ROAD TO REALITY, THE(ISBN=9780679776314)

　　Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive and comprehensible account of the physics of the universe. From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.

　　Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University. 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, Shadows of the Mind, and The Nature of Space and Time, which he wrote with Hawking. He has lectured extensively at universities throughout America. He lives in Oxford.

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; C1-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; abst-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 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 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 G?del’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 Welds 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 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 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 Schr?dinger’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: Schr?dinger 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 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 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 anthr...
　……