A Complete Guide to the
Laws of the Universe
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First published in Great Britain in 2004 by Jonathan Cape
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Cover
About the Book
About the Author
Also By Roger Penrose
Title Page
Dedication
Preface
Acknowledgements
Notation
Prologue
1 The roots of science
1.1 The quest for the forces that shape the world
1.2 Mathematical truth
1.3 Is Plato’s mathematical world ‘real’?
1.4 Three worlds and three deep mysteries
1.5 The Good, the True, and the Beautiful
2 An ancient theorem and a modern question
2.1 The Pythagorean theorem
2.2 Euclid’s postulates
2.3 Similar-areas proof of the Pythagorean theorem
2.4 Hyperbolic geometry: conformal picture
2.5 Other representations of hyperbolic geometry
2.6 Historical aspects of hyperbolic geometry
2.7 Relation to physical space
3 Kinds of number in the physical world
3.1 A Pythagorean catastrophe?
3.2 The real-number system
3.3 Real numbers in the physical world
3.4 Do natural numbers need the physical world?
3.5 Discrete numbers in the physical world
4 Magical complex numbers
4.1 The magic number ‘i’
4.2 Solving equations with complex numbers
4.3 Convergence of power series
4.4 Caspar Wessel’s complex plane
4.5 How to construct the Mandelbrot set
5 Geometry of logarithms, powers, and roots
5.1 Geometry of complex algebra
5.2 The idea of the complex logarithm
5.3 Multiple valuedness, natural logarithms
5.4 Complex powers
5.5 Some relations to modern particle physics
6 Real-number calculus
6.1 What makes an honest function?
6.2 Slopes of functions
6.3 Higher derivatives; C∞-smooth functions
6.4 The ‘Eulerian’ notion of a function?
6.5 The rules of differentiation
6.6 Integration
7 Complex-number calculus
7.1 Complex smoothness; holomorphic functions
7.2 Contour integration
7.3 Power series from complex smoothness
7.4 Analytic continuation
8 Riemann surfaces and complex mappings
8.1 The idea of a Riemann surface
8.2 Conformal mappings
8.3 The Riemann sphere
8.4 The genus of a compact Riemann surface
8.5 The Riemann mapping theorem
9 Fourier decomposition and hyperfunctions
9.1 Fourier series
9.2 Functions on a circle
9.3 Frequency splitting on the Riemann sphere
9.4 The Fourier transform
9.5 Frequency splitting from the Fourier transform
9.6 What kind of function is appropriate?
9.7 Hyperfunctions
10 Surfaces
10.1 Complex dimensions and real dimensions
10.2 Smoothness, partial derivatives
10.3 Vector fields and 1-forms
10.4 Components, scalar products
10.5 The Cauchy–Riemann equations
11 Hypercomplex numbers
11.1 The algebra of quaternions
11.2 The physical role of quaternions?
11.3 Geometry of quaternions
11.4 How to compose rotations
11.5 Clifford algebras
11.6 Grassmann algebras
12 Manifolds of n dimensions
12.1 Why study higher-dimensional manifolds?
12.2 Manifolds and coordinate patches
12.3 Scalars, vectors, and covectors
12.4 Grassmann products
12.5 Integrals of forms
12.6 Exterior derivative
12.7 Volume element; summation convention
12.8 Tensors; abstract-index and diagrammatic notation
12.9 Complex manifolds
13 Symmetry groups
13.1 Groups of transformations
13.2 Subgroups and simple groups
13.3 Linear transformations and matrices
13.4 Determinants and traces
13.5 Eigenvalues and eigenvectors
13.6 Representation theory and Lie algebras
13.7 Tensor representation spaces; reducibility
13.8 Orthogonal groups
13.9 Unitary groups
13.10 Symplectic groups
14 Calculus on manifolds
14.1 Differentiation on a manifold?
14.2 Parallel transport
14.3 Covariant derivative
14.4 Curvature and torsion
14.5 Geodesics, parallelograms, and curvature
14.6 Lie derivative
14.7 What a metric can do for you
14.8 Symplectic manifolds
15 Fibre bundles and gauge connections
15.1 Some physical motivations for fibre bundles
15.2 The mathematical idea of a bundle
15.3 Cross-sections of bundles
15.4 The Clifford-Hopf bundle
15.5 Complex vector bundles, (co)tangent bundles
15.6 Projective spaces
15.7 Non-triviality in a bundle connection
15.8 Bundle curvature
16 The ladder of infinity
16.1 Finite fields
16.2 A finite or infinite geometry for physics?
16.3 Different sizes of infinity
16.4 Cantor’s diagonal slash
16.5 Puzzles in the foundations of mathematics
16.6 Turing machines and Gödel’s theorem
16.7 Sizes of infinity in physics
17 Spacetime
17.1 The spacetime of Aristotelian physics
17.2 Spacetime for Galilean relativity
17.3 Newtonian dynamics in spacetime terms
17.4 The principle of equivalence
17.5 Cartan’s ‘Newtonian spacetime’
17.6 The fixed finite speed of light
17.7 Light cones
17.8 The abandonment of absolute time
17.9 The spacetime for Einstein’s general relativity
18 Minkowskian geometry
18.1 Euclidean and Minkowskian 4-space
18.2 The symmetry groups of Minkowski space
18.3 Lorentzian orthogonality; the ‘clock paradox’
18.4 Hyperbolic geometry in Minkowski space
18.5 The celestial sphere as a Riemann sphere
18.6 Newtonian energy and (angular) momentum
18.7 Relativistic energy and (angular) momentum
19 The classical fields of Maxwell and Einstein
19.1 Evolution away from Newtonian dynamics
19.2 Maxwell’s electromagnetic theory
19.3 Conservation and flux laws in Maxwell theory
19.4 The Maxwell field as gauge curvature
19.5 The energy–momentum tensor
19.6 Einstein’s field equation
19.7 Further issues: cosmological constant; Weyl tensor
19.8 Gravitational field energy
20 Lagrangians and Hamiltonians
20.1 The magical Lagrangian formalism
20.2 The more symmetrical Hamiltonian picture
20.3 Small oscillations
20.4 Hamiltonian dynamics as symplectic geometry
20.5 Lagrangian treatment of fields
20.6 How Lagrangians drive modern theory
21 The quantum particle
21.1 Non-commuting variables
21.2 Quantum Hamiltonians
21.3 Schrödinger’s equation
21.4 Quantum theory’s experimental background
21.5 Understanding wave–particle duality
21.6 What is quantum ‘reality’?
21.7 The ‘holistic’ nature of a wavefunction
21.8 The mysterious ‘quantum jumps’
21.9 Probability distribution in a wavefunction
21.10 Position states
21.11 Momentum-space description
22 Quantum algebra, geometry, and spin
22.1 The quantum procedures U and R
22.2 The linearity of U and its problems for R
22.3 Unitary structure, Hilbert space, Dirac notation
22.4 Unitary evolution: Schrödinger and Heisenberg
22.5 Quantum ‘observables’
22.6 YES/NO measurements; projectors
22.7 Null measurements; helicity
22.8 Spin and spinors
22.9 The Riemann sphere of two-state systems
22.10 Higher spin: Majorana picture
22.11 Spherical harmonics
22.12 Relativistic quantum angular momentum
22.13 The general isolated quantum object
23 The entangled quantum world
23.1 Quantum mechanics of many-particle systems
23.2 Hugeness of many-particle state space
23.3 Quantum entanglement; Bell inequalities
23.4 Bohm-type EPR experiments
23.5 Hardy’s EPR example: almost probability-free
23.6 Two mysteries of quantum entanglement
23.7 Bosons and fermions
23.8 The quantum states of bosons and fermions
23.9 Quantum teleportation
23.10 Quanglement
24 Dirac’s electron and antiparticles
24.1 Tension between quantum theory and relativity
24.2 Why do antiparticles imply quantum fields?
24.3 Energy positivity in quantum mechanics
24.4 Diffculties with the relativistic energy formula
24.5 The non-invariance of ∂/∂t
24.6 Clifford–Dirac square root of wave operator
24.7 The Dirac equation
24.8 Dirac’s route to the positron
25 The standard model of particle physics
25.1 The origins of modern particle physics
25.2 The zigzag picture of the electron
25.3 Electroweak interactions; reflection asymmetry
25.4 Charge conjugation, parity, and time reversal
25.5 The electroweak symmetry group
25.6 Strongly interacting particles
25.7 ‘Coloured quarks’
25.8 Beyond the standard model?
26 Quantum field theory
26.1 Fundamental status of QFT in modern theory
26.2 Creation and annihilation operators
26.3 Infinite-dimensional algebras
26.4 Antiparticles in QFT
26.5 Alternative vacua
26.6 Interactions: Lagrangians and path integrals
26.7 Divergent path integrals: Feynman’s response
26.8 Constructing Feynman graphs; the S-matrix
26.9 Renormalization
26.10 Feynman graphs from Lagrangians
26.11 Feynman graphs and the choice of vacuum
27 The Big Bang and its thermodynamic legacy
27.1 Time symmetry in dynamical evolution
27.2 Submicroscopic ingredients
27.3 Entropy
27.4 The robustness of the entropy concept
27.5 Derivation of the second law—or not?
27.6 Is the whole universe an ‘isolated system’?
27.7 The role of the Big Bang
27.8 Black holes
27.9 Event horizons and spacetime singularities
27.10 Black-hole entropy
27.11 Cosmology
27.12 Conformal diagrams
27.13 Our extraordinarily special Big Bang
28 Speculative theories of the early universe
28.1 Early-universe spontaneous symmetry breaking
28.2 Cosmic topological defects
28.3 Problems for early-universe symmetry breaking
28.4 Inflationary cosmology
28.5 Are the motivations for inflation valid?
28.6 The anthropic principle
28.7 The Big Bang’s special nature: an anthropic key?
28.8 The Weyl curvature hypothesis
28.9 The Hartle–Hawking ‘no-boundary’ proposal
28.10 Cosmological parameters: observational status?
29 The measurement paradox
29.1 The conventional ontologies of quantum theory
29.2 Unconventional ontologies for quantum theory
29.3 The density matrix
29.4 Density matrices for spin 
: the Bloch sphere
29.5 The density matrix in EPR situations
29.6 FAPP philosophy of environmental decoherence
29.7 Schrödinger’s cat with ‘Copenhagen’ ontology
29.8 Can other conventional ontologies resolve the ‘cat’?
29.9 Which unconventional ontologies may help?
30 Gravity’s role in quantum state reduction
30.1 Is today’s quantum theory here to stay?
30.2 Clues from cosmological time asymmetry
30.3 Time-asymmetry in quantum state reduction
30.4 Hawking’s black-hole temperature
30.5 Black-hole temperature from complex periodicity
30.6 Killing vectors, energy flow—and time travel!
30.7 Energy outflow from negative-energy orbits
30.8 Hawking explosions
30.9 A more radical perspective
30.10 Schrödinger’s lump
30.11 Fundamental conflict with Einstein’s principles
30.12 Preferred Schrödinger–Newton states?
30.13 FELIX and related proposals
30.14 Origin of fluctuations in the early universe
31 Supersymmetry, supra-dimensionality, and strings
31.1 Unexplained parameters
31.2 Supersymmetry
31.3 The algebra and geometry of supersymmetry
31.4 Higher-dimensional spacetime
31.5 The original hadronic string theory
31.6 Towards a string theory of the world
31.7 String motivation for extra spacetime dimensions
31.8 String theory as quantum gravity?
31.9 String dynamics
31.10 Why don’t we see the extra space dimensions?
31.11 Should we accept the quantum-stability argument?
31.12 Classical instability of extra dimensions
31.13 Is string QFT finite?
31.14 The magical Calabi–Yau spaces; M-theory
31.15 Strings and black-hole entropy
31.16 The ‘holographic principle’
31.17 The D-brane perspective
31.18 The physical status of string theory?
32 Einstein’s narrower path; loop variables
32.1 Canonical quantum gravity
32.2 The chiral input to Ashtekar’s variables
32.3 The form of Ashtekar’s variables
32.4 Loop variables
32.5 The mathematics of knots and links
32.6 Spin networks
32.7 Status of loop quantum gravity?
33 More radical perspectives; twistor theory
33.1 Theories where geometry has discrete elements
33.2 Twistors as light rays
33.3 Conformal group; compactified Minkowski space
33.4 Twistors as higher-dimensional spinors
33.5 Basic twistor geometry and coordinates
33.6 Geometry of twistors as spinning massless particles
33.7 Twistor quantum theory
33.8 Twistor description of massless fields
33.9 Twistor sheaf cohomology
33.10 Twistors and positive/negative frequency splitting
33.11 The non-linear graviton
33.12 Twistors and general relativity
33.13 Towards a twistor theory of particle physics
33.14 The future of twistor theory?
34 Where lies the road to reality?
34.1 Great theories of 20th century physics—and beyond?
34.2 Mathematically driven fundamental physics
34.3 The role of fashion in physical theory
34.4 Can a wrong theory be experimentally refuted?
34.5 Whence may we expect our next physical revolution?
34.6 What is reality?
34.7 The roles of mentality in physical theory
34.8 Our long mathematical road to reality
34.9 Beauty and miracles
34.10 Deep questions answered, deeper questions posed
Epilogue
Bibliography
Copyright
I dedicate this book to the memory of
DENNIS SCIAMA
who showed me the excitement of physics
Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at the University of Oxford. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics which he shared with Stephen Hawking for their joint contribution to our understanding of the universe. His books include The Emperor’s New Mind and Shadows of the Mind.
The Road to Reality is the most important and ambitious work of science for a generation. It provides nothing less than a comprehensive account of the physical universe and the essentials of its underlying mathematical theory. It assumes no particular specialist knowledge on the part of the reader, so that, for example, the early chapters give us the vital mathematical background to the physical theories explored later in the book.
Roger Penrose’s purpose is to describe as clearly as possible our present understanding of the universe and to convey a feeling for its deep beauty and philosophical implications, as well as its intricate logical interconnections.
The Road to Reality is rarely less than challenging, but the book is leavened by vivid descriptive passages, as well as hundreds of hand-drawn diagrams. In a single work of colossal scope one of the world’s greatest scientists has given us a complete and unrivalled guide to the glories of the universe that we all inhabit.
ALSO 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
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 difficult one, and real understanding has, for the most part, come but slowly. This inherent difficulty 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 find 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 difficulty.
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 significant 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 efforts 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 scientific matters. Perhaps the ‘stern face’ could have been a factor here, I do not know.) But on reflection, 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 first 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 find themselves insensitive to a difficulty that actually lurks behind this seemingly simple procedure. Perhaps many of those who find cancelling mysterious are seeing a certain profound issue more deeply than those of us who press onwards in a cavalier way, seeming to ignore it. What issue is this? It concerns the very way in which mathematicians can provide an existence to their mathematical entities and how such entities may relate to physical reality.
I recall that when at school, at the age of about 11, I was somewhat taken aback when the teacher asked the class what a fraction (such as 
) actually is! Various suggestions came forth concerning the dividing up of pieces of pie and the like, but these were rejected by the teacher on the (valid) grounds that they merely referred to imprecise physical situations to which the precise mathematical notion of a fraction was to be applied; they did not tell us what that clear-cut mathematical notion actually is. Other suggestions came forward, such as is ‘something with a 3 at the top and an 8 at the bottom with a horizontal line in between’ and I was distinctly surprised to find that the teacher seemed to be taking these suggestions seriously! I do not clearly recall how the matter was finally resolved, but with the hindsight gained from my much later experiences as a mathematics undergraduate, I guess my schoolteacher was making a brave attempt at telling us the definition of a fraction in terms of the ubiquitous mathematical notion of an equivalence class.
What is this notion? How can it be applied in the case of a fraction and tell us what a fraction actually is? Let us start with my classmate’s ‘something with a 3 at the top and an 8 on the bottom’. Basically, this is suggesting to us that a fraction is specified by an ordered pair of whole numbers, in this case the numbers 3 and 8. But we clearly cannot regard the fraction as being such an ordered pair because, for example, the fraction 
is the same number as the fraction , whereas the pair (6, 16) is certainly not the same as the pair (3, 8). This is only an issue of cancelling; for we can write as 
and then cancel the 2 from the top and the bottom to get . Why are we allowed to do this and thereby, in some sense, ‘equate’ the pair (6, 16) with the pair (3, 8)? The mathematician’s answer—which may well sound like a cop-out—has the cancelling rule just built in to the definition of a fraction: a pair of whole numbers (a × n, b × n) is deemed to represent the same fraction as the pair (a, b) whenever n is any non-zero whole number (and where we should not allow b to be zero either).
But even this does not tell us what a fraction is; it merely tells us something about the way in which we represent fractions. What is a fraction, then? According to the mathematician’s “equivalence class” notion, the fraction , for example, simply is the infinite collection of all pairs
(3, 8), ( – 3, – 8), (6, 16), ( – 6, – 16), (9, 24), ( – 9, – 24), (12, 32), . . . ,
where each pair can be obtained from each of the other pairs in the list by repeated application of the above cancellation rule.1 We also need definitions telling us how to add, subtract, and multiply such infinite collections of pairs of whole numbers, where the normal rules of algebra hold, and how to identify the whole numbers themselves as particular types of fraction.
This definition covers all that we mathematically need of fractions (such as being a number that, when added to itself, gives the number 1, etc.), and the operation of cancelling is, as we have seen, built into the definition. Yet it seems all very formal and we may indeed wonder whether it really captures the intuitive notion of what a fraction is. Although this ubiquitous equivalence class procedure, of which the above illustration is just a particular instance, is very powerful as a pure-mathematical tool for establishing consistency and mathematical existence, it can provide us with very top-heavy-looking entities. It hardly conveys to us the intuitive notion of what is, for example! No wonder my mother’s friend was confused.
In my descriptions of mathematical notions, I shall try to avoid, as far as I can, the kind of mathematical pedantry that leads us to define a fraction in terms of an ‘infinite class of pairs’ even though it certainly has its value in mathematical rigour and precision. In my descriptions here I shall be more concerned with conveying the idea—and the beauty and the magic—inherent in many important mathematical notions. The idea of a fraction such as is simply that it is some kind of an entity which has the property that, when added to itself 8 times in all, gives 3. The magic is that the idea of a fraction actually works despite the fact that we do not really directly experience things in the physical world that are exactly quantified by fractions—pieces of pie leading only to approximations. (This is quite unlike the case of natural numbers, such as 1, 2, 3, which do precisely quantify numerous entities of our direct experience.) One way to see that fractions do make consistent sense is, indeed, to use the ‘definition’ in terms of infinite collections of pairs of integers (whole numbers), as indicated above. But that does not mean that actually is such a collection. It is better to think of as being an entity with some kind of (Platonic) existence of its own, and that the infinite collection of pairs is merely one way of our coming to terms with the consistency of this type of entity. With familiarity, we begin to believe that we can easily grasp a notion like as something that has its own kind of existence, and the idea of an ‘infinite collection of pairs’ is merely a pedantic device—a device that quickly recedes from our imaginations once we have grasped it. Much of mathematics is like that.
To mathematicians (at least to most of them, as far as I can make out), mathematics is not just a cultural activity that we have ourselves created, but it has a life of its own, and much of it finds an amazing harmony with the physical universe. We cannot get any deep understanding of the laws that govern the physical world without entering the world of mathematics. In particular, the above notion of an equivalence class is relevant not only to a great deal of important (but confusing) mathematics, but a great deal of important (and confusing) physics as well, such as Einstein’s general theory of relativity and the ‘gauge theory’ principles that describe the forces of Nature according to modern particle physics. In modern physics, one cannot avoid facing up to the subtleties of much sophisticated mathematics. It is for this reason that I have spent the first 16 chapters of this work directly on the description of mathematical ideas.
What words of advice can I give to the reader for coping with this? There are four different levels at which this book can be read. Perhaps you are a reader, at one end of the scale, who simply turns off whenever a mathematical formula presents itself (and some such readers may have difficulty with coming to terms with fractions). If so, I believe that there is still a good deal that you can gain from this book by simply skipping all the formulae and just reading the words. I guess this would be much like the way I sometimes used to browse through the chess magazines lying scattered in our home when I was growing up. Chess was a big part of the lives of my brothers and parents, but I took very little interest, except that I enjoyed reading about the exploits of those exceptional and often strange characters who devoted themselves to this game. I gained something from reading about the brilliance of moves that they frequently made, even though I did not understand them, and I made no attempt to follow through the notations for the various positions. Yet I found this to be an enjoyable and illuminating activity that could hold my attention. Likewise, I hope that the mathematical accounts I give here may convey something of interest even to some profoundly non-mathematical readers if they, through bravery or curiosity, choose to join me in my journey of investigation of the mathematical and physical ideas that appear to underlie our physical universe. Do not be afraid to skip equations (I do this frequently myself) and, if you wish, whole chapters or parts of chapters, when they begin to get a mite too turgid! There is a great variety in the difficulty and technicality of the material, and something elsewhere may be more to your liking. You may choose merely to dip in and browse. My hope is that the extensive cross-referencing may sufficiently illuminate unfamiliar notions, so it should be possible to track down needed concepts and notation by turning back to earlier unread sections for clarification.
At a second level, you may be a reader who is prepared to peruse mathematical formulae, whenever such is presented, but you may not have the inclination (or the time) to verify for yourself the assertions that I shall be making. The confirmations of many of these assertions constitute the solutions of the exercises that I have scattered about the mathematical portions of the book. I have indicated three levels of difficulty by the icons –
very straight forward
needs a bit of thought
not to be undertaken lightly.
It is perfectly reasonable to take these on trust, if you wish, and there is no loss of continuity if you choose to take this position.
If, on the other hand, you are a reader who does wish to gain a facility with these various (important) mathematical notions, but for whom the ideas that I am describing are not all familiar, I hope that working through these exercises will provide a significant aid towards accumulating such skills. It is always the case, with mathematics, that a little direct experience of thinking over things on your own can provide a much deeper understanding than merely reading about them. (If you need the solutions, see the website www.roadsolutions.ox.ac.uk.)
Finally, perhaps you are already an expert, in which case you should have no difficulty with the mathematics (most of which will be very familiar to you) and you may have no wish to waste time with the exercises. Yet you may find that there is something to be gained from my own perspective on a number of topics, which are likely to be somewhat different (sometimes very different) from the usual ones. You may have some curiosity as to my opinions relating to a number of modern theories (e.g. supersymmetry, inflationary cosmology, the nature of the Big Bang, black holes, string theory or M-theory, loop variables in quantum gravity, twistor theory, and even the very foundations of quantum theory). No doubt you will find much to disagree with me on many of these topics. But controversy is an important part of the development of science, so I have no regrets about presenting views that may be taken to be partly at odds with some of the mainstream activities of modern theoretical physics.
It may be said that this book is really about the relation between mathematics and physics, and how the interplay between the two strongly influences those drives that underlie our searches for a better theory of the universe. In many modern developments, an essential ingredient of these drives comes from the judgement of mathematical beauty, depth, and sophistication. It is clear that such mathematical influences can be vitally important, as with some of the most impressively successful achievements of 20th-century physics: Dirac’s equation for the electron, the general framework of quantum mechanics, and Einstein’s general relativity. But in all these cases, physical considerations—ultimately observational ones—have provided the overriding criteria for acceptance. In many of the modern ideas for fundamentally advancing our understanding of the laws of the universe, adequate physical criteria—i.e. experimental data, or even the possibility of experimental investigation—are not available. Thus we may question whether the accessible mathematical desiderata are sufficient to enable us to estimate the chances of success of these ideas. The question is a delicate one, and I shall try to raise issues here that I do not believe have been sufficiently discussed elsewhere.
Although, in places, I shall present opinions that may be regarded as contentious, I have taken pains to make it clear to the reader when I am actually taking such liberties. Accordingly, this book may indeed be used as a genuine guide to the central ideas (and wonders) of modern physics. It is appropriate to use it in educational classes as an honest introduction to modern physics—as that subject is understood, as we move forward into the early years of the third millennium.
1 This is called an ‘equivalence class’ because it actually is a class of entities (the entities, in this particular case, being pairs of whole numbers), each member of which is deemed to be equivalent, in a specified sense, to each of the other members.
IT is inevitable, for a book of this length, which has taken me about eight years to complete, that there will be a great many to whom I owe my thanks. It is almost as inevitable that there will be a number among them, whose valuable contributions will go unattributed, owing to congenital disorganization and forgetfulness on my part. Let me first express my special thanks—and also apologies—to such people: who have given me their generous help but whose names do not now come to mind. But for various specific pieces of information and assistance that I can more clearly pinpoint, I thank Michael Atiyah, John Baez, Michael Berry, Dorje Brody, Robert Bryant, Hong-Mo Chan, Joy Christian, Andrew Duggins, Maciej Dunajski, Freeman Dyson, Artur Ekert, David Fowler, Margaret Gleason, Jeremy Gray, Stuart Hameroff, Keith Hannabuss, Lucien Hardy, Jim Hartle, Tom Hawkins, Nigel Hitchin, Andrew Hodges, Dipankar Home, Jim Howie, Chris Isham, Ted Jacobson, Bernard Kay, William Marshall, Lionel Mason, Charles Misner, Tristan Needham, Stelios Negre-pontis, Sarah Jones Nelson, Ezra (Ted) Newman, Charles Oakley, Daniel Oi, Robert Osserman, Don Page, Oliver Penrose, Alan Rendall, Wolfgang Rindler, Engelbert Schücking, Bernard Schutz, Joseph Silk, Christoph Simon, George Sparling, John Stachel, Henry Stapp, Richard Thomas, Gerard ’t Hooft, Paul Tod, James Vickers, Robert Wald, Rainer Weiss, Ronny Wells, Gerald Westheimer, John Wheeler, Nick Woodhouse, and Anton Zeilinger. Particular thanks go to Lee Smolin, Kelly Stelle, and Lane Hughston for numerous and varied points of assistance. I am especially indebted to Florence Tsou (Sheung Tsun) for immense help on matters of particle physics, to Fay Dowker for her assistance and judgement concerning various matters, most notably the presentation of certain quantum-mechanical issues, to Subir Sarkar for valuable information concerning cosmological data and the interpretation thereof, to Vahe Gurzadyan likewise, and for some advance information about his cosmological findings concerning the overall geometry of the universe, and particularly to Abhay Ashtekar, for his comprehensive information about loop-variable theory and also various detailed matters concerning string theory.
I thank the National Science Foundation for support under grants PHY 93-96246 and 00-90091, and the Leverhulme Foundation for the award of a two-year Leverhulme Emeritus Fellowship, during 2000–2002. Part-time appointments at Gresham College, London (1998–2001) and The Center for Gravitational Physics and Geometry at Penn State University, Pennsylvania, USA have been immensely valuable to me in the writing of this book, as has the secretarial assistance (most particularly Ruth Preston) and office space at the Mathematical Institute, Oxford University.
Special assistance on the editorial side has also been invaluable, under diffcult timetabling constraints, and with an author of erratic working habits. Eddie Mizzi’s early editorial help was vital in initiating the process of converting my chaotic writings into an actual book, and Richard Lawrence, with his expert efficiency and his patient, sensitive persistence, has been a crucial factor in bringing this project to completion. Having to fit in with such complicated reworking, John Holmes has done sterling work in providing a fine index. And I am particularly grateful to William Shaw for coming to our assistance at a late stage to produce excellent computer graphics (Figs. 1.2 and 2.19, and the implementation of the transformation involved in Figs. 2.16 and 2.19), used here for the Mandelbrot set and the hyperbolic plane. But all the thanks that I can give to Jacob Foster, for his Herculean achievement in sorting out and obtaining references for me and for checking over the entire manuscript in a remarkably brief time and filling in innumerable holes, can in no way do justice to the magnitude of his assistance. His personal imprint on a huge number of the end-notes gives those a special quality. Of course, none of the people I thank here are to blame for the errors and omissions that remain, the sole responsibility for that lying with me.
Special gratitude is expressed to The M.C. Escher Company, Holland for permission to reproduce Escher works in Figs. 2.11, 2.12, 2.16, and 2.22, and particularly to allow the modifications of Fig. 2.11 that are used in Figs. 2.12 and 2.16, the latter being an explicit mathematical transformation. All the Escher works used in this book are copyright (2004) The M.C. Escher Company. Thanks go also to the Institute of Theoretical Physics, University of Heidelberg and to Charles H. Lineweaver for permission to reproduce the respective graphs in Figs. 27.19 and 28.19.
Finally, my unbounded gratitude goes to my beloved wife Vanessa, not merely for supplying computer graphics for me on instant demand (Figs. 4.1, 4.2, 5.7, 6.2–6.8, 8.15, 9.1, 9.2, 9.8, 9.12, 21.3b, 21.10, 27.5, 27.14, 27.15, and the polyhedra in Fig. 1.1), but for her continued love and care, and her deep understanding and sensitivity, despite the seemingly endless years of having a husband who is mentally only half present. And Max, also, who in his entire life has had the chance to know me only in such a distracted state, gets my warmest gratitude—not just for slowing down the writing of this book (so that it could stretch its life, so as to contain at least two important pieces of information that it would not have done other-wise)—but for the continual good cheer and optimism that he exudes, which has helped to keep me going in good spirits. After all, it is through the renewal of life, such as he himself represents, that the new sources of ideas and insights needed for genuine future progress will come, in the search for those deeper laws that actually govern the universe in which we live.
(Not to be read until you are familiar with the concepts, but perhaps find the fonts confusing!)
I have tried to be reasonably consistent in the use of particular fonts in this book, but as not all of this is standard, it may be helpful to the reader to have the major usage that I have adopted made explicit.
Italic lightface (Greek or Latin) letters, such as in w2, pn, log z, cos θ, eiθ, or ex are used in the conventional way for mathematical variables which are numerical or scalar quantities; but established numerical constants, such as e, i, or π or established functions such as sin, cos, or log are denoted by upright letters. Standard physical constants such as c, G, h, ħ, g, or k are italic, however.
A vector or tensor quantity, when being thought of in its (abstract) entirety, is denoted by a boldface italic letter, such as R for the Riemann curvature tensor, while its set of components might be written with light-face italic letters (both for the kernel symbol its indices) as Rabcd. In accordance with the abstract-index notation, introduced here in §12.8, the quantity Rabcd may alternatively stand for the entire tensor R, if this interpretation is appropriate, and this should be clear from the text. Abstract linear transformations are kinds of tensors, and boldface italic letters such as T are used for such entities also. The abstract-index form Tab is also used here for an abstract linear transformation, where appropriate, the staggering of the indices making clear the precise connection with the ordering of matrix multiplication. Thus, the (abstract-)index expression SabTbc stands for the product ST of linear transformations. As with general tensors, the symbols Sab and Tbc could alternatively (according to context or explicit specification in the text) stand for the corresponding arrays of components—these being matrices—for which the corresponding bold upright letters S and T can also be used. In that case, ST denotes the corresponding matrix product. This ‘ambivalent’ interpretation of symbols such as Rabcd or Sab (either standing for the array of components or for the abstract tensor itself) should not cause confusion, as the algebraic (or differential) relations that these symbols are subject to are identical for both interpretations. A third notation for such quantities—the diagrammatic notation—is also sometimes used here, and is described in Figs. 12.17, 12.18, 14.6, 14.7, 14.21, 19.1 and elsewhere in the book.
There are places in this book where I need to distinguish the 4-dimensional spacetime entities of relativity theory from the corresponding ordinary 3-dimensional purely spatial entities. Thus, while a boldface italic notation might be used, as above, such as p or x, for the 4-momentum or 4-position, respectively, the corresponding 3-dimensional purely spatial entities would be denoted by the corresponding upright bold letters p or x. By analogy with the notation T for a matrix, above, as opposed to T for an abstract linear transformation, the quantities p and x would tend to be thought of as ‘standing for’ the three spatial components, in each case, whereas p and x might be viewed as having a more abstract component-free interpretation (although I shall not be particularly strict about this). The Euclidean ‘length’ of a 3-vector quantity a = (a1, a2, a3) may be written a, where 
, and the scalar product of a with b = (b1, b2, b3), written a • b = a1b1 + a2b2 + a3b3. This ‘dot’ notation for scalar products applies also in the general n-dimensional context, for the scalar (or inner) product α • ξ of an abstract covector α with a vector ξ.
A notational complication arises with quantum mechanics, however, since physical quantities, in that subject, tend to be represented as linear operators. I do not adopt what is a quite standard procedure in this context, of putting ‘hats’ (circumflexes) on the letters representing the quantum-operator versions of the familiar classical quantities, as I believe that this leads to an unnecessary cluttering of symbols. (Instead, I shall tend to adopt a philosophical standpoint that the classical and quantum entities are really the ‘same’—and so it is fair to use the same symbols for each—except that in the classical case one is justified in ignoring quantities of the order of ħ, so that the classical commutation properties ab = ba can hold, whereas in quantum mechanics, ab might differ from ba by something of order ħ.) For consistency with the above, such linear operators would seem to have to be denoted by italic bold letters (like T), but that would nullify the philosophy and the distinctions called for in the preceding paragraph. Accordingly, with regard to specific quantities, such as the momentum p or p, or the position x or x, I shall tend to use the same notation as in the classical case, in line with what has been said earlier in this paragraph. But for less specific quantum operators, bold italic letters such as Q will tend to be used.
The shell letters 
, 
, 
, 
, and 
q, respectively, for the system of natural numbers (i.e. non-negative integers), integers, real numbers, complex numbers, and the finite field with q elements (q being some power of a prime number, see §16.1), are now standard in mathematics, as are the corresponding n, n, n, n, 
, for the systems of ordered n-tuples of such numbers. These are canonical mathematical entities in standard use. In this book (as is not all that uncommon), this notation is extended to some other standard mathematical structures such as Euclidean 3-space 
3 or, more generally, Euclidean n-space n. In frequent use in this book is the standard flat 4-dimensional Minkowski spacetime, which is itself a kind of ‘pseudo-’ Euclidean space, so I use the shell letter 
for this space (with n to denote the n-dimensional version—a ‘Lorentzian’ spacetime with 1 time and (n – 1) space dimensions). Sometimes I use as an adjective, to denote ‘complexified’, so that we might consider the complex Euclidean n-space, for example, denoted by n. The shell letter 
can also be used as an adjective, to denote ‘projective’ (see §15.6), or as a noun, with n denoting projective n-space (or I use n or n if it is to be made clear that we are concerned with real or complex projective n-space, respectively). In twistor theory (Chapter 33), there is the complex 4-space 
, which is related to (or its complexification ) in a canonical way, and there is also the projective version . In this theory, there is also a space of null twistors (the double duty that this letter serves causing no conflict here), and its projective version .
The adjectival role of the shell letter should not be confused with that of the lightface sans serif C, which here stands for ‘complex conjugate of’ (as used in §13.1, 2). This is basically similar to another use of C in particle physics, namely charge conjugation, which is the operation which interchanges each particle with its antiparticle (see Chapters 25, 30). This operation is usually considered in conjunction with two other basic particle-physics operations, namely P for parity which refers to the operation of reflection in a mirror, and T, which refers to time-reversal. Sans serif letters which are bold serve a different purpose here, labelling vector spaces, the letters V, W, and H, being most frequently used for this purpose. The use of H, is specific to the Hilbert spaces of quantum mechanics, and Hn would stand for a Hilbert space of n complex dimensions. Vector spaces are, in a clear sense, flat. Spaces which are (or could be) curved are denoted by script letters, such as M, S, or T, where there is a special use for the particular script font 
to denote null infinity. In addition, I follow a fairly common convention to use script letters for Lagrangians (
) and Hamiltonians (
), in view of their very special status in physical theory.
AM-TEP was the King’s chief craftsman, an artist of consummate skills. It was night, and he lay sleeping on his workshop couch, tired after a handsomely productive evening’s work. But his sleep was restless—perhaps from an intangible tension that had seemed to be in the air. Indeed, he was not certain that he was asleep at all when it happened. Daytime had come—quite suddenly—when his bones told him that surely it must still be night.
He stood up abruptly. Something was odd. The dawn’s light could not be in the north; yet the red light shone alarmingly through his broad window that looked out northwards over the sea. He moved to the window and stared out, incredulous in amazement. The Sun had never before risen in the north! In his dazed state, it took him a few moments to realize that this could not possibly be the Sun. It was a distant shaft of a deep fiery red light that beamed vertically upwards from the water into the heavens.
As he stood there, a dark cloud became apparent at the head of the beam, giving the whole structure the appearance of a distant giant parasol, glowing evilly, with a smoky flaming staff. The parasol’s hood began to spread and darken—a daemon from the underworld. The night had been clear, but now the stars disappeared one by one, swallowed up behind this advancing monstrous creature from Hell.
Though terror must have been his natural reaction, he did not move, transfixed for several minutes by the scene’s perfect symmetry and awesome beauty. But then the terrible cloud began to bend slightly to the east, caught up by the prevailing winds. Perhaps he gained some comfort from this and the spell was momentarily broken. But apprehension at once returned to him as he seemed to sense a strange disturbance in the ground beneath, accompanied by ominous-sounding rumblings of a nature quite unfamiliar to him. He began to wonder what it was that could have caused this fury. Never before had he witnessed a God’s anger of such magnitude.
His first reaction was to blame himself for the design on the sacrificial cup that he had just completed—he had worried about it at the time. Had his depiction of the Bull-God not been sufficiently fearsome? Had that god been offended? But the absurdity of this thought soon struck him. The fury he had just witnessed could not have been the result of such a trivial action, and was surely not aimed at him specifically. But he knew that there would be trouble at the Great Palace. The Priest-King would waste no time in attempting to appease this Daemon-God. There would be sacrifices. The traditional offerings of fruits or even animals would not suffice to pacify an anger of this magnitude. The sacrifices would have to be human.