2.
BY ROGER PENROSE
The Emperor’s New Mind:
Concerning Computers, Minds,
and the Laws of Physics
Shadows of the Mind:
A Search for the Missing Science
of Consciousness
3.
Roger Penrose
T H E R O A D TO
REALITY
A Complete Guide to the Laws
of the Universe
JONATHAN CAPE
LONDON
4.
Published by Jonathan Cape 2004
2 4 6 8 10 9 7 5 3 1
Copyright ß Roger Penrose 2004
Roger Penrose has asserted his right under the Copyright, Designs
and Patents Act 1988 to be identified as the author of this work
This book is sold subject to the condition that it shall not,
by way of trade or otherwise, be lent, resold, hired out,
or otherwise circulated without the publisher’s prior
consent in any form of binding or cover other than that
in which it is published and without a similar condition
including this condition being imposed on the
subsequent purchaser
First published in Great Britain in 2004 by
Jonathan Cape
Random House, 20 Vauxhall Bridge Road,
London SW1V 2SA
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New South Wales 2061, Australia
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A CIP catalogue record for this book
is available from the British Library
ISBN 0–224–04447–8
Papers used by The Random House Group Limited are natural,
recyclable products made from wood grown in sustainable forests;
the manufacturing processes conform to the environmental
regulations of the country of origin
Printed and bound in Great Britain by
William Clowes, Beccles, Suffolk
5.
Contents
Preface
xv
Acknowledgements
xxiii
Notation
xxvi
Prologue
1
1 The roots of science
7
1.1
1.2
1.3
1.4
1.5
The quest for the forces that shape the world
Mathematical truth
Is Plato’s mathematical world ‘real’?
Three worlds and three deep mysteries
The Good, the True, and the Beautiful
2 An ancient theorem and a modern question
2.1
2.2
2.3
2.4
2.5
2.6
2.7
The Pythagorean theorem
Euclid’s postulates
Similar-areas proof of the Pythagorean theorem
Hyperbolic geometry: conformal picture
Other representations of hyperbolic geometry
Historical aspects of hyperbolic geometry
Relation to physical space
3 Kinds of number in the physical world
3.1
3.2
3.3
3.4
3.5
A Pythagorean catastrophe?
The real-number system
Real numbers in the physical world
Do natural numbers need the physical world?
Discrete numbers in the physical world
4 Magical complex numbers
4.1
4.2
7
9
12
17
22
25
25
28
31
33
37
42
46
51
51
54
59
63
65
71
The magic number ‘i’
Solving equations with complex numbers
v
71
74
6.
Contents
4.3
4.4
4.5
Convergence of power series
Caspar Wessel’s complex plane
How to construct the Mandelbrot set
5 Geometry of logarithms, powers, and roots
5.1
5.2
5.3
5.4
5.5
Geometry of complex algebra
The idea of the complex logarithm
Multiple valuedness, natural logarithms
Complex powers
Some relations to modern particle physics
6 Real-number calculus
6.1
6.2
6.3
6.4
6.5
6.6
What makes an honest function?
Slopes of functions
Higher derivatives; C1 -smooth functions
The ‘Eulerian’ notion of a function?
The rules of diVerentiation
Integration
7 Complex-number calculus
7.1
7.2
7.3
7.4
Complex smoothness; holomorphic functions
Contour integration
Power series from complex smoothness
Analytic continuation
8 Riemann surfaces and complex mappings
8.1
8.2
8.3
8.4
8.5
76
81
83
86
86
90
92
96
100
103
103
105
107
112
114
116
122
122
123
127
129
135
The idea of a Riemann surface
Conformal mappings
The Riemann sphere
The genus of a compact Riemann surface
The Riemann mapping theorem
135
138
142
145
148
9 Fourier decomposition and hyperfunctions
153
9.1
9.2
9.3
9.4
9.5
9.6
9.7
vi
Fourier series
Functions on a circle
Frequency splitting on the Riemann sphere
The Fourier transform
Frequency splitting from the Fourier transform
What kind of function is appropriate?
Hyperfunctions
153
157
161
164
166
168
172
7.
Contents
10 Surfaces
179
10.1
10.2
10.3
10.4
10.5
179
181
185
190
193
Complex dimensions and real dimensions
Smoothness, partial derivatives
Vector Welds and 1-forms
Components, scalar products
The Cauchy–Riemann equations
11 Hypercomplex numbers
11.1
11.2
11.3
11.4
11.5
11.6
The algebra of quaternions
The physical role of quaternions?
Geometry of quaternions
How to compose rotations
CliVord algebras
Grassmann algebras
12 Manifolds of n dimensions
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
Why study higher-dimensional manifolds?
Manifolds and coordinate patches
Scalars, vectors, and covectors
Grassmann products
Integrals of forms
Exterior derivative
Volume element; summation convention
Tensors; abstract-index and diagrammatic notation
Complex manifolds
13 Symmetry groups
13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
13.10
Groups of transformations
Subgroups and simple groups
Linear transformations and matrices
Determinants and traces
Eigenvalues and eigenvectors
Representation theory and Lie algebras
Tensor representation spaces; reducibility
Orthogonal groups
Unitary groups
Symplectic groups
14 Calculus on manifolds
14.1
14.2
14.3
14.4
DiVerentiation on a manifold?
Parallel transport
Covariant derivative
Curvature and torsion
198
198
200
203
206
208
211
217
217
221
223
227
229
231
237
239
243
247
247
250
254
260
263
266
270
275
281
286
292
292
294
298
301
vii
8.
Contents
14.5
14.6
14.7
14.8
Geodesics, parallelograms, and curvature
Lie derivative
What a metric can do for you
Symplectic manifolds
15 Fibre bundles and gauge connections
15.1
15.2
15.3
15.4
15.5
15.6
15.7
15.8
Some physical motivations for Wbre bundles
The mathematical idea of a bundle
Cross-sections of bundles
The CliVord bundle
Complex vector bundles, (co)tangent bundles
Projective spaces
Non-triviality in a bundle connection
Bundle curvature
16 The ladder of inWnity
16.1
16.2
16.3
16.4
16.5
16.6
16.7
Finite Welds
A Wnite or inWnite geometry for physics?
DiVerent sizes of inWnity
Cantor’s diagonal slash
Puzzles in the foundations of mathematics
¨
Turing machines and Godel’s theorem
Sizes of inWnity in physics
17 Spacetime
17.1
17.2
17.3
17.4
17.5
17.6
17.7
17.8
17.9
The spacetime of Aristotelian physics
Spacetime for Galilean relativity
Newtonian dynamics in spacetime terms
The principle of equivalence
Cartan’s ‘Newtonian spacetime’
The Wxed Wnite speed of light
Light cones
The abandonment of absolute time
The spacetime for Einstein’s general relativity
18 Minkowskian geometry
18.1
18.2
18.3
18.4
18.5
18.6
18.7
viii
Euclidean and Minkowskian 4-space
The symmetry groups of Minkowski space
Lorentzian orthogonality; the ‘clock paradox’
Hyperbolic geometry in Minkowski space
The celestial sphere as a Riemann sphere
Newtonian energy and (angular) momentum
Relativistic energy and (angular) momentum
303
309
317
321
325
325
328
331
334
338
341
345
349
357
357
359
364
367
371
374
378
383
383
385
388
390
394
399
401
404
408
412
412
415
417
422
428
431
434
9.
Contents
19 The classical Welds of Maxwell and Einstein
19.1
19.2
19.3
19.4
19.5
19.6
19.7
19.8
Evolution away from Newtonian dynamics
Maxwell’s electromagnetic theory
Conservation and Xux laws in Maxwell theory
The Maxwell Weld as gauge curvature
The energy–momentum tensor
Einstein’s Weld equation
Further issues: cosmological constant; Weyl tensor
Gravitational Weld energy
20 Lagrangians and Hamiltonians
20.1
20.2
20.3
20.4
20.5
20.6
The magical Lagrangian formalism
The more symmetrical Hamiltonian picture
Small oscillations
Hamiltonian dynamics as symplectic geometry
Lagrangian treatment of Welds
How Lagrangians drive modern theory
21 The quantum particle
21.1
21.2
21.3
21.4
21.5
21.6
21.7
21.8
21.9
21.10
21.11
Non-commuting variables
Quantum Hamiltonians
¨
Schrodinger’s equation
Quantum theory’s experimental background
Understanding wave–particle duality
What is quantum ‘reality’?
The ‘holistic’ nature of a wavefunction
The mysterious ‘quantum jumps’
Probability distribution in a wavefunction
Position states
Momentum-space description
22 Quantum algebra, geometry, and spin
22.1
22.2
22.3
22.4
22.5
22.6
22.7
22.8
22.9
22.10
22.11
The quantum procedures U and R
The linearity of U and its problems for R
Unitary structure, Hilbert space, Dirac notation
¨
Unitary evolution: Schrodinger and Heisenberg
Quantum ‘observables’
yes/no measurements; projectors
Null measurements; helicity
Spin and spinors
The Riemann sphere of two-state systems
Higher spin: Majorana picture
Spherical harmonics
440
440
442
446
449
455
458
462
464
471
471
475
478
483
486
489
493
493
496
498
500
505
507
511
516
517
520
521
527
527
530
533
535
538
542
544
549
553
559
562
ix
10.
Contents
22.12
22.13
Relativistic quantum angular momentum
The general isolated quantum object
23 The entangled quantum world
23.1
23.2
23.3
23.4
23.5
23.6
23.7
23.8
23.9
23.10
Quantum mechanics of many-particle systems
Hugeness of many-particle state space
Quantum entanglement; Bell inequalities
Bohm-type EPR experiments
Hardy’s EPR example: almost probability-free
Two mysteries of quantum entanglement
Bosons and fermions
The quantum states of bosons and fermions
Quantum teleportation
Quanglement
24 Dirac’s electron and antiparticles
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
Tension between quantum theory and relativity
Why do antiparticles imply quantum Welds?
Energy positivity in quantum mechanics
DiYculties with the relativistic energy formula
The non-invariance of ]=]t
CliVord–Dirac square root of wave operator
The Dirac equation
Dirac’s route to the positron
25 The standard model of particle physics
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
The origins of modern particle physics
The zigzag picture of the electron
Electroweak interactions; reXection asymmetry
Charge conjugation, parity, and time reversal
The electroweak symmetry group
Strongly interacting particles
‘Coloured quarks’
Beyond the standard model?
26 Quantum Weld theory
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
x
Fundamental status of QFT in modern theory
Creation and annihilation operators
InWnite-dimensional algebras
Antiparticles in QFT
Alternative vacua
Interactions: Lagrangians and path integrals
Divergent path integrals: Feynman’s response
Constructing Feynman graphs; the S-matrix
Renormalization
566
570
578
578
580
582
585
589
591
594
596
598
603
609
609
610
612
614
616
618
620
622
627
627
628
632
638
640
645
648
651
655
655
657
660
662
664
665
670
672
675
11.
Contents
26.10
26.11
Feynman graphs from Lagrangians
Feynman graphs and the choice of vacuum
27 The Big Bang and its thermodynamic legacy
27.1
27.2
27.3
27.4
27.5
27.6
27.7
27.8
27.9
27.10
27.11
27.12
27.13
Time symmetry in dynamical evolution
Submicroscopic ingredients
Entropy
The robustness of the entropy concept
Derivation of the second law—or not?
Is the whole universe an ‘isolated system’?
The role of the Big Bang
Black holes
Event horizons and spacetime singularities
Black-hole entropy
Cosmology
Conformal diagrams
Our extraordinarily special Big Bang
28 Speculative theories of the early universe
28.1
28.2
28.3
28.4
28.5
28.6
28.7
28.8
28.9
28.10
Early-universe spontaneous symmetry breaking
Cosmic topological defects
Problems for early-universe symmetry breaking
InXationary cosmology
Are the motivations for inXation valid?
The anthropic principle
The Big Bang’s special nature: an anthropic key?
The Weyl curvature hypothesis
The Hartle–Hawking ‘no-boundary’ proposal
Cosmological parameters: observational status?
29 The measurement paradox
29.1
29.2
29.3
29.4
29.5
29.6
29.7
29.8
29.9
The conventional ontologies of quantum theory
Unconventional ontologies for quantum theory
The density matrix
Density matrices for spin 1: the Bloch sphere
2
The density matrix in EPR situations
FAPP philosophy of environmental decoherence
¨
Schrodinger’s cat with ‘Copenhagen’ ontology
Can other conventional ontologies resolve the ‘cat’?
Which unconventional ontologies may help?
30 Gravity’s role in quantum state reduction
30.1
30.2
Is today’s quantum theory here to stay?
Clues from cosmological time asymmetry
680
681
686
686
688
690
692
696
699
702
707
712
714
717
723
726
735
735
739
742
746
753
757
762
765
769
772
782
782
785
791
793
797
802
804
806
810
816
816
817
xi
12.
Contents
30.3
30.4
30.5
30.6
30.7
30.8
30.9
30.10
30.11
30.12
30.13
30.14
Time-asymmetry in quantum state reduction
Hawking’s black-hole temperature
Black-hole temperature from complex periodicity
Killing vectors, energy Xow—and time travel!
Energy outXow from negative-energy orbits
Hawking explosions
A more radical perspective
¨
Schrodinger’s lump
Fundamental conXict with Einstein’s principles
¨
Preferred Schrodinger–Newton states?
FELIX and related proposals
Origin of Xuctuations in the early universe
31 Supersymmetry, supra-dimensionality, and strings
31.1
31.2
31.3
31.4
31.5
31.6
31.7
31.8
31.9
31.10
31.11
31.12
31.13
31.14
31.15
31.16
31.17
31.18
Unexplained parameters
Supersymmetry
The algebra and geometry of supersymmetry
Higher-dimensional spacetime
The original hadronic string theory
Towards a string theory of the world
String motivation for extra spacetime dimensions
String theory as quantum gravity?
String dynamics
Why don’t we see the extra space dimensions?
Should we accept the quantum-stability argument?
Classical instability of extra dimensions
Is string QFT Wnite?
The magical Calabi–Yau spaces; M-theory
Strings and black-hole entropy
The ‘holographic principle’
The D-brane perspective
The physical status of string theory?
32 Einstein’s narrower path; loop variables
32.1
32.2
32.3
32.4
32.5
32.6
32.7
Canonical quantum gravity
The chiral input to Ashtekar’s variables
The form of Ashtekar’s variables
Loop variables
The mathematics of knots and links
Spin networks
Status of loop quantum gravity?
33 More radical perspectives; twistor theory
33.1
33.2
xii
Theories where geometry has discrete elements
Twistors as light rays
819
823
827
833
836
838
842
846
849
853
856
861
869
869
873
877
880
884
887
890
892
895
897
902
905
907
910
916
920
923
926
934
934
935
938
941
943
946
952
958
958
962
13.
Contents
33.3
33.4
33.5
33.6
33.7
33.8
33.9
33.10
33.11
33.12
33.13
33.14
Conformal group; compactiWed Minkowski space
Twistors as higher-dimensional spinors
Basic twistor geometry and coordinates
Geometry of twistors as spinning massless particles
Twistor quantum theory
Twistor description of massless Welds
Twistor sheaf cohomology
Twistors and positive/negative frequency splitting
The non-linear graviton
Twistors and general relativity
Towards a twistor theory of particle physics
The future of twistor theory?
34 Where lies the road to reality?
34.1
34.2
34.3
34.4
34.5
34.6
34.7
34.8
34.9
34.10
Great theories of 20th century physics—and beyond?
Mathematically driven fundamental physics
The role of fashion in physical theory
Can a wrong theory be experimentally refuted?
Whence may we expect our next physical revolution?
What is reality?
The roles of mentality in physical theory
Our long mathematical road to reality
Beauty and miracles
Deep questions answered, deeper questions posed
968
972
974
978
982
985
987
993
995
1000
1001
1003
1010
1010
1014
1017
1020
1024
1027
1030
1033
1038
1043
Epilogue
1048
Bibliography
1050
Index
1081
xiii
14.
I dedicate this book to the memory of
DENNIS SCIAMA
who showed me the excitement of physics
15.
Preface
The purpose of this book is to convey to the reader some feeling for
what is surely one of the most important and exciting voyages of discovery
that humanity has embarked upon. This is the search for the underlying
principles that govern the behaviour of our universe. It is a voyage that
has lasted for more than two-and-a-half millennia, so it should not surprise us that substantial progress has at last been made. But this journey
has proved to be a profoundly diYcult one, and real understanding has,
for the most part, come but slowly. This inherent diYculty has led us
in many false directions; hence we should learn caution. Yet the 20th
century has delivered us extraordinary new insights—some so impressive
that many scientists of today have voiced the opinion that we may be
close to a basic understanding of all the underlying principles of physics.
In my descriptions of the current fundamental theories, the 20th century
having now drawn to its close, I shall try to take a more sober view.
Not all my opinions may be welcomed by these ‘optimists’, but I expect
further changes of direction greater even than those of the last century.
The reader will Wnd that in this book I have not shied away from
presenting mathematical formulae, despite dire warnings of the severe
reduction in readership that this will entail. I have thought seriously
about this question, and have come to the conclusion that what I have
to say cannot reasonably be conveyed without a certain amount of
mathematical notation and the exploration of genuine mathematical
concepts. The understanding that we have of the principles that actually
underlie the behaviour of our physical world indeed depends upon some
appreciation of its mathematics. Some people might take this as a cause
for despair, as they will have formed the belief that they have no
capacity for mathematics, no matter at how elementary a level. How
could it be possible, they might well argue, for them to comprehend the
research going on at the cutting edge of physical theory if they cannot
even master the manipulation of fractions? Well, I certainly see the
diYculty.
xv
16.
Preface
Yet I am an optimist in matters of conveying understanding. Perhaps I
am an incurable optimist. I wonder whether those readers who cannot
manipulate fractions—or those who claim that they cannot manipulate
fractions—are not deluding themselves at least a little, and that a good
proportion of them actually have a potential in this direction that they are
not aware of. No doubt there are some who, when confronted with a line
of mathematical symbols, however simply presented, can see only the stern
face of a parent or teacher who tried to force into them a non-comprehending parrot-like apparent competence—a duty, and a duty alone—and
no hint of the magic or beauty of the subject might be allowed to come
through. Perhaps for some it is too late; but, as I say, I am an optimist and
I believe that there are many out there, even among those who could never
master the manipulation of fractions, who have the capacity to catch some
glimpse of a wonderful world that I believe must be, to a signiWcant degree,
genuinely accessible to them.
One of my mother’s closest friends, when she was a young girl, was
among those who could not grasp fractions. This lady once told me so
herself after she had retired from a successful career as a ballet dancer. I
was still young, not yet fully launched in my activities as a mathematician,
but was recognized as someone who enjoyed working in that subject. ‘It’s
all that cancelling’, she said to me, ‘I could just never get the hang of
cancelling.’ She was an elegant and highly intelligent woman, and there is
no doubt in my mind that the mental qualities that are required in
comprehending the sophisticated choreography that is central to ballet
are in no way inferior to those which must be brought to bear on a
mathematical problem. So, grossly overestimating my expositional abilities, I attempted, as others had done before, to explain to her the simplicity and logical nature of the procedure of ‘cancelling’.
I believe that my eVorts were as unsuccessful as were those of others.
(Incidentally, her father had been a prominent scientist, and a Fellow of
the Royal Society, so she must have had a background adequate for the
comprehension of scientiWc matters. Perhaps the ‘stern face’ could have
been a factor here, I do not know.) But on reXection, I now wonder
whether she, and many others like her, did not have a more rational
hang-up—one that with all my mathematical glibness I had not noticed.
There is, indeed, a profound issue that one comes up against again and
again in mathematics and in mathematical physics, which one Wrst encounters in the seemingly innocent operation of cancelling a common
factor from the numerator and denominator of an ordinary numerical
fraction.
Those for whom the action of cancelling has become second nature,
because of repeated familiarity with such operations, may Wnd themselves
insensitive to a diYculty that actually lurks behind this seemingly simple
xvi
17.
Preface
procedure. Perhaps many of those who Wnd cancelling mysterious are
seeing a certain profound issue more deeply than those of us who press
on
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