Best Seller

“A comprehensive guide to physics’ big picture, and to the thoughts of one of the world’s most original thinkers.”—*The New York Times*

“Simply astounding. . . . Gloriously variegated. . . . Pure delight. . . . It is shocking that so much can be explained so well. . . . Penrose gives us something that has been missing from the public discourse on science lately–a reason to live, something to look forward to.” —*American Scientist*

“A remarkable book . . . teeming with delights.” —*Nature*

“This is his magnum opus, the culmination of an already stellar career and a comprehensive summary of the current state of physics and cosmology. It should be read by anyone entering the field and referenced by everyone working in it.” —*The New York Sun*

“Extremely comprehensive. . . . *The Road to Reality* unscores the fact that Penrose is one of the world’s most original thinkers.” —*Tucson Citizen*

“What a joy it is to read a book that doesn’t simplify, doesn’t dodge the difficult questions, and doesn’t always pretend to have answers. . . . Penrose’s appetite is heroic, his knowledge encyclopedic, his modesty a reminder that not all physicists claim to be able to explain the world in 250 pages.”

—*The Times *(London)

“For physics fans, the high point of the year will undoubtedly be *The Road to Reality.*”

—*The Guardian*

“A truly remarkable book…Penrose does much to reveal the beauty and subtlety that connects nature and the human imagination, demonstrating that the quest to understand the reality of our physical world, and the extent and limits of our mental capacities, is an awesome, never-ending journey rather than a one-way cul-de-sac.”—*London Sunday Times*

“Penrose’s work is genuinely magnificent, and the most stimulating book I have read in a long time.”—*Scotland on Sunday*

“Science needs more people like Penrose, willing and able to point out the flaws in fashionable models from a position of authority and to signpost alternative roads to follow.”—*The Independent*

*Preface Acknowledgements Notation Prologue *

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

2.1 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

3.1 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

4.1 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

5.1 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

6.1 What makes an honest function?

6.2 Slopes of functions

6.3 Higher derivatives; C1-smooth functions

6.4 The ‘Eulerian’ notion of a function?

6.5 The rules of differentiation

6.6 Integration

7.1 Complex smoothness; holomorphic functions

7.2 Contour integration

7.3 Power series from complex smoothness

7.4 Analytic continuation

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

9.1 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

10.1 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

11.1 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

12.1 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

13.1 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

14.1 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

15.1 Some physical motivations for fibre bundles

15.2 The mathematical idea of a bundle

15.3 Cross-sections of bundles

15.4 The Clifford 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

16.1 Finite fields

16.2 A Wnite or inWnite 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

17.1 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

19.1 Evolution away from Newtonian dynamics

19.2 Maxwell’s electromagnetic theory

19.3 Conservation and flux laws in Maxwell theory

19.4 The Maxwell Weld 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

20.1 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

21.1 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

22.1 The quantum procedures

22.2 The linearity of

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

23.1 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

24.1 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 d/dt

24.6 Clifford–Dirac square root of wave operator

24.7 The Dirac equation

24.8 Dirac’s route to the positron

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?

26.1 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

27.1 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

28.1 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?

29.1 The conventional ontologies of quantum theory

29.2 Unconventional ontologies for quantum theory

29.3 The density matrix

29.4 Density matrices for spin 1/2: 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?

30.1 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

31.1 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?

32.1 Canonical quantum gravity

32.2 The chiral input to Ashtekar’s variables

32.3 The form of Ashtekar’s variable

32.4 Loop variables

32.5 The mathematics of knots and links

32.6 Spin networks

32.7 Status of loop quantum gravity?

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?

34.1 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

Bibliography

Index

Contents