The Road to Reality: A Complete Guide to the Physical Universe

Cover
Contents
Preface
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
	Notes
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
	Notes
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
	Notes
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
	Notes
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
	Notes
6 Real-number calculus
	6.1 What makes an honest function?
	6.2 Slopes of functions
	6.3 Higher derivatives; C^{\\infty}-smooth functions
	6.4 The ‘Eulerian’ notion of a function?
	6.5 The rules of differentiation
	6.6 Integration
	Notes
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
	Notes
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
	Notes
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
	Notes
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
	Notes
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
	Notes
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
	Notes
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
	Notes
14 Calculus on manifolds
	14.1 DiVerentiation 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
	Notes
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 CliVord 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
	Notes
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 Godel’s theorem
	16.7 Sizes of infinity in physics
	Notes
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 of Einstein’s general relativity
	Notes
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
	Notes
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
	Notes
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
	Notes
21 The quantum particle
	21.1 Non-commuting variables
	21.2 Quantum Hamiltonians
	21.3 Schrodinger’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
	Notes
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: Schrodinger 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
	Notes
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
	Notes
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 Difficulties with the relativistic energy formula
	24.5 The non-invariance of \\partial / \\partial t
	24.6 Clifford–Dirac square root of wave operator
	24.7 The Dirac equation
	24.8 Dirac’s route to the positron
	Notes
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?
	Notes
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
	Notes
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
	Notes
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?
	Notes
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 1/2 : the Bloch sphere
	29.5 The density matrix in EPR situations
	29.6 FAPP philosophy of environmental decoherence
	29.7 Schrodinger’s cat with ‘Copenhagen’ ontology
	29.8 Can other conventional ontologies resolve the ‘cat’?
	29.9 Which unconventional ontologies may help?
	Notes
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 Schrodinger’s lump
	30.12 Preferred Schrodinger–Newton states?
	30.13 FELIX and related proposals
	30.14 Origin of fluctuations in the early universe
	Notes
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?
	Notes
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?
	Notes
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?
	Notes
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
	Notes
Epilogue
Bibliography
Index