The Road to Reality

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; 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 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 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 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 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 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 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 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 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 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 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 Index Contents