izations, if they existed.
SI units form a self-consistent system: the product or quotient of two SI units is also
an SI unit. This means that in principle, the same abbreviation, e.g. ‘SI’, could be used
for every unit.
The SI units are not the only possible set that could fulfil all these requirements, but they
are the only existing system that does so.*

The meaning of measurement
Every measurement is a comparison with a standard. Therefore, any measurement re-
Challenge 184 e quires matter to realize the standard (even for a speed standard), and radiation to achieve
the comparison. The concept of measurement thus assumes that matter and radiation ex-
ist and can be clearly separated from each other.
Every measurement is a comparison. Measuring thus implies that space and time ex-

Motion Mountain – The Adventure of Physics
ist, and that they differ from each other.
Every measurement produces a measurement result. Therefore, every measurement
implies the storage of the result. The process of measurement thus implies that the situ-
ation before and after the measurement can be distinguished. In other terms, every meas-
urement is an irreversible process.
Every measurement is a process. Thus every measurement takes a certain amount of
time and a certain amount of space.
All these properties of measurements are simple but important. Beware of anybody
who denies them.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Planck ’ s natural units
Since the exact form of many equations depends on the system of units used, theoretical
physicists often use unit systems optimized for producing simple equations. The chosen
units and the values of the constants of nature are related. In microscopic physics, the
system of Planck’s natural units is frequently used. They are defined by setting 𝑐 = 1, ℏ =
1, 𝐺 = 1, 𝑘 = 1, 𝜀0 = 1/4π and 𝜇0 = 4π. Planck units are thus defined from combinations
of fundamental constants; those corresponding to the fundamental SI units are given in
Table 29.** The table is also useful for converting equations written in natural units back
Challenge 185 e to SI units: just substitute every quantity 𝑋 by 𝑋/𝑋Pl.

* Apart from international units, there are also provincial units. Most provincial units still in use are of
Roman origin. The mile comes from milia passum, which used to be one thousand (double) strides of about
1480 mm each; today a nautical mile, once defined as minute of arc on the Earth’s surface, is defined as
exactly 1852 m. The inch comes from uncia/onzia (a twelfth – now of a foot). The pound (from pondere ‘to
weigh’) is used as a translation of libra – balance – which is the origin of its abbreviation lb. Even the habit
of counting in dozens instead of tens is Roman in origin. These and all other similarly funny units – like
the system in which all units start with ‘f’, and which uses furlong/fortnight as its unit of velocity – are now
officially defined as multiples of SI units.
** The natural units 𝑥Pl given here are those commonly used today, i.e., those defined using the constant
ℏ, and not, as Planck originally did, by using the constant ℎ = 2πℏ. The electromagnetic units can also be
defined with other factors than 4π𝜀0 in the expressions: for example, using 4π𝜀0 𝛼, with the fine-structure
Vol. IV, page 196 constant 𝛼, gives 𝑞Pl = 𝑒. For the explanation of the numbers between brackets, see below.
a units, measurements and constants 329

TA B L E 29 Planck’s (uncorrected) natural units.

Name Definition Va l u e

Basic units
the Planck length 𝑙Pl = √ℏ𝐺/𝑐3 = 1.616 0(12) ⋅ 10−35 m
the Planck time 𝑡Pl = √ℏ𝐺/𝑐5 = 5.390 6(40) ⋅ 10−44 s
the Planck mass 𝑚Pl = √ℏ𝑐/𝐺 = 21.767(16) μg
6
the Planck current 𝐼Pl = √4π𝜀0 𝑐 /𝐺 = 3.479 3(22) ⋅ 1025 A
the Planck temperature 𝑇Pl = √ℏ𝑐5 /𝐺𝑘2 = 1.417 1(91) ⋅ 1032 K

Trivial units
the Planck velocity 𝑣Pl = 𝑐 = 0.3 Gm/s
the Planck angular momentum 𝐿 Pl = ℏ = 1.1 ⋅ 10−34 Js

Motion Mountain – The Adventure of Physics
the Planck action 𝑆aPl = ℏ = 1.1 ⋅ 10−34 Js
the Planck entropy 𝑆ePl = 𝑘 = 13.8 yJ/K
Composed units
the Planck mass density 𝜌Pl = 𝑐5 /𝐺2 ℏ = 5.2 ⋅ 1096 kg/m3
the Planck energy 𝐸Pl = √ℏ𝑐5 /𝐺 = 2.0 GJ = 1.2 ⋅ 1028 eV
the Planck momentum 𝑝Pl = √ℏ𝑐3 /𝐺 = 6.5 Ns
5
the Planck power 𝑃Pl = 𝑐 /𝐺 = 3.6 ⋅ 1052 W
the Planck force 𝐹Pl = 𝑐4 /𝐺 = 1.2 ⋅ 1044 N

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
the Planck pressure 𝑝Pl = 𝑐7 /𝐺ℏ = 4.6 ⋅ 10113 Pa
the Planck acceleration 𝑎Pl = √𝑐7 /ℏ𝐺 = 5.6 ⋅ 1051 m/s2
the Planck frequency 𝑓Pl = √𝑐5 /ℏ𝐺 = 1.9 ⋅ 1043 Hz
the Planck electric charge 𝑞Pl = √4π𝜀0 𝑐ℏ = 1.9 aC = 11.7 e
4
the Planck voltage 𝑈Pl = √𝑐 /4π𝜀0 𝐺 = 1.0 ⋅ 1027 V
the Planck resistance 𝑅Pl = 1/4π𝜀0 𝑐 = 30.0 Ω
3
the Planck capacitance 𝐶Pl = 4π𝜀0 √ℏ𝐺/𝑐 = 1.8 ⋅ 10−45 F
the Planck inductance 𝐿 Pl = (1/4π𝜀0 )√ℏ𝐺/𝑐7 = 1.6 ⋅ 10−42 H
the Planck electric field 𝐸Pl = √𝑐7 /4π𝜀0 ℏ𝐺2 = 6.5 ⋅ 1061 V/m
the Planck magnetic flux density 𝐵Pl = √𝑐5 /4π𝜀0 ℏ𝐺2 = 2.2 ⋅ 1053 T

The natural units are important for another reason: whenever a quantity is sloppily called
‘infinitely small (or large)’, the correct expression is ‘as small (or as large) as the corres-
ponding corrected Planck unit’. As explained throughout the text, and especially in the
Vol. VI, page 37 final part, this substitution is possible because almost all Planck units provide, within
a correction factor of order 1, the extremal value for the corresponding observable –
some an upper and some a lower limit. Unfortunately, these correction factors are not
yet widely known. The exact extremal value for each observable in nature is obtained
330 a units, measurements and constants

when 𝐺 is substituted by 4𝐺 and 4π𝜀0 by 4π𝜀0 𝛼 in all Planck quantities. These extremal
values, or corrected Planck units, are the true natural units. To exceed the extremal values
Challenge 186 s is possible only for some extensive quantities. (Can you find out which ones?)

Other unit systems
A central aim of research in high-energy physics is the calculation of the strengths of
all interactions; therefore it is not practical to set the gravitational constant 𝐺 to unity,
as in the Planck system of units. For this reason, high-energy physicists often only set
𝑐 = ℏ = 𝑘 = 1 and 𝜇0 = 1/𝜀0 = 4π,* leaving only the gravitational constant 𝐺 in the
equations.
In this system, only one fundamental unit exists, but its choice is free. Often a stand-
ard length is chosen as the fundamental unit, length being the archetype of a measured
quantity. The most important physical observables are then related by

1/[𝑙2 ] = [𝐸]2 = [𝐹] = [𝐵] = [𝐸electric] ,

Motion Mountain – The Adventure of Physics
1/[𝑙] = [𝐸] = [𝑚] = [𝑝] = [𝑎] = [𝑓] = [𝐼] = [𝑈] = [𝑇] ,
1 = [𝑣] = [𝑞] = [𝑒] = [𝑅] = [𝑆action] = [𝑆entropy ] = ℏ = 𝑐 = 𝑘 = [𝛼] , (135)
[𝑙] = 1/[𝐸] = [𝑡] = [𝐶] = [𝐿] and
[𝑙]2 =1/[𝐸]2 = [𝐺] = [𝑃]

where we write [𝑥] for the unit of quantity 𝑥. Using the same unit for time, capacitance
and inductance is not to everybody’s taste, however, and therefore electricians do not
use this system.**
Often, in order to get an impression of the energies needed to observe an effect un-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
der study, a standard energy is chosen as fundamental unit. In particle physics the most
common energy unit is the electronvolt (eV), defined as the kinetic energy acquired by
an electron when accelerated by an electrical potential difference of 1 volt (‘protonvolt’
would be a better name). Therefore one has 1 eV = 1.6 ⋅ 10−19 J, or roughly
1
1 eV ≈ 6
aJ (136)

which is easily remembered. The simplification 𝑐 = ℏ = 1 yields 𝐺 = 6.9 ⋅ 10−57 eV−2 and
allows one to use the unit eV also for mass, momentum, temperature, frequency, time
Challenge 187 e and length, with the respective correspondences 1 eV ≡ 1.8 ⋅ 10−36 kg ≡ 5.4 ⋅ 10−28 Ns
≡ 242 THz ≡ 11.6 kK and 1 eV−1 ≡ 4.1 fs ≡ 1.2 μm.

* Other definitions for the proportionality constants in electrodynamics lead to the Gaussian unit system
often used in theoretical calculations, the Heaviside–Lorentz unit system, the electrostatic unit system, and
Ref. 276 the electromagnetic unit system, among others.
** In the list, 𝑙 is length, 𝐸 energy, 𝐹 force, 𝐸electric the electric and 𝐵 the magnetic field, 𝑚 mass, 𝑝 momentum,
𝑎 acceleration, 𝑓 frequency, 𝐼 electric current, 𝑈 voltage, 𝑇 temperature, 𝑣 speed, 𝑞 charge, 𝑅 resistance, 𝑃
power, 𝐺 the gravitational constant.
The web page www.chemie.fu-berlin.de/chemistry/general/units_en.html provides a tool to convert
various units into each other.
Researchers in general relativity often use another system, in which the Schwarzschild radius 𝑟s =
2𝐺𝑚/𝑐2 is used to measure masses, by setting 𝑐 = 𝐺 = 1. In this case, mass and length have the same
dimension, and ℏ has the dimension of an area.
a units, measurements and constants 331

To get some feeling for the unit eV, the following relations are useful. Room temper-
ature, usually taken as 20°C or 293 K, corresponds to a kinetic energy per particle of
0.025 eV or 4.0 zJ. The highest particle energy measured so far belongs to a cosmic ray
Ref. 277 with an energy of 3 ⋅ 1020 eV or 48 J. Down here on the Earth, an accelerator able to pro-
duce an energy of about 105 GeV or 17 nJ for electrons and antielectrons has been built,
and one able to produce an energy of 14 TeV or 2.2 μJ for protons will be finished soon.
Both are owned by CERN in Geneva and have a circumference of 27 km.
The lowest temperature measured up to now is 280 pK, in a system of rhodium
Ref. 278 nuclei held inside a special cooling system. The interior of that cryostat may even be
the coolest point in the whole universe. The kinetic energy per particle correspond-
ing to that temperature is also the smallest ever measured: it corresponds to 24 feV or
3.8 vJ = 3.8 ⋅ 10−33 J. For isolated particles, the record seems to be for neutrons: kinetic
energies as low as 10−7 eV have been achieved, corresponding to de Broglie wavelengths
of 60 nm.

Motion Mountain – The Adventure of Physics
Curiosities and fun challenges ab ou t units
The Planck length is roughly the de Broglie wavelength 𝜆 B = ℎ/𝑚𝑣 of a man walking
Ref. 279 comfortably (𝑚 = 80 kg, 𝑣 = 0.5 m/s); this motion is therefore aptly called the ‘Planck
stroll.’
∗∗
The Planck mass is equal to the mass of about 1019 protons. This is roughly the mass of
a human embryo at about ten days of age.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
∗∗
The most precisely measured quantities in nature are the frequencies of certain milli-
Ref. 280 second pulsars, the frequency of certain narrow atomic transitions, and the Rydberg
constant of atomic hydrogen, which can all be measured as precisely as the second is
defined. The caesium transition that defines the second has a finite line width that limits
the achievable precision: the limit is about 14 digits.
∗∗
The most precise clock ever built, using microwaves, had a stability of 10−16 during a
Ref. 281 running time of 500 s. For longer time periods, the record in 1997 was about 10−15 ; but
Ref. 282 values around 10−17 seem within technological reach. The precision of clocks is limited
for short measuring times by noise, and for long measuring times by drifts, i.e., by sys-
tematic effects. The region of highest stability depends on the clock type; it usually lies
between 1 ms for optical clocks and 5000 s for masers. Pulsars are the only type of clock
for which this region is not known yet; it certainly lies at more than 20 years, the time
elapsed at the time of writing since their discovery.
∗∗
The shortest times measured are the lifetimes of certain ‘elementary’ particles. In par-
Ref. 283 ticular, the lifetime of certain D mesons have been measured at less than 10−23 s. Such
times are measured using a bubble chamber, where the track is photographed. Can you
332 a units, measurements and constants

Challenge 188 s estimate how long the track is? (This is a trick question – if your length cannot be ob-
served with an optical microscope, you have made a mistake in your calculation.)
∗∗
The longest times encountered in nature are the lifetimes of certain radioisotopes, over
1015 years, and the lower limit of certain proton decays, over 1032 years. These times are
thus much larger than the age of the universe, estimated to be fourteen thousand million
Ref. 284 years.
∗∗
There is a unit for the spicy heat of chili peppers, officially called the pungency. The pun-
gency is due to an organic compound called capsaicin. If you multiply by 16 the capsaicin
concentration in parts per million, you get the Scoville heat unit for chili peppers. A few
extreme chili varieties exceed the value of 2 million Scoville units.

Motion Mountain – The Adventure of Physics
Precision and accuracy of measurements
Measurements are the basis of physics. Every measurement has an error. Errors are due
to lack of precision or to lack of accuracy. Precision means how well a result is reproduced
when the measurement is repeated; accuracy is the degree to which a measurement cor-
responds to the actual value.
Lack of precision is due to accidental or random errors; they are best measured by the
standard deviation, usually abbreviated 𝜎; it is defined through

1 𝑛
𝜎2 = ∑(𝑥 − 𝑥)̄ 2 ,

where 𝑥̄ is the average of the measurements 𝑥𝑖 . (Can you imagine why 𝑛 − 1 is used in
Challenge 189 s the formula instead of 𝑛?)
For most experiments, the distribution of measurement values tends towards a nor-
mal distribution, also called Gaussian distribution, whenever the number of measure-
ments is increased. The distribution, shown in Figure 182, is described by the expression

(𝑥−𝑥)̄ 2
𝑁(𝑥) ≈ e− 2𝜎2 . (138)

The square 𝜎2 of the standard deviation is also called the variance. For a Gaussian distri-
Challenge 190 e bution of measurement values, 2.35𝜎 is the full width at half maximum.
Lack of accuracy is due to systematic errors; usually these can only be estimated. This
estimate is often added to the random errors to produce a total experimental error, some-
Ref. 285 times also called total uncertainty. The relative error or uncertainty is the ratio between
the error and the measured value.
For example, a professional measurement will give a result such as 0.312(6) m. The
number between the parentheses is the standard deviation 𝜎, in units of the last digits.
As above, a Gaussian distribution for the measurement results is assumed. Therefore, a
Challenge 191 e value of 0.312(6) m implies that the actual value is expected to lie
a units, measurements and constants 333

N
number of measurements

standard deviation

full width at half maximum
(FWHM)

limit curve for a large number
of measurements: the
Gaussian distribution

x x
average value measured values

Motion Mountain – The Adventure of Physics
F I G U R E 182 A precision experiment and its measurement distribution. The precision is high if the
width of the distribution is narrow; the accuracy is high if the centre of the distribution agrees with the
actual value.

— within 1𝜎 with 68.3 % probability, thus in this example within 0.312 ± 0.006 m;
— within 2𝜎 with 95.4 % probability, thus in this example within 0.312 ± 0.012 m;
— within 3𝜎 with 99.73 % probability, thus in this example within 0.312 ± 0.018 m;
— within 4𝜎 with 99.9937 % probability, thus in this example within 0.312 ± 0.024 m;

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
— within 5𝜎 with 99.999 943 % probability, thus in this example within 0.312 ± 0.030 m;
— within 6𝜎 with 99.999 999 80 % probability, thus within 0.312 ± 0.036 m;
— within 7𝜎 with 99.999 999 999 74 % probability, thus within 0.312 ± 0.041 m.

However, these numbers are much too precise and should be taken with a grain of salt.
Note that standard deviations have one digit; you must be a world expert to use two,
and a fool to use more. If no standard deviation is given, a (1) is assumed. As a result,
among professionals, 1 km and 1000 m are not the same length!
What happens to the errors when two measured values 𝐴 and 𝐵 are added or subtrac-
ted? If the all measurements are independent – or uncorrelated – the standard deviation
of the sum and that of difference is given by 𝜎 = √𝜎𝐴2 + 𝜎𝐵2 . For both the product or ratio
of two measured and uncorrelated values 𝐶 and 𝐷, the result is 𝜌 = √𝜌𝐶2 + 𝜌𝐷2 , where the
𝜌 terms are the relative standard deviations.

Limits to precision
What are the limits to accuracy and precision? There is no way, even in principle, to
measure a length 𝑥 to a precision higher than about 61 digits, because in nature, the ratio
between the largest and the smallest measurable length is Δ𝑥/𝑥 > 𝑙Pl/𝑑horizon = 10−61 .
Challenge 192 e (Is this ratio valid also for force or for volume?) In the final volume of our text, studies
Vol. VI, page 94 of clocks and metre bars strengthen this theoretical limit.
But it is not difficult to deduce more stringent practical limits. No imaginable machine
334 a units, measurements and constants

can measure quantities with a higher precision than measuring the diameter of the Earth
within the smallest length ever measured, about 10−19 m; that is about 26 digits of preci-
sion. Using a more realistic limit of a 1000 m sized machine implies a limit of 22 digits.
If, as predicted above, time measurements really achieve 17 digits of precision, then they
are nearing the practical limit, because apart from size, there is an additional practical
restriction: cost. Indeed, an additional digit in measurement precision often means an
additional digit in equipment cost.

Physical constants
In physics, general observations are deduced from more fundamental ones. As a con-
sequence, many measurements can be deduced from more fundamental ones. The most
fundamental measurements are those of the physical constants.
The following tables give the world’s best values of the most important physical con-
stants and particle properties – in SI units and in a few other common units – as pub-
lished in the standard references. The values are the world averages of the best measure-

Motion Mountain – The Adventure of Physics
Ref. 286
ments made up to the present. As usual, experimental errors, including both random
and estimated systematic errors, are expressed by giving the standard deviation in the
last digits. In fact, behind each of the numbers in the following tables there is a long
Ref. 287 story which is worth telling, but for which there is not enough room here.
In principle, all quantitative properties of matter can be calculated with quantum the-
Page 261 ory – more precisely, equations of the standard model of particle – and a set of basic
physical constants that are given in the next table. For example, the colour, density and
elastic properties of any material can be predicted, in principle, in this way.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Constants that define the SI measurement units
Vacuum speed of light 𝑐 𝑐 299 792 458 m/s 0
Original Planck constant 𝑐 ℎ 6.626 070 15 ⋅ 10−34 Js 0
Reduced Planck constant, ℏ 1.054 571 817 ... ⋅ 10−34 Js 0
quantum of action
Positron charge 𝑐 𝑒 0.160 217 6634 aC 0
Boltzmann constant 𝑐 𝑘 1.380 649 ⋅ 10−23 J/K 0
Avogadro’s number 𝑁A 6.022 140 76 ⋅ 1023 1/mol 0
Constant that should define the SI measurement units
Gravitational constant 𝐺 6.674 30(15) ⋅ 10−11 Nm2 /kg2 2.2 ⋅ 10−5
Other fundamental constants
Number of space-time dimensions 3+1 0𝑏
2
Fine-structure constant 𝑑 or 𝛼 = 4π𝜀𝑒 ℏ𝑐 1/137.035 999 084(21) 1.5 ⋅ 10−10
0

e.m. coupling constant = 𝑔em (𝑚2e 𝑐2 ) = 0.007 297 352 5693(11) 1.5 ⋅ 10−10
Fermi coupling constant 𝑑 or 𝐺F /(ℏ𝑐)3 1.166 3787(6) ⋅ 10−5 GeV−2 5.1 ⋅ 10−7
weak coupling constant 𝛼w (𝑀Z ) = 𝑔w2 /4π 1/30.1(3) 1 ⋅ 10−2
a units, measurements and constants 335

TA B L E 30 (Continued) Basic physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t. 𝑎

Strong coupling constant 𝑑 𝛼s (𝑀Z ) = 𝑔s2 /4π 0.1179(10) 8.5 ⋅ 10−3
2
Weak mixing angle sin 𝜃W (𝑀𝑆) 0.231 22(4) 1.7 ⋅ 10−4
sin2 𝜃W (on shell) 0.222 90(30) 1.3 ⋅ 10−3
2
= 1 − (𝑚W /𝑚Z )
0.97383(24) 0.2272(10) 0.00396(9)
CKM quark mixing matrix |𝑉| ( 0.2271(10) 0.97296(24) 0.04221(80) )
0.00814(64) 0.04161(78) 0.999100(34)
Jarlskog invariant 𝐽 3.08(18) ⋅ 10−5
0.82(2) 0.55(4) 0.150(7)
PMNS neutrino mixing m. |𝑃| (0.37(13) 0.57(11) 0.71(7) )
0.41(13) 0.59(10) 0.69(7)

Motion Mountain – The Adventure of Physics
Electron mass 𝑚e 9.109 383 7015(28) ⋅ 10−31 kg 3.0 ⋅ 10−10
5.485 799 090 65(16) ⋅ 10−4 u 2.9 ⋅ 10−11
0.510 998 950 00(15) MeV 3.0 ⋅ 10−10
Muon mass 𝑚μ 1.883 531 627(42) ⋅ 10 kg 2.2 ⋅ 10−8
−28

105.658 3755(23) MeV 2.2 ⋅ 10−8
Tau mass 𝑚𝜏 1.776 82(12) GeV/𝑐2 6.8 ⋅ 10−5
El. neutrino mass 𝑚𝜈e < 2 eV/𝑐2
Muon neutrino mass 𝑚𝜈𝜇 < 2 eV/𝑐2
Tau neutrino mass 𝑚𝜈𝜏 < 2 eV/𝑐2

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Up quark mass 𝑢 21.6(+0.49/ − 0.26) MeV/𝑐2
Down quark mass 𝑑 4.67(+0.48/ − 0.17) MeV/𝑐2
Strange quark mass 𝑠 93(+11/ − 5) MeV/𝑐2
Charm quark mass 𝑐 1.27(2) GeV/𝑐2
Bottom quark mass 𝑏 4.18(3) GeV/𝑐2
Top quark mass 𝑡 172.9(0.4) GeV/𝑐2
Photon mass γ < 2 ⋅ 10−54 kg
W boson mass 𝑊± 80.379(12) GeV/𝑐2
Z boson mass 𝑍0 91.1876(21) GeV/𝑐2
Higgs mass H 125.10(14) GeV/𝑐2
Gluon mass g1...8 c. 0 MeV/𝑐2

𝑎. Uncertainty: standard deviation of measurement errors.
𝑏. Measured from to 10−19 m to 1026 m.
𝑐. Defining constant.
𝑑. All coupling constants depend on the 4-momentum transfer, as explained in the section on
Page 131 renormalization. Fine-structure constant is the traditional name for the electromagnetic coup-
ling constant 𝑔em in the case of a 4-momentum transfer of 𝑄2 = 𝑚2e 𝑐2 , which is the smallest
one possible. At higher momentum transfers it has larger values, e.g., 𝑔em (𝑄2 = 𝑀W
2 2
𝑐 ) ≈ 1/128.
In contrast, the strong coupling constant has lover values at higher momentum transfers; e.g.,
𝛼s (34 GeV) = 0.14(2).
336 a units, measurements and constants

Why do all these basic constants have the values they have? For any basic constant with
a dimension, such as the quantum of action ℏ, the numerical value has only historical
meaning. It is 1.054 ⋅ 10−34 Js because of the SI definition of the joule and the second.
The question why the value of a dimensional constant is not larger or smaller therefore
always requires one to understand the origin of some dimensionless number giving the
ratio between the constant and the corresponding natural unit that is defined with 𝑐, 𝐺,
Vol. IV, page 328 𝑘, 𝑁A and ℏ. Details and values for the natural units are given in the dedicated section.
In other words, understanding the sizes of atoms, people, trees and stars, the duration
of molecular and atomic processes, or the mass of nuclei and mountains, implies under-
standing the ratios between these values and the corresponding natural units. The key to
understanding nature is thus the understanding of all measurement ratios, and thus of
all dimensionless constants. This quest, including the understanding of the fine-structure
constant 𝛼 itself, is completed only in the final volume of our adventure.
The basic constants yield the following useful high-precision observations.

Motion Mountain – The Adventure of Physics
TA B L E 31 Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Vacuum permeability 𝜇0 1.256 637 062 12(19) μH/m 1.5 ⋅ 10−10
Vacuum permittivity 𝜀0 = 1/𝜇0 𝑐2 8.854 187 8128(13) pF/m 1.5 ⋅ 10−10
Vacuum impedance 𝑍0 = √𝜇0 /𝜀0 376.730 313 668(57) Ω 1.5 ⋅ 10−10
Loschmidt’s number 𝑁L 2.686 780 111... ⋅ 1025 1/m3 0
at 273.15 K and 101 325 Pa
Faraday’s constant 𝐹 = 𝑁A 𝑒 96 485.332 12... C/mol 0

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Universal gas constant 𝑅 = 𝑁A 𝑘 8.314 462 618... J/(mol K) 0
Molar volume of an ideal gas 𝑉 = 𝑅𝑇/𝑝 22.413 969 54... l/mol 0
at 273.15 K and 101 325 Pa
Rydberg constant 𝑎 𝑅∞ = 𝑚e 𝑐𝛼2 /2ℎ 10 973 731.568 160(21) m−1 1.9 ⋅ 10−12
Conductance quantum 𝐺0 = 2𝑒2 /ℎ 77.480 917 29... μS 0
Magnetic flux quantum 𝜑0 = ℎ/2𝑒 2.067 833 848... fWb 0
Josephson frequency ratio 2𝑒/ℎ 483.597 8484... THz/V 0
Von Klitzing constant ℎ/𝑒2 = 𝜇0 𝑐/2𝛼 25 812.807 45... Ω 0
Bohr magneton 𝜇B = 𝑒ℏ/2𝑚e 9.274 010 0783(28) yJ/T 3.0 ⋅ 10−10
Classical electron radius 𝑟e = 𝑒2 /4π𝜀0 𝑚e 𝑐2 2.817 940 3262(13) f m 4.5 ⋅ 10−10
Compton wavelength 𝜆 C = ℎ/𝑚e 𝑐 2.426 310 238 67(73) pm 3.0 ⋅ 10−10
of the electron 𝜆c = ℏ/𝑚e 𝑐 = 𝑟e /𝛼 0.386 159 267 96(12) pm 3.0 ⋅ 10−10
Bohr radius 𝑎 𝑎∞ = 𝑟e /𝛼2 52.917 721 0903(80) pm 1.5 ⋅ 10−10
Quantum of circulation ℎ/2𝑚e 3.636 947 5516(11) cm2 /s 3.0 ⋅ 10−10
Specific positron charge 𝑒/𝑚e 175.882 001 076(55) GC/kg 3.0 ⋅ 10−10
Cyclotron frequency 𝑓c /𝐵 = 𝑒/2π𝑚e 27.992 489 872(9) GHz/T 3.0 ⋅ 10−10
of the electron
Electron magnetic moment 𝜇e −9.284 764 7043(28) yJ/T 3.0 ⋅ 10−10
𝜇e /𝜇B −1.001 159 652 181 28(18) 1.7 ⋅ 10−13
𝜇e /𝜇N −1 838.281 971 88(11) ⋅ 103 6.0 ⋅ 10−11
a units, measurements and constants 337

TA B L E 31 (Continued) Derived physical constants.

Q ua nt i t y Symbol Va l u e i n S I u n i t s U n c e r t.

Electron g-factor 𝑔e −2.002 319 304 362 56(35) 1.7 ⋅ 10−13
Muon–electron mass ratio 𝑚μ /𝑚e 206.768 2830(46) 2.2 ⋅ 10−8
Muon magnetic moment 𝜇μ −4.490 448 30(10) ⋅ 10−26 J/T 2.2 ⋅ 10−8
Muon g-factor 𝑔μ −2.002 331 8418(13) 6.3 ⋅ 10−10
Atomic mass unit 1 u = 𝑚12C /12 1.660 539 066 60(50) ⋅ 10 kg 3.0 ⋅ 10−10
−27

Proton mass 𝑚p 1.672 621 923 69(51) ⋅ 10−27 kg 3.1 ⋅ 10−10
1.007 276 466 621(53) u 5.3 ⋅ 10−11
938.272 088 16(29) MeV 3.1 ⋅ 10−10
Proton–electron mass ratio 𝑚p /𝑚e 1 836.152 673 43(11) 6.0 ⋅ 10−11
Specific proton charge 𝑒/𝑚p 9.578 833 1560(29) ⋅ 10 C/kg 3.1 ⋅ 10−10
7

Proton Compton wavelength 𝜆 C,p = ℎ/𝑚p 𝑐 1.321 409 855 39(40) f m 3.1 ⋅ 10−10

Motion Mountain – The Adventure of Physics
Nuclear magneton 𝜇N = 𝑒ℏ/2𝑚p 5.050 783 7461(15) ⋅ 10 J/T 3.1 ⋅ 10−10
−27

Proton magnetic moment 𝜇p 1.410 606 797 36(60) ⋅ 10−26 J/T 4.2 ⋅ 10−10
𝜇p /𝜇B 1.521 032 202 30(46) ⋅ 10−3 3.0 ⋅ 10−10
𝜇p /𝜇N 2.792 847 344 63(82) 2.9 ⋅ 10−10
Proton gyromagnetic ratio 𝛾p = 2𝜇𝑝 /ℎ 42.577 478 518(18) MHz/T 4.2 ⋅ 10−10
Proton g factor 𝑔p 5.585 694 6893(16) 2.9 ⋅ 10−10
Neutron mass 𝑚n 1.674 927 498 04(95) ⋅ 10−27 kg 5.7 ⋅ 10−10
1.008 664 915 95(43) u 4.8 ⋅ 10−10

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
939.565 420 52(54) MeV 5.7 ⋅ 10−10
Neutron–electron mass ratio 𝑚n /𝑚e 1 838.683 661 73(89) 4.8 ⋅ 10−10
Neutron–proton mass ratio 𝑚n /𝑚p 1.001 378 419 31(49) 4.9 ⋅ 10−10
Neutron Compton wavelength 𝜆 C,n = ℎ/𝑚n 𝑐 1.319 590 905 81(75) f m 5.7 ⋅ 10−10
Neutron magnetic moment 𝜇n −0.966 236 51(23) ⋅ 10−26 J/T 2.4 ⋅ 10−7
𝜇n /𝜇B −1.041 875 63(25) ⋅ 10−3 2.4 ⋅ 10−7
𝜇n /𝜇N −1.913 042 73(45) 2.4 ⋅ 10−7
Stefan–Boltzmann constant 𝜎 = π2 𝑘4 /60ℏ3 𝑐2 56.703 744 19... nW/m K 2 4
0
Wien’s displacement constant 𝑏 = 𝜆 max 𝑇 2.897 771 955... mmK 0
58.789 257 57... GHz/K 0
Electron volt eV 0.160 217 6634... aJ 0
Bits to entropy conversion const. 𝑘 ln 2 1023 bit = 0.956 994... J/K 0
TNT energy content 3.7 to 4.0 MJ/kg 4 ⋅ 10−2

𝑎. For infinite mass of the nucleus.

Some useful properties of our local environment are given in the following table.
338 a units, measurements and constants

TA B L E 32 Astronomical constants.

Q ua nt it y Symbol Va l u e

Tropical year 1900 𝑎 𝑎 31 556 925.974 7 s
Tropical year 1994 𝑎 31 556 925.2 s
Mean sidereal day 𝑑 23ℎ 56󸀠 4.090 53󸀠󸀠
Average distance Earth–Sun 𝑏 149 597 870.691(30) km
Astronomical unit 𝑏 AU 149 597 870 691 m
Light year, based on Julian year 𝑏 al 9.460 730 472 5808 Pm
Parsec pc 30.856 775 806 Pm = 3.261 634 al
Earth’s mass 𝑀♁ 5.973(1) ⋅ 1024 kg
Geocentric gravitational constant 𝐺𝑀 3.986 004 418(8) ⋅ 1014 m3 /s2
2
Earth’s gravitational length 𝑙♁ = 2𝐺𝑀/𝑐 8.870 056 078(16) mm
Earth’s equatorial radius 𝑐 𝑅♁eq 6378.1366(1) km

Motion Mountain – The Adventure of Physics
Earth’s polar radius 𝑐 𝑅♁p 6356.752(1) km
Equator–pole distance 𝑐 10 001.966 km (average)
Earth’s flattening 𝑐 𝑒♁ 1/298.25642(1)
Earth’s av. density 𝜌♁ 5.5 Mg/m3
Earth’s age 𝑇♁ 4.54(5) Ga = 143(2) Ps
Earth’s normal gravity 𝑔 9.806 65 m/s2
Earth’s standard atmospher. pressure 𝑝0 101 325 Pa
Moon’s radius 𝑅v 1738 km in direction of Earth
Moon’s radius 𝑅h 1737.4 km in other two directions

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Moon’s mass 𝑀 7.35 ⋅ 1022 kg
Moon’s mean distance 𝑑 𝑑 384 401 km
Moon’s distance at perigee 𝑑 typically 363 Mm, historical minimum
359 861 km
Moon’s distance at apogee 𝑑 typically 404 Mm, historical maximum
406 720 km
Moon’s angular size 𝑒 average 0.5181° = 31.08 󸀠 , minimum
0.49°, maximum 0.55°
Moon’s average density 𝜌 3.3 Mg/m3
Moon’s surface gravity 𝑔 1.62 m/s2
Moon’s atmospheric pressure 𝑝 from 10−10 Pa (night) to 10−7 Pa (day)
Jupiter’s mass 𝑀 1.90 ⋅ 1027 kg
Jupiter’s radius, equatorial 𝑅 71.398 Mm
Jupiter’s radius, polar 𝑅 67.1(1) Mm
Jupiter’s average distance from Sun 𝐷 778 412 020 km
Jupiter’s surface gravity 𝑔 24.9 m/s2
Jupiter’s atmospheric pressure 𝑝 from 20 kPa to 200 kPa
Sun’s mass 𝑀⊙ 1.988 43(3) ⋅ 1030 kg
Sun’s gravitational length 2𝐺𝑀⊙ /𝑐2 2.953 250 08(5) km
Heliocentric gravitational constant 𝐺𝑀⊙ 132.712 440 018(8) ⋅ 1018 m3 /s2
a units, measurements and constants 339

TA B L E 32 (Continued) Astronomical constants.

Q ua nt it y Symbol Va l u e

Sun’s luminosity 𝐿⊙ 384.6 YW
Solar equatorial radius 𝑅⊙ 695.98(7) Mm
Sun’s angular size 0.53∘ average; minimum on fourth of July
(aphelion) 1888 󸀠󸀠 , maximum on fourth of
January (perihelion) 1952 󸀠󸀠
Sun’s average density 𝜌⊙ 1.4 Mg/m3
Sun’s average distance AU 149 597 870.691(30) km
Sun’s age 𝑇⊙ 4.6 Ga
Solar velocity 𝑣⊙g 220(20) km/s
around centre of galaxy
Solar velocity 𝑣⊙b 370.6(5) km/s

Motion Mountain – The Adventure of Physics
against cosmic background
Sun’s surface gravity 𝑔⊙ 274 m/s2
Sun’s lower photospheric pressure 𝑝⊙ 15 kPa
Distance to Milky Way’s centre 8.0(5) kpc = 26.1(1.6) kal
Milky Way’s age 13.6 Ga
Milky Way’s size c. 1021 m or 100 kal
Milky Way’s mass 1012 solar masses, c. 2 ⋅ 1042 kg
Most distant galaxy cluster known SXDF-XCLJ 9.6 ⋅ 109 al
0218-0510

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑎. Defining constant, from vernal equinox to vernal equinox; it was once used to define the
second. (Remember: π seconds is about a nanocentury.) The value for 1990 is about 0.7 s less,
Challenge 193 s corresponding to a slowdown of roughly 0.2 ms/a. (Watch out: why?) There is even an empirical
Ref. 288 formula for the change of the length of the year over time.
𝑏. The truly amazing precision in the average distance Earth–Sun of only 30 m results from time
averages of signals sent from Viking orbiters and Mars landers taken over a period of over twenty
years. Note that the International Astronomical Union distinguishes the average distance Earth–
Sun from the astronomical unit itself; the latter is defined as a fixed and exact length. Also the
light year is a unit defined as an exact number by the IAU. For more details, see www.iau.org/
public/measuring.
𝑐. The shape of the Earth is described most precisely with the World Geodetic System. The last
edition dates from 1984. For an extensive presentation of its background and its details, see the
www.wgs84.com website. The International Geodesic Union refined the data in 2000. The radii
and the flattening given here are those for the ‘mean tide system’. They differ from those of the
‘zero tide system’ and other systems by about 0.7 m. The details constitute a science in itself.
𝑑. Measured centre to centre. To find the precise position of the Moon at a given date, see
the www.fourmilab.ch/earthview/moon_ap_per.html page. For the planets, see the page www.
fourmilab.ch/solar/solar.html and the other pages on the same site.
𝑒. Angles are defined as follows: 1 degree = 1∘ = π/180 rad, 1 (first) minute = 1 󸀠 = 1°/60, 1 second
(minute) = 1 󸀠󸀠 = 1 󸀠 /60. The ancient units ‘third minute’ and ‘fourth minute’, each 1/60th of the
preceding, are not in use any more. (‘Minute’ originally means ‘very small’, as it still does in
340 a units, measurements and constants

modern English.)

Some properties of nature at large are listed in the following table. (If you want a chal-
Challenge 194 s lenge, can you determine whether any property of the universe itself is listed?)

TA B L E 33 Cosmological constants.

Q ua nt it y Symbol Va l u e

Cosmological constant Λ c. 1 ⋅ 10−52 m−2
𝑎
Age of the universe 𝑡0 4.333(53) ⋅ 1017 s = 13.8(0.1) ⋅ 109 a
(determined from space-time, via expansion, using general relativity)
Age of the universe 𝑎 𝑡0 over 3.5(4) ⋅ 1017 s = 11.5(1.5) ⋅ 109 a
(determined from matter, via galaxies and stars, using quantum theory)
Hubble parameter 𝑎 𝐻0 2.3(2) ⋅ 10−18 s−1 = 0.73(4) ⋅ 10−10 a−1
= ℎ0 ⋅ 100 km/s Mpc = ℎ0 ⋅ 1.0227 ⋅ 10−10 a−1

Motion Mountain – The Adventure of Physics
𝑎
Reduced Hubble parameter ℎ0 0.71(4)
𝑎 2
Deceleration parameter ̈ 0 /𝐻0 −0.66(10)
𝑞0 = −(𝑎/𝑎)
Universe’s horizon distance 𝑎 𝑑0 = 3𝑐𝑡0 40.0(6) ⋅ 1026 m = 13.0(2) Gpc
Universe’s topology trivial up to 1026 m
Number of space dimensions 3, for distances up to 1026 m
Critical density 𝜌c = 3𝐻02 /8π𝐺 ℎ20 ⋅ 1.878 82(24) ⋅ 10−26 kg/m3
of the universe = 0.95(12) ⋅ 10−26 kg/m3
(Total) density parameter 𝑎 Ω0 = 𝜌0 /𝜌c 1.02(2)
Baryon density parameter 𝑎 ΩB0 = 𝜌B0 /𝜌c 0.044(4)

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑎
Cold dark matter density parameter ΩCDM0 = 𝜌CDM0 /𝜌c 0.23(4)
Neutrino density parameter 𝑎 Ω𝜈0 = 𝜌𝜈0 /𝜌c 0.001 to 0.05
𝑎
Dark energy density parameter ΩX0 = 𝜌X0 /𝜌c 0.73(4)
Dark energy state parameter 𝑤 = 𝑝X /𝜌X −1.0(2)
Baryon mass 𝑚b 1.67 ⋅ 10−27 kg
Baryon number density 0.25(1) /m3
Luminous matter density 3.8(2) ⋅ 10−28 kg/m3
Stars in the universe 𝑛s 1022±1
Baryons in the universe 𝑛b 1081±1
𝑏
Microwave background temperature 𝑇0 2.725(1) K
Photons in the universe 𝑛𝛾 1089
2 4 4
Photon energy density 𝜌𝛾 = π 𝑘 /15𝑇0 4.6 ⋅ 10−31 kg/m3
Photon number density 410.89 /cm3 or 400 /cm3 (𝑇0 /2.7 K)3
Density perturbation amplitude √𝑆 5.6(1.5) ⋅ 10−6
Gravity wave amplitude √𝑇 < 0.71√𝑆
Mass fluctuations on 8 Mpc 𝜎8 0.84(4)
Scalar index 𝑛 0.93(3)
Running of scalar index d𝑛/d ln 𝑘 −0.03(2)
Planck length 𝑙Pl = √ℏ𝐺/𝑐3 1.62 ⋅ 10−35 m
a units, measurements and constants 341

TA B L E 33 (Continued) Cosmological constants.

Q ua nt it y Symbol Va l u e

Planck time 𝑡Pl = √ℏ𝐺/𝑐5 5.39 ⋅ 10−44 s
Planck mass 𝑚Pl = √ℏ𝑐/𝐺 21.8 μg
𝑎
Instants in history 𝑡0 /𝑡Pl 8.7(2.8) ⋅ 1060
Space-time points 𝑁0 = (𝑅0 /𝑙Pl )3 ⋅ 10244±1
inside the horizon 𝑎 (𝑡0 /𝑡Pl )
Mass inside horizon 𝑀 1054±1 kg

𝑎. The index 0 indicates present-day values.
𝑏. The radiation originated when the universe was 380 000 years old and had a temperature of
about 3000 K; the fluctuations Δ𝑇0 which led to galaxy formation are today about 16 ± 4 μK =
6(2) ⋅ 10−6 𝑇0 .

Motion Mountain – The Adventure of Physics
Vol. II, page 231

Useful numbers

π 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 375105
e 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 699959
γ 0.57721 56649 01532 86060 65120 90082 40243 10421 59335 939923
Ref. 289
ln 2 0.69314 71805 59945 30941 72321 21458 17656 80755 00134 360255
ln 10 2.30258 50929 94045 68401 79914 54684 36420 76011 01488 628772
√10 3.16227 76601 68379 33199 88935 44432 71853 37195 55139 325216

C OM P O SI T E PA RT IC L E PROPE RT I E S

T
he following table lists the most important composite particles.
he list has not changed much recently, mainly because of the vast progress
hat was achieved already in the middle of the twentieth century. In principle,
Page 261 using the standard model of particle physics, all properties of composite matter and ra-

Motion Mountain – The Adventure of Physics
diation can be deduced. In particular, all properties of objects encountered in everyday
life follow. (Can you explain how the size of an apple follows from the standard model?)
Challenge 195 s The most important examples of composites are grouped in the following table.

TA B L E 34 Properties of selected composites.

Composite M a s s 𝑚, q ua n t u m L i f e t i m e 𝜏, m a i n Size
numbers𝑎 d e c ay m o d e s (diam.)
Mesons (hadrons, bosons) (selected from over 130 known types)
̄ √2
Pion π0 (𝑢𝑢̄ − 𝑑𝑑)/ 134.976 4(6) MeV/𝑐2

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
84(6) as, 2𝛾 98.798(32) % ∼ 1 fm
𝐼𝐺 (𝐽𝑃𝐶 ) = 1− (0−+ ), 𝑆 = 𝐶 = 𝐵 = 0
Pion π+ (𝑢𝑑)̄ 139.569 95(35) MeV/𝑐2 26.030(5) ns, ∼ 1 fm
𝜇+ 𝜈𝜇 99.987 7(4) %
𝐼𝐺 (𝐽𝑃 ) = 1− (0− ), 𝑆 = 𝐶 = 𝐵 = 0
Kaon 𝐾𝑆0 𝑚𝐾𝑆0 89.27(9) ps ∼ 1 fm
Kaon 𝐾𝐿0 𝑚𝐾𝑆0 + 3.491(9) μeV/𝑐 2
51.7(4) ns ∼ 1 fm
± 2
Kaon 𝐾 (𝑢𝑠,̄ 𝑢𝑠)
̄ 493.677(16) MeV/𝑐 12.386(24) ns, ∼ 1 fm
𝜇+ 𝜈𝜇 63.51(18) %
π+ π0 21.16(14) %
Kaon 𝐾0 (ds̄) (50 % 𝐾𝑆 , 497.672(31) MeV/𝑐2 n.a. ∼ 1 fm
50 % 𝐾𝐿 )
All kaons 𝐾± , 𝐾0 , 𝐾𝑆0 , 𝐾𝐿0 : 𝐼(𝐽𝑃 ) = 12 (0− ), 𝑆 = ±1, 𝐵 = 𝐶 = 0
Baryons (hadrons, fermions) (selected from over 100 known types)
Proton 𝑝 or 𝑁+ (𝑢𝑢𝑑) 1.672 621 58(13) yg 𝜏total > 1.6 ⋅ 1025 a, 0.89(1) f m
= 1.007 276 466 88(13) u 𝜏(𝑝 → 𝑒+ π0 ) >5.5 ⋅ 1032 a Ref. 290
= 938.271 998(38) MeV/𝑐2
+
𝐼(𝐽𝑃 ) = 12 ( 12 ), 𝑆 = 0
gyromagnetic ratio 𝜇𝑝 /𝜇𝑁 = 2.792 847 337(29)
electric dipole moment 𝑑 = (−4 ± 6) ⋅ 10−26 𝑒 m
b composite particle properties 343

TA B L E 34 (Continued) Properties of selected composites.

Composite M a s s 𝑚, q ua n t u m L i f e t i m e 𝜏, m a i n Size
numbers𝑎 d e c ay m o d e s (diam.)
electric polarizability 𝛼e = 12.1(0.9) ⋅ 10−4 f m3
magnetic polarizability 𝛼m = 2.1(0.9) ⋅ 10−4 f m3
Neutron𝑏 𝑛 or 𝑁0 (𝑢𝑑𝑑) 1.674 927 16(13) yg 887.0(2.0) s, 𝑝𝑒− 𝜈𝑒̄ 100 % ∼ 1 fm
= 1.008 664 915 78(55) u = 939.565 330(38) MeV/𝑐2
+
𝐼(𝐽𝑃 ) = 12 ( 12 ), 𝑆 = 0
gyromagnetic ratio 𝜇𝑛 /𝜇𝑁 = −1.913 042 72(45)
electric dipole moment 𝑑𝑛 = (−3.3 ± 4.3) ⋅ 10−28 𝑒 m
electric polarizability 𝛼 = 0.98(23) ⋅ 10−3 f m3
Omega Ω− (𝑠𝑠𝑠) 1672.43(32) MeV/𝑐2 82.2(1.2) ps, ∼ 1 fm
Λ𝐾− 67.8(7) %,

Motion Mountain – The Adventure of Physics
Ξ0 π− 23.6(7) %
gyromagnetic ratio 𝜇Ω /𝜇𝑁 = −1.94(22)
composite radiation: glueballs
glueball candidate 𝑓0 (1500), 1503(11) MeV full width 120(19) MeV ∼ 1 fm
status unclear
𝐼𝐺 (𝐽𝑃𝐶 ) = 0+ (0++ )
Atoms (selected from 114 known elements with over 2000 known nuclides) Ref. 291
1
Hydrogen ( H) [lightest] 1.007 825 032(1) u = 1.6735 yg 2 ⋅ 53 pm

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Antihydrogen𝑐 1.007 u = 1.67 yg 2 ⋅ 53 pm
Helium (4 He) [smallest] 4.002 603250(1) u = 6.6465 yg 2 ⋅ 31 pm
Carbon (12 C) 12 u = 19.926 482(12) yg 2 ⋅ 77 pm
Bismuth (209 Bi∗ ) [shortest 209 u 0.1 ps Ref. 292
living and rarest]
Tantalum (180𝑚 Ta) [second 180 u > 1015 a Ref. 293
longest living radioactive]
Bismuth (209 Bi) [longest 209 u 1.9(2)1019 a Ref. 292
living radioactive]
Francium (223 Fr) [largest] 223 u 22 min 2 ⋅ 0.28 nm
Oganesson (289 Og) 294 u 0.9 ms
[heaviest]
Molecules𝑑 (selected from over 107 known types)
Hydrogen (H2 ) ∼ 2u > 1025 a
Water (H2 O) ∼ 18 u > 1025 a
ATP 507 u > 1010 a c. 3 nm
(adenosinetriphosphate)
Human Y chromosome 70 ⋅ 106 base pairs > 106 a c. 50 mm
(uncoiled)
344 b composite particle properties

TA B L E 34 (Continued) Properties of selected composites.

Composite M a s s 𝑚, q ua n t u m L i f e t i m e 𝜏, m a i n Size
numbers𝑎 d e c ay m o d e s (diam.)

Other composites
Blue whale nerve cell ∼ 1 kg ∼ 50 a 20 m
Cell (red blood) 0.1 ng 7 plus 120 days ∼ 10 μm
Cell (sperm) 10 pg not fecundated: ∼ 5 d length
60 μm,
head
3 μm ×
5 μm
Cell (ovule) 1 μg fecundated: over ∼ 120 μm
4000 million years

Motion Mountain – The Adventure of Physics
Cell (E. coli) 1 pg 4000 million years body: 2 μm
Apple 0.1 kg 4 weeks 0.1 m
Adult human 35 kg < 𝑚 < 350 kg 𝜏 ≈ 2.5 ⋅ 109 s Ref. 294 ∼ 1.7 m
≈ 600 million breaths
≈ 2 500 million heartbeats
< 122 a,
60 % H2 O and 40 % dust
Heaviest living thing: 6.6 ⋅ 106 kg > 130 a > 4 km
colony of aspen trees

Page 263 Notes (see also the notes of Table 9):
𝑎. The charge parity 𝐶 is defined only for certain neutral particles, namely those that are different
from their antiparticles. For neutral mesons, the charge parity is given by 𝐶 = (−1)𝐿+𝑆, where 𝐿
is the orbital angular momentum.
𝑃 is the parity under space inversion 𝑟 → −𝑟. For mesons, it is related to the orbital angular
momentum 𝐿 through 𝑃 = (−1)𝐿+1 .
The electric polarizability, defined on page 72 in volume III, is predicted to vanish for all ele-
mentary particles.
𝐺-parity is defined only for mesons and given by 𝐺 = (−1)𝐿+𝑆+𝐼 = (−1)𝐼 𝐶.
𝑏. Neutrons bound in nuclei have a lifetime of at least 1020 years.
𝑐. The first anti-atoms, made of antielectrons and antiprotons, were made in January 1996 at CERN
Ref. 296 in Geneva. All properties of antimatter checked so far are consistent with theoretical predictions.
𝑑. The number of existing molecules is several orders of magnitude larger than the number of
molecules that have been analysed and named.

The most important matter composites are the atoms. Their size, structure and interac-
tions determine the properties and colour of everyday objects. Atom types, also called
elements in chemistry, are most usefully set out in the so-called periodic table, which
groups together atoms with similar properties in rows and columns. It is given in Table 35
and results from the various ways in which protons, neutrons and electrons can combine
b composite particle properties 345

to form aggregates.
Comparable to the periodic table of the atoms, there are tables for the mesons (made
of two quarks) and the baryons (made of three quarks). Neither the meson nor the ba-
ryon table is included here; they can both be found in the Review of Particle Physics at
pdg.web.cern.ch. In fact, the baryon table still has a number of vacant spots. The miss-
ing baryons are extremely heavy and short-lived (which means expensive to make and
detect), and their discovery is not expected to yield deep new insights.

TA B L E 35 The periodic table of the elements, with their atomic numbers. Light blue: nonmetals,
orange: alkali metals, green: alkaline earth metals, grey: transition metals, dark blue: basic metals, light
orange: semimetals, yellow: halogens, brown: noble gases, red: lanthanoids, dark red: actinoids, black:
no data.

Group
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Motion Mountain – The Adventure of Physics
I II IIIa IVa Va VIa VIIa VIIIa Ia IIa III IV V VI VII VIII

Period

1 2
1
H He

3 4 5 6 7 8 9 10
2
Li Be B C N O F Ne

11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
4
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
5
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

55 56 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
6 ∗
Cs Ba Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

87 88 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
7 ∗∗
Fr Ra Rf Db Sg Bh Hs Mt Ds Rg Cn Nh Fl Mc Lv Ts Og

57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
Lanthanoids ∗
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
Actinoids ∗∗
Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr

More elaborate periodic tables can be found on the chemlab.pc.maricopa.edu/periodic
website. The most beautiful of them all can be found on page 60. The atomic number gives
346 b composite particle properties

the number of protons (and electrons) found in an atom of a given element. This number
determines the chemical behaviour of an element. Most – but not all – elements up to
92 are found on Earth; the others can be produced in laboratories. The highest element
discovered is element 118. In a famous case of research fraud, a scientist in the 1990s
tricked two whole research groups into claiming to have made and observed elements
116 and 118. Both elements were independently made and observed later on.
Ref. 297 Nowadays, extensive physical and chemical data are available for every element. Pho-
Page 61 tographs of the pure elements are shown in Figure 19. Elements in the same group behave
similarly in chemical reactions. The periods define the repetition of these similarities.
The elements of group 1 are the alkali metals (though the exceptional hydrogen is a
gas), those of group 2 are the alkaline earth metals. Also actinoids, lanthanoids are metals,
as are the elements of groups 3 to 12, which are called transition or heavy metals. The ele-
ments of group 16 are called chalkogens, i.e., ore-formers; group 17 are the halogens, i.e.,
the salt-formers, and group 18 are the inert noble gases, which form (almost) no chem-
ical compounds. The groups 13, 14 and 15 contain metals, semimetals, the only room-

Motion Mountain – The Adventure of Physics
temperature liquid – bromine – and a few gases and non-metals; these groups have no
special name. Groups 1 and 13 to 17 are central for the chemistry of life; in fact, 96 % of
living matter is made of C, O, N, H;* almost 4 % of P, S, Ca, K, Na, Cl; trace elements
such as Mg, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Cd, Pb, Sn, Li, Mo, Se, Si, I, F, As, B form the
rest. Over 30 elements are known to be essential for animal life. The full list is not yet
known; candidate elements to extend this list are Al, Br, Ge and W.
Many elements exist in versions with different numbers of neutrons in their nucleus,
and thus with different mass; these various isotopes – so called because they are found at
the same place in the periodic table – behave identically in chemical reactions. There are

TA B L E 36 The elements, with their atomic number, average mass, atomic radius and main properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Actinium𝑏 Ac 89 (227.0277(1)) (188) Highly radioactive metallic rare Earth
21.77(2) a (Greek aktis ray) 1899, used as alpha-
emitting source.
Aluminium Al 13 26.981 538 (8) 118c, Light metal (Latin alumen alum) 1827,
stable 143m used in machine construction and living
beings.
Americium𝑏 Am 95 (243.0614(1)) (184) Radioactive metal (Italian America from
7.37(2) ka Amerigo) 1945, used in smoke detectors.

* The ‘average formula’ of life is approximately C5 H40 O18 N.
b composite particle properties 347

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Antimony Sb 51 121.760(1)𝑓 137c, Toxic semimetal (via Arabic from Latin
stable 159m, stibium, itself from Greek, Egyptian for
205v one of its minerals) antiquity, colours
rubber, used in medicines, constituent of
enzymes.
Argon Ar 18 39.948(1)𝑓 (71n) Noble gas (Greek argos inactive, from an-
stable ergos without energy) 1894, third com-
ponent of air, used for welding and in

Motion Mountain – The Adventure of Physics
lasers.
Arsenic As 33 74.921 60(2) 120c, Poisonous semimetal (Greek arsenikon
stable 185v tamer of males) antiquity, for poisoning
pigeons and doping semiconductors.
Astatine𝑏 At 85 (209.9871(1)) (140) Radioactive halogen (Greek astatos un-
8.1(4) h stable) 1940, no use.
Barium Ba 56 137.327(7) 224m Earth-alkali metal (Greek bary heavy)
stable 1808, used in vacuum tubes, paint, oil in-
dustry, pyrotechnics and X-ray diagnosis.
Berkelium𝑏 Bk 97 (247.0703(1)) n.a. Made in lab, probably metallic (Berkeley,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
1.4(3) ka US town) 1949, no use because rare.
Beryllium Be 4 9.012 182(3) 106c, Toxic Earth-alkali metal (Greek beryllos,
stable 113m a mineral) 1797, used in light alloys, in
nuclear industry as moderator.
Bismuth Bi 83 208.980 40(1) 170m, Diamagnetic metal (Latin via German
stable 215v weisse Masse white mass) 1753, used in
magnets, alloys, fire safety, cosmetics, as
catalyst, nuclear industry.
Bohrium𝑏 Bh 107 (264.12(1)) n.a. Made in lab, probably metallic (after
0.44 s𝑔 Niels Bohr) 1981, found in nuclear reac-
tions, no use.
Boron B 5 10.811(7)𝑓 83c Semimetal, semiconductor (Latin borax,
stable from Arabic and Persian for brilliant)
1808, used in glass, bleach, pyrotechnics,
rocket fuel, medicine.
Bromine Br 35 79.904(1) 120c, Red-brown liquid (Greek bromos strong
stable 185v odour) 1826, fumigants, photography,
water purification, dyes, medicines.
348 b composite particle properties

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Cadmium Cd 48 112.411(8)𝑓 157m Heavy metal, cuttable and screaming
stable (Greek kadmeia, a zinc carbonate mineral
where it was discovered) 1817, electro-
plating, solder, batteries, TV phosphors,
dyes.
Caesium Cs 55 132.905 4519(2) 273m Alkali metal (Latin caesius sky blue) 1860,
stable getter in vacuum tubes, photoelectric
cells, ion propulsion, atomic clocks.

Motion Mountain – The Adventure of Physics
Calcium Ca 20 40.078(4)𝑓 197m Earth-alkali metal (Latin calcis chalk) an-
stable tiquity, pure in 1880, found in stones and
bones, reducing agent, alloying.
Californium𝑏 Cf 98 (251.0796(1)) n.a. Made in lab, probably metallic, strong
0.90(5) ka neutron emitter (Latin calor heat and for-
nicare have sex, the land of hot sex :-)
1950, used as neutron source, for well log-
ging.
Carbon C 6 12.0107(8)𝑓 77c Makes up coal and diamond (Latin carbo
stable coal) antiquity, used to build most life

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
forms.
Cerium Ce 58 140.116(1)𝑓 183m Rare Earth metal (after asteroid Ceres,
stable Roman goddess) 1803, cigarette lighters,
incandescent gas mantles, glass manu-
facturing, self-cleaning ovens, carbon-arc
lighting in the motion picture industry,
catalyst, metallurgy.
Chlorine Cl 17 35.453(2)𝑓 102c, Green gas (Greek chloros yellow-green)
stable 175v 1774, drinking water, polymers, paper,
dyes, textiles, medicines, insecticides,
solvents, paints, rubber.
Chromium Cr 24 51.9961(6) 128m Transition metal (Greek chromos colour)
stable 1797, hardens steel, makes steel stainless,
alloys, electroplating, green glass dye,
catalyst.
Cobalt Co 27 58.933 195(5) 125m Ferromagnetic transition metal (German
stable Kobold goblin) 1694, part of vitamin
B12 , magnetic alloys, heavy-duty alloys,
enamel dyes, ink, animal nutrition.
Copernicium𝑏 Cn 112 (285) 34 s𝑔 n.a. Made in lab, 1996, no use.
b composite particle properties 349

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Copper Cu 29 63.546(3)𝑓 128m Red metal (Latin cuprum from Cyprus is-
stable land) antiquity, part of many enzymes,
electrical conductors, bronze, brass and
other alloys, algicides, etc.
Curium𝑏 Cm 96 (247.0704(1)) n.a. Highly radioactive, silver-coloured (after
15.6(5) Ma Pierre and Marie Curie) 1944, used as ra-
dioactivity source.
Darmstadtium𝑏 Ds 110 (271) 1.6 min𝑔 n.a. Made in lab (after the German city) 1994,

Motion Mountain – The Adventure of Physics
no use.
Dubnium𝑏 Db 105 (262.1141(1)) n.a. Made in lab in small quantities, radio-
34(5) s active (Dubna, Russian city) 1967, no use
(once known as hahnium).
Dysprosium Dy 66 162.500(1)𝑓 177m Rare Earth metal (Greek dysprositos dif-
stable ficult to obtain) 1886, used in laser ma-
terials, as infrared source material, and in
nuclear industry.
Einsteinium𝑏 Es 99 (252.0830(1)) n.a. Made in lab, radioactive (after Albert
472(2) d Einstein) 1952, no use.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Erbium Er 68 167.259(3)𝑓 176m Rare Earth metal (Ytterby, Swedish town)
stable 1843, used in metallurgy and optical
fibres.
Europium Eu 63 151.964(1)𝑓 204m Rare Earth metal (named after the con-
stable tinent) 1901, used in red screen phosphor
for TV tubes.
Fermium𝑏 Fm 100 (257.0901(1)) n.a. Made in lab (after Enrico Fermi) 1952, no
100.5(2) d use.
Flerovium𝑏 Fl 114 (289) 2.7 s𝑔 1999, no use.
Fluorine F 9 18.998 4032(5) 62c, Gaseous halogen (from fluorine, a min-
stable 147v eral, from Greek fluo flow) 1886, used in
polymers and toothpaste.
Francium𝑏 Fr 87 (223.0197(1)) (278) Radioactive metal (from France) 1939, no
22.0(1) min use.
Gadolinium Gd 64 157.25(3)𝑓 180m Rare-earth metal (after Johan Gadolin)
stable 1880, used in lasers and phosphors.
Gallium Ga 31 69.723(1) 125c, Almost liquid metal (Latin for both the
stable 141m discoverer’s name and his nation, France)
1875, used in optoelectronics.
350 b composite particle properties

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Germanium Ge 32 72.64(1) 122c, Semiconductor (from Germania, as op-
stable 195v posed to gallium) 1886, used in electron-
ics.
Gold Au 79 196.966 569(4) 144m Heavy noble metal (Sanskrit jval to shine,
stable Latin aurum) antiquity, electronics, jew-
els.
Hafnium Hf 72 178.49(2)𝑐 158m Metal (Latin for Copenhagen) 1923, al-
stable loys, incandescent wire.

Motion Mountain – The Adventure of Physics
Hassium𝑏 Hs 108 (277) 16.5 min𝑔 n.a. Radioactive element (Latin form of Ger-
man state Hessen) 1984, no use .
Helium He 2 4.002 602(2)𝑓 (31n) Noble gas (Greek helios Sun) where it was
stable discovered 1895, used in balloons, stars,
diver’s gas and cryogenics.
Holmium Ho 67 164.930 32(2) 177m Metal (Stockholm, Swedish capital) 1878,
stable alloys.
Hydrogen H 1 1.007 94(7)𝑓 30c Reactive gas (Greek for water-former)
stable 1766, used in building stars and universe.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Indium In 49 114.818(3) 141c, Soft metal (Greek indikon indigo) 1863,
stable 166m used in solders and photocells.
Iodine I 53 126.904 47(3) 140c, Blue-black solid (Greek iodes violet) 1811,
stable 198v used in photography.
Iridium Ir 77 192.217(3) 136m Precious metal (Greek iris rainbow) 1804,
stable electrical contact layers.
Iron Fe 26 55.845(2) 127m Metal (Indo-European ayos metal, Latin
stable ferrum) antiquity, used in metallurgy.
Krypton Kr 36 83.798(2)𝑓 (88n) Noble gas (Greek kryptos hidden) 1898,
stable used in lasers.
Lanthanum La 57 138.905 47(7)𝑐,𝑓 188m Reactive rare Earth metal (Greek
stable lanthanein to be hidden) 1839, used in
lamps and in special glasses.
Lawrencium𝑏 Lr 103 (262.110 97(1)) n.a. Appears in reactions (after Ernest
3.6(3) h Lawrence) 1961, no use.
Lead Pb 82 207.2(1)𝑐,𝑓 175m Poisonous, malleable heavy metal (Latin
stable plumbum) antiquity, used in car batteries,
radioactivity shields, paints.
Lithium Li 3 6.941(2)𝑓 156m Light alkali metal with high specific heat
stable (Greek lithos stone) 1817, used in batter-
ies, anti-depressants, alloys, nuclear fu-
sion and many chemicals.
b composite particle properties 351

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Livermorium𝑏 Lv 116 (293) 61 ms𝑔 False discovery claim from 1999, correct
claim from 2000, no use.
Lutetium Lu 71 174.967(1)𝑓 173m Rare-earth metal (Latin Lutetia for Paris)
stable 1907, used as catalyst.
Magnesium Mg 12 24.3050(6) 160m Light common alkaline Earth metal
stable (from Magnesia, a Greek district in Thes-
salia) 1755, used in alloys, pyrotechnics,
chemical synthesis and medicine, found

Motion Mountain – The Adventure of Physics
in chlorophyll.
Manganese Mn 25 54.938 045(5) 126m Brittle metal (Italian manganese, a
stable mineral) 1774, used in alloys, colours
amethyst and permanganate.
Meitnerium𝑏 Mt 109 (268.1388(1)) n.a. Appears in nuclear reactions (after Lise
0.070 s𝑔 Meitner) 1982, no use.
Mendelevium𝑏 Md 101 (258.0984(1)) n.a. Appears in nuclear reactions (after
51.5(3) d Дмитрии Иванович Менделеев
Dmitriy Ivanovich Mendeleyev) 1955, no
use.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Mercury Hg 80 200.59(2) 157m Liquid heavy metal (Latin god Mer-
stable curius, Greek hydrargyrum liquid sil-
ver) antiquity, used in switches, batteries,
lamps, amalgam alloys.
Molybdenum Mo 42 95.94(2)𝑓 140m Metal (Greek molybdos lead) 1788, used
stable in alloys, as catalyst, in enzymes and lub-
ricants.
Moscovium𝑏 Mc 115 (288) 8 ms 𝑔
2004 (Moscow), no use.
Neodymium Nd 60 144.242(3)𝑐,𝑓 182m (Greek neos and didymos new twin) 1885,
stable used in magnets.
Neon Ne 10 20.1797(6)𝑓 (36n) Noble gas (Greek neos new) 1898, used in
stable lamps, lasers and cryogenics.
Neptunium𝑏 Np 93 (237.0482(1)) n.a. Radioactive metal (planet Neptune, after
2.14(1) Ma Uranus in the solar system) 1940, appears
in nuclear reactors, used in neutron de-
tection and by the military.
Nickel Ni 28 58.6934(2) 125m Metal (German Nickel goblin) 1751, used
stable in coins, stainless steels, batteries, as cata-
lyst.
Nihonium𝑏 Nh 113 (284) 0.48 s𝑔 2003 (Nihon is Japan in Japanese), no use.
352 b composite particle properties

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Niobium Nb 41 92.906 38(2) 147m Ductile metal (Greek Niobe, mythical
stable daughter of Tantalos) 1801, used in arc
welding, alloys, jewellery, superconduct-
ors.
Nitrogen N 7 14.0067(2)𝑓 70c, Diatomic gas (Greek for nitre-former)
stable 155v 1772, found in air, in living organisms,
Viagra, fertilizers, explosives.
Nobelium𝑏 No 102 (259.1010(1)) n.a. (after Alfred Nobel) 1958, no use.

Motion Mountain – The Adventure of Physics
58(5) min
Oganesson𝑏 Og 118 (294) 0.9 ms𝑔 False discovery claim in 1999, correct
claim from 2006 (after Yuri Oganessian),
no use.
Osmium Os 76 190.23(3)𝑓 135m Heavy metal (from Greek osme odour)
stable 1804, used for fingerprint detection and
in very hard alloys.
Oxygen O 8 15.9994(3)𝑓 66c, Transparent, diatomic gas (formed from
stable 152v Greek to mean ‘acid former’) 1774, used
for combustion, blood regeneration, to

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
make most rocks and stones, in countless
compounds, colours auroras red.
Palladium Pd 46 106.42(1)𝑓 138m Heavy metal (from asteroid Pallas, after
stable the Greek goddess) 1802, used in alloys,
white gold, catalysts, for hydride storage.
Phosphorus P 15 30.973 762(2) 109c, Poisonous, waxy, white solid (Greek
stable 180v phosphoros light bearer) 1669, fertilizers,
glasses, porcelain, steels and alloys, living
organisms, bones.
Platinum Pt 78 195.084(9) 139m Silvery-white, ductile, noble heavy
stable metal (Spanish platina little silver)
pre-Columbian, again in 1735, used
in corrosion-resistant alloys, magnets,
furnaces, catalysts, fuel cells, cathodic
protection systems for large ships and
pipelines; being a catalyst, a fine plat-
inum wire glows red hot when placed in
vapour of methyl alcohol, an effect used
in hand warmers.
b composite particle properties 353

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Plutonium Pu 94 (244.0642(1)) n.a. Extremely toxic alpha-emitting metal
80.0(9) Ma (after the planet) synthesized 1940, found
in nature 1971, used as nuclear explosive,
and to power space equipment, such as
satellites and the measurement equip-
ment brought to the Moon by the Apollo
missions.
Polonium Po 84 (208.9824(1)) (140) Alpha-emitting, volatile metal (from Po-

Motion Mountain – The Adventure of Physics
102(5) a land) 1898, used as thermoelectric power
source in space satellites, as neutron
source when mixed with beryllium; used
in the past to eliminate static charges in
factories, and on brushes for removing
dust from photographic films.
Potassium K 19 39.0983(1) 238m Reactive, cuttable light metal (German
stable Pottasche, Latin kalium from Arabic
quilyi, a plant used to produce potash)
1807, part of many salts and rocks, essen-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tial for life, used in fertilizers, essential to
chemical industry.
Praeseodymium Pr 59 140.907 65(2) 183m White, malleable rare Earth metal (Greek
stable praesos didymos green twin) 1885, used in
cigarette lighters, material for carbon arcs
used by the motion picture industry for
studio lighting and projection, glass and
enamel dye, darkens welder’s goggles.
Promethium𝑏 Pm 61 (144.9127(1)) 181m Radioactive rare Earth metal (from the
17.7(4) a Greek mythical figure of Prometheus)
1945, used as β source and to excite phos-
phors.
Protactinium Pa 91 (231.035 88(2)) n.a. Radioactive metal (Greek protos first, as
32.5(1) ka it decays into actinium) 1917, found in
nature, no use.
Radium Ra 88 (226.0254(1)) (223) Highly radioactive metal (Latin radius
1599(4) a ray) 1898, no use any more; once used
in luminous paints and as radioactive
source and in medicine.
354 b composite particle properties

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Radon Rn 86 (222.0176(1)) (130n) Radioactive noble gas (from its old name
3.823(4) d ‘radium emanation’) 1900, no use (any
more), found in soil, produces lung can-
cer .
Rhenium Re 75 186.207(1)𝑐 138m TRansition metal (Latin rhenus for Rhine
stable river) 1925, used in filaments for mass
spectrographs and ion gauges, supercon-
ductors, thermocouples, flash lamps, and

Motion Mountain – The Adventure of Physics
as catalyst.
Rhodium Rh 45 102.905 50(2) 135m White metal (Greek rhodon rose) 1803,
stable used to harden platinum and palladium
alloys, for electroplating, and as catalyst.
Roentgenium𝑏 Rg 111 (272.1535(1)) n.a. Made in lab (after Conrad Roentgen)
1.5 ms𝑔 1994, no use.
Rubidium Rb 37 85.4678(3)𝑓 255m Silvery-white, reactive alkali metal (Latin
stable rubidus red) 1861, used in photocells, op-
tical glasses, solid electrolytes.
Ruthenium Ru 44 101.107(2)𝑓 134m White metal (Latin Rhuthenia for Rus-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
stable sia) 1844, used in platinum and palla-
dium alloys, superconductors, as catalyst;
the tetroxide is toxic and explosive.
Rutherfordium𝑏 Rf 104 (261.1088(1)) n.a. Radioactive transactinide (after Ernest
1.3 min𝑔 Rutherford) 1964, no use.
Samarium Sm 62 150.36(2)𝑐,𝑓 180m Silver-white rare Earth metal (from
stable the mineral samarskite, after Wassily
Samarski) 1879, used in magnets, optical
glasses, as laser dopant, in phosphors, in
high-power light sources.
Scandium Sc 21 44.955 912(6) 164m Silver-white metal (from Latin Scansia
stable Sweden) 1879, the oxide is used in high-
intensity mercury vapour lamps, a radio-
active isotope is used as tracer.
Seaborgium𝑏 Sg 106 266.1219(1) n.a. Radioactive transurane (after Glenn
21 s𝑔 Seaborg) 1974, no use.
Selenium Se 34 78.96(3)𝑓 120c, Red or black or grey semiconductor
stable 190v (Greek selene Moon) 1818, used in xer-
ography, glass production, photographic
toners, as enamel dye.
b composite particle properties 355

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Silicon Si 14 28.0855(3)𝑓 105c, Grey, shiny semiconductor (Latin silex
stable 210v pebble) 1823, Earth’s crust, electronics,
sand, concrete, bricks, glass, polymers,
solar cells, essential for life.
Silver Ag 47 107.8682(2)𝑓 145m White metal with highest thermal and
stable electrical conductivity (Latin argentum,
Greek argyros) antiquity, used in photo-
graphy, alloys, to make rain.

Motion Mountain – The Adventure of Physics
Sodium Na 11 22.989 769 28(2) 191m Light, reactive metal (Arabic souwad
soda, Egyptian and Arabic natrium) com-
stable ponent of many salts, soap, paper, soda,
salpeter, borax, and essential for life.
Strontium Sr 38 87.62(1)𝑓 215m Silvery, spontaneously igniting light
stable metal (Strontian, Scottish town) 1790,
used in TV tube glass, in magnets, and
in optical materials.
Sulphur S 16 32.065(5)𝑓 105c, Yellow solid (Latin) antiquity, used in
stable 180v gunpowder, in sulphuric acid, rubber vul-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
canization, as fungicide in wine produc-
tion, and is essential for life; some bac-
teria use sulphur instead of oxygen in
their chemistry.
Tantalum Ta 73 180.947 88(2) 147m Heavy metal (Greek Tantalos, a mythical
stable figure) 1802, used for alloys, surgical in-
struments, capacitors, vacuum furnaces,
glasses.
Technetium𝑏 Tc 43 (97.9072(1)) 136m Radioactive (Greek technetos artificial)
6.6(10) Ma 1939, used as radioactive tracer and in
nuclear technology.
Tellurium Te 52 127.60(3)𝑓 139c, Brittle, garlic-smelling semiconductor
stable 206v (Latin tellus Earth) 1783, used in alloys
and as glass component.
Tennessine𝑏 Ts 117 (294) 78 ms𝑔 2010 (Tennessee), no use.
Terbium Tb 65 158.925 35(2) 178m Malleable rare Earth metal (Ytterby,
stable Swedish town) 1843, used as dopant in
optical material.
Thallium Tl 81 204.3833(2) 172m Soft, poisonous heavy metal (Greek thal-
stable los branch) 1861, used as poison and for
infrared detection.
356 b composite particle properties

TA B L E 36 (Continued) The elements, with their atomic number, average mass, atomic radius and main
properties.

Name Sym- At. Aver. mass 𝑎 Ato- Main properties, (naming) ℎ dis-
bol n. in u (error), mic 𝑒 covery date and use
longest ra-
lifetime dius
in pm
Thorium Th 90 232.038 06(2)𝑑,𝑓 180m Radioactive (Nordic god Thor, as in
14.0(1) Ga ‘Thursday’) 1828, found in nature, heats
Earth, used as oxide in gas mantles for
campers, in alloys, as coating, and in nuc-
lear energy.
Thulium Tm 69 168.934 21(2) 175m Rare Earth metal (Thule, mythical name
stable for Scandinavia) 1879, found in monazite,
used in lasers and radiation detectors.

Motion Mountain – The Adventure of Physics
𝑓
Tin Sn 50 118.710(7) 139c, Grey metal that, when bent, allows one
stable 210v, to hear the ‘tin cry’ (Latin stannum) an-
162m tiquity, used in paint, bronze and super-
conductors.
Titanium Ti 22 47.867(1) 146m Metal (Greek hero Titanos) 1791, alloys,
stable fake diamonds.
Tungsten W 74 183.84(1) 141m Heavy, highest-melting metal (Swedish
stable tung sten heavy stone, German name
Wolfram) 1783, lightbulbs.
Uranium U 92 238.028 91(3)𝑑,𝑓 156m Radioactive and of high density (planet

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
4.468(3) ⋅ 109 a Uranus, after the Greek sky god) 1789,
found in pechblende and other minerals,
used for nuclear energy.
Vanadium V 23 50.9415(1) 135m Metal (Vanadis, scandinavian goddess of
stable beauty) 1830, used in steel.
Xenon Xe 54 131.293(6)𝑓 (103n) Noble gas (Greek xenos foreign) 1898,
stable 200v used in lamps and lasers.
𝑓
Ytterbium Yb 70 173.04(3) 174m Malleable heavy metal (Ytterby, Swedish
stable town) 1878, used in superconductors.
Yttrium Y 39 88.905 85(2) 180m Malleable light metal (Ytterby, Swedish
stable town) 1794, used in lasers.
Zinc Zn 30 65.409(4) 139m Heavy metal (German Zinke protuber-
stable ance) antiquity, iron rust protection.
Zirconium Zr 40 91.224(2)𝑓 160m Heavy metal (from the mineral zircon,
stable after Arabic zargum golden colour) 1789,
chemical and surgical instruments, nuc-
lear industry.

1
𝑎. The atomic mass unit is defined as 1 u = 12 𝑚(12 C), making 1 u = 1.660 5402(10) yg. For
elements found on Earth, the average atomic mass for the naturally occurring isotope mixture is
Ref. 298 given, with the error in the last digit in brackets. For elements not found on Earth, the mass of
b composite particle properties 357

the longest living isotope is given; as it is not an average, it is written in brackets, as is customary
in this domain.
𝑏. The element is not found on Earth because of its short lifetime.
𝑐. The element has at least one radioactive isotope.
𝑑. The element has no stable isotopes.
𝑒. Strictly speaking, the atomic radius does not exist. Because atoms are clouds, they have no
boundary. Several approximate definitions of the ‘size’ of atoms are possible. Usually, the radius is
defined in such a way as to be useful for the estimation of distances between atoms. This distance
Ref. 299 is different for different bond types. In the table, radii for metallic bonds are labelled m, radii
Ref. 299 for (single) covalent bonds with carbon c, and Van der Waals radii v. Noble gas radii are labelled
n. Note that values found in the literature vary by about 10 %; values in brackets lack literature
references.
The covalent radius can be up to 0.1 nm smaller than the metallic radius for elements on the
(lower) left of the periodic table; on the (whole) right side it is essentially equal to the metallic
radius. In between, the difference between the two decreases towards the right. Can you explain
Challenge 196 s why? By the way, ionic radii differ considerably from atomic ones, and depend both on the ionic

Motion Mountain – The Adventure of Physics
charge and the element itself.
All these values are for atoms in their ground state. Excited atoms can be hundreds of times
larger than atoms in the ground state; however, excited atoms do not form solids or chemical
compounds.
𝑓. The isotopic composition, and thus the average atomic mass, of the element varies depending
on the place where it was mined or on subsequent human treatment, and can lie outside the
values given. For example, the atomic mass of commercial lithium ranges between 6.939 and
Ref. 291 6.996 u. The masses of isotopes are known in atomic mass units to nine or more significant digits,
and usually with one or two fewer digits in kilograms. The errors in the atomic mass are thus
Ref. 298 mainly due to the variations in isotopic composition.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑔. The lifetime errors are asymmetric or not well known.
ℎ. Extensive details on element names can be found on elements.vanderkrogt.net.
Appendix C

A L G E BR A S , SHA PE S A N D G ROU P S

M
athematicians are fond of generalizing concepts. One of the
ost generalized concepts of all is the concept of space. Understanding
athematical definitions and generalizations means learning to think with
precision. The appendix of the previous, fourth volume provided a simple introduc-

Motion Mountain – The Adventure of Physics
tion to the types of spaces that are of importance in physics; this appendix provides an
introduction to the algebras that are of importance in physics.

al gebras
The term algebra is used in mathematics with three different, but loosely related, mean-
ings. First, it denotes a part of mathematics, as in ‘I hated algebra at school’. Secondly,
it denotes a set of formal rules that are obeyed by abstract objects, as in the expression
‘tensor algebra’. Finally – and this is the only meaning used here – an algebra denotes a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
specific type of mathematical structure.
Intuitively, an algebra is a set of vectors with a vector multiplication defined on it.
More precisely, a (unital, associative) algebra is a vector space (over a field 𝐾) that is also a
(unital) ring. (The concept is due to Benjamin Peirce (b. 1809 Salem, d. 1880 Cambridge),
father of Charles Sanders Peirce.) A ring is a set for which an addition and a multiplic-
Vol. IV, page 223 ation is defined – like the integers. Thus, in an algebra, there are (often) three types of
multiplications:
— the (main) algebraic multiplication: the product of two vectors 𝑥 and 𝑦 is another
vector 𝑧 = 𝑥𝑦;
— the scalar multiplication: the 𝑐-fold multiple of a vector 𝑥 is another vector 𝑦 = 𝑐𝑥;
— if the vector space is a inner product space, the scalar product: the scalar product of
two algebra elements (vectors) 𝑥 and 𝑦 is a scalar 𝑐 = 𝑥 ⋅ 𝑦;
A precise definition of an algebra thus only needs to define properties of the (main) mul-
tiplication and to specify the number field 𝐾. An algebra is defined by the following ax-
ioms

𝑥(𝑦 + 𝑧) = 𝑥𝑦 + 𝑥𝑧 , (𝑥 + 𝑦)𝑧 = 𝑥𝑧 + 𝑦𝑧 distributivity of multiplication
𝑐(𝑥𝑦) = (𝑐𝑥)𝑦 = 𝑥(𝑐𝑦) bilinearity (139)
algebras 359

for all vectors 𝑥, 𝑦, 𝑧 and all scalars 𝑐 ∈ K. To stress their properties, algebras are also
called linear algebras.
For example, the set of all linear transformations of an 𝑛-dimensional linear space
(such as the translations on a plane, in space or in time) is a linear algebra, if the com-
position is taken as multiplication. So is the set of observables of a quantum mechanical
system.*
An associative algebra is an algebra whose multiplication has the additional property
that
𝑥(𝑦𝑧) = (𝑥𝑦)𝑧 associativity . (141)

Most algebras that arise in physics are associative** and unital. Therefore, in mathemat-
ical physics, a linear unital associative algebra is often simply called an algebra.
The set of multiples of the unit 1 of the algebra is called the field of scalars scal(A) of
the algebra A. The field of scalars is also a subalgebra of A. The field of scalars and the
scalars themselves behave in the same way.

Motion Mountain – The Adventure of Physics
We explore a few examples. The set of all polynomials in one variable (or in several
Challenge 198 e variables) forms an algebra. It is commutative and infinite-dimensional. The constant
polynomials form the field of scalars.
The set of 𝑛 × 𝑛 matrices, with the usual operations, also forms an algebra. It is 𝑛2 -
dimensional. Those diagonal matrices (matrices with all off-diagonal elements equal to
zero) whose diagonal elements all have the same value form the field of scalars. How is
Challenge 199 ny the scalar product of two matrices defined?
The set of all real-valued functions over a set also forms an algebra. Can you specify
Challenge 200 s the multiplication? The constant functions form the field of scalars.

* Linear transformations are mappings from the vector space to itself, with the property that sums and scalar
multiples of vectors are transformed into the corresponding sums and scalar multiples of the transformed
Challenge 197 s vectors. Can you specify the set of all linear transformations of the plane? And of three-dimensional space?
And of Minkowski space?
All linear transformations transform some special vectors, called eigenvectors (from the German word
eigen meaning ‘self’) into multiples of themselves. In other words, if 𝑇 is a transformation, 𝑒 a vector, and

𝑇(𝑒) = 𝜆𝑒 (140)

where 𝜆 is a scalar, then the vector 𝑒 is called an eigenvector of 𝑇, and 𝜆 is associated eigenvalue. The set of all
eigenvalues of a transformation 𝑇 is called the spectrum of 𝑇. Physicists did not pay much attention to these
Vol. IV, page 88 mathematical concepts until they discovered quantum theory. Quantum theory showed that observables are
transformations in Hilbert space, because any measurement interacts with a system and thus transforms it.
Quantum-mechanical experiments also showed that a measurement result for an observable must be an
eigenvalue of the corresponding transformation. The state of the system after the measurement is given by
Vol. IV, page 157 the eigenvector corresponding to the measured eigenvalue. Therefore every expert on motion must know
what an eigenvalue is.
** Note that a non-associative algebra does not possess a matrix representation.
360 c algebras, shapes and groups

which there is a mapping ∗ : 𝐴 → 𝐴, 𝑥 󳨃→ 𝑥∗ , called an involution, with the properties

(𝑥∗ )∗ =𝑥
(𝑥 + 𝑦)∗ = 𝑥∗ + 𝑦∗
(𝑐𝑥)∗ = 𝑐𝑥∗ for all 𝑐∈ℂ
(𝑥𝑦)∗ = 𝑦∗ 𝑥∗ (142)

valid for all elements 𝑥, 𝑦 of the algebra 𝐴. The element 𝑥∗ is called the adjoint of 𝑥.
Star algebras are the main type of algebra used in quantum mechanics, since quantum-
mechanical observables form a ∗-algebra.
A C∗-algebra is a Banach algebra over the complex numbers with an involution ∗ (a
function that is its own inverse) such that the norm ‖𝑥‖ of an element 𝑥 satisfies

‖𝑥‖2 = 𝑥∗ 𝑥 . (143)

Motion Mountain – The Adventure of Physics
(A Banach algebra is a complete normed algebra; an algebra is complete if all Cauchy se-
quences converge.) In short, C∗-algebra is a nicely behaved algebra whose elements form
a continuous set and a complex vector space. The name C comes from ‘continuous func-
tions’. Indeed, the bounded continuous functions form such an algebra, with a properly
Challenge 201 s defined norm. Can you find it?
Every C∗-algebra (pronounced ‘Cee-star’) contains a space of Hermitean elements
(which have a real spectrum), a set of normal elements, a multiplicative group of unitary
elements and a set of positive elements (with non-negative spectrum). In quantum the-
ory, a physical system is described by a C∗-algebra, and its Hermitean elements are the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
observables.
We should mention one important type of algebra used in mathematics. A division
algebra is an algebra for which the equations 𝑎𝑥 = 𝑏 and 𝑦𝑎 = 𝑏 are uniquely solvable in
𝑥 or 𝑦 for all 𝑏 and all 𝑎 ≠ 0. Obviously, all type of continuous numbers must be division
algebras. Division algebras are thus one way to generalize the concept of a number. One
of the important results of modern mathematics states that (finite-dimensional) division
algebras can only have dimension 1, like the reals, dimension 2, like the complex numbers,
dimension 4, like the quaternions, or dimension 8, like the octonions. There is thus no
way to generalize the concept of (continuous) ‘number’ to other dimensions.
And now for some fun. Imagine a ring A which contains a number field K as a subring
(or ‘field of scalars’). If the ring multiplication is defined in such a way that a general ring
element multiplied with an element of K is the same as the scalar multiplication, then A
is a vector space, and thus an algebra – provided that every element of K commutes with
every element of A. (In other words, the subring K must be central.)
For example, the quaternions ℍ are a four-dimensional real division algebra, but al-
though ℍ is a two-dimensional complex vector space, it is not a complex algebra, because
𝑖 does not commute with 𝑗 (one has 𝑖𝑗 = −𝑗𝑖 = 𝑘). In fact, there are no finite-dimensional
complex division algebras, and the only finite-dimensional real associative division al-
gebras are ℝ, ℂ and ℍ.
Now, if you are not afraid of getting a headache, think about this remark: every K-
algebra is also an algebra over its field of scalars. For this reason, some mathematicians
algebras 361

prefer to define an (associative) K-algebra simply as a ring which contains K as a central
subfield.
In physics, it is the algebras related to symmetries which play the most important role.
We study them next.

Lie algebras
A Lie algebra is special type of algebra (and thus of vector space). Lie algebras are the
most important type of non-associative algebras. A vector space 𝐿 over the field ℝ (or
ℂ) with an additional binary operation [ , ], called Lie multiplication or the commutator,
is called a real (or complex) Lie algebra if this operation satisfies

[𝑋, 𝑌] = −[𝑌, 𝑋] antisymmetry
[𝑎𝑋 + 𝑏𝑌, 𝑍] = 𝑎[𝑋, 𝑍] + 𝑏[𝑌, 𝑍] (left-)linearity
[𝑋, [𝑌,𝑍]] + [𝑌, [𝑍, 𝑋]] + [𝑍, [𝑋, 𝑌]] = 0 Jacobi identity (144)

Motion Mountain – The Adventure of Physics
for all elements 𝑋, 𝑌, 𝑍 ∈ 𝐿 and for all 𝑎, 𝑏 ∈ ℝ (or ℂ). (Lie algebras are named after
Challenge 202 e Sophus Lie.) The first two conditions together imply bilinearity. A Lie algebra is called
commutative if [𝑋, 𝑌] = 0 for all elements 𝑋 and 𝑌. The dimension of the Lie algebra is
the dimension of the vector space. A subspace 𝑁 of a Lie algebra 𝐿 is called an ideal* if
[𝐿, 𝑁] ⊂ 𝑁; any ideal is also a subalgebra. A maximal ideal 𝑀 which satisfies [𝐿, 𝑀] = 0
is called the centre of 𝐿.
A Lie algebra is called a linear Lie algebra if its elements are linear transformations of
another vector space 𝑉 (intuitively, if they are ‘matrices’). It turns out that every finite-
dimensional Lie algebra is isomorphic to a linear Lie algebra. Therefore, there is no loss

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of generality in picturing the elements of finite-dimensional Lie algebras as matrices.
The name ‘Lie algebra’ was chosen because the generators, i.e., the infinitesimal ele-
Page 371 ments of every Lie group, form a Lie algebra. Since all important symmetries in nature
form Lie groups, Lie algebras appear very frequently in physics. In mathematics, Lie al-
gebras arise frequently because from any associative finite-dimensional algebra (in which
the symbol ⋅ stands for its multiplication) a Lie algebra appears when we define the com-
mutator by
[𝑋, 𝑌] = 𝑋 ⋅ 𝑌 − 𝑌 ⋅ 𝑋 . (145)

(This fact gave the commutator its name.) Lie algebras are non-associative in general;
but the above definition of the commutator shows how to build one from an associative
algebra.
Since Lie algebras are vector spaces, the elements 𝑇𝑖 of a basis of the Lie algebra always
obey a relation of the form:
[𝑇𝑖 , 𝑇𝑗 ] = ∑ 𝑐𝑖𝑗𝑘 𝑇𝑘 . (146)
𝑘

The numbers 𝑐𝑖𝑗𝑘 are called the structure constants of the Lie algebra. They depend on

Challenge 203 ny * Can you explain the notation [𝐿, 𝑁]? Can you define what a maximal ideal is and prove that there is only
one?
362 c algebras, shapes and groups

the choice of basis. The structure constants determine the Lie algebra completely. For
example, the algebra of the Lie group SU(2), with the three generators defined by 𝑇𝑎 =
Vol. IV, page 231 𝜎𝑎 /2𝑖, where the 𝜎𝑎 are the Pauli spin matrices, has the structure constants 𝐶𝑎𝑏𝑐 = 𝜀𝑎𝑏𝑐 .*

Classification of Lie algebras
Finite-dimensional Lie algebras are classified as follows. Every finite-dimensional Lie al-
gebra is the (semidirect) sum of a semisimple and a solvable Lie algebra.
A Lie algebra is called solvable if, well, if it is not semisimple. Solvable Lie algebras
have not yet been classified completely. They are not important in physics.
A semisimple Lie algebra is a Lie algebra which has no non-zero solvable ideal. Other
equivalent definitions are possible, depending on your taste:
— a semisimple Lie algebra does not contain non-zero Abelian ideals;
— its Killing form is non-singular, i.e., non-degenerate;
— it splits into the direct sum of non-Abelian simple ideals (this decomposition is

Motion Mountain – The Adventure of Physics
unique);
— every finite-dimensional linear representation is completely reducible;
— the one-dimensional cohomology of 𝑔 with values in an arbitrary finite-dimensional
𝑔-module is trivial.
Finite-dimensional semisimple Lie algebras have been completely classified. They de-
compose uniquely into a direct sum of simple Lie algebras. Simple Lie algebras can be
complex or real.
The simple finite-dimensional complex Lie algebras all belong to four infinite classes
and to five exceptional cases. The infinite classes are also called classical, and are: 𝐴 𝑛 for

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑛 ⩾ 1, corresponding to the Lie groups SL(𝑛 + 1) and their compact ‘cousins’ SU(𝑛 + 1);
𝐵𝑛 for 𝑛 ⩾ 1, corresponding to the Lie groups SO(2𝑛 + 1); 𝐶𝑛 for 𝑛 ⩾ 1, corresponding
to the Lie groups Sp(2𝑛); and 𝐷𝑛 for 𝑛 ⩾ 4, corresponding to the Lie groups SO(2𝑛).
Thus 𝐴 𝑛 is the algebra of all skew-Hermitean matrices; 𝐵𝑛 and 𝐷𝑛 are the algebras of the

* Like groups, Lie algebras can be represented by matrices, i.e., by linear operators. Representations of Lie
algebras are important in physics because many continuous symmetry groups are Lie groups.
The adjoint representation of a Lie algebra with basis 𝑎1 ...𝑎𝑛 is the set of matrices ad(𝑎) defined for each
element 𝑎 by
[𝑎, 𝑎𝑗 ] = ∑ ad(𝑎)𝑐𝑗 𝑎𝑐 . (147)
𝑐

The definition implies that ad(𝑎𝑖 )𝑗𝑘 = 𝑐𝑖𝑗𝑘 , 𝑐𝑖𝑗𝑘
where are the structure constants of the Lie algebra. For a real
Lie algebra, all elements of ad(𝑎) are real for all 𝑎 ∈ 𝐿.
Note that for any Lie algebra, a scalar product can be defined by setting

𝑋 ⋅ 𝑌 = Tr( ad𝑋 ⋅ ad𝑌 ) . (148)

This scalar product is symmetric and bilinear. (Can you show that it is independent of the representation?)
The corresponding bilinear form is also called the Killing form, after the mathematician Wilhelm Killing
(b. 1847 Burbach, d. 1923 Münster), the discoverer of the ‘exceptional’ Lie groups. The Killing form is in-
variant under the action of any automorphism of the Lie algebra L. In a given basis, one has

𝑋 ⋅ 𝑌 = Tr( (ad𝑋) ⋅ (ad𝑌)) = 𝑐𝑙𝑘𝑖 𝑐𝑠𝑖𝑘 𝑥𝑙 𝑦𝑠 = 𝑔𝑙𝑠 𝑥𝑙 𝑦𝑠 (149)

where 𝑔𝑙𝑠 = 𝑐𝑙𝑘𝑖 𝑐𝑠𝑖𝑘 is called the Cartan metric tensor of L.
topology – what shapes exist? 363

F I G U R E 183 Which of the two situations can be untied without cutting?

Motion Mountain – The Adventure of Physics
symmetric matrices; and 𝐶𝑛 is the algebra of the traceless matrices.
The exceptional Lie algebras are 𝐺2 , 𝐹4 , 𝐸6 , 𝐸7 , 𝐸8 . In all cases, the index gives the
number of so-called roots of the algebra. The dimensions of these algebras are 𝐴 𝑛 : 𝑛(𝑛 +
2); 𝐵𝑛 and 𝐶𝑛 : 𝑛(2𝑛 + 1); 𝐷𝑛 : 𝑛(2𝑛 − 1); 𝐺2 : 14; 𝐹4 : 32; 𝐸6 : 78; 𝐸7 : 133; 𝐸8 : 248.
The simple and finite-dimensional real Lie algebras are more numerous; their classi-
fication follows from that of the complex Lie algebras. Moreover, corresponding to each
Ref. 300 complex Lie group, there is always one compact real one. Real Lie algebras are not so

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
important in fundamental physics.
Of the large number of infinite-dimensional Lie algebras, only one is important in
physics, the Poincaré algebra. A few other such algebras only appeared in failed attempts
for unification.

top ol o gy – what shapes exist?

“ ”
Topology is group theory.
The Erlangen program

In a simplified view of topology that is sufficient for physicists, only one type of entity
can possess shape: manifolds. Manifolds are generalized examples of pullovers: they are
locally flat, can have holes and boundaries, and can often be turned inside out.
Pullovers are subtle entities. For example, can you turn your pullover inside out while
Challenge 204 s your hands are tied together? (A friend may help you.) By the way, the same feat is also
possible with your trousers, while your feet are tied together. Certain professors like to
demonstrate this during topology lectures – of course with a carefully selected pair of
underpants.
Ref. 301 Another good topological puzzle, the handcuff puzzle, is shown in Figure 183. Which
Challenge 205 s of the two situations can be untied without cutting the ropes?
For a mathematician, pullovers and ropes are everyday examples of manifolds, and
364 c algebras, shapes and groups

the operations that are performed on them are examples of deformations. Let us look
at some more precise definitions. In order to define what a manifold is, we first need to
define the concept of topological space.

Topolo gical spaces

“
En Australie, une mouche qui marche au
plafond se trouve dans le même sens qu’une

”
vache chez nous.
Philippe Geluck, La marque du chat.

Ref. 302 The study of shapes requires a good definition of a set made of ‘points’. To be able to talk
about shape, these sets must be structured in such a way as to admit a useful concept
of ‘neighbourhood’ or ‘closeness’ between the elements of the set. The search for the
most general type of set which allows a useful definition of neighbourhood has led to
the concept of topological space. There are two ways to define a topology: one can define

Motion Mountain – The Adventure of Physics
the concept of open set and then define the concept of neighbourhood with their help, or
the other way round. We use the second option, which is somewhat more intuitive.
A topological space is a finite or infinite set 𝑋 of elements, called points, together with
the neighbourhoods for each point. A neighbourhood 𝑁 of a point 𝑥 is a collection of
subsets 𝑌𝑥 of 𝑋 with the properties that
— 𝑥 is in every 𝑌𝑥 ;
— if 𝑁 and 𝑀 are neighbourhoods of 𝑥, so is 𝑁 ∩ 𝑀;
— anything containing a neighbourhood of 𝑥 is itself a neighbourhood of 𝑥.
The choice of the subsets 𝑌𝑥 is free. The subsets 𝑌𝑥 for all points 𝑥, chosen in a particular

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
definition, contain a neighbourhood for each of their points; they are called open sets. (A
neighbourhood and an open set usually differ, but all open sets are also neighbourhoods.
Neighbourhoods of 𝑥 can also be described as subsets of 𝑋 that contain an open set that
contains 𝑥.)
One also calls a topological space a ‘set with a topology’. In effect, a topology specifies
the systems of ‘neighbourhoods’ of every point of the set. ‘Topology’ is also the name of
the branch of mathematics that studies topological spaces.
For example, the real numbers together with all open intervals form the usual topo-
logy of ℝ. Mathematicians have generalized this procedure. If one takes all subsets of ℝ
– or any other basis set – as open sets, one speaks of the discrete topology. If one takes
only the full basis set and the empty set as open sets, one speaks of the trivial or indiscrete
topology.
The concept of topological space allows us to define continuity. A mapping from one
topological space 𝑋 to another topological space 𝑌 is continuous if the inverse image
of every open set in 𝑌 is an open set in 𝑋. You may verify that this condition is not
Challenge 206 e satisfied by a real function that makes a jump. You may also check that the term ‘inverse’
is necessary in the definition; otherwise a function with a jump would be continuous, as
such a function may still map open sets to open sets.*

* The Cauchy–Weierstass definition of continuity says that a real function 𝑓(𝑥) is continuous at a point 𝑎
if (1) 𝑓 is defined on an open interval containing 𝑎, (2) 𝑓(𝑥) tends to a limit as 𝑥 tends to 𝑎, and (3) the
topology – what shapes exist? 365

F I G U R E 184 Examples of orientable and non-orientable
manifolds of two dimensions: a disc, a Möbius strip, a
sphere and a Klein bottle.

We thus need the concept of topological space, or of neighbourhood, if we want to ex-

Motion Mountain – The Adventure of Physics
press the idea that there are no jumps in nature. We also need the concept of topological
space in order to be able to define limits.
Of the many special kinds of topological spaces that have been studied, one type is par-
ticularly important. A Hausdorff space is a topological space in which for any two points
𝑥 and 𝑦 there are disjoint open sets 𝑈 and 𝑉 such that 𝑥 is in 𝑈 and 𝑦 is in 𝑉. A Haus-
dorff space is thus a space where, no matter how ‘close’ two points are, they can always
be separated by open sets. This seems like a desirable property; indeed, non-Hausdorff
spaces are rather tricky mathematical objects. (At Planck energy, it seems that vacuum
appears to behave like a non-Hausdorff space; however, at Planck energy, vacuum is not

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
really a space at all. So non-Hausdorff spaces play no role in physics.) A special case of
Hausdorff space is well-known: the manifold.

Manifolds
In physics, the most important topological spaces are differential manifolds. Loosely
speaking, a differential manifold – physicists simply speak of a manifold – is a set of points
that looks like ℝ𝑛 under the microscope – at small distances. For example, a sphere and
a torus are both two-dimensional differential manifolds, since they look locally like a
plane. Not all differential manifolds are that simple, as the examples of Figure 184 show.
A differential manifold is called connected if any two points can be joined by a path
lying in the manifold. (The term has a more general meaning in topological spaces. But
the notions of connectedness and pathwise connectedness coincide for differential mani-
folds.) We focus on connected manifolds in the following discussion. A manifold is called
simply connected if every loop lying in the manifold can be contracted to a point. For
example, a sphere is simply connected. A connected manifold which is not simply con-
nected is called multiply connected. A torus is multiply connected.
Manifolds can be non-orientable, as the well-known Möbius strip illustrates. Non-
orientable manifolds have only one surface: they do not admit a distinction between

limit is 𝑓(𝑎). In this definition, the continuity of 𝑓 is defined using the intuitive idea that the real numbers
form the basic model of a set that has no gaps. Can you see the connection with the general definition given
Challenge 207 ny above?
366 c algebras, shapes and groups

Motion Mountain – The Adventure of Physics
F I G U R E 185 Compact (left) and non-compact (right) manifolds of various dimensions.

Challenge 208 e front and back. If you want to have fun, cut a paper Möbius strip into two along a centre
line. You can also try this with paper strips with different twist values, and investigate the
regularities.
In two dimensions, closed manifolds (or surfaces), i.e., surfaces that are compact and

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
without boundary, are always of one of three types:
— The simplest type are spheres with 𝑛 attached handles; they are called n-tori or surfaces
of genus 𝑛. They are orientable surfaces with Euler characteristic 2 − 2𝑛.
— The projective planes with 𝑛 handles attached are non-orientable surfaces with Euler
characteristic 1 − 2𝑛.
— The Klein bottles with 𝑛 attached handles are non-orientable surfaces with Euler char-
acteristic −2𝑛.
Therefore Euler characteristic and orientability describe compact surfaces up
to homeomorphism (and if surfaces are smooth, then up to diffeomorphism).
Page 366 Homeomorphisms are defined below.
The two-dimensional compact manifolds or surfaces with boundary are found by re-
moving one or more discs from a surface in this list. A compact surface can be embedded
in ℝ3 if it is orientable or if it has non-empty boundary.
In physics, the most important manifolds are space-time and Lie groups of observ-
ables. We study Lie groups below. Strangely enough, the topology of space-time is not
known. For example, it is unclear whether or not it is simply connected. Obviously, the
reason is that it is difficult to observe what happens at large distances form the Earth.
However, a similar difficulty appears near Planck scales.
If a manifold is imagined to consist of rubber, connectedness and similar global prop-
erties are not changed when the manifold is deformed. This fact is formalized by saying
topology – what shapes exist? 367

F I G U R E 186 Simply connected (left), multiply connected (centre) and disconnected (right) manifolds of
one (above) and two (below) dimensions.

Motion Mountain – The Adventure of Physics
copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
F I G U R E 187 Examples of homeomorphic pairs of
manifolds.

that two manifolds are homeomorphic (from the Greek words for ‘same’ and ‘shape’) if
between them there is a continuous, one-to-one and onto mapping with a continuous
inverse. The concept of homeomorphism is somewhat more general than that of rub-
ber deformation, as can be seen from Figure 187. If the mapping and the manifolds are
differentiable, one says that the two manifolds are diffeomorphic.

Holes, homotopy and homolo gy
Only ‘well-behaved’ manifolds play a role in physics: namely those which are orient-
able and connected. In addition, the manifolds associated with observables, are always
compact. The main non-trivial characteristic of connected compact orientable mani-
folds is that they contain ‘holes’ (see Figure 188). It turns out that a proper descrip-
tion of the holes of manifolds allows us to distinguish between all different, i.e., non-
homeomorphic, types of manifold.
There are three main tools to describe holes of manifolds and the relations among
them: homotopy, homology and cohomology. These tools play an important role in the
study of gauge groups, because any gauge group defines a manifold.
In other words, through homotopy and homology theory, mathematicians can clas-
368 c algebras, shapes and groups

F I G U R E 188 The ﬁrst four two-dimensional compact connected orientable manifolds: 0-, 1-, 2- and
3-tori.

sify manifolds. Given two manifolds, the properties of the holes in them thus determine
whether they can be deformed into each other.
Physicists are now extending these results of standard topology. Deformation is a clas-
sical idea which assumes continuous space and time, as well as arbitrarily small action.
In nature, however, quantum effects cannot be neglected. It is speculated that quantum
effects can transform a physical manifold into one with a different topology: for example,
Challenge 209 d a torus into a sphere. Can you find out how this can be achieved?
Topological changes of physical manifolds happen via objects that are generalizations
of manifolds. An orbifold is a space that is locally modelled by ℝ𝑛 modulo a finite group.

Motion Mountain – The Adventure of Physics
Examples are the tear-drop or the half-plane. Orbifolds were introduced by Satake Ichiro
in 1956; the name was coined by William Thurston. Orbifolds are heavily studied in string
theory.

t ypes and cl assification of groups
Vol. I, page 272 We introduced mathematical groups early on because groups, especially symmetry
groups, play an important role in many parts of physics, from the description of solids,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
molecules, atoms, nuclei, elementary particles and forces up to the study of shapes, cycles
and patterns in growth processes.
Group theory is also one of the most important branches of modern mathematics, and
is still an active area of research. One of the aims of group theory is the classification of all
groups. This has been achieved only for a few special types. In general, one distinguishes
between finite and infinite groups. Finite groups are better understood.
Every finite group is isomorphic to a subgroup of the symmetric group 𝑆𝑁 , for some
number 𝑁. Examples of finite groups are the crystalline groups, used to classify crystal
structures, or the groups used to classify wallpaper patterns in terms of their symmetries.
The symmetry groups of Platonic and many other regular solids are also finite groups.
Finite groups are a complex family. Roughly speaking, a general (finite) group can be
seen as built from some fundamental bricks, which are groups themselves. These fun-
damental bricks are called simple (finite) groups. One of the high points of twentieth-
century mathematics was the classification of the finite simple groups. It was a collab-
orative effort that took around 30 years, roughly from 1950 to 1980. The complete list of
Ref. 303 finite simple groups consists of
1) the cyclic groups Z𝑝 of prime group order;
2) the alternating groups A𝑛 of degree 𝑛 at least five;
3) the classical linear groups, PSL(𝑛; 𝑞), PSU(𝑛; 𝑞), PSp(2𝑛; 𝑞) and PΩ𝜀 (𝑛; 𝑞);
4) the exceptional or twisted groups of Lie type 3 D4 (𝑞), E6 (𝑞), 2 E6 (𝑞), E7 (𝑞), E8 (𝑞),
F4 (𝑞), 2 F4 (2𝑛), G2 (𝑞), 2 G2 (3𝑛 ) and 2 B2 (2𝑛);
5) the 26 sporadic groups, namely M11 , M12 , M22 , M23 , M24 (the Mathieu groups), J1 ,
types and classification of groups 369

J2 , J3 , J4 (the Janko groups), Co1 , Co2 , Co3 (the Conway groups), HS, Mc, Suz (the Co1
‘babies’), Fi22 , Fi23 , Fi󸀠24 (the Fischer groups), F1 = M (the Monster), F2 , F3 , F5 , He (= F7 )
(the Monster ‘babies’), Ru, Ly, and ON.
The classification was finished in the 1980s after over 10 000 pages of publications.
The proof is so vast that a special series of books has been started to summarize and ex-
plain it. The first three families are infinite. The last family, that of the sporadic groups,
is the most peculiar; it consists of those finite simple groups which do not fit into the
other families. Some of these sporadic groups might have a role in particle physics: pos-
sibly even the largest of them all, the so-called Monster group. This is still a topic of
research. (The Monster group has about 8.1 ⋅ 1053 elements; more precisely, its order is
808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000 or 246 ⋅ 320 ⋅ 59 ⋅
76 ⋅ 112 ⋅ 133 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71.)
Of the infinite groups, only those with some finiteness condition have been studied.
It is only such groups that are of interest in the description of nature. Infinite groups
are divided into discrete groups and continuous groups. Discrete groups are an active

Motion Mountain – The Adventure of Physics
area of mathematical research, having connections with number theory and topology.
Continuous groups are divided into finitely generated and infinitely generated groups.
Finitely generated groups can be finite-dimensional or infinite-dimensional.
The most important class of finitely generated continuous groups are the Lie groups.

Lie groups
In nature, the Lagrangians of the fundamental forces are invariant under gauge trans-
formations and under continuous space-time transformations. These symmetry groups
are examples of Lie groups, which are a special type of infinite continuous group.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
They are named after the great mathematician Sophus Lie (b. 1842 Nordfjordeid,
d. 1899 Kristiania). His name is pronounced like ‘Lee’.
A (real) Lie group is an infinite symmetry group, i.e., a group with infinitely many
elements, which is also an analytic manifold. Roughly speaking, this means that the ele-
ments of the group can be seen as points on a smooth (hyper-) surface whose shape
can be described by an analytic function, i.e., by a function so smooth that it can be ex-
pressed as a power series in the neighbourhood of every point where it is defined. The
points of the Lie group can be multiplied according to the group multiplication. Further-
more, the coordinates of the product have to be analytic functions of the coordinates of
the factors, and the coordinates of the inverse of an element have to be analytic func-
tions of the coordinates of the element. In fact, this definition is unnecessarily strict: it
can be proved that a Lie group is just a topological group whose underlying space is a
finite-dimensional, locally Euclidean manifold.
A complex Lie group is a group whose manifold is complex and whose group opera-
tions are holomorphic (instead of analytical) functions in the coordinates.
In short, a Lie group is a well-behaved manifold in which points can be multiplied
(and technicalities). For example, the circle 𝑇 = {𝑧 ∈ ℂ : |𝑧| = 1}, with the usual complex
1
multiplication, is a real Lie group. It is Abelian. This group is also called S , as it is the one-
dimensional sphere, or U(1), which means ‘unitary group of one dimension’. The other
one-dimensional Lie groups are the multiplicative group of non-zero real numbers and
its subgroup, the multiplicative group of positive real numbers.
370 c algebras, shapes and groups

So far, in physics, only linear Lie groups have played a role – that is, Lie groups which
act as linear transformations on some vector space. (The cover of SL(2,ℝ) or the complex
compact torus are examples of non-linear Lie groups.) The important linear Lie groups
for physics are the Lie subgroups of the general linear group GL(N,K), where 𝐾 is a num-
ber field. This is defined as the set of all non-singular, i.e., invertible, N×N real, complex
or quaternionic matrices. All the Lie groups discussed below are of this type.
Every complex invertible matrix 𝐴 can be written in a unique way in terms of a unitary
matrix 𝑈 and a Hermitean matrix 𝐻:

𝐴 = 𝑈e𝐻 . (150)

Challenge 210 s (𝐻 is given by 𝐻 = 12 ln 𝐴† 𝐴, and 𝑈 is given by 𝑈 = 𝐴e−𝐻 .)
The simple Lie groups U(1) and SO(2,ℝ) and the Lie groups based on the real and
complex numbers are Abelian (see Table 37); all others are non-Abelian.
Lie groups are manifolds. Therefore, in a Lie group one can define the distance

Motion Mountain – The Adventure of Physics
between two points, the tangent plane (or tangent space) at a point, and the notions of
integration and differentiations. Because Lie groups are manifolds, Lie groups have the
same kind of structure as the objects of Figures 184, 185 and 186. Lie groups can have any
number of dimensions. Like for any manifold, their global structure contains important
information; let us explore it.

C onnectedness
It is not hard to see that the Lie groups SU(N) are simply connected for all N = 2, 3 . . . ;
they have the topology of a 2N-dimensional sphere. The Lie group U(1), having the to-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
pology of the 1-dimensional sphere, or circle, is multiply connected.
The Lie groups SO(N) are not simply connected for any N = 2, 3 . . . . In gen-
eral, SO(N,K) is connected, and GL(N,ℂ) is connected. All the Lie groups SL(N,K) are
connected; and SL(N,ℂ) is simply connected. The Lie groups Sp(N,K) are connected;
Sp(2N,ℂ) is simply connected. Generally, all semi-simple Lie groups are connected.
The Lie groups O(N,K), SO(N,M,K) and GL(N,ℝ) are not connected; they contain
two connected components.
Note that the Lorentz group is not connected: it consists of four separate pieces. Like
the Poincaré group, it is not compact, and neither is any of its four pieces. Broadly speak-
ing, the non-compactness of the group of space-time symmetries is a consequence of the
non-compactness of space-time.

C ompactness
A Lie group is compact if it is closed and bounded when seen as a manifold. For a given
parametrization of the group elements, the Lie group is compact if all parameter ranges
are closed and finite intervals. Otherwise, the group is called non-compact. Both compact
and non-compact groups play a role in physics. The distinction between the two cases
is important, because representations of compact groups can be constructed in the same
simple way as for finite groups, whereas for non-compact groups other methods have
to be used. As a result, physical observables, which always belong to a representation of
a symmetry group, have different properties in the two cases: if the symmetry group is
types and classification of groups 371

compact, observables have discrete spectra; otherwise they do not.
All groups of internal gauge transformations, such as U(1) and SU(𝑛), form compact
groups. In fact, field theory requires compact Lie groups for gauge transformations. The
only compact Lie groups are the torus groups T𝑛 , O(𝑛), U(𝑛), SO(𝑛) and SU(𝑛), their
double cover Spin(𝑛) and the Sp(𝑛). In contrast, SL(𝑛,ℝ), GL(𝑛,ℝ), GL(𝑛,ℂ) and all oth-
ers are not compact.
Besides being manifolds, Lie groups are obviously also groups. It turns out that most
of their group properties are revealed by the behaviour of the elements which are very
close (as points on the manifold) to the identity.
Every element of a compact and connected Lie group has the form exp(𝐴) for some 𝐴.
The elements 𝐴 arising in this way form an algebra, called the corresponding Lie algebra.
For any linear Lie group, every element of the connected subgroup can be expressed as
a finite product of exponentials of elements of the corresponding Lie algebra. Mathem-
atically, the vector space defined by the Lie algebra is tangent to the manifold defined
by the Lie group, at the location of the unit element. In short, Lie algebras express the

Motion Mountain – The Adventure of Physics
local properties of Lie groups near the identity. That is the reason for their importance
Page 361 in physics.

TA B L E 37 Properties of the most important real and complex Lie groups.

Lie Descrip- Properties 𝑎 Lie al- Description of Dimen-
group tion gebra Lie algebra sion
1. Real groups real
𝑛 𝑛
ℝ Euclidean Abelian, simply ℝ Abelian, thus Lie 𝑛

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
space with connected, not bracket is zero; not
addition compact; simple
π0 = π1 = 0
ℝ× non-zero real Abelian, not ℝ Abelian, thus Lie 1
numbers with connected, not bracket is zero
multiplica- compact; π0 = ℤ2 ,
tion no π1
ℝ>0 positive real Abelian, simply ℝ Abelian, thus Lie 1
numbers with connected, not bracket is zero
multiplica- compact;
tion π0 = π1 = 0
S1 = ℝ/ℤ complex Abelian, connected, ℝ Abelian, thus Lie 1
= U(1) = numbers of not simply bracket is zero
T = SO(2) absolute value connected,
= Spin(2) 1, with multi- compact; π0 = 0,
plication π1 = ℤ
ℍ× non-zero simply connected, ℍ quaternions, with Lie 4
quaternions not compact; bracket the
with multi- π0 = π1 = 0 commutator
plication
372 c algebras, shapes and groups

TA B L E 37 (Continued) Properties of the most important real and complex Lie groups.

Lie Descrip- Properties 𝑎 Lie al- Description of Dimen-
group tion gebra Lie algebra sion
S3 quaternions simply connected, Im(ℍ) quaternions with zero 3
of absolute compact; real part, with Lie
value 1, with isomorphic to bracket the
multiplica- SU(2), Spin(3) and commutator; simple
tion, also to double cover of and semi-simple;
known as SO(3); π0 = π1 = 0 isomorphic to real
Sp(1); 3-vectors, with Lie
topologically bracket the cross
a 3-sphere product; also
isomorphic to su(2)
and to so(3)
GL(𝑛, ℝ) general linear not connected, not M(𝑛, ℝ) 𝑛-by-𝑛 matrices, with 𝑛2

Motion Mountain – The Adventure of Physics
group: compact; π0 = ℤ2 , Lie bracket the
invertible no π1 commutator
𝑛-by-𝑛 real
matrices
GL+ (𝑛, ℝ) 𝑛-by-𝑛 real simply connected, M(𝑛, ℝ) 𝑛-by-𝑛 matrices, with 𝑛2
matrices with not compact; Lie bracket the
positive π0 = 0, for 𝑛 = 2: commutator
determinant π1 = ℤ, for 𝑛 ≥ 2:
π1 = ℤ2 ; GL+ (1, ℝ)
isomorphic to ℝ>0

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
SL(𝑛, ℝ) special linear simply connected, sl(𝑛, ℝ) 𝑛-by-𝑛 matrices with 𝑛2 − 1
group: real not compact if = A𝑛−1 trace 0, with Lie
matrices with 𝑛 > 1; π0 = 0, for bracket the
determinant 1 𝑛 = 2: π1 = ℤ, for commutator
𝑛 ≥ 2: π1 = ℤ2 ;
SL(1, ℝ) is a single
point, SL(2, ℝ) is
isomorphic to
SU(1, 1) and
Sp(2, ℝ)
O(𝑛, ℝ) orthogonal not connected, so(𝑛, ℝ) skew-symmetric 𝑛(𝑛 − 1)/2
= O(𝑛) group: real compact; π0 = ℤ2 , 𝑛-by-𝑛 real matrices,
orthogonal no π1 with Lie bracket the
matrices; commutator; so(3, ℝ)
symmetry of is isomorphic to su(2)
hypersphere and to ℝ3 with the
cross product
types and classification of groups 373

TA B L E 37 (Continued) Properties of the most important real and complex Lie groups.

Lie Descrip- Properties 𝑎 Lie al- Description of Dimen-
group tion gebra Lie algebra sion
SO(𝑛, ℝ) special connected, so(𝑛, ℝ) skew-symmetric 𝑛(𝑛 − 1)/2
= SO(𝑛) orthogonal compact; for 𝑛 ⩾ 2 = B 𝑛−1 or 𝑛-by-𝑛 real matrices,
2
group: real not simply D𝑛 with Lie bracket the
2
orthogonal connected; π0 = 0, commutator; for 𝑛 = 3
matrices with for 𝑛 = 2: π1 = ℤ, and 𝑛 ⩾ 5 simple and
determinant 1 for 𝑛 ≥ 2: π1 = ℤ2 semisimple; SO(4) is
semisimple but not
simple
Spin(𝑛) spin group; simply connected so(𝑛, ℝ) skew-symmetric 𝑛(𝑛 − 1)/2
double cover for 𝑛 ⩾ 3, compact; 𝑛-by-𝑛 real matrices,
of SO(𝑛); for 𝑛 = 3 and 𝑛 ⩾ 5 with Lie bracket the
Spin(1) is simple and commutator

Motion Mountain – The Adventure of Physics
isomorphic to semisimple; for
ℚ2 , Spin(2) to 𝑛 > 1: π0 = 0, for
S1 𝑛 > 2: π1 = 0
Sp(2𝑛, ℝ) symplectic not compact; sp(2𝑛, ℝ) real matrices 𝐴 that 𝑛(2𝑛 + 1)
group: real π0 = 0, π1 = ℤ = C𝑛 satisfy 𝐽𝐴 + 𝐴𝑇 𝐽 = 0
symplectic where 𝐽 is the
matrices standard
skew-symmetric
matrix;𝑏 simple and
semisimple

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Sp(𝑛) for compact compact, simply sp(𝑛) 𝑛-by-𝑛 quaternionic 𝑛(2𝑛 + 1)
𝑛⩾3 symplectic connected; matrices 𝐴 satisfying
group: π0 = π1 = 0 𝐴 = −𝐴∗ , with Lie
quaternionic bracket the
𝑛 × 𝑛 unitary commutator; simple
matrices and semisimple
U(𝑛) unitary not simply u(𝑛) 𝑛-by-𝑛 complex 𝑛2
group: connected, matrices 𝐴 satisfying
complex 𝑛 × 𝑛 compact; it is not a 𝐴 = −𝐴∗ , with Lie
unitary complex Lie bracket the
matrices group/algebra; commutator
π0 = 0, π1 = ℤ;
isomorphic to 𝑆1 for
𝑛=1
SU(𝑛) special simply connected, su(𝑛) 𝑛-by-𝑛 complex 𝑛2 − 1
unitary compact; it is not a matrices 𝐴 with trace
group: complex Lie 0 satisfying 𝐴 = −𝐴∗ ,
complex 𝑛 × 𝑛 group/algebra; with Lie bracket the
unitary π0 = π1 = 0 commutator; for 𝑛 ⩾ 2
matrices with simple and
determinant 1 semisimple
2. Complex groups𝑐 complex
374 c algebras, shapes and groups

TA B L E 37 (Continued) Properties of the most important real and complex Lie groups.

Lie Descrip- Properties 𝑎 Lie al- Description of Dimen-
group tion gebra Lie algebra sion
ℂ𝑛 group Abelian, simply ℂ𝑛 Abelian, thus Lie 𝑛
operation is connected, not bracket is zero
addition compact;
π0 = π1 = 0
×
ℂ non-zero Abelian, not simply ℂ Abelian, thus Lie 1
complex connected, not bracket is zero
numbers with compact; π0 = 0,
multiplica- π1 = ℤ
tion
GL(𝑛, ℂ) general linear simply connected, M(𝑛, ℂ) 𝑛-by-𝑛 matrices, with 𝑛2
group: not compact; π0 = 0, Lie bracket the
invertible π1 = ℤ; for 𝑛 = 1 commutator

Motion Mountain – The Adventure of Physics
𝑛-by-𝑛 isomorphic to ℂ×
complex
matrices
SL(𝑛, ℂ) special linear simply connected; sl(𝑛, ℂ) 𝑛-by-𝑛 matrices with 𝑛2 − 1
group: for 𝑛 ⩾ 2 not trace 0, with Lie
complex compact; bracket the
matrices with π0 = π1 = 0; commutator; simple,
determinant 1 SL(2, ℂ) is semisimple; sl(2, ℂ) is
isomorphic to isomorphic to
Spin(3, ℂ) and su(2, ℂ) ⊗ ℂ

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Sp(2, ℂ)
PSL(2, ℂ) projective not compact; sl(2, ℂ) 2-by-2 matrices with 3
special linear π0 = 0, π1 = ℤ2 trace 0, with Lie
group; bracket the
isomorphic to commutator; sl(2, ℂ)
the Möbius is isomorphic to
group, to the su(2, ℂ) ⊗ ℂ
restricted
Lorentz
group
SO+ (3, 1, ℝ)
and to
SO(3, ℂ)
O(𝑛, ℂ) orthogonal not connected; for so(𝑛, ℂ) skew-symmetric 𝑛(𝑛 − 1)/2
group: 𝑛 ⩾ 2 not compact; 𝑛-by-𝑛 complex
complex π0 = ℤ2 , no π1 matrices, with Lie
orthogonal bracket the
matrices commutator
mathematical curiosities and fun challenges 375

TA B L E 37 (Continued) Properties of the most important real and complex Lie groups.

Lie Descrip- Properties 𝑎 Lie al- Description of Dimen-
group tion gebra Lie algebra sion
SO(𝑛, ℂ) special for 𝑛 ⩾ 2 not so(𝑛, ℂ) skew-symmetric 𝑛(𝑛 − 1)/2
orthogonal compact; not simply 𝑛-by-𝑛 complex
group: connected; π0 = 0, matrices, with Lie
complex for 𝑛 = 2: π1 = ℤ, bracket the
orthogonal for 𝑛 ≥ 2: π1 = ℤ2 ; commutator; for 𝑛 = 3
matrices with non-Abelian for and 𝑛 ⩾ 5 simple and
determinant 1 𝑛 > 2, SO(2, ℂ) is semisimple
Abelian and
isomorphic to ℂ×
Sp(2𝑛, ℂ) symplectic not compact; sp(2𝑛, ℂ) complex matrices that 𝑛(2𝑛 + 1)
group: π0 = π1 = 0 satisfy 𝐽𝐴 + 𝐴𝑇 𝐽 = 0
complex where 𝐽 is the

Motion Mountain – The Adventure of Physics
symplectic standard
matrices skew-symmetric
matrix;𝑏 simple and
semi-simple

𝑎. The group of components π0 of a Lie group is given; the order of π0 is the number of components
of the Lie group. If the group is trivial (0), the Lie group is connected. The fundamental group π1
of a connected Lie group is given. If the group π1 is trivial (0), the Lie group is simply connected.
This table is based on that in the Wikipedia, at en.wikipedia.org/wiki/Table_of_Lie_groups.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝑏. The standard skew-symmetric matrix 𝐽 of rank 2𝑛 is 𝐽𝑘𝑙 = 𝛿𝑘,𝑛+𝑙 − 𝛿𝑘+𝑛,𝑙 .
𝑐. Complex Lie groups and Lie algebras can be viewed as real Lie groups and real Lie algebras of
twice the dimension.

mathematical curiosities and fun challenges
Challenge 211 ny A theorem of topology says: you cannot comb a hairy football. Can you prove it?
∗∗
Topology is fun. If you want to laugh for half an hour, fix a modified pencil, as shown in
Figure 189, to a button hole and let people figure out how to get it off again.
∗∗
There are at least six ways to earn a million dollars with mathematical research. The Clay
Mathematics Institute at www.claymath.org offered such a prize for major advances in
seven topics:
— proving the Birch and Swinnerton–Dyer conjecture about algebraic equations;
— proving the Poincaré conjecture about topological manifolds;
— solving the Navier–Stokes equations for fluids;
376 c algebras, shapes and groups

loop is dinner final result
shorter jacket
than the with
pencil button hole

Motion Mountain – The Adventure of Physics
F I G U R E 189 A well-known magic dexterity trick: make your friend go mad by adding a pencil to his
dinner jacket.

— finding criteria distinguishing P and NP numerical problems;
— proving the Riemann hypothesis stating that the non-trivial zeros of the zeta function
lie on a line;
— proving the Hodge conjectures;
— proving the connection between Yang–Mills theories and a mass gap in quantum field

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Vol. VI, page 345 theory.
The Poincaré conjecture was solved in 2002 by Grigori Perelman; on each of the other
six topics, substantial progress can buy you a house.
C HA L L E NG E H I N T S A N D S OLU T ION S

Challenge 1, page 10: Do not hesitate to be demanding and strict. The next edition of the text
will benefit from it.
Challenge 2, page 17: A virus is an example. It has no own metabolism. By the way, the ability
of some viruses to form crystals is not a proof that they are not living beings, in contrast to what
is often said. Apart from viruses, also prions, viroids and virusoids are examples of non-living

Motion Mountain – The Adventure of Physics
systems that can reproduce. Another, quite different border example between life and non-living
matter is provided by the tardigrades. These little animals, about 1 mm in size, can loose all their
water, remain in this dry – or ‘‘dead’’ – state for years, and then start living again when a drop
of water is added to them.
Challenge 3, page 18: The navigation systems used by flies are an example.
Challenge 6, page 22: The thermal energy 𝑘𝑇 is about 4 zJ and a typical relaxation time is 0.1 ps.
Challenge 10, page 30: The argument is correct.
Challenge 8, page 30: This is not possible at present. If you know a way, publish it. It would help
a sad single mother who has to live without financial help from the father, despite a lawsuit, as it

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
was yet impossible to decide which of the two candidates is the right one.
Challenge 9, page 30: Also identical twins count as different persons and have different fates.
Imprinting in the womb is different, so that their temperament will be different. The birth exper-
ience will be different; this is the most intense experience of every human, strongly determining
his fears and thus his character. A person with an old father is also quite different from that with a
young father. If the womb is not that of his biological mother, a further distinction of the earliest
and most intensive experiences is given.
Challenge 11, page 30: Be sure to publish your results.
Challenge 12, page 31: Yes, but only very young ones. Why?
Challenge 13, page 31: The reason of animal symmetry is simple: without symmetry, they would
not be able to move in a straight line.
Challenge 14, page 31: Life’s chemicals are synthesized inside the body; the asymmetry has
been inherited along the generations. The common asymmetry thus shows that all life has a com-
mon origin.
Challenge 15, page 31: Well, men are more similar to chimpanzees than to women. More seri-
ously, the above data, even though often quoted, are wrong. Newer measurements by Roy Britten
in 2002 have shown that the difference in genome between humans and chimpanzees is about
5 % (See R. J. Britten, Divergence between samples of chimpanzee and human DNA sequences is
5 %, counting indels, Proceedings of the National Academy of Sciences 99, pp. 13633–13635, 15th
of October, 2002.) In addition, though the difference between man and woman is smaller than
one whole chromosome, the large size of the X chromosome, compared with the small size of the
Y chromosome, implies that men have about 3 % less genetic material than women. However, all
378 challenge hints and solutions

men have an X chromosome as well. That explains that still other measurements suggest that all
humans share a pool of at least 99.9 % of common genes.
Challenge 18, page 34: Chemical processes, including diffusion and reaction rates, are strongly
temperature dependent. They affect the speed of motion of the individual and thus its chance of
survival. Keeping temperature in the correct range is thus important for evolved life forms.
Challenge 19, page 34: The first steps are not known at all. Subsequent processes that added the
complexity of cells are better understood.
Challenge 20, page 34: Since all the atoms we are made of originate from outer space, the answer
is yes. But if one means that biological cells came to Earth from space, the answer is no, as most
cells do not like vacuum. The same is true for DNA.
In fact, life and reproduction are properties of complex systems. In other words, asking
whether life comes from outer space is like asking: ‘Could car insurance have originated in outer
space?’
Challenge 23, page 40: Haven’t you tried yet? Physics is an experimental science.
Challenge 31, page 48: Exponential decays occur when the probability of decay is constant over

Motion Mountain – The Adventure of Physics
time. For humans, this is not the case. Why not?
Challenge 32, page 51: There are no non-physical processes: anything that can be observed is a
physical process. Consciousness is due to processes in the brain, thus inside matter; thus it is a
quantum process. At body temperature, coherence has lifetimes much smaller than the typical
thought process.
Challenge 33, page 52: Radioactive dating methods can be said to be based on the nuclear in-
teractions, even though the detection is again electromagnetic.
Challenge 34, page 53: All detectors of light can be called relativistic, as light moves with max-
imal speed. Touch sensors are not relativistic following the usual sense of the word, as the speeds
involved are too small. The energies are small compared to the rest energies; this is the case even

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
if the signal energies are attributed to electrons only.
Challenge 35, page 53: The noise is due to the photoacoustic effect; the periodic light period-
ically heats the air in the jam glass at the blackened surface and thus produces sound. See
M. Euler, Kann man Licht hören?, Physik in unserer Zeit 32, pp. 180–182, 2001.
Challenge 36, page 55: It implies that neither resurrection nor reincarnation nor eternal life are
possible.
Challenge 39, page 65: The ethanol disrupts the hydrogen bonds between the water molecules
so that, on average, they can get closer together. A video of the experiment is found at www.
youtube.com/watch?v=LUW7a7H-KuY.
Challenge 42, page 66: You get an intense yellow colour due to the formation of lead iodide
(PbI2 ).
Challenge 44, page 73: The usual way to pack oranges on a table is the densest way to pack
spheres.
Challenge 45, page 74: Just use a paper drawing. Draw a polygon and draw it again at latter
times, taking into account how the sides grow over time. You will see by yourself how the faster
growing sides disappear over time.
Challenge 47, page 84: With a combination of the methods of Table 7, this is indeed possible. In
fact, using cosmic rays to search for unknown chambers in the pyramids has been already done
Ref. 304 in the 1960s. The result was that no additional chambers exist.
Challenge 49, page 88: For example, a heavy mountain will push down the Earth’s crust into
the mantle, makes it melt on the bottom side, and thus lowers the position of the top.
challenge hints and solutions 379

Challenge 50, page 88: These developments are just starting; the results are still far from the ori-
ginal one is trying to copy, as they have to fulfil a second condition, in addition to being a ‘copy’
of original feathers or of latex: the copy has to be cheaper than the original. That is often a much
tougher request than the first.
Challenge 51, page 88: About 0.2 m.
Challenge 53, page 89: Since the height of the potential is always finite, walls can always be over-
come by tunnelling.
Challenge 54, page 90: The lid of a box can never be at rest, as is required for a tight closure, but
is always in motion, due to the quantum of action.
Challenge 56, page 90: The unit of thermal conductance is 𝑇π2 𝑘2 /3ℏ, where 𝑇 is temperature
and 𝑘 is the Boltzmann constant.
Challenge 57, page 91: Extremely slender structures are not possible for two reasons: First, be-
cause structures built of homogeneous materials do not to achieve such ratios; secondly, the
bending behaviour of plants is usually not acceptable in human-built structures.
Challenge 58, page 93: The concentrations and can be measured from polar ice caps, by measur-

Motion Mountain – The Adventure of Physics
ing how the isotope concentration changes over depth. Both in evaporation and in condensation
of water, the isotope ratio depends on the temperature. The measurements in Antarctica and in
Greenland coincide, which is a good sign of their trustworthiness.
Challenge 59, page 94: In the summer, tarmac is soft.
Challenge 62, page 113: The one somebody else has thrown away. Energy costs about
10 cents/kWh. For new lamps, the fluorescent lamp is the best for the environment, even
though it is the least friendly to the eye and the brain, due to its flickering.
Challenge 63, page 118: This old dream depends on the precise conditions. How flexible does
the display have to be? What lifetime should it have? The newspaper like display is many years

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
away and maybe not even possible.
Challenge 64, page 118: There is only speculation on the answer; the tendency of most research-
ers is to say no.
Challenge 65, page 118: The challenge here is to find a cheap way to deflect laser beams in a
controlled way. Cheap lasers are already available.
Challenge 66, page 118: No, as it is impossible because of momentum conservation and because
of the no-cloning theorem.
Challenge 67, page 119: There are companies trying to sell systems based on quantum crypto-
logy; but despite the technical interest, the commercial success is questionable.
Challenge 68, page 119: I predicted since the year 2000 that mass-produced goods using this
technology (at least 1 million pieces sold) will not be available before 2025.
Challenge 69, page 119: Maybe, but for extremely high prices.
Challenge 70, page 120: The set-up is affordable: it uses a laser at 3.39 μm, a detector and some
optics on a tripod. Sensitivity to alcohol absorption is excellent. Only future will tell.
Challenge 72, page 125: For example, you could change gravity between two mirrors.
Challenge 73, page 125: As usual in such statements, either group or phase velocity is cited, but
not the corresponding energy velocity, which is always below 𝑐.
Challenge 75, page 128: Echoes do not work once the speed of sound is reached and do not work
well when it is approached. Both the speed of light and that of sound have a finite value. Moving
with a mirror still gives a mirror image. This means that the speed of light cannot be reached. If
it cannot be reached, it must be the same for all observers.
380 challenge hints and solutions

Challenge 76, page 129: Mirrors do not usually work for matter; in addition, if they did, matter,
because of its rest energy, would require much higher acceleration values.
Challenge 79, page 131: The classical radius of the electron, which is the size at which the field
energy would make up the full electron mass, is about 137 times smaller, thus much smaller, than
the Compton wavelength of the electron.
Challenge 80, page 133: The overhang can have any value whatsoever. There is no limit. Taking
the indeterminacy relation into account introduces a limit as the last brick or card must not allow
the centre of gravity, through its indeterminacy, to be over the edge of the table.
Challenge 81, page 133: A larger charge would lead to a field that spontaneously generates elec-
tron positron pairs, the electron would fall into the nucleus and reduce its charge by one unit.
Challenge 83, page 133: The Hall effect results from the deviation of electrons in a metal due to
an applied magnetic field. Therefore it depends on their speed. One gets values around 1 mm.
Inside atoms, one can use Bohr’s atomic model as approximation.
Challenge 84, page 133: The steps are due to the particle nature of electricity and all other mov-
ing entities.

Motion Mountain – The Adventure of Physics
Challenge 85, page 134: If we could apply the Banach–Tarski paradox to vacuum, it seems that
Vol. I, page 57 we could split, without any problem, one ball of vacuum into two balls of vacuum, each with
the same volume as the original. In other words, one ball with vacuum energy 𝐸 could not be
distinguished from two balls of vacuum energy 2𝐸.
We used the Banach–Tarski paradox in this way to show that chocolate (or any other matter)
Vol. I, page 333 possesses an intrinsic length. But it is not clear that we can now deduce that the vacuum has an
intrinsic length. Indeed, the paradox cannot be applied to vacuum for two reasons. First, there
indeed is a maximum energy and minimum length in nature. Secondly, there is no place in nature
without vacuum energy; so there is no place were we could put the second ball. We thus do not
know why the Banach–Tarski paradox for vacuum cannot be applied, and thus cannot use it to

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
deduce the existence of a minimum length in vacuum.
It is better to argue in the following way for a minimum length in vacuum. If there were no
intrinsic length cut-off, the vacuum energy would be infinite. Experiments however, show that it
is finite.
Challenge 86, page 134: Mud is a suspension of sand; sand is not transparent, even if made of
clear quartz, because of the scattering of light at the irregular surface of its grains. A suspension
cannot be transparent if the index of refraction of the liquid and the suspended particles is differ-
ent. It is never transparent if the particles, as in most sand types, are themselves not transparent.
Challenge 87, page 134: The first answer is probably no, as composed systems cannot be smaller
than their own compton wavelength; only elementary systems can. However, the universe is not
a system, as it has no environment. As such, its length is not a precisely defined concept, as
an environment is needed to measure and to define it. (In addition, gravity must be taken into
account in those domains.) Thus the answer is: in those domains, the question makes no sense.
Challenge 88, page 134: Methods to move on perfect ice from mechanics:

— if the ice is perfectly flat, rest is possible only in one point – otherwise you oscillate around
that point, as shown in challenge 26;
— do nothing, just wait that the higher centrifugal acceleration at body height pulls you away;
— to rotate yourself, just rotate your arm above your head;
— throw a shoe or any other object away;
— breathe in vertically, breathing out (or talking) horizontally (or vice versa);
— wait to be moved by the centrifugal acceleration due to the rotation of the Earth (and its
oblateness);
challenge hints and solutions 381

— jump vertically repeatedly: the Coriolis acceleration will lead to horizontal motion;
— wait to be moved by the Sun or the Moon, like the tides are;
— ‘swim’ in the air using hands and feet;
— wait to be hit by a bird, a flying wasp, inclined rain, wind, lava, earthquake, plate tectonics,
or any other macroscopic object (all objects pushing count only as one solution);
— wait to be moved by the change in gravity due to convection in Earth’s mantle;
— wait to be moved by the gravitation of some comet passing by;
— counts only for kids: spit, sneeze, cough, fart, pee; or move your ears and use them as wings.
Note that gluing your tongue is not possible on perfect ice.
Challenge 89, page 135: Methods to move on perfect ice using thermodynamics and electrody-
namics:
— use the radio/TV stations nearby to push you around;
— use your portable phone and a mirror;
— switch on a pocket lamp, letting the light push you;
— wait to be pushed around by Brownian motion in air;

Motion Mountain – The Adventure of Physics
— heat up one side of your body: black body radiation will push you;
— heat up one side of your body, e.g. by muscle work: the changing airflow or the evaporation
will push you;
— wait for one part of the body to be cooler than the other and for the corresponding black
body radiation effects;
— wait for the magnetic field of the Earth to pull on some ferromagnetic or paramagnetic metal
piece in your clothing or in your body;
— wait to be pushed by the light pressure, i.e. by the photons, from the Sun or from the stars,
maybe using a pocket mirror to increase the efficiency;
— rub some polymer object to charge it electrically and then move it in circles, thus creating a

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
magnetic field that interacts with the one of the Earth.
Note that perfect frictionless surfaces do not melt.
Challenge 90, page 135: Methods to move on perfect ice using general relativity:

— move an arm to emit gravitational radiation;
— deviate the cosmic background radiation with a pocket mirror;
— wait to be pushed by gravitational radiation from star collapses;
— wait for the universe to contract.

Challenge 91, page 135: Methods to move on perfect ice using quantum effects:

— wait for your wave function to spread out and collapse at the end of the ice surface;
— wait for the pieces of metal in the clothing to attract to the metal in the surrounding through
the Casimir effect;
— wait to be pushed around by radioactive decays in your body.

Challenge 92, page 135: Methods to move on perfect ice using materials science, geophysics and
astrophysics:
— be pushed by the radio waves emitted by thunderstorms and absorbed in painful human
joints;
— wait to be pushed around by cosmic rays;
— wait to be pushed around by the solar wind;
— wait to be pushed around by solar neutrinos;
382 challenge hints and solutions

— wait to be pushed by the transformation of the Sun into a red giant;
— wait to be hit by a meteorite.

Challenge 93, page 135: A method to move on perfect ice using self-organization, chaos theory,
and biophysics:
— wait that the currents in the brain interact with the magnetic field of the Earth by controlling
your thoughts.

Challenge 94, page 135: Methods to move on perfect ice using quantum gravity:

— accelerate your pocket mirror with your hand;
— deviate the Unruh radiation of the Earth with a pocket mirror;
— wait for proton decay to push you through the recoil.

Challenge 96, page 142: This is a trick question: if you can say why, you can directly move to
the last volume of this adventure and check your answer. The gravitational potential changes the
phase of a wave function, like any other potential does; but the reason why this is the case will

Motion Mountain – The Adventure of Physics
only become clear in the last volume of this series.
Challenge 101, page 144: No. Bound states of massless particles are always unstable.
Challenge 102, page 146: This is easy only if the black hole size is inserted into the entropy
bound by Bekenstein. A simple deduction of the black hole entropy that includes the factor 1/4
is not yet at hand; more on this in the last volume.
Challenge 103, page 146: An entropy limit implies an information limit; only a given informa-
tion can be present in a given region of nature. This results in a memory limit.
Challenge 104, page 146: In natural units, the exact expression for entropy is 𝑆 = 0.25𝐴. If each
Planck area carried one bit (degree of freedom), the entropy would be 𝑆 = ln 𝑊 = ln(2𝐴 ) =

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
𝐴 ln 2 = 0.693𝐴. This close to the exact value.
Challenge 108, page 152: The universe has about 1022 stars; the Sun has a luminosity of about
1026 W; the total luminosity of the visible matter in the universe is thus about 1048 W. A gamma-
ray burster emits up to 3 ⋅ 1047 W.
Challenge 114, page 154: They are carried away by the gravitational radiation.
Challenge 120, page 158: No system is known in nature which emits or absorbs only one grav-
iton at a time. This is another point speaking against the existence of gravitons.
Challenge 124, page 167: Two stacked foils show the same effect as one foil of the same total
thickness. Thus the surface plays no role.
Challenge 126, page 172: The electron is held back by the positive charge of the nucleus, if the
number of protons in the nucleus is sufficient, as is the case for those nuclei we are made of.
Challenge 128, page 180: The half-time 𝑡1/2 is related to the life-time 𝜏 by 𝑡1/2 = 𝜏 ln 2.
Challenge 129, page 181: The number is small compared with the number of cells. However, it
is possible that the decays are related to human ageing.
Challenge 131, page 184: By counting decays and counting atoms to sufficient precision.
Challenge 133, page 185: The radioactivity necessary to keep the Earth warm is low; lava is only
slightly more radioactive than usual soil.
Challenge 134, page 197: There is no way to conserve both energy and momentum in such a
decay.
Challenge 135, page 197: The combination of high intensity X-rays and UV rays led to this ef-
fect.
challenge hints and solutions 383

Challenge 139, page 209: The nuclei of nitrogen and carbon have a high electric charge which
strongly repels the protons.
Challenge 141, page 218: See the paper by C.J. Hogan mentioned in Ref. 270.
Challenge 142, page 221: Touching something requires getting near it; getting near means a
small time and position indeterminacy; this implies a small wavelength of the probe that is used
for touching; this implies a large energy.
Challenge 145, page 228: The processes are electromagnetic in nature, thus electric charges give
the frequency with which they occur.
Challenge 146, page 237: Designing a nuclear weapon is not difficult. University students can do
it, and even have done so a few times. The first students who did so were two physics graduates in
1964, as told on www.guardian.co.uk/world/2003/jun/24/usa.science. It is not hard to conceive a
design and even to build it. By far the hardest problem is getting or making the nuclear mater-
ial. That requires either an extensive criminal activity or a vast technical effort, with numerous
large factories, extensive development, and coordination of many technological activities. Most
importantly, such a project requires a large financial investment, which poor countries cannot

Motion Mountain – The Adventure of Physics
afford without great sacrifices for all the population. The problems are thus not technical, but
financial.
Challenge 150, page 268: In 2008, an estimated 98 % of all physicists agreed. Time will tell
whether they are right.
Challenge 152, page 280: A mass of 100 kg and a speed of 8 m/s require 43 m2 of wing surface.
Challenge 155, page 287: The issue is a red herring. The world has three dimensions.
Challenge 156, page 290: The largest rotation angle Δ𝜑 that can be achieved in one stroke 𝐶 is
found by maximizing the integral

𝑎2

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Δ𝜑 = − ∫ d𝜃 (151)
𝐶 𝑎2 + 𝑏2
Since the path 𝐶 in shape space is closed, we can use Stokes’ theorem to transform the line in-
tegral to a surface integral over the surface 𝑆 enclosed by 𝐶 in shape space:

2𝑎𝑏2
Δ𝜑 = ∫ d𝑎 d𝜃 . (152)
𝑆 (𝑎2 + 𝑏2 )2
The maximum angle is found by noting that 𝜃 can vary at most between 0 and π, and that 𝑎 can
vary at most between 0 and ∞. This yields
π ∞
2𝑎𝑏2
Δ𝜑max = ∫ ∫ d𝑎 d𝜃 = π . (153)
𝜃=0 𝑎=0 (𝑎2 + 𝑏2 )2

Challenge 158, page 294: A cloud is kept afloat and compact by convection currents. Clouds
without convection can often be seen in the summer: they diffuse and disappear. The details of
the internal and external air currents depend on the cloud type and are a research field on its
own.
Challenge 163, page 300: Lattices are not isotropic, lattices are not Lorentz invariant.
Challenge 165, page 303: The infinite sum is not defined for numbers; however, it is defined for
a knotted string.
Challenge 166, page 304: The research race for the solution is ongoing, but the goal is still far.
Challenge 167, page 305: This is a simple but hard question. Find out!
384 challenge hints and solutions

Challenge 170, page 308: Large raindrops are pancakes with a massive border bulge. When the
size increases, e.g. when a large drop falls through vapour, the drop splits, as the central mem-
brane is then torn apart.
Challenge 171, page 308: It is a drawing; if it is interpreted as an image of a three-dimensional
object, it either does not exist, or is not closed, or is an optical illusion of a torus.
Challenge 172, page 308: See T. Fink & Y. Mao, The 85 Ways to Tie a Tie, Broadway Books,
2000.
Challenge 173, page 308: See T. Clarke, Laces high, Nature Science Update 5th of December,
2002, or www.nature.com/nsu/021202/021202-4.html.
Challenge 176, page 310: In fact, nobody has even tried to do so yet. It may also be that the
problem makes no sense.
Challenge 178, page 316: Most macroscopic matter properties fall in this class, such as the
change of water density with temperature.
Challenge 180, page 322: Before the speculation can be fully tested, the relation between
particles and black holes has to be clarified first.

Motion Mountain – The Adventure of Physics
Challenge 181, page 323: Never expect a correct solution for personal choices. Do what you
yourself think and feel is correct.
Challenge 186, page 330: Planck limits can be exceeded for extensive observables for which
many particle systems can exceed single particle limits, such as mass, momentum, energy or
electrical resistance.
Challenge 188, page 332: Do not forget the relativistic time dilation.
Challenge 189, page 332: The formula with 𝑛 − 1 is a better fit. Why?
Challenge 193, page 339: The slowdown goes quadratically with time, because every new slow-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
down adds to the old one!
Challenge 194, page 340: No, only properties of parts of the universe are listed. The universe
Vol. VI, page 112 itself has no properties, as shown in the last volume.
Challenge 195, page 342: The gauge coupling constants, via the Planck length, determine the
size of atoms, the strength of chemical bonds and thus the size of all things.
Challenge 196, page 357: Covalent bonds tend to produce full shells; this is a smaller change on
the right side of the periodic table.
Challenge 197, page 359: The solution is the set of all two by two matrices, as each two by two
matrix specifies a linear transformation, if one defines a transformed point as the product of the
point and this matrix. (Only multiplication with a fixed matrix can give a linear transformation.)
Can you recognize from a matrix whether it is a rotation, a reflection, a dilation, a shear, or a
stretch along two axes? What are the remaining possibilities?
Challenge 200, page 359: The (simplest) product of two functions is taken by point-by-point
multiplication.
Challenge 201, page 360: The norm ‖𝑓‖ of a real function 𝑓 is defined as the supremum of its
absolute value:
‖𝑓‖ = sup |𝑓(𝑥)| . (154)
𝑥∈R

In simple terms: the maximum value taken by the absolute of the function is its norm. It is also
called ‘sup’-norm. Since it contains a supremum, this norm is only defined on the subspace of
bounded continuous functions on a space X, or, if X is compact, on the space of all continuous
functions (because a continuous function on a compact space must be bounded).
challenge hints and solutions 385

Challenge 204, page 363: Take out your head, then pull one side of your pullover over the cor-
responding arm, continue pulling it over the over arm; then pull the other side, under the first,
to the other arm as well. Put your head back in. Your pullover (or your trousers) will be inside
out.
Challenge 205, page 363: Both can be untied.
Challenge 209, page 368: The transformation from one manifold to another with different topo-
logy can be done with a tiny change, at a so-called singular point. Since nature shows a minimum
action, such a tiny change cannot be avoided.
Challenge 210, page 370: The product 𝑀† 𝑀 is Hermitean, and has positive eigenvalues. Thus
𝐻 is uniquely defined and Hermitean. 𝑈 is unitary because 𝑈† 𝑈 is the unit matrix.

“
Gedanken sind nicht stets parat. Man schreibt

”
auch, wenn man keine hat.*
Wilhelm Busch, Aphorismen und Reime.

1 The use of radioactivity for breeding of new sorts of wheat, rice, cotton, roses, pineapple and
many more is described by B. S. Ahloowalia & M. Maluszynski, Induced mutations

Motion Mountain – The Adventure of Physics
– a new paradigm in plant breeding, Euphytica 11, pp. 167–173, 2004. Cited on page 21.
2 See John T. B onner, Why Size Matters: From Bacteria to Blue Whales, Princeton Uni-
versity Press, 2011. Cited on page 21.
3 See the book by Peter Läuger, Electrogenic Ion Pumps, Sinauer, 1991. Cited on page 21.
4 The motorized screw used by viruses was described by A.A. Simpson & al., Structure
of the bacteriophage phi29 DNA packaging motor, Nature 408, pp. 745–750, 2000. Cited on
page 22.
5 S. M. Block, Real engines of creation, Nature 386, pp. 217–219, 1997. Cited on page 22.
6 Early results and ideas on molecular motors are summarised by B. Goss Levi, Measured

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
steps advance the understanding of molecular motors, Physics Today pp. 17–19, April 1995.
Newer results are described in R. D. Astumian, Making molecules into motors, Scientific
American pp. 57–64, July 2001. Cited on pages 22 and 23.
7 R. Bartussek & P. Hänggi, Brownsche Motoren, Physikalische Blätter 51, pp. 506–507,
1995. See also R. Alt-Haddou & W. Herzog, Force and motion generation of myosin
motors: muscle contraction, Journal of Electromyography and kinesiology 12, pp. 435–445,
2002. Cited on page 24.
8 N. Hirokawa, S. Niwa & Y. Tanaka, Molecular motors in neurons: transport mechan-
isms and roles in brain function, development, and disease, Neuron 68, pp. 610–638, 2010.
Cited on page 25.
9 J. Weber & A. E. Senior, ATP synthesis driven by proton transport in F1Fo-ATP synthase,
FEBS letters 545, pp. 61–70, 2003. Cited on page 26.
10 This truly fascinating research result, worth a Nobel Prize, is summarized in
N. Hirokawa, Y. Tanaka & Y. Okada, Left-right determination: involvement of
molecular motor KIF3, cilia, and nodal flow, Cold Spring Harbor Perspectives in Bio-
logy 1, p. a000802, 2009, also available at www.cshperspectives.org. The website also
links to numerous captivating films of the involved microscopic processes, found at beta.
cshperspectives.cshlp.org. Cited on page 28.
11 R. J. Cano & M. K. B orucki, Revival and identification of bacterial spores in 25- to 40-
million-year-old Dominican amber, Science 26, pp. 1060–1064, 1995. Cited on page 31.

* ‘Thoughts are not always available. Many write even without them.’
bibliography 387

12 The first papers on bacteria from salt deposits were V. R. Ott & H. J. Dombrwoski,
Mikrofossilien in den Mineralquellen zu Bad Nauheim, Notizblatt des Hessischen Landes-
amtes für Bodenforschung 87, pp. 415–416, 1959, H. J. Dombrowski, Bacteria from Pa-
leozoic salt deposits, Annals of the New York Academy of Sciences 108, pp. 453–460, 1963. A
recent confirmation is R. H. Vreeland, W. D. Rosenzweig & D. W. Powers, Isola-
tion of a 250 million-year-old halotolerant bacterium from a primary salt crystal, Nature 407,
pp. 897–899, 2000. Cited on page 31.
13 This is explained in D. Graur & T. Pupko, The permian bacterium that isn’t, Mo-
lecular Biology and Evolution 18, pp. 1143–1146, 2001, and also in M. B. Hebsgaard,
M. J. Phillips & E. Willerslev, Geologically ancient DNA: fact or artefact?, Trends
in Microbiology 13, pp. 212–220, 2005. Cited on page 31.
14 Gabriele Walker, Snowball Earth – The Story of the Great Global Catastrophe That
Spawned Life as We Know It, Crown Publishing, 2003. No citations.
15 J. D. Rummel, J. H. Allton & D. Morrison, A microbe on the moon? Surveyor III and
lessons learned for future sample return missions, preprint at www.lpi.usra.edu/meetings/

Motion Mountain – The Adventure of Physics
sssr2011/pdf/5023.pdf. Cited on page 31.
16 The table and the evolutionary tree are taken from J. O. McInerney, M. Mullarkey,
M. E. Wernecke & R. Powell, Bacteria and archaea: molecular techniques reveal as-
tonishing diversity, Biodiversity 3, pp. 3–10, 2002. The evolutionary tree might still change
a little in the coming years. Cited on page 32.
17 The newest estimate is by R. Sender, S. Fuchs & R. Milo, Revised estimates for the
number of human and bacteria cells in the body, PLOS Biology 14, p. e1002533, 2016, free pre-
print at biorxiv.org/content/early/2016/01/06/036103. The first professional estimate was by
E. Bianconi, A. Piovesan, F. Facchin, A. Beraudi, R. Casadei, F. Frabetti,
L. Vitale, M. C. Pelleri, S. Tassani, F. Piva, S. Perez-Amodio, P. Strippoli

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
& S. Canaider, An estimation of the number of cells in the human body, Annals of Hu-
man Biology 40, pp. 463–471, 2013, also available for free online. A typical older and
more optimistic estimate is E. K. Costello, C. L. Lauber, M. Hamady, N. Fierer,
J. I. Gordon & R. Knight, Bacterial community variation in human body habitats across
space and time, Science Express 5 November 2009. Cited on page 33.
18 E. Puttonen, C. Briese, G. Mandlburger, M. Wieser, M. Pfennigbauer,
A. Zlinsky & N. Pfeifer, Quantification of overnight movement of birch (Betula pen-
dula) branches and foliage with short interval terrestrial laser scanning, Frontiers in Plant
Science 7, 2016, free download at journal.frontiersin.org. A video of the tree motion is also
available there. Cited on page 37.
19 This is taken from the delightful children text Hans J. Press, Spiel das Wissen schafft,
Ravensburger Buchverlag 1964, 2004. Cited on page 38.
20 The discovery of a specific taste for fat was published by F. Laugerette, P. Passilly-
Degrace, B. Patris, I. Niot, M. Febbraio, J. P. Montmayeur & P. Besnard,
CD36 involvement in orosensory detection of dietary lipids, spontaneous fat preference, and
digestive secretions, Journal of Clinical Investigation 115, pp. 3177–3184, 2005. Cited on
page 40.
21 There is no standard procedure to learn to enjoy life to the maximum. A good foundation
can be found in those books which teach the ability to those which have lost it.
The best experts are those who help others to overcome traumas. Peter A. Levine
& Ann Frederick, Waking the Tiger – Healing Trauma – The Innate Capacity to Trans-
form Overwhelming Experiences, North Atlantic Books, 1997. Geoff Graham, How to
388 bibliography

Become the Parent You Never Had - a Treatment for Extremes of Fear, Anger and Guilt,
Real Options Press, 1986. A good complement to these texts is the approach presen-
ted by Bert Hellinger, Zweierlei Glück, Carl Auer Verlag, 1997. Some of his books
are also available in English. The author presents a simple and efficient technique for
reducing entanglement with one’s family past. Another good book is Phil Stutz &
Barry Michels, The Tools – Transform Your Problems into Courage, Confidence, and
Creativity, Random House, 2012.
The next step, namely full mastery in the enjoyment of life, can be found in any book
written by somebody who has achieved mastery in any one topic. The topic itself is not
important, only the passion is. A few examples:
A. de la Garanderie, Le dialogue pédagogique avec l’élève, Centurion, 1984,
A. de la Garanderie, Pour une pédagogie de l’intelligence, Centurion, 1990,
A. de la Garanderie, Réussir ça s’apprend, Bayard, 1994. De la Garanderie explains
how the results of teaching and learning depend in particular on the importance of evoca-
tion, imagination and motivation.
Plato, Phaedrus, Athens, 380 bce.

Motion Mountain – The Adventure of Physics
Françoise Dolto, La cause des enfants, Laffont, 1985, and her other books. Dolto
(b. 1908 Paris, d. 1988 Paris), a child psychiatrist, is one of the world experts on the growth
of the child; her main theme was that growth is only possible by giving the highest possible
responsibility to every child during its development.
In the domain of art, many had the passion to achieve full pleasure. A good piece
of music, a beautiful painting, an expressive statue or a good film can show it. On a
smaller scale, the art to typeset beautiful books, so different from what many computer
programs do by default, the best introduction are the works by Jan Tschichold (b. 1902
Leipzig, d. 1974 Locarno), the undisputed master of the field. Among the many books he
designed are the beautiful Penguin books of the late 1940s; he also was a type designer,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
e.g. of the Sabon typeface. A beautiful summary of his views is the short but condensed
text Jan Tschichold, Ausgewählte Aufsätze über Fragen der Gestalt des Buches und der
Typographie, Birkhäuser Verlag, Basel, 1993. An extensive and beautiful textbook on the
topic is Hans Peter Willberg & Friedrich Forssman, Lesetypographie, Verlag
Hermann Schmidt, Mainz, 1997. See also Robert Bringhurst, The Elements of Typo-
graphic Style, Hartley & Marks, 2004.
Many scientists passionately enjoyed their occupation. Any biography of Charles Dar-
win will purvey his fascination for biology, of Friedrich Bessel for astronomy, of Albert
Einstein for physics and of Linus Pauling for chemistry. Cited on page 41.
22 The group of John Wearden in Manchester has shown by experiments with humans that the
accuracy of a few per cent is possible for any action with a duration between a tenth of a
second and a minute. See J. McCrone, When a second lasts forever, New Scientist pp. 53–
56, 1 November 1997. Cited on page 42.
23 The chemical clocks in our body are described in John D. Palmer, The Living Clock,
Oxford University Press, 2002, or in A. Ahlgren & F. Halberg, Cycles of Nature: An
Introduction to Biological Rhythms, National Science Teachers Association, 1990. See also
the www.msi.umn.edu/~halberg/introd website. Cited on page 42.
24 D.J Morré & al., Biochemical basis for the biological clock, Biochemistry 41, pp. 11941–
11945, 2002. Cited on page 42.
25 An introduction to the sense of time as result of clocks in the brain is found in R. B. Ivry
& R. Spencer, The neural representation of time, Current Opinion in Neurobiology 14,
pp. 225–232, 2004. The interval timer is explain in simple words in K. Wright, Times
in our lives, Scientific American pp. 40–47, September 2002. The MRI research used is
bibliography 389

S. M. Rao, A. R. Mayer & D. L. Harrington, The evolution of brain activation dur-
ing temporal processing, Nature Neuroscience 4, pp. 317–323, 2001. Cited on page 44.
26 See, for example, Jan Hilgevoord, Time in quantum mechanics, American Journal of
Physics 70, pp. 301–306, 2002. Cited on page 44.
27 E. J. Zimmerman, The macroscopic nature of space-time, American Journal of Physics 30,
pp. 97–105, 1962. Cited on page 45.
28 See P.D. Peşić, The smallest clock, European Journal of Physics 14, pp. 90–92, 1993. Cited
on page 46.
29 The possibilities for precision timing using single-ion clocks are shown in W.H. Oskay &
al., Single-atom clock with high accuracy, Physical Review Letters 97, p. 020801, 2006. Cited
on page 46.
30 A pretty example of a quantum mechanical system showing exponential behaviour at all
times is given by H. Nakazato, M. Namiki & S. Pascazio, Exponential behaviour
of a quantum system in a macroscopic medium, Physical Review Letters 73, pp. 1063–1066,
1994. Cited on page 48.

Motion Mountain – The Adventure of Physics
31 See the delightful book about the topic by Paolo Facchi & Saverio Pascazio,
La regola d’oro di Fermi, Bibliopolis, 1999. An experiment observing deviations at short
times is S. R. Wilkinson, C. F. Bharucha, M. C. Fischer, K. W. Madison,
P. R. Morrow, Q. Niu, B. Sundaram & M. G. Raizen, Nature 387, p. 575, 1997.
Cited on page 48.
32 See, for example, R. Efron, The duration of the present, Annals of the New York Academy
of Sciences 138, pp. 713–729, 1967. Cited on page 49.
33 W. M. Itano, D. J. Heinzen, J. J. B ollinger & D. J. Wineland, Quantum Zeno
effect, Physical Review A 41, pp. 2295–2300, 1990. M. C. Fischer, B. Gutiérrez-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Medina & M. G. Raizen, Observation of the Quantum Zeno and Anti-Zeno effects in
an unstable system, Physical Review Letters 87, p. 040402, 2001, also www-arxiv.org/abs/
quant-ph/0104035. Cited on page 50.
34 See P. Facchi, Z. Hradil, G. Krenn, S. Pascazio & J. Řeháček Quantum
Zeno tomography, Physical Review A 66, p. 012110, 2002. Cited on page 50.
35 See P. Facchi, H. Nakazato & S. Pascazio From the quantum Zeno to the inverse
quantum Zeno effect, Physical Review Letters 86, pp. 2699–2702, 2001. Cited on page 50.
36 H. Kobayashi & S. Kohshima, Unique morphology of the human eye, Nature 387,
pp. 767–768, 1997. No citations.
37 James W. Prescott, Body pleasure and the origins of violence, The Futurist Bethseda,
1975, also available at www.violence.de/prescott/bullettin/article.html. Cited on page 54.
38 Frances Ashcroft, The Spark of Life: Electricity in the Human Body, Allen Lane, 2012.
Cited on page 54.
39 Felix Tretter & Margot Albus, Einführung in die Psychopharmakotherapie –
Grundlagen, Praxis, Anwendungen, Thieme, 2004. Cited on page 54.
40 See the talk on these experiments by Helen Fisher on www.ted.org and the information on
helenfisher.com. Cited on page 55.
41 See for example P. Py ykkö, Relativity, gold, closed-shell interactions, and CsAu.NH3, An-
gewandte Chemie, International Edition 41, pp. 3573–3578, 2002, or L. J. Norrby, Why
is mercury liquid? Or, why do relativistic effects not get into chemistry textbooks?, Journal of
Chemical Education 68, pp. 110–113, 1991. Cited on page 58.
390 bibliography

42 On the internet, the ‘spherical’ periodic table is credited to Timothy Stowe; but there is no
reference for that claim, except an obscure calendar from a small chemical company. The
original table (containing a number errors) used to be found at the chemlab.pc.maricopa.
edu/periodic/stowetable.html website; it is now best found by searching for images called
‘stowetable’ with any internet search engine. Cited on page 59.
43 For good figures of atomic orbitals, take any modern chemistry text. Or go to csi.chemie.
tu-darmstadt.de/ak/immel/. Cited on page 60.
44 For experimentally determined pictures of the orbitals of dangling bonds, see for example
F. Giessibl & al., Subatomic features on the silicon (111)-(7x7) surface observed by atomic
force microscopy, Science 289, pp. 422–425, 2000. Cited on page 60.
45 L. Gagliardi & B. O. Roos, Quantum chemical calculations show that the uranium mo-
lecule U2 has a quintuple bond, Nature 433, pp. 848–851, 2005. B. O. Roos, A. C. B orin
& L. Gagliardi, Reaching the maximum multiplicity of the covalent chemical bond, An-
gewandte Chemie, International Edition 46, pp. 1469–1472, 2007. Cited on page 62.
46 H. -W. Fink & C. Escher, Zupfen am Lebensfaden – Experimente mit einzelnen DNS-

Motion Mountain – The Adventure of Physics
Molekülen, Physik in unserer Zeit 38, pp. 190–196, 2007. Cited on page 62.
47 This type of atomic bond became well-known through the introduction by D. Kleppner,
The most tenuous of molecules, Physics Today 48, pp. 11–12, 1995. Cited on page 66.
48 Exploring the uses of physics and chemistry in forensic science is fascinating. For a beau-
tiful introduction to this field with many stories and a lot of physics, see the book by
Patrick Voss-de Haan, Physik auf der Spur, Wiley-VCH, 2005. Cited on page 67.
49 See, for example, the extensive explanations by Rafal Swiecki on his informatvie website
www.minelinks.com. Cited on page 69.
50 T. Gu, M. Li, C. McCammon & K. K. M. Lee, Redox-induced lower mantle density con-
trast and effect on mantle structure and primitive oxygen, Nature Geoscience 9, pp. 723–727,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
2016. Cited on page 71.
51 See the website by Tom Hales on www.math.lsa.umich.edu/~hales/countdown. An earlier
book on the issue is the delightful text by Max Leppmaier, Kugelpackungen von Kepler
bis heute, Vieweg Verlag, 1997. Cited on page 73.
52 For a short overview, also covering binary systems, see C. Bechinger, H. -
H. von Grünberg & P. Leiderer, Entropische Kräfte - warum sich repulsiv wech-
selwirkende Teilchen anziehen können, Physikalische Blätter 55, pp. 53–56, 1999. Cited on
page 73.
53 H. -W. Fink & G. Ehrlich, Direct observation of three-body interactions in adsorbed lay-
ers: Re on W(110), Physical Review Letters 52, pp. 1532–1534, 1984, and H. -W. Fink &
G. Ehrlich, Pair and trio interactions between adatoms: Re on W(110), Journal of Chem-
ical Physics 61, pp. 4657–4665, 1984. Cited on pages 73 and 74.
54 See for example the textbook by A. Pimpinelli & J. Villain, Physics of Crystal Growth,
Cambridge University Press, 1998. Cited on page 74.
55 Y. Furukawa, Faszination der Schneekristalle - wie ihre bezaubernden Formen entstehen,
Chemie in unserer Zeit 31, pp. 58–65, 1997. His website www.lowtem.hokudai.ac.jp/~frkw/
index_e.html gives more information on the topic of snow crystals. Cited on page 74.
56 L. Bindi, P. J. Steinhardt, N. Yai & P. J. Lu, Natural Quasicrystal, Science 324,
pp. 1306–1309, 2009. Cited on page 81.
57 The present attempts to build cheap THz wave sensors – which allow seeing through clothes
– is described by D. Clery, Brainstorming their way to an imaging revolution, Science 297,
pp. 761–763, 2002. Cited on page 84.
bibliography 391

58 The switchable mirror effect was discovered by J. N. Huiberts, R. Griessen,
J. H. Rector, R. J. Wijngarden, J. P. Dekker, D. G. de Groot & N. J. Koeman,
Yttrium and lanthanum hydride films with switchable optical properties, Nature 380, pp. 231–
234, 1996. A good introduction is R. Griessen, Schaltbare Spiegel aus Metallhydriden,
Physikalische Blätter 53, pp. 1207–1209, 1997. Cited on page 86.
59 V. F. Weisskopf, Of atoms, mountains and stars: a study in qualitative physics, Science
187, p. 605, 1975. Cited on page 87.
60 K. Schwab, E. A. Henriksen, J. M. Worlock & M. L. Roukes, Measurement of
the quantum of thermal conductance, Nature 404, pp. 974–977, 2000. Cited on page 90.
61 K. Autumn, Y. Liang, T. Hsich, W. Zwaxh, W. P. Chan, T. Kenny, R. Fearing
& R. J. Full, Adhesive force of a single gecko foot-hair, Nature 405, pp. 681–685, 2000. Cited
on page 90.
62 A. B. Kesel, A. Martin & T. Seidl, Getting a grip on spider attachment: an AFM ap-
proach to microstructure adhesion in arthropods, Smart Materials and Structures 13, pp. 512–
518, 2004. Cited on page 91.

Motion Mountain – The Adventure of Physics
63 See the article on www.bio-pro.de/magazin/wissenschaft/archiv_2005/index.html?
lang=en&artikelid=/artikel/03049/index.html. Cited on page 91.
64 See the beautiful article by R. Ritchie, M. J. Buehler & P. Hansma, Plasticity and
toughness in bones, Physics Today 82, pp. 41–47, 2009. See also the fascinating review art-
icles P. Fratzl, Von Knochen, Holz und Zähnen, Physik Journal 1, pp. 49–55, 2002, and
P. Fratzl & R. Weinkamer, Nature’s hierarchical materials, Progress in Material Sci-
ence 52, pp. 1263–1334, 2007. Cited on page 91.
65 H. Inaba, T. Saitou, K. -I. Tozaki & H. Hayashi, Effect of the magnetic field on the
melting transition of H2O and D2O measured by a high resolution and supersensitive differ-

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
ential scanning calorimeter, Journal of Applied Physics 96, pp. 6127–6132, 2004. Cited on
page 91.
66 The graph for temperature is taken from J. Jouzel & al., Orbital and millennial Antarctic
climate variability over the past 800,000 years, Science 317, pp. 793–796, 2007. The graph for
CO2 is combined from six publications: D.M. Etheridge & al., Natural and anthropo-
genic changes in atmospheric CO2 over the last 1000 years from air in Antarctic ice and firm,
Journal of Geophysical Research 101, pp. 4115–4128, 1996, J.R. Petit & al., Climate and
atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature 399,
pp. 429–436, 1999, A. Indermühle & al., Atmospheric CO2 concentration from 60 to 20
kyr BP from the Taylor Dome ic cores, Antarctica, Geophysical Research Letters 27, pp. 735–
738, 2000, E. Monnin & al., Atmospheric CO2 concentration over the last glacial termina-
tion, Science 291, pp. 112–114, 2001, U. Siegenthaler & al., Stable carbon cycle-climate
relationship during the late pleistocene, Science 310, pp. 1313–1317, 2005, D. Lüthi & al.,
High-resolution carbon dioxide concentration record 650,000-800,000 years before present,
Nature 453, pp. 379–382, 2008. Cited on page 93.
67 H. Oerlemans, De afkoeling van de aarde, Nederlands tijdschrift voor natuurkunde
pp. 230–234, 2004. This excellent paper by one of the experts of the field also explains that
the Earth has two states, namely today’s tropical state and the snowball state, when it is
covered with ice. And there is hysteresis for the switch between the two states. Cited on
page 93.
68 R. Hemley & al., presented these results at the International Conference on New
Diamond Science in Tsukuba (Japan) 2005. See figures at www.carnegieinstitution.org/
diamond-13may2005. Cited on page 93.
392 bibliography

69 Ingo Dierking, Textures of Liquid Crystals, Wiley-VCH, 2003. Cited on page 95.
70 The photon hall effect was discovered by Geert Rikken, Bart van Tiggelen &
Anja Sparenberg, Lichtverstrooiing in een magneetveld, Nederlands tijdschrift voor
natuurkunde 63, pp. 67–70, maart 1998. Cited on page 96.
71 C. Strohm, G. L. J. A. Rikken & P. Wyder, Phenomenological evidence for the phonon
Hall effect, Physical Review Letters 95, p. 155901, 2005. Cited on page 97.
72 E. Chibowski, L. Hołysz, K. Terpiłowski, Effect of magnetic field on deposition and
adhesion of calcium carbonate particles on different substrates, Journal of Adhesion Science
and Technology 17, pp. 2005–2021, 2003, and references therein. Cited on page 97.
73 K. S. Novoselov, D. Jiang, F. Schedin, T. B ooth, V. V. Khotkevich,
S. V. Morozov & A. K. Geim, Two Dimensional Atomic Crystals, Proceedings of the Na-
tional Academy of Sciences 102, pp. 10451–10453, 2005. See also the talk by Andre Geim,
QED in a pencil trace, to be downloaded at online.kitp.ucsb.edu/online/graphene_m07/
geim/ or A. C. Neto, F. Guinea & N. M. Peres, Drawing conclusions from graphene,
Physics World November 2006. Cited on pages 97, 98, and 414.

Motion Mountain – The Adventure of Physics
74 R.R. Nair & al., Fine structure constant defines visual transparency of graphene, Science
320, p. 1308, 2008. Cited on page 99.
75 The first paper was J. Bardeen, L. N. Cooper & J. R. Schrieffer, Microscopic
Theory of Superconductivity, Physical Review 106, pp. 162–164, 1957. The full, so-
called BCS theory, is then given in the masterly paper J. Bardeen, L. N. Cooper &
J. R. Schrieffer, Theory of Superconductivity, Physical Review 108, pp. 1175–1204, 1957.
Cited on page 103.
76 A. P. Finne, T. Araki, R. Blaauwgeers, V. B. Eltsov, N. B. Kopnin,
M. Krusius, L. Skrbek, M. Tsubota & G. E. Volovik, An intrinsic velocity-
independent criterion for superfluid turbulence, Nature 424, pp. 1022–1025, 2003. Cited

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
on page 105.
77 The original prediction, which earned Laughlin a Nobel Prize in Physics, is in
R. B. Laughlin, Anomalous quantum Hall effect: an incompressible quantum fluid with
fractionally charged excitations, Physical Review Letters 50, pp. 1395–1398, 1983. Cited on
page 107.
78 The first experimental evidence is by V. J. Goldman & B. Su, Resonance tunnelling
in the fractional quantum Hall regime: measurement of fractional charge, Science 267,
pp. 1010–1012, 1995. The first unambiguous measurements are by R. de Picciotto
& al., Direct observation of a fractional charge, Nature 389, pp. 162–164, 1997, and
L. Saminadayar, D. C. Glattli, Y. Jin & B. Etienne, Observation of the e,3 frac-
tionally charged Laughlin quasiparticle/ Physical Review Letters 79, pp. 2526–2529, 1997.
or arxiv.org/abs/cond-mat/9706307. Cited on page 107.
79 The discovery of the original quantum Hall effect, which earned von Klitzing the Nobel
Prize in Physics, was published as K. von Klitzing, G. Dorda & M. Pepper, New
method for high-accuracy determination of the fine-structure constant based on quantised
Hall resistance, Physical Review Letters 45, pp. 494–497, 1980. Cited on page 107.
80 The fractional quantum Hall effect was discovered by D. C. Tsui, H. L. Stormer &
A. C. Gossard, Two-dimensional magnetotransport in the extreme quantum limit, Phys-
ical Review Letters 48, pp. 1559–1562, 1982. Tsui and Stormer won the Nobel Prize in Physics
with this paper. Cited on page 107.
81 K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L. Stormer,
U. Zeitler, J. C. Mann, G. S. B oebinger, P. Kim & A. K. Geim, Room-
bibliography 393

temperature quantum Hall effect in graphene, Science 315, p. 1379, 2007. Cited on page
107.
82 See for example the overview by M. B orn & T. Jüstel, Umweltfreundliche Lichtquellen,
Physik Journal 2, pp. 43–49, 2003. Cited on page 108.
83 K. An, J. J. Childs, R. R. Desari & M. S. Field, Microlaser: a laser with one atom in
an optical resonator, Physical Review Letters 73, pp. 3375–3378, 19 December 1994. Cited
on page 114.
84 J. Kasparian, Les lasers femtosecondes : applications atmosphériques, La Recherche
pp. RE14 1–7, February 2004/ See also the www.teramobile.org website. Cited on page 115.
85 A. Takita, H. Yamamoto, Y. Hayasaki, N. Nishida & H. Misawa, Three-
dimensional optical memory using a human fingernail, Optics Express 13, pp. 4560–
4567, 2005. They used a Ti:sapphire oscillator and a Ti:sapphire amplifier. Cited on page
115.
86 The author predicted laser umbrellas in 2011. Cited on page 115.
87 W. Guo & H. Maris, Observations of the motion of single electrons in liquid helium,

Motion Mountain – The Adventure of Physics
Journal of Low Temperature Physics 148, pp. 199–206, 2007. Cited on page 117.
88 The story is found on many places in the internet. Cited on page 119.
89 S. L. B oersma, Aantrekkende kracht tussen schepen, Nederlands tidschrift voor
natuuurkunde 67, pp. 244–246, Augustus 2001. See also his paper S. L. B oersma, A
maritime analogy of the Casimir effect, American Journal of Physics 64, pp. 539–541, 1996.
Cited on page 122.
90 P. Ball, Popular physics myth is all at sea – does the ghostly Casimir effect really cause ships
to attract each other?, online at www.nature.com/news/2006/060501/full/news060501-7.
html. Cited on pages 122 and 125.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
91 A. Larraza & B. Denardo, An acoustic Casimir effect, Physics Letters A 248, pp. 151–
155, 1998. See also A. Larraza, A demonstration apparatus for an acoustic analog of the
Casimir effect, American Journal of Physics 67, pp. 1028–1030, 1999. Cited on page 124.
92 H. B. G. Casimir, On the attraction between two perfectly conducting bodies, Proceedings
of the Koninklijke Nederlandse Akademie van Wetenschappen 51, pp. 793–795, 1948. Cited
on page 124.
93 H. B. G. Casimir & D. Polder, The influence of retardation on the London–Van der
Waals forces, Physical Review 73, pp. 360–372, 1948. Cited on page 124.
94 B. W. Derjaguin, I. I. Abrikosova & E. M. Lif shitz, Direct measurement of mo-
lecular attraction between solids separated by a narrow gap, Quaterly Review of the Chem-
ical Society (London) 10, pp. 295–329, 1956. Cited on page 124.
95 M. J. Sparnaay, Attractive forces between flat plates, Nature 180, pp. 334–335, 1957.
M. J. Sparnaay, Measurements of attractive forces between flat plates, Physica (Utrecht)
24, pp. 751–764, 1958. Cited on page 124.
96 S. K. Lamoreaux, Demonstration of the Casimir force in the 0.6 to 6 𝜇m range, Physical
Review Letters 78, pp. 5–8, 1997. U. Mohideen & A. Roy, Precision measurement of the
Casimir force from 0.1 to 0.9 μm, Physical Review Letters 81, pp. 4549–4552, 1998. Cited on
page 124.
97 A. Lambrecht, Das Vakuum kommt zu Kräften, Physik in unserer Zeit 36, pp. 85–91,
2005. Cited on page 124.
98 T. H. B oyer, Van der Waals forces and zero-point energy for dielectric and permeable ma-
terials, Physical Review A 9, pp. 2078–2084, 1974. This was taken up again in O. Kenneth,
394 bibliography

I. Klich, A. Mann & M. Revzen, Repulsive Casimir forces, Physical Review Letters 89,
p. 033001, 2002. However, none of these effects has been verified yet. Cited on page 125.
99 The effect was discovered by K. Scharnhorst, On propagation of light in the vacuum
between plates, Physics Letters B 236, pp. 354–359, 1990. See also G. Barton, Faster-
than-c light between parallel mirrors: the Scharnhorst effect rederived, Physics Letters B 237,
pp. 559–562, 1990, and P. W. Milonni & K. Svozil, Impossibility of measuring faster-
than-c signalling by the Scharnhorst effect, Physics Letters B 248, pp. 437–438, 1990. The
latter corrects an error in the first paper. Cited on page 125.
100 D.L. Burke & al., Positron production in multiphoton light-by-light scattering, Phys-
ical Review Letters 79, pp. 1626–1629, 1997. A simple summary is given in Ber-
tram Schwarzschild, Gamma rays create matter just by plowing into laser light,
Physics Today 51, pp. 17–18, February 1998. Cited on page 130.
101 M. Kardar & R. Golestanian, The ‘friction’ of the vacuum, and other fluctuation-
induced forces, Reviews of Modern Physics 71, pp. 1233–1245, 1999. Cited on page 131.
102 G. Gour & L. Sriramkumar, Will small particles exhibit Brownian motion in the

Motion Mountain – The Adventure of Physics
quantum vacuum?, arxiv.org/abs/quant/phys/9808032 Cited on page 131.
103 P. A. M. Dirac, The requirements of fundamental physical theory, European Journal of
Physics 5, pp. 65–67, 1984. This article transcribes a speech by Paul Dirac just before his
death. Disturbed by the infinities of quantum electrodynamics, he rejects as wrong the the-
ory which he himself found, even though it correctly describes all experiments. The humil-
ity of a great man who is so dismissive of his main and great achievement was impressive
when hearing him, and still is impressive when reading the talk. Cited on page 132.
104 H. Euler & B. Kockel, Naturwissenschaften 23, p. 246, 1935, H. Euler, Annalen der
Physik 26, p. 398, 1936, W. Heisenberg & H. Euler, Folgerung aus der Diracschen The-
orie des Electrons, Zeitschrift für Physik 98, pp. 714–722, 1936. Cited on page 133.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
105 Th. Stoehlker & al., The 1s Lamb shift in hydrogenlike uranium measured on cooled,
decelerated ions beams, Physical Review Letters 85, p. 3109, 2000. Cited on page 133.
106 M. Roberts & al., Observation of an electric octupole transition in a single ion, Physical
Review Letters 78, pp. 1876–1879, 1997. A lifetime of 3700 days was determined. Cited on
page 133.
107 See the instructive paper by J. Młyńczak, J. Kubicki & K. Kopczyński Stand-off
detection of alcohol in car cabins, Journal of Applied Remote Sensing 8, p. 083627, 2014.
Cited on page 119.
108 The fluid helium based quantum gyroscope was first used to measure the rotation of
the Earth by the French research group O. Avenel & E. Varoquaux, Detection of
the Earth rotation with a superfluid double-hole resonator, Czechoslovak Journal of Phys-
ics (Supplement S6) 48, pp. 3319–3320, 1996. The description of the set-up is found in
Yu. Mukharsky, O. Avenel & E. Varoquaux, Observation of half-quantum defects
in superfluid 3 He − B, Physical Review Letters 92, p. 210402, 2004. The gyro experiment
was later repeated by the Berkeley group, K. Schwab, N. Bruckner & R. E. Packard,
Detection of the Earth’s rotation using superfluid phase coherence, Nature 386, pp. 585–587,
1997. For an review of this topic, see O. Avenel, Yu. Mukharsky & E. Varoquaux,
Superfluid gyrometers, Journal of Low Temperature Physics 135, p. 745, 2004. Cited on page
120.
109 K. Schwab, E. A. Henriksen, J. M. Worlock & M. L. Roukes, Measurement
of the quantum of thermal conductance, Nature 404, p. 974, 2000. For the optical ana-
log of the experiment, see the beautiful paper by E. A. Montie, E. C. Cosman,
bibliography 395

G. W. ’ t Hooft, M. B. van der Mark & C. W. J. Beenakker, Observation of the
optical analogue of quantized conductance of a point contact, Nature 350, pp. 594–595, 18
April 1991, and the longer version E. A. Montie, E. C. Cosman, G. W. ’ t Hooft,
M. B. van der Mark & C. W. J. Beenakker, Observation of the optical analogue of
the quantised conductance of a point contact, Physica B 175, pp. 149–152, 1991. For more de-
tails on this experiment, see the volume Light, Charges and Brains of the Motion Mountain
series. Cited on page 133.
110 Research on how humans move is published in the journals Human Movement Science and
also in Human Movement. Cited on page 137.
111 See, for example, the paper by D. A. Bryant, Electron acceleration in the aurora, Con-
temporary Physics 35, pp. 165–179, 1994. A recent development is D. L. Chandler,
Mysterious electron acceleration explained, found at web.mit.edu/newsoffice/2012/
plasma-phenomenon-explained-0227.html; it explains the results of the paper by
J. Egedal, W. Daughton & A. Le, Large-scale acceleration by parallel electric fields
during magnetic reconnection, Nature Physics 2012. Cited on page 137.

Motion Mountain – The Adventure of Physics
112 The is a recent research field; discharges above clouds are described, e.g., on www.
spritesandjets.com and on en.wikipedia.org/wiki/Sprite_(lightning). See also eurosprite.
blogspot.com. Cited on page 137.
113 About Coulomb explosions and their importance for material processing with lasers, see
for example D. Müller, Picosecond lasers for high-quality industrial micromachining,
Photonics Spectra pp. 46–47, November 2009. Cited on page 137.
114 See, e.g., L.O ’ C. Drury, Acceleration of cosmic rays, Contemporary Physics 35, pp. 231–
242, 1994. Cited on page 137.
115 See Dirk Kreimer, New mathematical structures in renormalizable quantum field the-
ories, arxiv.org/abs/hep-th/0211136 or Annals of Physics 303, pp. 179–202, 2003, and the

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
erratum ibid., 305, p. 79, 2003. Cited on page 137.
116 See, for example, the paper by M. Urban, A particle mechanism for the index of refraction,
preprint at arxiv.org/abs/0709.1550. Cited on page 138.
117 S. Fray, C. Alvarez Diez, T. W. Hänsch & M. Weitz, Atomic interferometer with
amplitude gratings of light and its applications to atom based tests of the equivalence principle,
Physical Review Letters 93, p. 240404, 2004, preprint at arxiv.org/abs/physics/0411052. The
idea for the experiment is old and can be traced back to the proposal, made in the 1950s, to
build an atomic fountain. Cited on page 140.
118 The original papers on neutron behaviour in the gravitational field are all worth
reading. The original feat was reported in V. V. Nesvizhevsky, H. G. B oerner,
A. K. Petoukhov, H. Abele, S. Baessler, F. Ruess, Th. Stoeferle,
A. Westphal, A. M. Gagarski, G. A. Petrov & A. V. Strelkov, Quantum
states of neutrons in the Earth’s gravitational field, Nature 415, pp. 297–299, 17 Janu-
ary 2002. Longer explanations are found in V. V. Nesvizhevsky, H. G. B oerner,
A. M. Gagarsky, A. K. Petoukhov, G. A. Petrov, H. Abele, S. Baessler,
G. Divkovic, F. J. Ruess, Th. Stoeferle, A. Westphal, A. V. Strelkov,
K. V. Protasov & A. Yu. Voronin, Measurement of quantum states of neutrons in the
Earth’s gravitational field, Physical Review D 67, p. 102002, 2003, preprint at arxiv.org/abs/
hep-ph/0306198, and in V. V. Nesvizhevsky, A. K. Petukhov, H. G. B oerner,
T. A. Baranova, A. M. Gagarski, G. A. Petrov, K. V. Protasov, A. Yu. Voronin,
S. Baessler, H. Abele, A. Westphal & L. Lucovac, Study of the neutron quantum
states in the gravity field, European Physical Journal C 40, pp. 479–491, 2005, preprint
396 bibliography

at arxiv.org/abs/hep-ph/0502081 and A. Yu. Voronin, H. Abele, S. Baessler,
V. V. Nesvizhevsky, A. K. Petukhov, K. V. Protasov & A. Westphal,
Quantum motion of a neutron in a wave-guide in the gravitational field, Physical Re-
view D 73, p. 044029, 2006, preprint at arxiv.org/abs/quant-ph/0512129. Cited on page
141.
119 H. Rauch, W. Treimer & U. B onse, Test of a single crystal neutron interferometer,
Physics Letters A 47, pp. 369–371, 1974. Cited on page 142.
120 R. Colella, A. W. Overhauser & S. A. Werner, Observation of gravitationally in-
duced quantum mechanics, Physical Review Letters 34, pp. 1472–1475, 1975. This experiment
is usually called the COW experiment. Cited on page 142.
121 Ch. J. B ordé & C. Lämmerzahl, Atominterferometrie und Gravitation, Physikalische
Blätter 52, pp. 238–240, 1996. See also A. Peters, K. Y. Chung & S. Chu, Observation
of gravitational acceleration by dropping atoms, Nature 400, pp. 849–852, 26 August 1999.
Cited on page 142.
122 J. D. Bekenstein, Black holes and entropy, Physical Review D 7, pp. 2333–2346, 1973.

Motion Mountain – The Adventure of Physics
Cited on page 145.
123 R. B ousso, The holographic principle, Review of Modern Physics 74, pp. 825–874, 2002,
also available as arxiv.org/abs/hep-th/0203101. The paper is an excellent review; however, it
has some argumentation errors, as explained on page 146. Cited on page 146.
124 This is clearly argued by S. Carlip, Black hole entropy from horizon conformal field theory,
Nuclear Physics B Proceedings Supplement 88, pp. 10–16, 2000. Cited on page 146.
125 S. A. Fulling, Nonuniqueness of canonical field quantization in Riemannian space-time,
Physical Review D 7, pp. 2850–2862, 1973, P. Davies, Scalar particle production in Schwar-
zchild and Rindler metrics, Journal of Physics A: General Physics 8, pp. 609–616, 1975,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
W. G. Unruh, Notes on black hole evaporation, Physical Review D 14, pp. 870–892, 1976.
Cited on page 146.
126 About the possibility to measure Fulling–Davies–Unruh radiation directly, see for example
the paper by H. Rosu, On the estimates to measure Hawking effect und Unruh effect in the
laboratory, International Journal of Modern Physics D3 p. 545, 1994. arxiv.org/abs/gr-qc/
9605032 or the paper P. Chen & T. Tajima, Testing Unruh radiation with ultra-intense
lasers, Physical Review Letters 83, pp. 256–259, 1999. Cited on page 147.
127 E. T. Akhmedov & D. Singleton, On the relation between Unruh and Sokolov–
Ternov effects, preprint at arxiv.org/abs/hep-ph/0610391; see also E. T. Akhmedov &
D. Singleton, On the physical meaning of the Unruh effect, preprint at arxiv.org/abs/
0705.2525. Unfortunately, as Bell and Leinaas mention, the Sokolov–Ternov effect cannot
be used to confirm the expression of the black hole entropy, because the radiation is not
thermal, and so a temperature of the radiation is hard to define. Nevertheless, C. Schiller
predicts (October 2009) that careful analysis of the spectrum could be used (1) to check the
proportionality of entropy and area in black holes, and (2) to check the transformation of
temperature in special relativity. Cited on page 147.
128 W. G. Unruh & R. M. Wald, Acceleration radiation and the generalised second law of
thermodynamics, Physical Review D 25, pp. 942–958 , 1982. Cited on page 148.
129 R. M. Wald, The thermodynamics of black holes, Living Reviews of Relativity 2001,
www-livingreviews.org/lrr-2001-6. Cited on page 149.
130 For example, if neutrinos were massless, they would be emitted by black holes more fre-
quently than photons. For a delightful popular account from a black hole expert, see
bibliography 397

Igor Novikov, Black Holes and the Universe, Cambridge University Press 1990. Cited
on pages 150 and 154.
131 The original paper is W. G. Unruh, Experimental black hole evaporation?, Physical Review
Letters 46, pp. 1351–1353, 1981. A good explanation with good literature overview is the
one by Matt Visser, Acoustical black holes: horizons, ergospheres and Hawking radiation,
arxiv.org/abs/gr-qc/9712010. Cited on page 151.
132 Optical black holes are explored in W. G. Unruh & R. Schützhold, On slow light as a
black hole analogue, Physical Review D 68, p. 024008, 2003, preprint at arxiv.org/abs/gr-qc/
0303028. Cited on page 151.
133 T. Damour & R. Ruffini, Quantum electrodynamical effects in Kerr–Newman geomet-
ries, Physical Review Letters 35, pp. 463–466, 1975. Cited on page 153.
134 These were the Vela satellites; their existence and results were announced officially only in
1974, even though they were working already for many years. Cited on page 152.
135 An excellent general introduction into the topic of gamma ray bursts is S. Klose,
J. Greiner & D. Hartmann, Kosmische Gammastrahlenausbrüche – Beobachtungen

Motion Mountain – The Adventure of Physics
und Modelle, Teil I und II, Sterne und Weltraum March and April 2001. Cited on pages
152 and 153.
136 When the gamma-ray burst encounters the matter around the black hole, it is broadened.
The larger the amount of matter, the broader the pulse is. See G. Preparata, R. Ruffini
& S. -S. Xue, The dyadosphere of black holes and gamma-ray bursts, Astronomy and As-
trophysics 338, pp. L87–L90, 1998, R. Ruffini, J. D. Salmonson, J. R. Wilson & S. -
S. Xue, On the pair electromagnetic pulse of a black hole with electromagnetic structure,
Astronomy and Astrophysics 350, pp. 334–343, 1999, R. Ruffini, J. D. Salmonson,
J. R. Wilson & S. -S. Xue, On the pair electromagnetic pulse from an electromagnetic

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
black hole surrounded by a baryonic remnant, Astronomy and Astrophysics 359, pp. 855–
864, 2000, and C. L. Bianco, R. Ruffini & S. -S. Xue, The elementary spike produced
by a pure e+ e− pair-electromagnetic pulse from a black hole: the PEM pulse, Astronomy and
Astrophysics 368, pp. 377–390, 2001. For a very personal account by Ruffini on his involve-
ment in gamma-ray bursts, see his paper Black hole formation and gamma ray bursts, arxiv.
org/abs/astro-ph/0001425. Cited on page 153.
137 See the publication by D.W. Fox & al., Early optical emission from the γ-ray burst of 4 Oc-
tober 2002, Nature 422, pp. 284–286, 2003. See also arxiv.org/abs/astro-ph/0301377, arxiv.
org/abs/astro-ph/0301262 and arxiv.org/abs/astro-ph/0303539. Cited on page 152.
138 Negative heat capacity has also been found in atom clusters and in nuclei. See, e.g., M.
Schmidt & al., Negative heat capacity for a cluster of 147 sodium atoms, Physical Review
Letters 86, pp. 1191–1194, 2001. Cited on page 154.
139 H. -P. Nollert, Quasinormal modes: the characteristic ‘sound’ of black holes and neutron
stars, Classical and Quantum Gravity 16, pp. R159–R216, 1999. Cited on page 154.
140 On the membrane description of black holes, see Kip S. Thorne, Richard H. Price
& Douglas A. MacDonald, editors, Black Holes: the Membrane Paradigm, Yale Uni-
versity Press, 1986. Cited on page 154.
141 Page wrote a series of papers on the topic; a beautiful summary is Don N. Page, How fast
does a black hole radiate information?, International Journal of Modern Physics 3, pp. 93–
106, 1994, which is based on his earlier papers, such as Information in black hole radi-
ation, Physical Review Letters 71, pp. 3743–3746, 1993. See also his preprint at arxiv.org/
abs/hep-th/9305040. Cited on page 155.
398 bibliography

142 See Don N. Page, Average entropy of a subsystem, Physical Review Letters 71, pp. 1291–
1294, 1993. The entropy formula of this paper, used above, was proven by S. K. Foong &
S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem, Physical Review
Letters 72, pp. 1148–1151, 1994. Cited on page 156.
143 R. Lafrance & R. C. Myers, Gravity’s rainbow: limits for the applicability of the equi-
valence principle, Physical Review D 51, pp. 2584–2590, 1995, arxiv.org/abs/hep-th/9411018.
Cited on page 157.
144 M. Yu. Kuchiev & V. V. Flambaum, Scattering of scalar particles by a black hole, arxiv.
org/abs/gr-qc/0312065. See also M. Yu. Kuchiev & V. V. Flambaum, Reflection on
event horizon and escape of particles from confinement inside black holes, arxiv.org/abs/
gr-qc/0407077. Cited on page 157.
145 See the widely cited but wrong paper by G. C. Ghirardi, A. Rimini & T. Weber, Uni-
fied dynamics for microscopic and macroscopic systems, Physical Review D 34, pp. 470–491,
1986. Cited on page 143.
146 I speculate that this version of the coincidences could be original; I have not found it in the

Motion Mountain – The Adventure of Physics
literature. Cited on page 144.
147 Steven Weinberg, Gravitation and Cosmology, Wiley, 1972. See equation 16.4.3 on page
619 and also page 620. Cited on page 144.
148 It could be that knot theory provides a relation between a local knot invariant, related to
particles, and a global one. Cited on page 144.
149 This point was made repeatedly by Steven Weinberg, namely in Derivation of gauge
invariance and the equivalence principle from Lorentz invariance of the S-matrix , Phys-
ics Letters 9, pp. 357–359, 1964, in Photons and gravitons in S-matrix theory: derivation
of charge conservation and equality of gravitational and inertial mass, Physical Review B
135, pp. 1049–1056, 1964, and in Photons and gravitons in perturbation theory: derivation

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
of Maxwell’s and Einstein’s equations, Physical Review B 138, pp. 988–1002, 1965. Cited on
page 158.
150 E. Joos, Why do we observe a classical space-time?, Physics Letters A 116, pp. 6–8, 1986.
Cited on page 159.
151 L. H. Ford & T. A. Roman, Quantum field theory constrains traversable wormhole geo-
metries, arxiv.org/abs/gr-qc/9510071 or Physical Review D 53, pp. 5496–5507, 1996. Cited
on page 159.
152 B. S. Kay, M. Radzikowski & R. M. Wald, Quantum field theory on spacetimes with
a compactly generated Cauchy horizon, arxiv.org/abs/gr-qc/9603012 or Communications in
Mathematical Physics 183 pp. 533–556, 1997. Cited on page 159.
153 M. J. Pfenning & L. H. Ford, The unphysical nature of ‘warp drive’, Classical and
Quantum Gravity 14, pp. 1743–1751, 1997. Cited on page 159.
154 A excellent technical introduction to nuclear physics is B ogdan Povh, Klaus Rith,
Christoph Scholz & Frank Zetsche, Teilchen und Kerne, Springer, 5th edition,
1999. It is also available in English translation. Cited on page 162.
155 For magnetic resonance imaging films of the heart beat, search for ‘cardiac MRI’ on the
internet. See for example www.youtube.com/watch?v=58l6oFhfZU. Cited on page 164.
156 See the truly unique paper by C. Bamberg, G. Rademacher, F. Güttler,
U. Teichgräber, M. Cremer, C. Bührer, C. Spies, L. Hinkson, W. Henrich,
K. D. Kalache & J. W. Dudenhausen, Human birth observed in real-time open mag-
netic resonance imaging, American Journal of Obstetrics & Gynecology 206, p. 505e1-505e6,
2012. Cited on page 164.
bibliography 399

Motion Mountain – The Adventure of Physics
F I G U R E 190 The wonderful origin of human life (© W.C.M. Weijmar Schultz).

157 W.C.M. Weijmar Schultz & al., Magnetic resonance imaging of male and female gen-
itals during coitus and female sexual arousal, British Medical Journal 319, pp. 1596–1600,
December 18, 1999, available online as www.bmj.com/cgi/content/full/319/7225/1596. Cited
on page 164.
158 M. Chantell, T. C. Weekes, X. Sarazin & M. Urban, Antimatter and the moon,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Nature 367, p. 25, 1994. M. Amenomori & al., Cosmic ray shadow by the moon observed
with the Tibet air shower array, Proceedings of the 23rd International Cosmic Ray Confer-
ence, Calgary 4, pp. 351–354, 1993. M. Urban, P. Fleury, R. Lestienne & F. Plouin,
Can we detect antimatter from other galaxies by the use of the Earth’s magnetic field and the
Moon as an absorber?, Nuclear Physics, Proceedings Supplement 14B, pp. 223–236, 1990.
Cited on page 177.
159 Joseph Magill & Jean Galy, Radioactivity Radionuclides Radiation, Springer, 2005.
This dry but dense book contains most data about the topic, including much data on all the
known nuclides. Cited on pages 181 and 198.
160 An good summary on radiometric dating is by R. Wiens, Radiometric dating – a christian
perpective, www.asa3.org/ASA/resources/Wiens.html. The absurd title is due to the habit in
many religious circles to put into question radiometric dating results. Putting apart the few
religious statements in the review, the content is well explained. Cited on pages 182 and 184.
161 See for example the excellent and free lecture notes by Heinz-Günther Stosch, Ein-
führung in die Isotopengeochemie, 2004, available on the internet. Cited on page 182.
162 G. B. Dalrymple, The age of the Earth in the twentieth century: a problem (mostly) solved,
Special Publications, Geological Society of London 190, pp. 205–221, 2001. Cited on page
184.
163 A good overview is given by A. N. Halliday, Radioactivity, the discovery of time and the
earliest history of the Earth, Contemporary Physics 38, pp. 103–114, 1997. Cited on page 185.
164 J. Dudek, A. God, N. Schunck & M. Mikiewicz, Nuclear tetrahedral symmetry:
possibly present throughout the periodic table, Physical Review Letters 88, p. 252502, 24 June
400 bibliography

2002. Cited on page 186.
165 A good introduction is R. Clark & B. Wodsworth, A new spin on nuclei, Physics
World pp. 25–28, July 1998. Cited on page 187.
166 John Horgan, The End of Science – Facing the Limits of Knowledge in the Twilight of the
Scientific Age, Broadway Books, 1997, chapter 3, note 1. Cited on page 191.
167 G. Charpak & R. L. Garwin, The DARI, Europhysics News 33, pp. 14–17, Janu-
ary/February 2002. Cited on page 193.
168 M. Brunetti, S. Cecchini, M. Galli, G. Giovannini & A. Pagliarin,
Gamma-ray bursts of atmospheric origin in the MeV energy range, Geophysical Research
Letters 27, p. 1599, 2000. Cited on page 195.
169 For a recent image of Lake Karachay, see www.google.com/maps/@55.6810205,60.796688,
3519m/data=!3m1!1e3. Cited on page 196.
170 A book with nuclear explosion photographs is Michael Light, 100 Suns, Jonathan
Cape, 2003. Cited on page 197.
171 A conversation with Peter Mansfield, Europhysics Letters 37, p. 26, 2006. Cited on page 193.

Motion Mountain – The Adventure of Physics
172 An older but still fascinating summary of solar physics is R. Kippenhahn, Hundert Mil-
liarden Sonnen, Piper, 1980. It was a famous bestseller and is available also in English trans-
lation. Cited on page 200.
173 H. Bethe, On the formation of deuterons by proton combination, Physical Review 54,
pp. 862–862, 1938, and H. Bethe, Energy production in stars, Physical Review 55, pp. 434–
456, 1939. Cited on page 200.
174 M. Nauenberg & V. F. Weisskopf, Why does the sun shine?, American Journal of
Physics 46, pp. 23–31, 1978. Cited on page 202.
175 The slowness of the speed of light inside the Sun is due to the frequent scattering of photons

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
by solar matter. The most serious estimate is by R. Mitalas & K. R. Sills, On the photon
diffusion time scale for the Sun, The Astrophysical Journal 401, pp. 759–760, 1992. They
give a photon escape time of 0.17 Ma, an average photon free mean path of 0.9 mm and an
average speed of 0.97 cm/s. See also the interesting paper by M. Stix, On the time scale of
energy transport in the sun, Solar Physics 212, pp. 3–6, 2003, which comes to the conclusion
that the speed of energy transport is 30 Ma, two orders of magnitude higher than the photon
diffusion time. Cited on page 203.
176 See the freely downloadable book by John Wesson, The Science of JET - The Achievements
of the Scientists and Engineers Who Worked on the Joint European Torus 1973-1999, JET Joint
Undertaking, 2000, available at www.jet.edfa.org/documents/wesson/wesson.html. Cited
on page 211.
177 J. D. Lawson, Some criteria for a power producing thermonuclear reactor, Proceedings of
the Physical Society, London B 70, pp. 6–10, 1957. The paper had been kept secret for two
years. However, the result was already known, before Lawson, to all Russian nuclear phys-
icists several years earlier. Cited on page 212.
178 The classic paper is R. A. Alpher, H. Bethe & G. Gamow, The Origin of Chemical
Elements, Physical Review 73, pp. 803–804, 1948. Cited on page 213.
179 The famous overview of nucleosyntehis, over 100 pages long, is the so-called B2 FH paper
by M. Burbidge, G. Burbidge, W. Fowler & F. Hoyle, Synthesis of the elements in
stars, Reviews of Modern Physics 29, pp. 547–650, 1957. Cited on page 213.
180 The standard reference is E. Anders & N. Grevesse, Abundances of the elements – met-
eoritic and solar, Geochimica et Cosmochimica Acta 53, pp. 197–214, 1989. Cited on page
215.
bibliography 401

181 S. Goriely, A. Bauswein & H. T. Janka, R-process nucleosynthesis in dynamically
ejected matter of neutron star mergers, Astrophysical Journal 738, p. L38, 2011, preprint at
arxiv.org/abs/1107.0899. Cited on pages 215 and 216.
182 Kendall, Friedman and Taylor received the 1990 Nobel Prize in Physics for a series of ex-
periments they conducted in the years 1967 to 1973. The story is told in the three Nobel
lectures R. E. Taylor, Deep inelastic scattering: the early years, Review of Modern Phys-
ics 63, pp. 573–596, 1991, H. W. Kendall, Deep inelastic scattering: Experiments on the
proton and the observation of scaling, Review of Modern Physics 63, pp. 597–614, 1991, and
J. I. Friedman, Deep inelastic scattering: Comparisons with the quark model, Review of
Modern Physics 63, pp. 615–620, 1991. Cited on page 219.
183 G. Zweig, An SU3 model for strong interaction symmetry and its breaking II, CERN Report
No. 8419TH. 412, February 21, 1964. Cited on page 219.
184 About the strange genius of Gell-Mann, see the beautiful book by George Johnson,
Murray Gell-Mann and the Revolution in Twentieth-Century Physics, Knopf, 1999. Cited on
page 219.

Motion Mountain – The Adventure of Physics
185 The best introduction might be the wonderfully clear text by Donald H. Perkins, In-
troduction to High Energy Physics, Cambridge University Press, fourth edition, 2008. Also
beautiful, with more emphasis on the history and more detail, is Kurt Gottfried &
Victor F. Weisskopf, Concepts of Particle Physics, Clarendon Press, Oxford, 1984. Vic-
tor Weisskopf was one of the heroes of the field, both in theoretical research and in the
management of CERN, the European organization for particle research. Cited on pages 221,
231, and 401.
186 The official reference for all particle data, worth a look for every physicist, is the massive
collection of information compiled by the Particle Data Group, with the website pdg.web.
cern.ch containing the most recent information. A printed review is published about every

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
two years in one of the major journals on elementary particle physics. See for example C.
Amsler & al., The Review of Particle Physics, Physics Letters B 667, p. 1, 2008. For some
measured properties of these particles, the official reference is the set of so-called CODATA
values given in reference Ref. 286. Cited on pages 221, 232, 233, 235, 245, 251, 252, 255,
and 262.
187 H. Fritsch, M. Gell-Mann & H. Leutw yler, Advantages of the color octet picture,
Physics Letters B 47, pp. 365–368, 1973. Cited on page 223.
188 Quantum chromodynamics can be explored in books of several levels. For the first, pop-
ular level, see the clear longseller by one of its founders, Harald Fritzsch, Quarks –
Urstoff unserer Welt, Piper Verlag, 2006, or, in English language, Quarks: the Stuff of Mat-
ter, Penguin Books, 1983. At the second level, the field can be explored in texts on high
energy physics, as those of Ref. 185. The third level contains books like Kerson Huang,
Quarks, Leptons and Gauge Fields, World Scientific, 1992, or Felix J. Ynduráin, The
Theory of Quark and Gluon Interactions, Springer Verlag, 1992, or Walter Greiner &
Andreas Schäfer, Quantum Chromodynamics, Springer Verlag, 1995, or the modern
and detailed text by Stephan Narison, QCD as a Theory of Hadrons, Cambridge Uni-
versity Press, 2004. As always, a student has to discover by himself or herself which text is
most valuable. Cited on pages 224, 230, and 232.
189 S. Dürr & al., Ab initio determination of the light hadron masses, Science 322, pp. 1224–
1227, 2008. Cited on page 228.
190 See for example C. Bernard & al., Light hadron spectrum with Kogut–Susskind quarks,
Nuclear Physics, Proceedings Supplement 73, p. 198, 1999, and references therein. Cited on
page 228.
402 bibliography

191 R. Brandelik & al., Evidence for planar events in 𝑒+ 𝑒− annihilation at high energies, Phys-
ics Letters B 86, pp. 243–249, 1979. Cited on page 236.
192 For a pedagogical introduction to lattice QCD calculations, see R. Gupta, Introduction to
Lattice QCD, preprint at arxiv.org/abs/hep-lat/9807028, or the clear introduction by Mi-
chael Creutz, Quarks, Gluons and Lattices, Cambridge University Press, 1983. Cited
on page 231.
193 S. Bethke, Experimental tests of asymptotic freedom, Progress in Particle and Nuclear
Physics 58, pp. 351–368, 2007, preprint at arxiv.org/abs/hep-ex/0606035. Cited on page
232.
194 F. Abe & al., Measurement of dijet angular distributions by the collider detector at Fermilab,
Physical Review Letters 77, pp. 5336–5341, 1996. Cited on page 233.
195 The approximation of QCD with zero mass quarks is described by F. Wilczek, Getting its
from bits, Nature 397, pp. 303–306, 1999. It is also explained in F. Wilczek, Asymptotic
freedom, Nobel lecture 2004. The proton’s mass is set by the energy scale at which the strong
coupling, coming from its value at Planck energy, becomes of order unity. Cited on pages

Motion Mountain – The Adventure of Physics
233, 234, and 258.
196 A. J. Buchmann & E. M. Henley, Intrinsic quadrupole moment of the nucleon, Physical
Review C 63, p. 015202, 2000. Alfons Buchmann also predicts that the quadrupole moment
of the other, strange 𝐽 = 1/2 octet baryons is positive, and predicts a prolate structure for
all of them (private communication). For the decuplet baryons, with 𝐽 = 3/2, the quadru-
pole moment can often be measured spectroscopically, and is always negative. The four Δ
baryons are thus predicted to have a negative intrinsic quadrupole moment and thus an ob-
late shape. This explained in A. J. Buchmann & E. M. Henley, Quadrupole moments of
baryons, Physical Review D 65, p. 073017, 2002. For recent updates, see A. J. Buchmann,
Charge form factors and nucleon shape, pp. 110–125, in the Shape of Hadrons Workshop

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Conference, Athens Greece, 27-29 April 2006, AIP Conference Proceedings 904, Eds. C.N.
Papanicolas, Aron Bernstein. For updates on other baryons, see A. J. Buchmann, Struc-
ture of strange baryons, Hyp 2006, International Conference on Hypernuclear and Strange
Particle Physics, Oct.10-14, Mainz, Germany, to be published in European Physics Journal
A 2007. The topic is an active field of research; for example, results on magnetic octupole
moments are expected soon. Cited on page 234.
197 S. Strauch & al., Polarization transfer in the 4 He (e,e’p) 3 H reaction up to 𝑄2 = 2.6(GeV/
c)2 , Physical Review Letters 91, p. 052301, 2003. Cited on page 234.
198 F. Close, Glueballs and hybrids: new states of matter, Contemporary Physics 38, pp. 1–12,
1997. See also the next reference. Cited on page 237.
199 A tetraquark is thought to be the best explanation for the f0 (980) resonance at 980 MeV.
The original proposal of this explanation is due to R. L. Jaffe, Multiquark hadrons I: phe-
2
nomenology of 𝑄2 𝑄 mesons, Physical Review D 15, pp. 267–280, 1977, R. L. Jaffe, Mul-
tiquark hadrons II: methods, Physical Review D 15, pp. 281–289, 1977, and R. L. Jaffe,
2
Physical Review D 𝑄2 𝑄 resonances in the baryon-antibaryon system,17, pp. 1444–1458,
1978. For a clear and detailed modern summary, see the excellent review by E. Klempt
& A. Zaitsev, Glueballs, hybrids, multiquarks: experimental facts versus QCD inspired
concepts, Physics Reports 454, pp. 1–202, 2007, preprint at arxiv.org/abs/0708.4016. See
also F. Giacosa, Light scalars as tetraquarks, preprint at arxiv.org/abs/0711.3126, and
V. Crede & C. A. Meyer, The experimental status of glueballs, preprint at arxiv.org/abs/
0812.0600. However, other researchers argue against this possibility; see, e.g., arxiv.org/abs/
1404.5673v2. The issue is not closed. Cited on page 237.
bibliography 403

200 Pentaquarks were first predicted by Maxim Polyakov, Dmitri Diakonov, and Victor Petrov
in 1997. Two experimental groups in 2003 claimed to confirm their existence, with a mass
of 1540 MeV; see K. Hicks, An experimental review of the Θ+ pentaquark, arxiv.org/abs/
hep-ex/0412048. Results from 2005 and later, however, ruled out that the 1540 MeV particle
is a pentaquark. Cited on page 237.
201 See, for example, Ya. B. Zel ’ dovich & V. S. Popov, Electronic structure of superheavy
atoms, Soviet Physics Uspekhi 17, pp. 673–694, 2004 in the English translation. Cited on
page 134.
202 J. Tran Thanh Van, editor, CP violation in Particle Physics and Astrophysics, Proc. Conf.
Chateau de Bois, France, May 1989, Editions Frontières, 1990. Cited on page 245.
203 P.L. Anthony & al., Observation of parity nonconservation in Møller scattering, Physical
Review Letters 92, p. 181602, 2004. Cited on page 246.
204 M. A. B ouchiat & C. C. B ouchiat, Weak neutral currents in atomic physics, Physics
Letters B 48, pp. 111–114, 1974. U. Amaldi, A. B öhm, L. S. Durkin, P. Langacker,
A. K. Mann, W. J. Marciano, A. Sirlin & H. H. Williams, Comprehensive analy-

Motion Mountain – The Adventure of Physics
sis of data pertaining to the weak neutral current and the intermediate-vector-boson masses,
Physical Review D 36, pp. 1385–1407, 1987. Cited on page 246.
205 M. C. Noecker, B. P. Masterson & C. E. Wiemann, Precision measurement of par-
ity nonconservation in atomic cesium: a low-energy test of electroweak theory, Physical Re-
view Letters 61, pp. 310–313, 1988. See also D.M. Meekhof & al., High-precision meas-
urement of parity nonconserving optical rotation in atomic lead, Physical Review Letters 71,
pp. 3442–3445, 1993. Cited on page 247.
206 S. C. Bennet & C. E. Wiemann, Measurement of the 6S – 7S transition polarizability
in atomic cesium and an improved test of the standard model, Physical Review Letters 82,
pp. 2484–2487, 1999. The group has also measured the spatial distribution of the weak

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
charge, the so-called the anapole moment; see C.S. Wood & al., Measurement of parity
nonconservation and an anapole moment in cesium, Science 275, pp. 1759–1763, 1997. Cited
on page 247.
207 C. Jarlskog, Commutator of the quark mass matrices in the standard electroweak model
and a measure of maximal CP nonconservation, Physical Review Letters 55, pp. 1039–1042,
1985. Cited on page 251.
208 The correct list of citations is a topic of intense debate. It surely includes Y. Nambu &
G. Jona–Lasinio, Dynamical model of the elementary particles based on an analogy with
superconductivity - I, Physical Review 122, p. 345-358, 1961, P. W. Anderson, Plasmons,
gauge invariance, and mass, Physical Review 130, pp. 439–442, 1963, The list then continues
with reference Ref. 210. Cited on pages 253 and 404.
209 K. Grotz & H. V. Klapdor, Die schwache Wechselwirkung in Kern-, Teilchen- und As-
trophysik, Teubner Verlag, Stuttgart, 1989. Also available in English and in several other
languages. Cited on page 255.
210 P. W. Higgs, Broken symmetries, massless particles and gauge fields, Physics Letters 12,
pp. 132–133, 1964, P. W. Higgs, Broken symmetries and the masses of the gauge bosons,
Physics Letters 13, pp. 508–509, 1964. He then expanded the story in P. W. Higgs, Spon-
taneous symmetry breakdown without massless bosons, Physical Review 145, pp. 1156–1163,
1966. Higgs gives most credit to Anderson, instead of to himself; he also mentions Brout
and Englert, Guralnik, Hagen, Kibble and ’t Hooft. These papers are F. Englert &
R. Brout, Broken symmetry and the mass of the gauge vector mesons, Physics Review Let-
ters 13, pp. 321–323, 1964, G. S. Guralnik, C. R. Hagen & T. W. B. Kibble, Global
404 bibliography

conservations laws and massless particles, Physical Review Letters 13, pp. 585–587, 1964,
T. W. B. Kibble, Symmetry breaking in non-Abelian gauge theories, Physical Review 155,
pp. 1554–1561, 1967. For the ideas that inspired all these publications, see Ref. 208. Cited on
pages 258 and 403.
211 D. Treille, Particle physics from the Earth and from the sky: Part II, Europhysics News
35, no. 4, 2004. Cited on page 256.
212 Rumination is studied in P. Jordan & de Laer Kronig, in Nature 120, p. 807, 1927.
Cited on page 259.
213 K.W.D. Ledingham & al., Photonuclear physics when a multiterawatt laser pulse inter-
acts with solid targets, Physical Review Letters 84, pp. 899–902, 2000. K.W.D. Leding-
ham & al., Laser-driven photo-transmutation of Iodine-129 – a long lived nuclear waste
product, Journal of Physics D: Applied Physics 36, pp. L79–L82, 2003. R. P. Singhal,
K. W. D. Ledingham & P. McKenna, Nuclear physics with ultra-intense lasers – present
status and future prospects, Recent Research Developments in Nuclear Physics 1, pp. 147–
169, 2004. Cited on page 259.

Motion Mountain – The Adventure of Physics
214 The electron radius limit is deduced from the 𝑔 − 2 measurements, as explained in the No-
bel Prize talk by Hans Dehmelt, Experiments with an isolated subatomic particle at rest,
Reviews of Modern Physics 62, pp. 525–530, 1990, or in Hans Dehmelt, Is the electron
a composite particle?, Hyperfine Interactions 81, pp. 1–3, 1993. Cited on page 264.
215 G. Gabrielse, H. Dehmelt & W. Kells, Observation of a relativistic, bistable hyster-
esis in the cyclotron motion of a single electron, Physical Review Letters 54, pp. 537–540,
1985. Cited on page 264.
216 For the bibliographic details of the latest print version of the Review of Particle Physics,
see Appendix B. The online version can be found at pdg.web.cern.ch. The present status
on grand unification can also be found in the respective section of the overview. Cited on

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
pages 268, 269, and 270.
217 Slides of a very personal review talk by H. Georgi, The future of grand unification, can
be found at www2.yukawa.kyoto-u.ac.jp/~yt100sym/files/yt100sym_georgi.pdf. A modern
research approach is S. Raby, Grand unified theories, arxiv.org/abs/hep-ph/0608183. Cited
on page 269.
218 H. Jeon & M. Longo, Search for magnetic monopoles trapped in matter, Physical Review
Letters 75, pp. 1443–1447, 1995. Cited on page 269.
219 The quantization of charge as a consequence of the existence of magnetic monopoles is due
to P. Dirac, Quantised singularities in the electromagnetic Field, Proceedings of the Royal
Society (London) A 133, pp. 60–72, 1931. Cited on page 269.
220 On proton decay rates, see the latest data of the Particle Data Group, at pdg.web.cern.ch.
Cited on page 270.
221 U. Amaldi, de B oer & H. Fürstenau, Comparison of grand unified theories with elec-
troweak and strong coupling constants measured at LEP, Physics Letters 260, pp. 447–455,
1991. This widely cited paper is the standard reference for this issue. An update is found
in de B oer & C. Sander, Global electroweak fits and gauge coupling unification, Phys-
ics Letters B 585 pp. 276–286, 2004, or preprint at arxiv.org/abs/hep-ph/0307049. The
figure is taken with permission from the home page of Wim de Boer www-ekp.physik.
uni-karlsruhe.de/~deboer/html/Forschung/forschung.html. Cited on page 270.
222 Peter G. O. Freund, Introduction to Supersymmetry, Cambridge 1988. Julius Wess &
Jonathan Bagger, Supersymmetry and Supergravity, Princeton University Press, 1992.
This widely cited book contains a lot of mathematics but little physics. Cited on page 271.
bibliography 405

223 S. Coleman & J. Mandula, All possible symmetries of the S matrix, Physical Review
159, pp. 1251–1256, 1967. Cited on page 272.
224 P. C. Argyres, Dualities in supersymmetric field theories, Nuclear Physics Proceedings
Supplement 61A, pp. 149–157, 1998, preprint available at arxiv.org/abs/hep-th/9705076.
Cited on page 274.
225 Michael Stone editor, Bosonization, World Scientific, 1994. R. Rajaraman, Solitons
and Instantons, North Holland, 1987. However, the hope of explaining the existence of fer-
mions as the result of an infinite number of interacting bosons – this is what ‘bosonization’
means – has not been successful. Cited on page 275.
226 In 1997, the smallest human-made flying object was the helicopter built by a group of the
Institut für Mikrotechnik in Mainz, in Germany. A picture is available at their web page, to
be found at www.imm-mainz.de/English/billboard/f_hubi.html. The helicopter is 24 mm
long, weighs 400 mg and flies (though not freely) using two built-in electric motors driv-
ing two rotors, running at between 40 000 and 100 000 revolutions per minute. See also
the helicopter from Stanford University at www-rpl.stanford.edu/RPL/htmls/mesoscopic/

Motion Mountain – The Adventure of Physics
mesicopter/mesicopter.html, with an explanation of its battery problems. Cited on page
279.
227 Henk Tennekes, De wetten van de vliegkunst – over stijgen, dalen, vliegen en zweven,
Aramith Uitgevers, 1992. This clear and interesting text is also available in English. Cited
on page 279.
228 The most recent computational models of lift still describe only two-dimensional wing mo-
tion, e.g., Z. J. Wang, Two dimensional mechanism for insect hovering, Physical Review
Letters 85 pp. 2216–2219, 2000. A first example of a mechanical bird has been constructed
by Wolfgang Send; it can be studied on the www.aniprop.de website. See also W. Send,
Physik des Fliegens, Physikalische Blätter 57, pp. 51–58, June 2001. Cited on page 280.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
229 R. B. Srygley & A. L. R. Thomas, Unconventional lift-generating mechanisms in free-
flying butterflies, Nature 420, pp. 660–664, 2002. Cited on page 280.
230 The book by John Brackenbury, Insects in Flight, 1992. is a wonderful introduction
into the biomechanics of insects, combining interesting science and beautiful photographs.
Cited on page 282.
231 The simulation of insect flight using enlarged wing models flapping in oil instead of air
is described for example in www.dickinson.caltech.edu/research_robofly.html. The higher
viscosity of oil allows achieving the same Reynolds number with larger sizes and lower
frequencies than in air. See also the 2013 video of the talk on insect flight at www.ted.com/
talks/michael_dickinson_how_a_fly_flies.html. Cited on page 282.
232 A summary of undulatory swimming that includes the two beautiful illustrations in-
cluded in the text, Figure 157 and Figure 158, is M. Gazzola, M. Argentina &
L. Mahadevan, Scaling macroscopic aquatic locomotion, Nature Physics 10, pp. 758–761,
2014. Cited on pages 283 and 284.
233 E. Purcell, Life at low Reynolds number, American Journal of Physics 45, p. 3, 1977. See
also the review E. Lauga & T. R. Powers, The hydrodynamics of swimming microorgan-
isms, Reports on Progress in Physics 72, p. 096601, 2009, preprint at arxiv.org/abs/0812.
2887. Cited on page 285.
234 A short but informative review is by S. Vogel, Modes and scaling in aquatic loco-
motion, Integrative and Comparative Biology 48, pp. 702–712, 2008, available online at icb.
oxfordjournals.org/content/48/6/702.full. Cited on page 284.
406 bibliography

235 Most bacteria are flattened, ellipsoidal sacks kept in shape by the membrane enclosing the
cytoplasma. But there are exceptions; in salt water, quadratic and triangular bacteria have
been found. More is told in the corresponding section in the interesting book by Bern-
ard Dixon, Power Unseen – How Microbes Rule the World, W.H. Freeman, New York,
1994. Cited on page 286.
236 S. Pitnick, G. Spicer & T. A. Markow, How long is a giant sperm?, Nature 375, p. 109,
1995. Cited on page 286.
237 M. Kawamura, A. Sugamoto & S. Nojiri, Swimming of microorganisms viewed from
string and membrane theories, Modern Journal of Physics Letters A 9, pp. 1159–1174, 1994.
Also available as arxiv.org/abs/hep-th/9312200. Cited on page 286.
238 W. Nutsch & U. Rüffer, Die Orientierung freibeweglicher Organismen zum Licht,
dargestellt am Beispiel des Flagellaten Chlamydomonas reinhardtii, Naturwissenschaften 81,
pp. 164–174, 1994. Cited on page 286.
239 They are also called prokaryote flagella. See for example S. C. Schuster & S. Khan,
The bacterial flagellar motor, Annual Review of Biophysics and Biomolecular Structure

Motion Mountain – The Adventure of Physics
23, pp. 509–539, 1994, or S. R. Caplan & M. Kara-Ivanov, The bacterial flagellar
motor, International Review of Cytology 147, pp. 97–164, 1993. See also the information
on the topic that can be found on the website www.id.ucsb.edu:16080/fscf/library/origins/
graphics-captions/flagellum.html. Cited on page 286.
240 For an overview of the construction and the motion of coli bacteria, see H. C. Berg, Motile
behavior of bacteria, Physics Today 53, pp. 24–29, January 2000. Cited on page 287.
241 J. W. Shaevitz, J. Y. Lee & D. A. Fletcher, Spiroplasma swim by a processive change
in body helicity, Cell 122, pp. 941–945, 2005. Cited on page 287.
242 This is from the book by David Dusenbery, Life at a Small Scale, Scientific American

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Library, 1996. Cited on page 287.
243 F. Wilczek & A. Zee, Appearance of gauge structures in simple dynamical systems, Phys-
ical Review Letters 52, pp. 2111–2114, 1984, A. Shapere & F. Wilczek, Self-propulsion
at low Reynold number, Physical Review Letters 58, pp. 2051–2054, 1987, A. Shapere
& F. Wilczek, Gauge kinematics of deformable bodies, American Journal of Physics
57, pp. 514–518, 1989, A. Shapere & F. Wilczek, Geometry of self-propulsion at low
Reynolds number, Journal of Fluid Mechanics 198, pp. 557–585, 1989, A. Shapere &
F. Wilczek, Efficiencies of self-propulsion at low Reynolds number, Journal of Fluid Mech-
anics 198, pp. 587–599, 1989. See also R. Montgomery, Gauge theory of the falling cat,
Field Institute Communications 1, pp. 75–111, 1993. Cited on page 289.
244 E. Putterman & O. Raz, The square cat, American Journal of Physics 76, pp. 1040–
1045, 2008. Cited on page 289.
245 J. Wisdom, Swimming in spacetime: motion by cyclic changes in body shape, Science 299,
pp. 1865–1869, 21st of March, 2003. A similar effect was discovered later on by E. Guéron
& R. A. Mosna, The relativistic glider, Physical Review D 75, p. 081501(R), 2007, preprint
at arxiv.org/abs/gr-qc/0612131. Cited on page 291.
246 S. Smale, A classification of immersions of the two-sphere, Transactions of the American
Mathematical Society 90, pp. 281–290, 1958. Cited on page 292.
247 G. K. Francis & B. Morin, Arnold Shapiro’s Eversion of the Sphere, Mathematical
Intelligencer pp. 200–203, 1979. See also the unique manual for drawing manifolds by
George Francis, The Topological Picturebook, Springer Verlag, 1987. It also contains a
chapter on sphere eversion. Cited on page 292.
bibliography 407

248 B. Morin & J. -P. Petit, Le retournement de la sphere, Pour la Science 15, pp. 34–41, 1979.
See also the clear article by A. Phillips, Turning a surface inside out, Scientific American
pp. 112–120, May 1966. Cited on page 292.
249 S. Lev y, D. Maxwell & T. Munzner, Making Waves – a Guide to the Ideas Behind
Outside In, Peters, 1995. Cited on page 293.
250 George K. Batchelor, An Introduction to Fluid Mechanics, Cambridge University
Press, 1967, and H. Hashimoto, A soliton on a vortex filament, Journal of Fluid Mech-
anics 51, pp. 477–485, 1972. A summary is found in H. Zhou, On the motion of slender
vortex filaments, Physics of Fluids 9, p. 970-981, 1997. Cited on page 295.
251 V. P. Dmitriyev, Helical waves on a vortex filament, American Journal of Physics 73,
pp. 563–565, 2005, and V. P. Dmitriyev, Mechanical analogy for the wave-particle: helix
on a vortex filament, arxiv.org/abs/quant-ph/0012008. Cited on page 296.
252 T. Jacobson, Thermodynamics of spacetime: the Einstein equation of state, Physical Re-
view Letters 75, pp. 1260–1263, 1995, or arxiv.org/abs/gr-qc/9504004. Cited on page 298.
253 J. Frenkel & T. Kontorowa, Über die Theorie der plastischen Verformung, Physikalis-

Motion Mountain – The Adventure of Physics
che Zeitschrift der Sowietunion 13, pp. 1–10, 1938. F. C. Frank, On the equations
of motion of crystal dislocations, Proceedings of the Physical Society A 62, pp. 131–
134, 1949, J. Eshelby, Uniformly moving dislocations, Proceedings of the Physical
Society A 62, pp. 307–314, 1949. See also G. Leibfried & H. Dietze, Zeitschrift
für Physik 126, p. 790, 1949. A general introduction can be found in A. Seeger &
P. Schiller, Kinks in dislocations lines and their effects in internal friction in crystals,
Physical Acoustics 3A, W. P. Mason, ed., Academic Press, 1966. See also the textbooks by
Frank R. N. Nabarro, Theory of Crystal Dislocations, Oxford University Press, 1967,
or J. P. Hirth & J. Lothe, Theory of Dislocations, McGraw Hills Book Company, 1968.
Cited on page 299.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
254 Enthusiastic introductions into the theoretical aspects of polymers are the books by Alex-
ander Yu. Grosberg & Alexei R. Khokhlov, Statistical Physics of Macromolec-
ules, AIP, 1994, and Pierre-Gilles de Gennes, Scaling Concepts in Polymer Physics,
Cornell University Press, 1979. See also the review by P. -G. de Gennes, Soft matter, Re-
views of Modern Physics 63, p. 645, 1992. Cited on page 300.
255 The master of combining research and enjoyment in mathematics is Louis Kauffman. An
example is his art is the beautiful text Louis H. Kauffman, Knots and Physics, World
Scientific, third edition, 2001. It gives a clear introduction to the mathematics of knots and
their applications. Cited on page 302.
256 A good introduction to knot tabulation is the paper by J. Hoste, M. Thistlethwaite
& J. Weeks, The first 1,701,936 knots, The Mathematical Intelligencer 20, pp. 33–47, 1998.
Cited on page 303.
257 I. Stewart, Game, Set and Math, Penguin Books, 1989, pp. 58–67. Cited on page 303.
258 P. Pieranski, S. Przybyl & A. Stasiak, Tight open knots, arxiv.org/abs/physics/
0103016. No citations.
259 T. Ashton, J. Cantarella, M. Piatek & E. Rawdon, Self-contact sets for 50 tightly
knotted and linked tubes, arxiv.org/abs/math/0508248. Cited on page 304.
260 J. Cantarella, J. H. G. Fu, R. Kusner, J. M. Sullivan & N. C. Wrinkle, Critic-
ality for the Gehring link problem, Geometry and Topology 10, pp. 2055–2116, 2006, preprint
at arxiv.org/abs/math/0402212. Cited on page 305.
261 W. R. Taylor, A deeply knotted protein structure and how it might fold, Nature 406,
pp. 916–919, 2000. Cited on page 307.
408 bibliography

262 Alexei Sossinsky, Nœuds – histoire d’une théorie mathématique, Editions du Seuil,
1999. D. Jensen, Le poisson noué, Pour la science, dossier hors série, pp. 14–15, April 1997.
Cited on page 306.
263 D. M. Raymer & D. E. Smith, Spontaneous knotting of an agitated string, Proceedings
of the National Academy of Sciences (USA) 104, pp. 16432–16437, 2007, or www.pnas.org/
cgi/doi/10.1073/pnas.0611320104. This work won the humorous Ignobel Prize in Physics in
2008; seeimprobable.com/ig. Cited on page 307.
264 A. C. Hirshfeld, Knots and physics: Old wine in new bottles, American Journal of Physics
66, pp. 1060–1066, 1998. Cited on page 307.
265 For some modern knot research, see P. Holdin, R. B. Kusner & A. Stasiak, Quant-
ization of energy and writhe in self-repelling knots, New Journal of Physics 4, pp. 20.1–20.11,
2002. Cited on page 308.
266 H. R. Pruppacher & J. D. Klett, Microphysics of Clouds and Precipitation, Reidel,
1978, pp. 316–319. Falling drops are flattened and look like a pill, due to the interplay
between surface tension and air flow. See also U. Thiele, Weine nicht, wenn der Regen

Motion Mountain – The Adventure of Physics
zerfällt, Physik Journal 8, pp. 16–17, 2009. Cited on page 308.
267 J. J. Socha, Becoming airborne without legs: the kinematics of take-off in a flying
snake, Chrysopelea paradisi, Journal of Experimental Biology 209, pp. 3358–3369, 2006,
J. J. Socha, T. O ’ Dempsey & M. LaBarbera, A three-dimensional kinematic analy-
sis of gliding in a flying snake, Chrysopelea paradisi, Journal of Experimental Biology 208,
pp. 1817–1833, 2005, J. J. Socha & M. LaBarbera, Effects of size and behavior on aerial
performance of two species of flying snakes (Chrysopelea), Journal of Experimental Biology
208, pp. 1835–1847, 2005. A full literature list on flying snakes can be found on the website
www.flyingsnake.org. Cited on page 309.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
268 An informative account of the world of psychokinesis and the paranormal is given by
the famous professional magician James Randi, Flim-flam!, Prometheus Books, Buffalo
1987, as well as in several of his other books. See also the www.randi.org website. No cita-
tions.
269 James Clerk Maxwell, Scientific Papers, 2, p. 244, October 1871. Cited on page 320.
270 A good introduction is C. J. Hogan, Why the universe is just so, Reviews of Modern Phys-
ics 72, pp. 1149–1161, 2000. Most of the material of Table 27 is from the mighty book by
John D. Barrow & Frank J. Tipler, The Anthropic Cosmological Principle, Oxford
University Press, 1986. Discarding unrealistic options is also an interesting pastime. See
for example the reasons why life can only be carbon-based, as explained in the essay by
I. Asimov, The one and only, in his book The Tragedy of the Moon, Doubleday, Garden
City, New York, 1973. Cited on pages 320 and 383.
271 L. Smolin, The fate of black hole singularities and the parameters of the standard models of
particle physics and cosmology, arxiv.org/abs/gr-qc/9404011. Cited on page 322.
272 Aristotle, Treaty of the heaven, III, II, 300 b 8. See Jean-Paul Dumont, Les écoles
présocratiques, Folio Essais, Gallimard, p. 392, 1991. Cited on page 324.
273 Le Système International d’Unités, Bureau International des Poids et Mesures, Pavillon de
Breteuil, Parc de Saint Cloud, 92310 Sèvres, France. All new developments concerning SI
units are published in the journal Metrologia, edited by the same body. Showing the slow
pace of an old institution, the BIPM launched a website only in 1998; it is now reachable at
www.bipm.fr. See also the www.utc.fr/~tthomass/Themes/Unites/index.html website; this
includes the biographies of people who gave their names to various units. The site of its
bibliography 409

British equivalent, www.npl.co.uk/npl/reference, is much better; it provides many details
as well as the English-language version of the SI unit definitions. Cited on page 325.
274 The bible in the field of time measurement is the two-volume work by J. Vanier &
C. Audoin, The Quantum Physics of Atomic Frequency Standards, Adam Hilge, 1989. A
popular account is Tony Jones, Splitting the Second, Institute of Physics Publishing, 2000.
The site opdaf1.obspm.fr/www/lexique.html gives a glossary of terms used in the field.
For precision length measurements, the tools of choice are special lasers, such as mode-
locked lasers and frequency combs. There is a huge literature on these topics. Equally large
is the literature on precision electric current measurements; there is a race going on for the
best way to do this: counting charges or measuring magnetic forces. The issue is still open.
On mass and atomic mass measurements, see page 71 in volume II. On high-precision tem-
perature measurements, see page 548 in volume I. Cited on page 326.
275 The unofficial SI prefixes were first proposed in the 1990s by Jeff K. Aronson of the Uni-
versity of Oxford, and might come into general usage in the future. See New Scientist 144,
p. 81, 3 December 1994. Other, less serious proposals also exist. Cited on page 327.
276 For more details on electromagnetic unit systems, see the standard text by John

Motion Mountain – The Adventure of Physics
David Jackson, Classical Electrodynamics, 3rd edition, Wiley, 1998. Cited on page
330.
277 D.J. Bird & al., Evidence for correlated changes in the spectrum and composition of cosmic
rays at extremely high energies, Physical Review Letters 71, pp. 3401–3404, 1993. Cited on
page 331.
278 P. J. Hakonen, R. T. Vuorinen & J. E. Martikainen, Nuclear antiferromagnetism
in rhodium metal at positive and negative nanokelvin temperatures, Physical Review Letters
70, pp. 2818–2821, 1993. See also his article in Scientific American, January 1994. Cited on
page 331.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
279 A. Zeilinger, The Planck stroll, American Journal of Physics 58, p. 103, 1990. Can you
Challenge 212 e find another similar example? Cited on page 331.
280 An overview of this fascinating work is given by J. H. Taylor, Pulsar timing and relativ-
istic gravity, Philosophical Transactions of the Royal Society, London A 341, pp. 117–134,
1992. Cited on page 331.
281 The most precise clock built in 2004, a caesium fountain clock, had a precision of one
part in 1015 . Higher precision has been predicted to be possible soon, among others
by M. Takamoto, F. -L. Hong, R. Higashi & H. Katori, An optical lattice clock,
Nature 435, pp. 321–324, 2005. Cited on page 331.
282 J. Bergquist, ed., Proceedings of the Fifth Symposium on Frequency Standards and Met-
rology, World Scientific, 1997. Cited on page 331.
283 See the information on D±𝑠 mesons from the particle data group at pdg.web.cern.ch/pdg.
Cited on page 331.
284 About the long life of tantalum 180, see D. Belic & al., Photoactivation of 180 Tam and
its implications for the nucleosynthesis of nature’s rarest naturally occurring isotope, Physical
Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 332.
285 The various concepts are even the topic of a separate international standard, ISO 5725, with
the title Accuracy and precision of measurement methods and results. A good introduction is
John R. Taylor, An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements, 2nd edition, University Science Books, Sausalito, 1997. Cited on page 332.
286 P. J. Mohr, B. N. Taylor & D. B. Newell, CODATA recommended values of the fun-
damental physical constants: 2010, preprint at arxiv.org/abs/1203.5425. This is the set of
410 bibliography

constants resulting from an international adjustment and recommended for international
use by the Committee on Data for Science and Technology (CODATA), a body in the In-
ternational Council of Scientific Unions, which brings together the International Union of
Pure and Applied Physics (IUPAP), the International Union of Pure and Applied Chemistry
(IUPAC) and other organizations. The website of IUPAC is www.iupac.org. Cited on pages
334 and 401.
287 Some of the stories can be found in the text by N. W. Wise, The Values of Precision,
Princeton University Press, 1994. The field of high-precision measurements, from which
the results on these pages stem, is a world on its own. A beautiful introduction to it
is J. D. Fairbanks, B. S. Deaver, C. W. Everitt & P. F. Michaelson, eds., Near
Zero: Frontiers of Physics, Freeman, 1988. Cited on page 334.
288 For details see the well-known astronomical reference, P. Kenneth Seidelmann, Ex-
planatory Supplement to the Astronomical Almanac, 1992. Cited on page 339.
289 See the corresponding reference in the first volume. Cited on page 341.
290 The proton charge radius was determined by measuring the frequency of light emitted by

Motion Mountain – The Adventure of Physics
hydrogen atoms to high precision by T. Udem, A. Huber, B. Gross, J. Reichert,
M. Prevedelli, M. Weitz & T. W. Hausch, Phase-coherent measurement of the hy-
drogen 1S–2S transition frequency with an optical frequency interval divider chain, Physical
Review Letters 79, pp. 2646–2649, 1997. Cited on page 342.
291 For a full list of isotopes, see R. B. Firestone, Table of Isotopes, Eighth Edition, 1999 Up-
date, with CD-ROM, John Wiley & Sons, 1999. For a list of isotopes on the web, see the Corean
website by J. Chang, atom.kaeri.re.kr. For a list of precise isotope masses, see the csnwww.
in2p3.fr website. Cited on pages 343, 346, and 357.
292 The ground state of bismuth 209 was thought to be stable until early 2003. It was
then discovered that it was radioactive, though with a record lifetime, as reported by

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
P. de Marcillac, N. Coron, G. Dambier, J. Leblanc & J. -P. Moalic, Exper-
imental detection of α-particles from the radioactive decay of natural bismuth, Nature 422,
pp. 876–878, 2003. By coincidence, the excited state 83 MeV above the ground state of the
same bismuth 209 nucleus is the shortest known radioactive nuclear state. Cited on page
343.
293 For information on the long life of tantalum 180, see D. Belic & al., Photoactivation of
180
Tam and its implications for the nucleosynthesis of nature’s rarest naturally occurring iso-
tope, Physical Review Letters 83, pp. 5242–5245, 20 December 1999. Cited on page 343.
294 Stephen J. Gould, The Panda’s Thumb, W.W. Norton & Co., 1980. This is one of several
interesting and informative books on evolutionary biology by the best writer in the field.
Cited on page 344.
295 F. Marques & al., Detection of neutron clusters, Physical Review C 65, p. 044006,
2002. Opposite results have been obtained by B. M. Sherrill & C. A. Bertulani,
Proton-tetraneutron elastic scattering, Physical Review C 69, p. 027601, 2004, and D.V.
Aleksandrov & al., Search for resonances in the three- and four-neutron systems in the
7Li(7Li, 11C)3n and 7Li(7Li, 10C)4n reactions, JETP Letters 81, p. 43, 2005. No citations.
296 For a good review, see the article by P. T. Greenland, Antimatter, Contemporary Physics
38, pp. 181–203, 1997. Cited on page 344.
297 Almost everything known about each element and its chemistry can be found in the en-
cyclopaedic Gmelin, Handbuch der anorganischen Chemie, published from 1817 onwards.
There are over 500 volumes, now all published in English under the title Handbook of Inor-
ganic and Organometallic Chemistry, with at least one volume dedicated to each chemical
bibliography 411

element. On the same topic, an incredibly expensive book with an equally bad layout is
Per Enhag, Encyclopedia of the Elements, Wiley–VCH, 2004. Cited on page 346.
298 The atomic masses, as given by IUPAC, can be found in Pure and Applied Chemistry 73,
pp. 667–683, 2001, or on the www.iupac.org website. For an isotope mass list, see the
csnwww.in2p3.fr website. Cited on pages 346, 356, and 357.
299 The metallic, covalent and Van der Waals radii are from Nathaniel W. Alcock, Bond-
ing and Structure, Ellis Horwood, 1999. This text also explains in detail how the radii are
defined and measured. Cited on page 357.
300 M. Flato, P. Sally & G. Zuckerman (editors), Applications of Group Theory in Phys-
ics and Mathematical Physics, Lectures in applied mathematics, volume 21, American Math-
ematical Society, 1985. This interesting and excellent book is well worth reading. Cited on
page 363.
301 For more puzzles, see the excellent book James Tanton, Solve This – Math Activities for
Studends and Clubs, Mathematical Association of America, 2001. Cited on page 363.
302 For an introduction to topology, see for example Mikio Nakahara, Geometry, Topology

Motion Mountain – The Adventure of Physics
and Physics, IOP Publishing, 1990. Cited on page 364.
303 An introduction to the classification theorem is R. Solomon, On finite simple groups and
their classification, Notices of the AMS 42, pp. 231–239, 1995, also available on the web as
www.ams.org/notices/199502/solomon.ps Cited on page 368.
304 A pedagogical explanation is given by C. G. Wohl, Scientist as detective: Luis Alvarez and
the pyramid burial chambers, the JFK assassination, and the end of the dinosaurs, American
Journal of Physics 75, pp. 968–977, 2007. Cited on page 378.

Acknowled gements
Many people who have kept their gift of curiosity alive have helped to make this project come
true. Most of all, Peter Rudolph and Saverio Pascazio have been – present or not – a constant
reference for this project. Fernand Mayné, Ata Masafumi, Roberto Crespi, Serge Pahaut, Luca
Bombelli, Herman Elswijk, Marcel Krijn, Marc de Jong, Martin van der Mark, Kim Jalink, my

Motion Mountain – The Adventure of Physics
parents Peter and Isabella Schiller, Mike van Wijk, Renate Georgi, Paul Tegelaar, Barbara and
Edgar Augel, M. Jamil, Ron Murdock, Carol Pritchard, Richard Hoffman, Stephan Schiller, Franz
Aichinger and, most of all, my wife Britta have all provided valuable advice and encouragement.
Many people have helped with the project and the collection of material. Most useful was the
help of Mikael Johansson, Bruno Barberi Gnecco, Lothar Beyer, the numerous improvements by
Bert Sierra, the detailed suggestions by Claudio Farinati, the many improvements by Eric Shel-
don, the detailed suggestions by Andrew Young, the continuous help and advice of Jonatan Kelu,
the corrections of Elmar Bartel, and in particular the extensive, passionate and conscientious
help of Adrian Kubala.
Important material was provided by Bert Peeters, Anna Wierzbicka, William Beaty, Jim Carr,

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
John Merrit, John Baez, Frank DiFilippo, Jonathan Scott, Jon Thaler, Luca Bombelli, Douglas
Singleton, George McQuarry, Tilman Hausherr, Brian Oberquell, Peer Zalm, Martin van der
Mark, Vladimir Surdin, Julia Simon, Antonio Fermani, Don Page, Stephen Haley, Peter Mayr,
Allan Hayes, Norbert Dragon, Igor Ivanov, Doug Renselle, Wim de Muynck, Steve Carlip, Tom
Bruce, Ryan Budney, Gary Ruben, Chris Hillman, Olivier Glassey, Jochen Greiner, squark, Mar-
tin Hardcastle, Mark Biggar, Pavel Kuzin, Douglas Brebner, Luciano Lombardi, Franco Bagnoli,
Lukas Fabian Moser, Dejan Corovic, Paul Vannoni, John Haber, Saverio Pascazio, Klaus Finken-
zeller, Leo Volin, Jeff Aronson, Roggie Boone, Lawrence Tuppen, Quentin David Jones, Arnaldo
Uguzzoni, Frans van Nieuwpoort, Alan Mahoney, Britta Schiller, Petr Danecek, Ingo Thies, Vi-
taliy Solomatin, Carl Offner, Nuno Proença, Elena Colazingari, Paula Henderson, Daniel Darre,
Wolfgang Rankl, John Heumann, Joseph Kiss, Martha Weiss, Antonio González, Antonio Mar-
tos, André Slabber, Ferdinand Bautista, Zoltán Gácsi, Pat Furrie, Michael Reppisch, Enrico Pasi,
Thomas Köppe, Martin Rivas, Herman Beeksma, Tom Helmond, John Brandes, Vlad Tarko, Na-
dia Murillo, Ciprian Dobra, Romano Perini, Harald van Lintel, Andrea Conti, François Belfort,
Dirk Van de Moortel, Heinrich Neumaier, Jarosław Królikowski, John Dahlman, Fathi Namouni,
Paul Townsend, Sergei Emelin, Freeman Dyson, S.R. Madhu Rao, David Parks, Jürgen Janek,
Daniel Huber, Alfons Buchmann, William Purves, Pietro Redondi, Douglas Singleton, Emil
Akhmedov Damoon Saghian, Zach Joseph Espiritu, Miles Mutka, plus a number of people who
wanted to remain unnamed.
The software tools were refined with extensive help on fonts and typesetting by Michael Zedler
and Achim Blumensath and with the repeated and valuable support of Donald Arseneau; help
came also from Ulrike Fischer, Piet van Oostrum, Gerben Wierda, Klaus Böhncke, Craig Up-
right, Herbert Voss, Andrew Trevorrow, Danie Els, Heiko Oberdiek, Sebastian Rahtz, Don Story,
credits 413

Vincent Darley, Johan Linde, Joseph Hertzlinger, Rick Zaccone, John Warkentin, Ulrich Diez,
Uwe Siart, Will Robertson, Joseph Wright, Enrico Gregorio, Rolf Niepraschk, Alexander Grahn
and Paul Townsend.
The typesetting and book design is due to the professional consulting of Ulrich Dirr. The
typography was much improved with the help of Johannes Küster and his Minion Math font.
The design of the book and its website also owe much to the suggestions and support of my wife
Britta.
I also thank the lawmakers and the taxpayers in Germany, who, in contrast to most other
countries in the world, allow residents to use the local university libraries.
From 2007 to 2011, the electronic edition and distribution of the Motion Mountain text was
generously supported by the Klaus Tschira Foundation.

Film credits
The animation of the actin–myosin system on page 22 is copyright and courtesy by San Diego
State University, Jeff Sale and Roger Sabbadini. It can be found on the website www.sci.sdsu.
edu/movies/actin_myosin.html. The film on page 118, showing single electrons moving through

Motion Mountain – The Adventure of Physics
liquid helium, is copyright and courtesy of Humphrey Maris; it can be found on his website
physics.brown.edu/physics/researchpages/cme/bubble. The film of the spark chamber showing
the cosmic rays on page 176 is courtesy and copyright of Wolfgang Rueckner and Allen Crockett
and found on isites.harvard.edu/icb. The film of the solar flare on page 206 was taken by NASA’s
TRACE satellite, whose website is trace.lmsal.com. The film of a flying snake on page 309 is copy-
right and courtesy by Jake Socha. It can be found on his website www.flyingsnake.org.

Image credits
The photograph of the east side of the Langtang Lirung peak in the Nepalese Himalayas, shown

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
on the front cover, is courtesy and copyright by Kevin Hite and found on his blog thegettingthere.
com. The photograph of a soap bubble on page 14 is courtesy and copyright of Jason Tozer for
Creative Review/Sony; its making is described on creativereview.co.uk/crblog. The whale photo-
graph on page 16 is courtesy by the NOAA in the USA. The photograph of a baobab with elephant
is courtesy and copyright by Ferdinand Reus. The picture of the orchid on page 19 is courtesy
and copyright of Elmar Bartel. The sea urchin photograph on page 23 was taken by Kristina
Yu, and is copyright and courtesy of the Exploratorium, found at www.exploratorium.edu. The
photograph of the piezoelectric motor on page 25 is courtesy and copyright of Piezomotor AB
and is found on their website www.piezomotor.se. The beautiful illustration of ATP synthase on
page 26 is courtesy and copyright of Joachim Weber and is found on his website www.depts.ttu.
edu/chemistry/Faculty/weber. The images of embronic nodes on page 27 are courtesy and copy-
right by Hirokawa Nobutaka and found in the cited paper. The illustration of the pleasure system
in the human brain on page 56 is courtesy of the NIH and can be found, e.g., on the website irp.
drugabuse.gov/NRB/technical/RFCMRI.php. The wonderful table of the elements on page 61 is
courtesy and copyright of Theodore Gray and is for sale at www.theodoregray.com. The image of
the water molecule on page 62 is copyright of Benjah-bmm27 and is found on Wikimedia. The
illustrations of DNA on page 64 are copyright and courtesy by David Deerfield and dedicated
to his memory. They are taken from his website www.psc.edu/~deerfiel/Nucleic_Acid-SciVis.
html. The images of DNA molecules on page 65 are copyright and courtesy by Hans-Werner
Fink and used with permission of Wiley VCH. The beautiful illustration of star sizes on page 68
is courtesy and copyright of Dave Jarvis and found on Wikimedia. The photographs of rocks
on page 70 are copyright and courtesy of Siim Sepp; they can be found on his wonderful web-
site www.sandatlas.org. The photograph of the marble is a detail of the Rape of Proserpina by
414 credits

Gian Lorenzo Bernini and is courtesy of Wikimedia; the marble can be admired in the Gal-
leria Borghese in Rome. The rhenium photographs on page 74 are courtesy of Hans-Werner Fink
and copyright and courtesy of the American Physical Society; they are found at journals.aps.
org/prl/abstract/10.1103/PhysRevLett.52.1532. The pictures of snow flakes on page 74 are courtesy
and copyright of Furukawa Yoshinori; his website is www.lowtem.hokudai.ac.jp/~frkw/index_e.
html. The mineral photographs on page 75 onwards are copyright and courtesy of Rob Lavin-
sky at irocks.com, and taken from his large and beautiful collection there and at www.mindat.
org/photo-49529.html. The photograph of natural corundum on page 76, the malachite photo-
graph on page 173 and the magnetite photograph on page page 175 are copyright and courtesy
of Stephan Wolfsried and taken from his beautiful collection of microcrystals at www.mindat.
org/user-1664.html. The photograph of synthetic corundum on page 77 is copyright of the Mor-
ion Company at www.motioncompany.com and courtesy of Uriah Prichard. The photograph of
Paraiba tourmalines on page page 77 is copyright and courtesy of Manfred Fuchs and found on
his website at www.tourmalinavitalis.de. The photograph of the large corundum single crystal,
usually falsely called ‘sapphire’, is cortesy and copyright of GT Advanced and found on www.
gtat.com. The synthetic crystals photographs on page 78 and later are copyright of Northrop

Motion Mountain – The Adventure of Physics
Grumman and courtesy of Scott Griffin. The alexandrite photographs on page 78 are copyright
and courtesy of John Veevaert and Trinity Mineral Co., trinityminerals.com. The photograph of
PZT – lead zirconium titanate – products on page 79 is copyright and courtesy of Ceramtec. The
photograph of synthetic diamond on page 80 is copyright and courtesy of Diamond Materials
GmbH. The photograph of a diamond knife on page 80 is copyright and courtesy of Diamat-
rix. The photographs of the silicon crystal growing machines and of the resulting silicon crystals
on page 81 are copyright of PVATePla at www.PVATePla.com, and are used by kind permission
of Gert Fisahn. The photograph of hydroxylapatite on page 82 is copyright and courtesy of Ak-
sel Österlöf. The shark teeth photograph on page 82 is courtesy and copyright of Peter Doe and
found at www.flickr.com/photos/peteredin. The photograph of a copper single crystal on page 82

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
is courtesy and copyright of Lachlan Cranswick. The photographs of the natural quasicrystal
and its propeties on page 82 are copyright and courtesy of Luca Bindi and Paul Steinhardt. The
photograph and drawings of a switchable mirror on page 85 are courtesy and copyright Ronald
Griessen and found on his website www.nat.vu.nl/~griessen. The drawing of an Ashby chart on
page 89 is courtesy and copyright of Carol Livermore; it is found in the MIT OpenCourseWare
course materials for 6.777J / 2.372J Design and Fabrication. The insect leg images on page 90
are copyright and courtesy of the Max Planck Gesellschaft and found in the cited article. The
photographs of bone structure on page 92 is courtesy of Peter Fratzl and copyright of him and
Physik Journal. The photograph of a bystander with light and terahertz waves on page 93 are
courtesy of the Jefferson Lab. The climate graph on page 95 is copyright and courtesy of Dieter
Lüthi. The images of crystal growth techniques on page 96 are copyright of the Ivan Golota and
used with his permission and that of Andrey Vesselovsky. The photograph of a Hall probe on
page 97 is courtesy and copyright Metrolab, and found on their website at www.metrolab.com.
The collection of images of single atom sheets on page 98 is taken from Ref. 73, copyright of
the National Academy of Sciences, and are courtesy of the authors and of Tiffany Millerd. The
graphene photograph on page 98 is copyright and courtesy of Andre Geim. The photographs on
page 99 are copyright and courtesy of tapperboy and Diener Electronics. The photograph of a
piece of aerogel on page 100 is courtesy of NASA. The photograph of transistors on page 101 is
copyright and courtesy of Benedikt Seidl and found on Wikimedia; the chip image is copyright
and courtesy of the Ioactive blog at blog.ioactive.com. The photograph of the fountain effect on
page 104 is courtesy and copyright of the Pacific Institute of Theoretical Physics. The photograph
of a vortex lattice on page 105 is copyright of Andre Schirotzek, courtesy of Wolfgang Ketterle,
and found on their research website at cua.mit.edu/ketterle_group. The photograph of a laser
credits 415

delay line on page 109 is courtesy and copyright of the Laser Zentrum Hannover. The photo-
graph of the violin on page 116 is copyright Franz Aichinger and courtesy of EOS. The figure of
the floating display is copyright and courtesy of Burton Inc. and found on www.burton-jp.com.
The photograph of the guide star laser on page 117 is courtesy and copyright of ESO and Babak
Tafreshi. The photograph of the helium gyroscope on page 119 is courtesy and copyright of Eric
Varoquaux. The pictures of rainbows on page 138 are courtesy and copyright of ed g2s, found
on Wikimedia, and of Christophe Afonso, found on his beautiful site www.flickr.com/photos/
chrisafonso21. The picture of neutron interferometers on page 142 is courtesy and copyright of
Helmut Rauch and Erwin Seidl, and found on the website of the Atominstitut in Vienna at www.
ati.ac.at. The simulated view of a black hole on page 145 is copyright and courtesy of Ute Kraus
and can be found on her splendid website www.tempolimit-lichtgeschwindigkeit.de. The map
with the γ ray bursts on page 152 is courtesy of NASA and taken from the BATSE website at
www.batse.msfc.nasa.gov/batse. The baobab photograph on page 160 is copyright and courtesy
of Bernard Gagnon and found on Wikimedia. The photograph of the MRI machine on page 163
is courtesy and copyright of Royal Philips Electronics. The MRI images of the head and the spine
on page 163 are courtesy and copyright of Joseph Hornak and taken from his website www.cis.

Motion Mountain – The Adventure of Physics
rit.edu/htbooks/mri. The image of the birth on page 164is courtesy and copyright of C. Bamberg.
The photograph of the Big European Bubble Chamber on page 168 is courtesy and copyright of
CERN. On page 169, the photograph of the Wilson cloud chamber is courtesy and copyright of
Wiemann Lehrmittel; the picture of α rays taken by Patrick Blackett is courtesy and copyright of
the Royal Society. The electroscope photograph on page 173 is courtesy and copyright of Harald
Chmela and taken from his website www.hcrs.at. The photograph of Vicor Hess on page 174 has
no known copyright. The photograph of the Geigerpod on page 176 is courtesy and copyright of
Joseph Reinhardt, and found on his website flickr.com/photos/javamoose. The photograph of the
Moon shadow on page 177 is courtesy and copyright of CERN and found on cerncourier.com/
cws/article/cern/28658. The photograph of an aurora on page 178 is courtesy and copyright of

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Jan Curtis and taken from his website climate.gi.alaska.edu/Curtis/curtis.html. The photograph
of the spectrometer on page 182 is courtesy and copyright of the Hungarian Academy of Sciences
and found at www.atomki.hu/hekal/muszaki_hatter/ams_en.html. The figure of the Erta Ale vol-
cano on page 185 is courtesy and copyright of Marco Fulle and part of the wonderful photograph
collection at www.stromboli.net. The photograph of the NMR machine on page 193 is courtesy
and copyright of Bruker and found on their website www.bruker.com. The stunning photographs
of the solar corona on page 205 are copyright of Miloslav Druckmüller and his team, including
Peter Aniol, Vojtech Rušin, Ľubomír Klocok, Karel Martišek and Martin Dietzel. The images are
part of the fascinating site www.zam.fme.vutbr.cz/~druck/Eclipse/Index.htm and used here with
the permission of Miloslav Druckmüller. The drawing on nucleic binding energies on page 209is
courtesy and copyright of the Max Planck Institute for Gravitational Physics. The drawing of the
JET reactor on page 211 is courtesy and copyright of EFDA-JET. On page 214, the photograph of
the Crab nebula is courtesy NASA and ESA; the photograph of the Dumbbell nebula is courtesy
and copyright Bill Snyder and was featured on apod.nasa.gov/apod/ap111227.html The graph of
the nuclide abundances on page 216 is courtesy and copyright of Thomas Janka. The photograph
of the Proton Synchroton on page 221 is copyright and courtesy of CERN. The drawing of the
strong coupling constant on page page 232 is courtesy and copyright of Siegfried Bethke. The
photograph of the quark jet on page 236 is courtesy and copyright of DESY. On page 241, the
photograph of the watch is courtesy and copyright of Traser and found at www.traser.com; the
spectrum is courtesy of the Katrin collaboration. The photographs of the neutrino collision on
page 243 and of the W and Z boson experiments on page 244 and page 245 are courtesy and copy-
right ofCERN. On page 249, the photograph of the experimental set-up is copyright and cour-
tesy of the Brookhaven National Laboratory; the measurement result is copyright and courtesy
416 credits

of the Nobel Foundation. The photograph of the Sudbury Neutrino Observatory on page 257 is
courtesy and copyright SNO, and found on their website at www.sno.phy.queensu.ca/sno/images.
The unification graph on page 271 is courtesy and copyright of W. de Boer and taken from his
home page at www-ekp.physik.uni-karlsruhe.de/~deboer. The graph on unification on page 276
is courtesy and copyright of CERN. The picture of the butterfly in a wind tunnel on page 279 is
courtesy and copyright of Robert Srygley and Adrian Thomas. On page 281, the picture of the
vulture is courtesy and copyright of S.L. Brown from the website SLBrownPhoto.com; the pic-
ture of the hummingbird is courtesy and copyright of the Pennsylvania Game Commission and
Joe Kosack and taken from the website www.pgc.state.pa.us; the picture of the dragonfly is cour-
tesy and copyright of nobodythere and found on his website xavier.favre2.free.fr/coppermine.
The pictures of feathers insects on page 282 are material used with kind permission of Hort-
NET, a product of The Horticulture and Food Research Institute of New Zealand. The figures on
page 283 and page 284 are copyright of L. Mahadevan and courtesy of Macmillan. The picture
of the scallop on page 285 is courtesy and copyright of Dave Colwell. The historical cat photo-
graphs on page 288 are by Etienne-Jules Marey and have no copyright. The picture of cliff diver
Artem Silchenko on page 288 is copyright of the World High Diving Federation, found at the web

Motion Mountain – The Adventure of Physics
site www.whdf.com, and courtesy of Frederic Weill. The picture of the eversion of the sphere on
page 292 is courtesy and copyright of John Sullivan and found on his Optiverse website on new.
math.uiuc.edu/optiverse. The photograph of a waterspout on page 294 is copyright and cour-
tesy of Zé Nogueira and found on his website at www.flickr.com/photos/zenog. The drawing of
a mixed dislocation on page 298 has been produced especially for this text and is copyright and
courtesy of Ulrich Kolberg. The knot and link diagrams on pages 302 and 306 are courtesy and
copyright of Robert Scharein and taken from his website on www.knotplot.com. The images of
the tight knots on page 304 are copyright and courtesy of Piotr Pieranski. The clasp images on
page 305 is courtesy and copyright of Jason Cantarella. The photograph of a hagfish on page 306 is
courtesy and copyright of Christine Ortlepp; it is also found on the web page www.lemonodor.

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
com/scruz-2003-01-24/17m.shtml. The disordered knot images on page 306 are courtesy and
copyright of PNAS. The MRI image on page 399 of a married couple is courtesy and copyright of
Willibrord Weijmar Schultz. The photograph on the back cover, of a basilisk running over wa-
ter, is courtesy and copyright by the Belgian group TERRA vzw and found on their website www.
terravzw.org. All drawings are copyright by Christoph Schiller. If you suspect that your copyright
is not correctly given or obtained, this has not been done on purpose; please contact me in this
case.
SU B J E C T I N DE X

Symbols agate 75 antihydrogen
*-algebra 359 age determination 182–184 properties 343
MRI, dangers of 193 ageing 42 antiknot 303
α decay 180 aging 36 antimony 347

Motion Mountain – The Adventure of Physics
α particle 180 aircraft antiscreening 233
α particles 172 why does it fly? 280 antisymmetry 361
β decay 180 alexandrite 78 apes 210
β particle 180 AlGaAs laser 113 aphelion 339
γ decay 181 algebra 358, 359 apogee 338
γ particle 181 algebra, linear 359 apple 344
alkali metals 59, 346 APS 414
A alkaline earth metals 346 Arabidopsis 43
α-ray dating 184 Allen belt, van 197 Archilochus colubris 281
acceleration alpha decay see 𝛼 decay argon 185, 347

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Planck 329 definition 180 Armillaria mellea 110
accelerator mass spectroscopy alpha particle see 𝛼 particle arsenic 347
184 alpha rays see 𝛼 rays Ashby chart 88, 89
accuracy 332 alumina 76 associative algebra 359
limits to 333 aluminium 346 astatine 347
aces 219 aluminium amalgam 66 astronaut see cosmonaut
Acetabularia 43 Alzheimer patients 67 astronomical unit 339
acne 33 Alzheimer’s disease 67 astronomy 262
actin 23 amalgam 351 asymmetry
actinium 346 americium 346 right-left of human body
actinoids 346 amethyst 75 27
action amoeba 287 asymptotic freedom 232, 233
Planck 329 amount of radioactive atmosphere
action, quantum of, ℏ material 193 pressure 338
physics and 8 ampere atom
Adansonia grandidieri 160 definition 325 discovery of its structure
adenosine triphosphate 23, 25 amphiboles 72 169
adjoint representation 362 Anagrus 282 falling 140
aerodynamics 280 anapole moment 403 atom interferometers 143
aerogels 99 angels 312 atomic 331
aeroplane angler fish 110 atomic mass unit 263, 337
why does it fly? 280 angular momentum 314 atomic number 345
aeroplane, model 279 anti-atoms 344 atomic radius 357
418 subject index

atoms bats 41 black holes die 151
and elementary particles battery black-hole temperature 149
262 using the weak interaction blood 193
history of 213 256 BN 97
atoms and reproduction 19 BCS theory 392 body
atoms and swimming 282 beauty 264 human, asymmetry of 27
atoms are rare 179 beauty quark 223 Bohr atom, gravitational 143
atoms, matter is not made of becquerel 327 Bohr magneton 336
179 becquerel (SI unit) 193 Bohr radius 336
ATP 23, 25, 343 beech, fighting 36 bohrium 347
ATP consumption of beer 114 Boltzmann constant 264
A molecular motors 22
ATP synthase 22
being, living 15
Bekenstein–Hawking
Boltzmann constant 𝑘 334
physics and 8
structure of 26 temperature 149 bomb
atoms ATP synthase 26 beliefs 322 Hiroshima 194
atto 327 BeppoSAX satellite 152 bomb, nuclear 191

Motion Mountain – The Adventure of Physics
aurora australis 179 berkelium 347 bombs 153
aurora borealis 179 beryllium 347 in nature 217
aurora, artificial 197 beta decay see 𝛽 decay Bombus terrestris 279
aurum 350 definition 180 bond
autoradiography 196 beta particle see electron angle of 62
Avogadro’s number 334 beta rays see 𝛽 rays chemical, illustration of 61,
awe 323 Bethe-Weizsäcker cycle 208, 62
209 chemical, measurement of
B Bikini atoll 217 63
β decay 240 biology 262 bonds, chemical 59

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
bacteria 33, 46 biomass bones
gliding 285 of species 36 seeing own 197
number of 33 biotite 72 books 99
bacterium BIPM 325 boron 347
lifetime 31 bird Bos taurus 36
swimming 284 navel 31 Bose–Einsein condensation
badminton 88 birds 206 315
balsa wood 88 birth Bose–Einstein condensate
Banach–Tarski paradox or video of 164 105, 108
theorem 134 bismuth 347, 410 bosonization 405
bananas, knotted 305 properties 343 bottom quark 223, 263
barium 347 Bi2 Sr2 CaCu2 Ox 97 mass 335
baryon bit bottomness 264
diagram 221 to entropy conversion 337 box tightness 90
observed number of 317 bitartrate 75 braid 305
baryon number black body radiation constant brain 41
definition 189 150 and molecular motors 25
baryon number density 340 black hole 69, 146 size, in whales 41
baryon table 345 illustration of 145 brain’s interval timer 44
baryons 222 black hole observations 151 brain, clock in 44
base units 325 black hole radiation 147 branching ratios 229
basis 361 black holes 322 Bridgmanite 71
bath, vacuum as 131 black holes are born 151 bromine 347
subject index 419

Bronshtein cube 8 CERN 174, 242, 256, 273, 331 climate change 93
Brownian motors 24 CGPM 326 cloak of invisibility 86
bubble chamber 167 chain reaction 189 clock
bulb in everyday life 195 biological 42–44
light, scams 114–115 in fission 189 does not exist 45
bumblebee 279 in nuclear devices 191 living 42–44
Bureau International des chalkogens 346 clock in brain 44
Poids et Mesures 325 challenge clock oscillator 46
Burgers vector 299 classification 9 clocks 44, 52
butterfly 278 change clone
quantum of, precise value human 30
B C
C violation 248
264, 334
charge
clothes
see through 390
C*-algebra 360 elementary 𝑒, physics and clothes, seeing through 91
Bronshtein Cabibbo angle 251 8 cloud chamber 167
Cabibbo–Kobayashi– positron or electron, value clouds 293, 357

Motion Mountain – The Adventure of Physics
Maskawa mixing matrix of 264, 334 cluster emission 181
254 charge conjugation violation CNO cycle 208, 209
cadmium 348 248 cobalt 348
caesium 348 charged weak current CODATA 410
caesium and parity interaction 254 CODATA 401
non-conservation 247 charm quark 223, 263 coefficient of local
calcium 348 mass 335 self-induction 295
californium 348 chemistry 262 coherent 114
candela Chernobyl disaster 194 cold working 80
definition 326 Chew-Frautschi plots 230, 231 Coleman–Mandula theorem

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
candle 109 chiral symmetry 225 272
candy floss 169 chirality in nature 307 colour 263
capsaicin 40, 332 chirality of knots in nature as source of the strong
carbon 348 307 interaction 227
properties 343 Chlamydomonas 286 strong charge 223
cardiopulmonary Chlamys 285 unknown origin of 323
resuscitation 36 chlorine 348 colour, evidence for three 228
Cartan metric tensor 362 chloroplasts 25 Commission Internationale
Casimir effect 124, 314 cholera 22 des Poids et Mesures 325
cat chromium 348 commutative 361
cloned 30 chromosome 31, 343 commutator 361
falling 287 chrysoberyl 78 compact discs 17
square 289 Chrysopelea paradisii 309 compactness 370
Cathartes aura 281 cilia 286 completeness 360
cats 38 cilia, nodal 29 complex Lie group 369
cell 344 citrine 75 complex numbers 360
first biological 34 CKM matrix 251 Compton Gamma Ray
cell motility 22 clasp 305 Observatory 152
centi 327 classical 362 Compton wavelength 314, 336
centre 361 classifications conductance 133
Cepheids 217 in biology 32 conductance quantum 336
ceramics 76 Clay Mathematics Institute conduction electrons 87
cerium 348 375 conductivity, electrical 315
420 subject index

cones 38 cosmonaut 119 dating, radiometric 182–184
Conférence Générale des and body rotation 289 day
Poids et Mesures 325 and cosmic rays 178 length measurement by
confinement of quarks 224, eye flashes 179 plants 56
231 lifetime of 178 sidereal 338
conformal field theory 268 coulomb 327 time unit 327
conformal symmetry 225, 273 Coulomb explosion 137 death 55, 86
Conférence Générale des counter 42 deca 327
Poids et Mesures 326 coupling constant unification decay 47, 48
connected manifold 365 270 as nuclear motion 188
consciousness 51 cows, ruminating 259 decay time
C definition 51
constants
CP violation 248
CPT invariance 250
definition 180
decay, alpha see 𝛼 decay
table of astronomical 338 creation decay, beta see 𝛽 decay
cones table of basic physical 264, none in nature 322 deci 327
334 creation of light 113 degenerate matter 186

Motion Mountain – The Adventure of Physics
table of cosmological 340 cromosome, human Y 343 degree
table of derived physical crystal angle unit 327
336 face formation 74 degree Celsius 327
constituent quark mass 233 formation of 72 delusion
continuity 364 maximum density 73 about unification 320
Convention du Mètre 325 maximum entropy 73 Demodex brevis 33
Conway groups 369 virus 377 density
Cooper pairs 87, 103 crystal database 82 Planck 329
copernicium 348 crystal shapes 82 deoxyribonucleic acid 62
copper 80, 349 crystallization dating 184 Desmodium gyrans 43

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
copper single crystal 80 crystals 71–82 Desoxyribonukleinsäure 62
copycat 30 Cu 80 deuterium 211
core 58 cube deviation
cork 88 Bronshtein 8 standard, illustration 333
corona 202, 208 physics 8 devils 312
photograph of solar 205 the physics 318 dextrose 66
temperature of 208 cumulonimbus 293 diamond 79, 269
corrected Planck units 330 cuprum 349 from moss 100
corundum 76 curie (unit) 194 harder than 100
cosmic radiation 174 curium 349 diamonds 93
neutrinos 256 current Dicomorpha 282
cosmic rays 153, 174 Planck 329 diffeomorphism
composition 175, 177 current quark mass 233 definition 367
cosmonauts and 178 curve difference
danger for cosmonauts 178 closed time-like 159 man and chimpanzee 31
discovery 172 cyclotron frequency 336 differential manifold 365
evolution and 179 diffusion 287
extragalactic origin 178 D digital versatile discs, or DVD
lightning 178 daemons 312 17
types of 175 dangers of MRI 193 dimension 361
cosmological constant 340 dark energy 317 dimensionless 336
cosmological constant Λ darmstadtium 349 Dirac equation 253
as millennium issue 317 dating, radiocarbon 184 and Sokolov–Ternov effect
subject index 421

147 flattening 338 lack of 252
Dirac equation and chemistry gravitational length 338 element, adjoint 360
58 mass 338 elementary particle
disclinations 86, 88 normal gravity 338 properties 262
dislocation loop 298 radius 338 table 262
dislocations 86, 88, 299 rotation, and superfluidity elementary particles, electric
distribution 120 polarizability 344
Gaussian 332 snowball 391 elements 344, 345
normal 332 earthquakes 83 embryo 22
division algebra 360 echo 128 emission
DNA eddies 280 spontaneous 125
D and genes 53
illustrations of 64
edge dislocations 299
effective width 299
emotions
inspired by quantum field
images of 65 egg theory 320
Dirac DNA 62, 63 picture of 23 Encarsia 282
DNA molecules 307 Eiffel tower 88 energy

Motion Mountain – The Adventure of Physics
DNS 62 eigengrau 53 Planck 329
dolphins 41, 282 eigenvalue 359 energy of the universe 145
donate eigenvector 359 energy width 263
to this book 10 einsteinium 349 engineering 262
dopamine 54 electric field, critical 133 entity
dose electrical conductance 133 wobbly 309, 310
radioactive 193 electricity entropy
down quark 223, 263 solar storms and 208 Planck 329
mass 335 electrification 137 to bit conversion 337
Drosophila bifurca 286 electromagnetic unit system entropy, state of highest 154

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
Drosophila melanogaster 43, 330 enzymes 349
278, 286 electrometer 172 erbium 349
duality 273–274 electron 242, 262 error
electromagnetic 273 classical radius 336 in measurements 332
duality of coupling constants filming a single 117 random 332
268 g-factor 337 relative 332
dubnium 349 magnetic moment 336 systematic 332
Duckburg 24 mass 335 total 332
DVD 17 electron holes 87 Erta Ale 185
dyadosphere 153 electron neutrino 242 Escherichia coli 40, 286
DyI3 109 electron radius 264 etanercept 67
dynamical Casimir effect 129 electron volt ethene 36, 56
dyons 275 value 337 Ethiopia 185
dysprosium 349 electron, weak charge 246 Euler characteristic 366
electrons 84 Euphasia superba 36
E electronvolt 330 europium 349
E. coli 344 electroscope 172 evaporation 155
ear 38 electrostatic unit system 330 Evarcha arcuata 90
Earth electroweak coupling 252 eversion 292
age 338 electroweak interaction evolution 46
age of 184 does not exist 252 biological 29
average density 338 electroweak mixing 252 three principles of 29
equatorial radius 338 electroweak unification tree of 32
422 subject index

Exa 327 fire 195 francium 349
exciton 87 fire tornados 295 properties 343
explosions 153 fire whirls 295 Franz Aichinger 412
exposure firefly 110 fraud 346
radiation 194 fireworks 195 free energy 301
extension sensors 40 Fischer groups 369 freedom
extraterrestrial life 34 fish asymptotic 275
extraterrestrials 34 and propellers 283 fruit flies 278
eye 38 fission fruit fly 17
eye sensitivity 38 nuclear 188 full width at half maximum
Sun and 200 332
E F
F. spectabilis 43
flagella
prokaryote 286, 406
Fulling–Davies–Unruh effect
129, 146, 298
F. suspensa 43 flagellum 286 Fulling–Davies–Unruh
Exa F. viridissima 43 flavour symmetry 225 radiation 157
faeces 52 flavours 223 fundamental group 375

Motion Mountain – The Adventure of Physics
farad 327 flerovium 349 fur 29
Faraday’s constant 336 floor fusion
Fe stability of 67 challenge of confined 213
fission and 208 flow confined 211
fusion and 208 nodal 29 in stars 208
feathers 88 turbulent 281 inside Sun 200
femto 327 flower stems 305 reactors 211
femtosecond laser 115 fluctuations
Fermi constant 252 zero-point 124 G
Fermi coupling constant 265, fluorine 349 γ ray burst 152

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
334 fly G-parity 263
fermion, composite 107 common Musca domestica GABA 54
fermium 349 18 gadolinium 349
ferromagnetism 315 flying systems 278 gait
ferrum 350 foam undulatory swimming 283
Feynman diagram 135 as origin of life 34 galaxies as clouds 293
field of scalars 359 food quality 192 Galileo Galilei 140
field theory, conformal 268 football, hairy 375 gallium 349
Fields medal 292 force gamma ray burst 152
figure-eight 303 entropic 73 gamma ray bursts 397
fine structure constant 107, van der Waals, at feet of gamma-ray burst
127 geckos and spiders 90 locations of 152
graphene and 98 formula of life 346 gamma-ray bursts 153, 397
limits number of chemical formulae 134 GaN laser 113
elements 133 Forsythia europaea 43 garlic-smelling
rainbow and 138 Foucault pendulum 120 semiconductor 355
fine structure constant, limit fountain effect 104 garnet 77
on 134 foxfire 110 gas constant, universal 336
fine tuning 322 fraction gauge
fine-structure constant 264, brittle 88 symmetry 317
265, 328, 334, 335 ductile 88 gauge groups 268
finger print 67 fractional quantum Hall effect gauge symmetry 369
and radioactivity 196 107 gauge theory
subject index 423

and shape change 289 graphene 97, 98, 108 hassium 350
from falling cats 287 grasshopper 17 Hausdorff space 365
gauge transformations 371 gravitational Bohr radius 143 heart
gauge-dependent 290 gravitational constant 264 position 27
gauge-invariant 290 geocentric 338 heat capacity of diatomic gas
Gaussian distribution 332 heliocentric 338 315
Gaussian unit system 330 gravitational constant 𝐺 334 heat capacity of metals 315
gecko 90 physics and 8 Heaviside–Lorentz unit
Geiger–Müller counters 174 gravitational coupling system 330
Geigerpod 176 constant 264 heavy ion emission 181
Gell-Mann matrices 227 graviton hecto 327
G gemstones 75
general relativity
definition of 158
gravity measurement with a
helioseismology 210, 217
helium 120, 320, 321, 343, 350
millennium issues and thermometer 147 helium burning 210
gauge 317–318 gray 327 helix 297
open quastions 316 gray (SI unit) 193 hell

Motion Mountain – The Adventure of Physics
generators 361 greenhouse effect 93 hotness of 184
genes 22, 53 group of components 375 henry 327
geologist 69 group, monster 369 hertz 327
geology 69, 262 group, simple 368 Higgs 263
geosmin 56 groups Higgs boson 253, 258
germanium 350 gauge and Lie 268 Higgs mass 335
ghosts 312 growth Hiroshima bomb 194
Giant’s Causeway 71 in living beings 16 history
giant, red 68 growth of trees 34 of matter 200
Giga 327 growth rings 99 HoI3 109

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
global warming 93 GUT 269 holes in manifolds 368
glucose 66 gypsum 75 holmium 350
glueball 236, 343 gyromagnetic ratio of the homeomorphism
definition 237 electron 315 definition 367
gluinos 272 Homo sapiens 36, 43
gluon 262, 335 H horizon 143
absorption 224 hadron hormones 63
definition 224 large number of 222 hornblende 72
emission 224 hadrons 222 hour 327
scattering 224 hafnium 350 Hubble parameter 340
gluon jets 236 hagfish 306 human
goblin 348, 351 hahnium 349 properties 344
god 356 half-life human energy consumption
goddess 312, 348, 352, 356 definition 180 199
gods 312 relation to lifetime 263 hummingbirds 281
gold 80, 99, 350 Hall effect 95, 380 hydrargyrum 351
gold foil experiment 167 Hall effect, fractional hydrodynamics 279
golden rule 48 quantum 107 hydrogen 350
grand unification 268–271 Hall effect, phonon 97 properties 343
grand unified theory 269 Hall effect, photonic 96 hydrogen–hydrogen cycle
grandmother: a hard problem Hall probes 95 201, 208
304 halogens 346 hydroxylapatite 79
grape sugar 66 handcuff puzzle 363 Hypericum 67
424 subject index

hypernova 153 IUPAC 410 krypton 350
IUPAC 411
I IUPAP 410 L
ideal 361 lady’s dress 292
igneous rocks 69 J Lagrangian,QED 126
ignition 212 Jacobi identity 361 Lamb shift 125–126, 133, 315
illusion Janko groups 369 lamp 108
of motion 39 Jarlskog invariant 251, 265, 335 lamp, ideal 110
imaging JET, Joint European Torus 211 lamps 109
magnetic resonance 51, 162 Joint European Torus 211 gas discharge 108
indium 350 Josephson constant 315 incandescent 108
H infinite-dimensional 363
infrasound 41
Josephson frequency ratio 336
joule 327
recombination 108
lamps, sodium 110
InGaAsP laser 113 junk DNA 53 lamps, xenon 110
hypernova ink fish 286 Jupiter 68 land mines, detection of 85
insects 279 properties 338 lanthanoids 346

Motion Mountain – The Adventure of Physics
inside 167 lanthanum 350
instanton 275 K large number hypothesis 144
interaction kaon 172 laser 114, 315
strong nuclear 200 properties 342 list of types 109
weak 240–260 Karachay, Lake 35, 195 laser mosquito killers 115
weak, curiosities 255 kefir grains 30 laser sword 130
weak, summary 259 kelvin laser umbrella 115
interference 130 definition 325 laser weapon 111
interferometer, neutron 142 Killing form 362 laser, CO2 111
interferometers 142 kilo 327 laser, argon 110

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
internal conversion 181 kilogram laser, beer 112
International Astronomical definition 325 laser, cadmium 111
Union 339 kilonova 215 laser, copper 111
International Geodesic Union kinesin 25 laser, gold 111
339 KJ 66 laser, helium-neon 110
intrinsic properties 312 Klein bottles 366 laser, krypton 111
invariant, link 307 Klitzing, von – constant 315, laser, lead salt 113
invariant, topological 307 336 laser, nitrogen 111
inversion 293 knot laser, quantum cascade 113
invisibility 85–86 and particles 309 laser, semiconductor 113
invisibility cloak 85 in plants 305 laser, vodka 112
involution 360 no in octopus arms 305 laser, water 111
iodine 350 tight laser, xenon 111
ion channel 54 illustration of 304 latex 88
ionic radii 357 Knot Atlas 302 lattice QCD 231
iridium 350 knot fish 306 lattice gauge theory 231
iron 350 knot invariants 303 lava 184
fission and 208 knot problem, simplest 304 radioactivity of 185
fusion and 208 knot shapes 304 lawrencium 350
isomeric transition 181 knot theory 137 Lawson criterion 212
isotope knot, mathematical 302 lead 350
definition 172 KnotPlot 302 from Roman times 196
isotopes 346, 410 knotted protein 307 radioactivity of natural 196
subject index 425

learning linear motors 22 manifolds 363
best method for 9 link 305 manta 286
without markers 9 classification 305 marble 69
without screens 9 links, long 305 marker
length lipoid pneumonia 64 bad for learning 9
Planck 329 liquid crystals, colours in 95 Mars 88
lepton number lithium 213, 350, 357 Mars trip 178
definition 189 litre 327 masers 114
levitation livermorium 351 mass
neutron 171 living being 15 Planck 329
lie construction plan 18 mass ratio
L on invisibility 86
Lie algebra 361, 371
definition of 17
living thing, heaviest 344
muon–electron 337
neutron–electron 337
Lie algebra, localization neutron–proton 337
learning finite-dimensional 362 limits to particle 313 proton–electron 337
Lie algebra, solvable 362 looking through matter 83 materials science 262

Motion Mountain – The Adventure of Physics
Lie algebras 371 Lorentz group 370 Mathieu groups 368
Lie group 369 Loschmidt’s number 336 matter
Lie group, compactness of 370 lotus effect 99 birth of 200
Lie group, connectedness of love history of 213
370 romantic 55 looking through 83
Lie group, linear 370 love, making 164 matter is not made of atoms
Lie groups 268 lumen 327 179
Lie multiplication 361 lung cancer 195 matter, composite 342
lie, biggest in the world 258 lutetium 351 Mauna Kea 88
life 15, 19 lux 327 maximal ideal 361

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
definition of 17 Maxwell equations 254
life time M Mayak 195
definition 180 machine measurement
life’s basic processes 18 definition of 19 comparison 328
life’s chemical formula 346 molecular 21 definition 325, 328
lifetime machines, quantum 19 error definition 332
relation to half-life 263 magma irreversibility 328
lifetime, atomic 315 radioactivity of 185 meaning 328
light magmatites 71 process 328
speed inside the Sun 203 magnesium 351 medicine 262
light bulb magnetic charge 273 holistic 34
scams 114–115 magnetic domain walls 87 medicines 63
light can hit light 130 magnetic field, critical 133 Mega 327
light emitting diodes 110 magnetic flux quantum 336 meitnerium 351
light swords 130 magnetic monopoles 275 Melanophila acuminata 41
light year 338, 339 magnetic resonance imaging memory
lightning 51, 84, 162 and reproduction 19
cosmic rays and 178 magneton, nuclear 337 Mendel’s ‘laws’ of heredity 18
lightning rods, laser 115 magnons 87 mendelevium 351
limit, definition of 365 manganese 351 menthol 40
limits manifold 365 mercury 66, 351
to precision 333 analytic 369 mercury lamps 109
line 303 manifold, connected 365 meson
426 subject index

diagram 222 of quarks 251 molecular 21
meson table 345 mixing matrix ultrasound 25
mesons 222 CKM quark 265, 335 motors, molecular 19
metabolism 18 PMNS neutrino 252, 265, MRI 162
metacentric height 122 335 multiverse 322
metal halogenide lamps 109 MnO 108 muon 172, 242, 263
metals 346 mobile g-factor 337
heavy 346 neurochemical 55 muon magnetic moment 337
transition 346 moduli space 273 muon mass 335
metamaterials 86 molar volume 336 muon neutrino 242, 263
metamorphic rocks 69 molecular motors 23 muons 84
M metamorphites 71
metastability 180
molecular pumps 22
molecule
Musca domestica 43, 279
muscle
metre mirror 31 working of 21
meson definition 325 most tenuous 66 muscle motion 21
micro 327 molecule size 315 muscovite 72

Motion Mountain – The Adventure of Physics
microorganism molybdenum 351 music and mathematics 277
swimming 285–287 momentum mycoplasmas 46
microwave background Planck 329 myosin 23
temperature 340 monopole, magnetic 269 Myxine glutinosa 306
migration 206 Monster group 369 Möbius strip 365
mile 328 monster group 369
military 151 Moon N
Milky Way density 338 Na 109, 110
age 339 properties 338 NaI 109
mass 339 MoS2 97 nano 327

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
size 339 moscovium 351 NASA 119
millennium list 316 moss 100 natural unit 336
of open issues 316 motion navel
milli 327 and measurement units in birds 31
mineral 326 NbSe2 97
rock-forming 71 as illusion 52 neighbourhood 364
minerals 71 is fundamental 326 Nelumbo nucifera 99
mines, detection of 85 reasons for existence 51 neodymium 351
miniaturization reasons for observability 51 neon 351
feats of living beings 17 symmetry 31 Neonothopanus gardneri 110
Minion Math font 413 through strokes 289 neptunium 351
minute 327 with limbs 31 nerve cell
definition 339 wobbly 309, 310 blue whale 344
mirror 147 motion inversion violation neurology 51
accelerated 129 250 neurosciences 262
light emission from 129 Motion Mountain Neurospora crassa 43
mirror and source aims of book series 7 neurotransmitters
motion of 128 helping the project 10 important types 54
mirror molecules 31 supporting the project 10 neutral weak current
mirrors 124, 128 top of 324 interaction 254
mitochondria 25 motor neutrinium 258
mixing ciliary 30 neutrino 196, 258
of neutrinos 251 linear, film of 22 atmospheric 256
subject index 427

cosmic 256 nova 215 takes time 49
Earth 256 nuclear magnetic resonance ocean floors 71
fossil 256 162, 164 octonions 360
man-made 256 nuclear magneton 337 octopus
masses 335 nuclear motion and knots 305
PMNS mixing matrix 252, bound 188 Oganesson 343
265, 335 nuclear physics 162 oganesson 352
prediction 241 nuclear reaction 238 ohm 327
solar 256 nuclear reactor oil tanker 169
neutrino flux on Earth 256 as power plant 196 Oklo 196
neutrino mixing natural 196 olivine 72
N definition 251
neutrino oscillations 257
nuclei
history of 200
Olympus mons 88
omega
neutrino, electron 263 nucleon properties 343
neu trino neutrinos 84, 396 definition 172 one million dollar prize 375
neutron is composed 219 one-body problem 134

Motion Mountain – The Adventure of Physics
Compton wavelength 337 nucleosynthesis 213–216 onyx 75
is composed 219 primordial 213 open questions
levitation 171 nucleus in quantum theory and
magnetic moment 337 colour of 186 general relativity 316
mass 337 discovery of 169 open questions in QED 137
properties 171, 343 fission 188 open set 364
quark content 222 free motion of 172 opiorphin 54
neutron capture 208, 213 in cosmic rays 172 optical black holes 151
neutron emission 181 is usually composed 169 optical coherence
neutron interferomtry 142 mass limit 185 tomography 115

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
neutron mass 233 shape of 186–187 orbifold 368
neutron star 68, 185–186 shape oscillations 199 orchid 19
size of 186 size of 164 ore-formers 346
neutron star mergers 215 spin 172 organelles 22
neutron stars 210 spin of 164 orientation sensors 40
neutron trap 171 strong force in 172 orthoclase 72
neutron, magnetic moment tranformation with lasers oscillator 42
227 259 osmium 352
neutrons 84, 170 nucleus accumbens 54, 55 ovule
and table tennis 140–142 nuclide picture of 23
newton 327 definition 172 oxygen 352
nickel 351 nuclides 343
nihonium 351 nymphs 312 P
niobium 352 P violation 245
nitrogen 352 O p–p cycle 201, 208
NMR 162 oak, fighting 36 packing of spheres 73
Nobel Prizes, scientists with object paddle wheel 26
two 103 wobbly 309, 310 pain sensors 40
nobelium 352 object, full list of properties pair creation 314
noble gases 59, 346 262 palladium 352
node, on embryo 28 objects are made of particles paraffin
non-singular matrix 370 311 dangers of 64
nose 40 observation Paramecium 286
428 subject index

parameter space 273 314 pleasure system 54
parity 263 definition 131 illustration of 56
parity violation 245, 246 Peta 327 plumbum 350
parity violation in electrons PETRA collider 236 plutonium 195, 353
246 phanerophyte, monopodal 33 Poincaré algebra 363
parsec 338 phase of wave function in points 364
particle gravity 142 poise 281
elementary, definition 312 pheasants 280 poisons 63
limit to localization 313 Philips 124 polaritons 87
transformation 240 phonon Hall effect 97 polarons 87
virtual, and Lamb shift 125 phonons 86 poliomyelitis 259
P zoo 220
Particle Data Group 401
phosphorus 352
photino 272
pollen 282
polonium 64, 195, 353
particle pairs photoacoustic effect 378 polymer
parameter virtual 130 photon 254, 262 electroactive 33
particle reactions 313 hitting photon 130 polymer, DNA as 62

Motion Mountain – The Adventure of Physics
particle transformations 313 mass 335 Pontecorvo–Maki–
particle, alpha see 𝛼 particle number density 340 Nakagawa–Sakata mixing
particle, beta see electron photon hall effect 392 matrix 251, 254
particle, virtual photon-photon scattering 315 positron charge
in nuclear physics 188 photonic Hall effect 96 specific 336
particles, virtual 122 photoperiodism 43, 56 value of 264, 334
pascal 327 physics positron tomography 51
Pauli pressure 68, 210 map of 8 positrons 84
Pauli spin matrices 225 mathematical, limits of 277 potassium 185, 353
Pauli’s exclusion principle 67 nuclear 162 potatoes irradiation 198

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
PbNO 66 physics cube 8, 318 praeseodymium 353
PbI 378 phytochrome system 57 precision 332
pencil 375 pico 327 limits to 333
pencils 98 pigeons 41 predictions
Penrose process 153 pigs 210 difficulties for 46
pentaquarks 237, 403 pion 172 prefixes 327, 409
people 210 properties 342 SI, table 327
perception research 38 plagioclase 72 prefixes, SI 327
perigee 338 Planaria 31 present
perihelion 339 Planck constant takes time 49
periodic table 344 value of 264, 334 Zeno and the absence of
with videos 59 Planck length 128 the 49
periodic table of the elements Planck stroll 331 pressure, negative
59, 345 Planck time 144 in trees 117
permeability Planck units primal scream of a black hole
vacuum 336 as limits 329 153
permeability, vacuum 264 table of 329 prime knots 303
permittivity Planck units, corrected 330 principal quantum number 58
vacuum 336 Planck’s natural units 328 principle
permittivity, vacuum 264 plankton 286 anthropic 322
perovskite 78 plasma 91, 212 simian 322
perturbation theory 136 plasmons 86 prions 377
and quantum field theory platinum 352 Prochlorococcus 36
subject index 429

projective planes 366 essence 313 quartz 72, 75
promethium 353 essence of 224 quartz, transparency of 86
propeller 282 Lagrangian 224 quasars 153
fish have none 283 quantum electrodynamics 126 quasicrystal, natural 81
protactinium 353 essence 313 quasiparticle 86–87
proton 170 quantum field theory definition of 86
Compton wavelength 337 collective aspects 274–275 quaternions 360
g factor 337 definition 313
gyromagnetic ratio 337 emotions of 320 R
in magnetic resonance essence of 274 r-process 215
imaging 162 intensity of 320 Rad (unit) 194
P is composed 219
lifetime 315
perturbative aspect of 314
topological 273
radian 326
radiation 342
magnetic moment 337 quantum groups 273 cosmic see grand
projective mass 315, 337 quantum Hall effect 107 unification, 174
properties 171, 342 quantum machines 19 radiation exposure 178

Motion Mountain – The Adventure of Physics
quark content 222 quantum number radiation pressure 210
specific charge 337 principal, illustration of 59 radiative decay 314
proton decay 269 quantum numbers radioactive dose 193
proton emission 181 list of 263 radioactive material
proton lifetime 269 quantum of action amount of 193
proton mass 233 precise value 264, 334 radioactivity 165, 180, 193
proton shape, variation of 234 quantum of circulation 336 and hell 185
proton, magnetic moment 227 quantum particle dangers of 195, 197
protons 170 elementary 312 discovery of 165
protonvolt 330 quantum physics measurement of 193

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
psychology 262 in a nutshell 311–324 of Earth 185
Puffinus griseus 30 quantum systems in gravity of human body 181
pullovers 363 140 of lava 185
pump quantum theory types of 166
molecular 21 in three sentences 311 units 194
pungency 332 millennium issues and radioactivity, artificial 192
puzzle 316–317 Radiocarbon dating 184
animal symmetry 31 open questions 316 radiocarbon dating 184
pyrite 80 precision of 314 radiocarbon dating method
pyroxenes 72 quantum Zeno effect 184
radioactivity and 180 radiometric dating 182–184
Q quark radium 353
QAD 259 mixing matrix 265, 335 radius, covalent 357
QED 126 table of 223 radius, ionic 357
QED, open questions in 137 types 223 radon 354
quality factor 122 quark confinement 224, 229 rain on demand 309
quanton, elementary 312 quark mass 233 rainbow
quantons 311 quark masses 233 and fine structure constant
quantum asthenodynamics quark mixing 138
259 definition 251 photograph of 138
essence 313 quark model 220–223 rainbow due to gravity 157
quantum chromodynamics quark stars 69 raindrops 308
223 quarks 108, 219, 223 rainforest 99
430 subject index

ratchet 24 S sense of taste 40
classical 25 S duality 274 sensors
picture of 24 s-process of touch, illustration of 39
ratchet, quantum 25 definition 213 sensors, animal 40
rays, alpha see 𝛼 rays Salmonella 286 sexes, number of 30
rays, beta see 𝛽 rays salt-formers 346 shadow of the Moon by
rays, cosmic see cosmic rays Salticidae 90 cosmic rays 177
reaction samarium 354 sharks 41
nuclear 188 sand 380 sheets, thinnest, in nature 97
reaction rate sapphire 76 shells 58
chemical 315 satellites 151 shoe laces 308
R reactor
for nuclear power 196
scaling 97
scallop
showers
cosmic ray 178
natural nuclear 196 swimming 285 shroud, Turin 184
ratchet reactor, nuclear 191 scallop theorem 285 shuttlecocks 88
red giants 68 scandium 354 SI

Motion Mountain – The Adventure of Physics
red-shift values 152 scattering prefixes
Regge trajectory 230 nuclear 187 table of 327
relativity, special, and scattering experiment 167 units 325, 334
dislocations 299 Schrödinger equation, SI units
rem (unit) 194 complex numbers in 297 definition 325
renormalization 127 Schrödinger equation, for prefixes 327
of quantum field theory extended entities 296 supplementary 326
275 Schrödinger’s equation 58 siemens 327
reproduction 18 Schwarzschild radius as sievert 194, 327
reproduction as proof of length unit 330 sievert (SI unit) 193

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
existence of atoms 19 ScI3 109 signal distribution 42
research fraud 346 science fiction, not in silent holes 151
reset mechanism 42 quantum gravity 159 silica 71
Reynolds number 281, 405 Scoville heat unit 332 silicon 79, 355
definition 285 screw dislocations 299 silver 80, 355
rhenium 62, 73, 354 seaborgium 354 simple 362
rhodium 354 second 327 simply connected 365
Rhodospirillum rubrum 287 definition 325, 339 single atom 46, 114
rock cycle 69 second principle of singular point 385
rock types 71 thermodynamics 149 skatole 52
rocks 69 secret service 91 skyrmions 275
rods 38 sedimentary rocks 69 slime eel 306
roentgenium 354 sedimentites 71 smartphone
rotational motors 22 see bad for learning 9
rotons 87 through clothes 390 smell
rubber 301 selectron 272 sense of 52
rubidium 354 selenium 354 smoking
ruby 76 self-acceleration 157 cancer due to radioactivity
ruthenium 354 self-reproduction 18 194, 195
rutherfordium 354 semiconductor smoky quartz 75
Rydberg atom 294 garlic-smelling 355 snakes 41
Rydberg constant 315, 331, 336 semisimple 362 sneeze 30
röntgen (unit) 194 sense of smell 40 snow flakes 74
subject index 431

sodium 355 standard model collapse of 210
sodium lamps 109 open questions 265 convection inside the 204
Sokolov–Ternov effect 147, summary 261–265 corona photograph 205
396 standard quantum limit for energy source in 200
solar constant clocks 46 formation 216
variation 210 stannum 356 images at different
solar cycle 210 star wavelengths 201
solar flare 204 collapse of 210 lifetime remaining 202
solar storms 204 neutron 68, 69 motion in 204
soliton 87 pressure in 210 neutrino flux 217
solitons 275 quark 69 pressure 68
S soul 312
space-time
shining of 208
size 68
Sun’s age 339
Sun’s lower photospheric
non-commutative 273 surface 68 pressure 339
sodium space-time duality 274 temperature sensitivity 210 Sun’s luminosity 339
space-time foam 159 star algebra 359 Sun’s mass 338

Motion Mountain – The Adventure of Physics
space-time, fluid 298 stardust 216 Sun’s surface gravity 339
space-time, solid 299 stars 109, 110 superconducting
space-time, swimming Stefan–Boltzmann black body supercollider 258
through curved 291 radiation constant 150, 315, superconductivity 103, 235,
spark chambers 174 337 316
sparticles 272 steradian 326 superfluidity 103, 120, 315
special relativity 128 stibium 347 supergravity 272
special relativity and stimulated emission 114 supernova 153, 215
dislocations 299 stokes (unit) 281 cosmic radiation and 175,
spectrum 359 stone 177, 178

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
spectrum of hot objects 315 age of a 69 debris dating 183
speed stone formation 69 definition 216
of light 𝑐 stones 311 matter distribution and
physics and 8 strange quark 223, 263 216
speed of light mass 335 neutron star and 186
inside the Sun 203 Streptococcus mitis 31 Sun, Earth and 216
sperm 286 striatum 44 supersymmetry 268, 270–273
sphere packing 73 stroke support
spinach 56 motion 289 this book 10
spinor strong coupling constant 225, surface 303
as conserved quantity 272 265, 335 of a star 68
spirits 312 strong CP problem 238 surface, compact 366
spirochaetes 287 strong interaction surfaces of genus 𝑛 366
Spiroplasma 287 feeble side 219 swimming 282–287
sponsor introduction 219 and atoms 282
this book 10 strontium 355 interfacial 287
spontaneous fission 181 structure constants 226, 361 lift-based 284
spores 282 SU(3) macroscopic 284
squark 412 in nuclei 199 microscopic 284
squid 110 subalgebra 361 science of human 306
stalk 91 sulfates 75 swimming through curved
standard deviation 332 sulphur 355 space-time 291
illustration 333 Sun 200–208 swords in science fiction 130
432 subject index

symmetry terahertz 33
and unification 268 waves 390 tree 344
beyond the standard terahertz waves 84, 91 definition 33
model 271–277 terbium 355 fighting 36
symmetry, conformal 273 tesla 327 growth 33
symmetry, external 272 tetrahedral skeletons 60 image of 160
symmetry, internal 272 tetraquark 402 motion of 36, 37
synapses 41 tetraquarks 237 tree growth 34
Système International thallium 355 trees
d’Unités (SI) 325 thermometer 147 and gravity 143
system thorium 185, 356 trefoil knot 303
S metastable 180 three-body problem 134
thrips 282
triple-α process 210
tritium 211, 240
T thulium 356 trivial knot 302
symmetry T duality 274 thunderstorms 195 tropical year 338
T violation 250 Ti:sapphire laser 393 trousers 363

Motion Mountain – The Adventure of Physics
table tie knots 308 trout 41
periodic, illustration of 60 time truth 264
periodic, with of observation 49 truth quark 223
photographs 61 Planck 329 tungsten 62, 73, 109, 356
table tennis with neutrons 140 time inversion violation 250 atoms, images of 74
Talitrus saltator 43 time machines 159 tuning, fine 322
Taningia danae 110 tin 356 tunnelling rate 314
tantalum 355, 410 titanium 356 Turin shroud 184
properties 343 TmI5 109 twins as father(s) 30
tape, adhesive 97 TNT energy content 337 two-body problem 134

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
tape, sticky 97 toilet brushes 282 two-dimensional crystals 98
tardigrade 377 tokamak 212
taste 40 tongue 40 U
tau 242, 263 tonne, or ton 327 udeko 327
tau mass 335 tooth decay 83 Udekta 327
tau neutrino 242, 263 tooth paste 311 ultrasound 41
tax collection 325 top quark 223, 263 ultrasound imaging 84
teaching mass 335 umami 40
best method for 9 topness 264 uncertainty
teapot, unknown properties topoisomerases 307 relative 332
of 323 topological invariant 307 total 332
technetium 355 topological space 364 unification
teeth 79 topology 364 delusions about 320
and plasmas 91 tornado 295 incorrect graph on 275
telepathy 118 torus, 𝑛- 366 lack of electroweak 252
teleportation 118 touch sensors 38 search for 268
tellurium 355 tourmaline 76 the dreom of 268–277
telomeres 43 transformation what for 323
temperature of particles 240 unit
Planck 329 transformations, linear 359 astronomical 338
temperature, human 34 transistor 100 natural 336
tennessine 355 transpiration-cohesion- units 325
Tera 327 tension model natural 328
subject index 433

non-SI 328 virus crystallization 377 whale brain size 41
Planck’s 328 virusoids 377 whales 41
provincial 328 viscosity 281 wheel, paddle 26
SI, definition 325 kinematic 285 wheels and propellers 283
units, true natural 330 viscosity, kinematic 281 Wien’s displacement constant
universe 134 viscosity,dynamic 281 315, 337
multiple 322 vitamin B12 348 Wikipedia 375
unpredictability vodka 114 wine 75
practical 46 volt 327 wine, dating of 197
Unruh effect 146 vortex 294 wing
Unruh radiation 129 in superfluids 105 membrane 281
U illustration about 148
up quark 223, 263
vortex evolution 295
vortex filaments 295
wings
fixed 281
mass 335 vortex tubes 295 flapping 281
units uranium 165, 185, 356 wood 88
W World Geodetic System 339

Motion Mountain – The Adventure of Physics
V W boson 243, 262 worm holes 159
vacuum introduction 241 wound healing 22
energy density 317 mass 335 writhe
impedance 336 Waals, van der definition 308
permeability 336 force in living being 90 quasi quantization 308
permittivity 336 warming, global 93
vacuum as bath 131 warp drive 159 X
vacuum permeability 264 water X bosons 269
vacuum permittivity 264 properties 343 X-ray binaries 217
vacuum polarization 133 water density 315 X-rays 84

copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
vacuum temperature 146 water drops and droplets 308 Xe 110
vacuum, swimming through water waves 123 xenno 327
291 waterspout 295 xenon 109, 356
Van-der-Waals interaction watt 327 Xenta 327
220 wave
vanadium 356 terahertz 390 Y
vanilla ice cream 53 wave function phase in Yang–Mills theory 224
variance 332 gravity 142 yawning 54
vector boson waves, terahertz 84, 91 yocto 327
weak, introduction 241 weak charge 250, 255, 264 Yotta 327
vector coupling 255 weak interaction ytterbium 356
Vela satellites 397 curiosities 255 yttrium 356
velocity weakness of 241 Yukawa coupling 253
Planck 329 weak intermediate bosons 242
vendeko 327 weak isospin 250, 255, 264 Z
Vendekta 327 weak mixing angle 252, 265, Z boson 243, 262
ventral tegmental area 54, 55 269, 335 introduction 241
Viagra 352 weak vector bosons 242 mass 335
video weapons Z boson mass 315
bad for learning 9 nuclear 197 Zeno effect
viroids 377 weber 327 quantum 50
virtual particles 122 weko 327 zepto 327
virus 17 Wekta 327 zero-body problem 134
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–September 2021 free pdf ﬁle available at www.motionmountain.net
subject index

particle 220
zoo
zirconium 356
zinc 356
zero-point fluctuations 124
Zetta 327
434

zero-point
Z
MOTION MOUNTAIN
The Adventure of Physics – Vol. V
Motion inside Matter –
Pleasure, Technology and Stars

Which quantum effects are at the basis of life?
How many motors does a human contain?
How do our senses work?
What crystals are used in everyday life?
What is vacuum energy?
How does magnetic resonance work?
Where do the atoms in our body come from?
Why do stars shine?
Which problems in physics are unsolved?

Answering these and other questions on motion,
this series gives an entertaining and mind-twisting
introduction into modern physics – one that is
surprising and challenging on every page.
Starting from everyday life, the adventure provides
an overview of modern results in mechanics,
heat, electromagnetism, relativity,
quantum physics and unification.

Christoph Schiller, PhD Université Libre de Bruxelles,
is a physicist and physics popularizer. He wrote this
book for his children and for all students, teachers and
readers interested in physics, the science of motion.

Pdf file available free of charge at
www.motionmountain.net