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Penrose R. The Road to Reality: A Complete Guide to the Laws of the Universe

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Penrose R. The Road to Reality: A Complete Guide to the Laws of the Universe
Jonathan Cape 2004. — 1136 p.
Preface
Acknowledgements
Notation
Prologue
The roots of science
The quest for the forces that shape the world
Mathematical truth
Is Plato’s mathematical world ‘real’?
Three worlds and three deep mysteries
The Good, the True, and the Beautiful
An ancient theorem and a modern question
The Pythagorean theorem
Euclid’s postulates
Similar-areas proof of the Pythagorean theorem
Hyperbolic geometry: conformal picture
Other representations of hyperbolic geometry
Historical aspects of hyperbolic geometry
Relation to physical space
Kinds of number in the physical world
A Pythagorean catastrophe?
The real-number system
Real numbers in the physical world
Do natural numbers need the physical world?
Discrete numbers in the physical world
Magical complex numbers
The magic number ‘i’
Solving equations with complex numbers
Convergence of power series
Caspar Wessel’s complex plane
How to construct the Mandelbrot set
Geometry of logarithms, powers, and roots
Geometry of complex algebra
The idea of the complex logarithm
Multiple valuedness, natural logarithms
Complex powers
Some relations to modern particle physics
Real-number calculus
What makes an honest function?
Slopes of functions
Higher derivatives; C-smooth functions
The ‘Eulerian’ notion of a function?
The rules of differentiation
Integration
Complex-number calculus
Complex smoothness; holomorphic functions
Contour integration
Power series from complex smoothness
Analytic continuation
Riemann surfaces and complex mappings
The idea of a Riemann surface
Conformal mappings
The Riemann sphere
The genus of a compact Riemann surface
The Riemann mapping theorem
Fourier decomposition and hyperfunctions
Fourier series
Functions on a circle
Frequency splitting on the Riemann sphere
The Fourier transform
Frequency splitting from the Fourier transform
What kind of function is appropriate?
Hyperfunctions
Surfaces
Complex dimensions and real dimensions
Smoothness, partial derivatives
Vector Welds and 1-forms
Components, scalar products
The Cauchy-Riemann equations
Hypercomplex numbers
The algebra of quaternions
The physical role of quaternions?
Geometry of quaternions
How to compose rotations
Clifford algebras
Grassmann algebras
Manifolds of n dimensions
Why study higher-dimensional manifolds?
Manifolds and coordinate patches
Scalars, vectors, and covectors
Grassmann products
Integrals of forms
Exterior derivative
Volume element; summation convention
Tensors; abstract-index and diagrammatic notation
Complex manifolds
Symmetry groups
Groups of transformations
Subgroups and simple groups
Linear transformations and matrices
Determinants and traces
Eigenvalues and eigenvectors
Representation theory and Lie algebras
Tensor representation spaces; reducibility
Orthogonal groups
Unitary groups
Symplectic groups
Calculus on manifolds
Differentiation on a manifold?
Parallel transport
Covariant derivative
Curvature and torsion
Geodesics, parallelograms, and curvature
Lie derivative
What a metric can do for you
Symplectic manifolds
Fibre bundles and gauge connections
Some physical motivations for Wbre bundles
The mathematical idea of a bundle
Cross-sections of bundles
The Clifford bundle
Complex vector bundles, (co)tangent bundles
Projective spaces
Non-triviality in a bundle connection
Bundle curvature
The ladder of infinity
Finite Welds
A finite or infinite geometry for physics?
Different sizes of infinity
Cantor’s diagonal slash
Puzzles in the foundations of mathematics
Turing machines and Godel’s theorem
Sizes of infinity in physics
Spacetime
The spacetime of Aristotelian physics
Spacetime for Galilean relativity
Newtonian dynamics in spacetime terms
The principle of equivalence
Cartan’s ‘Newtonian spacetime’
The fixed finite speed of light
Light cones
The abandonment of absolute time
The spacetime for Einstein’s general relativity
Minkowskian geometry
Euclidean and Minkowskian 4-space
The symmetry groups of Minkowski space
Lorentzian orthogonality; the ‘clock paradox’
Hyperbolic geometry in Minkowski space
The celestial sphere as a Riemann sphere
Newtonian energy and (angular) momentum
Relativistic energy and (angular) momentum
The classical Welds of Maxwell and Einstein
Evolution away from Newtonian dynamics
Maxwell’s electromagnetic theory
Conservation and flux laws in Maxwell theory
The Maxwell Weld as gauge curvature
The energy-momentum tensor
Einstein’s field equation
Further issues: cosmological constant; Weyl tensor
Gravitational field energy
Lagrangians and Hamiltonians
The magical Lagrangian formalism
The more symmetrical Hamiltonian picture
Small oscillations
Hamiltonian dynamics as symplectic geometry
Lagrangian treatment of fields
How Lagrangians drive modern theory
The quantum particle
Non-commuting variables
Quantum Hamiltonians
Schrödinger’s equation
Quantum theory’s experimental background
Understanding wave-particle duality
What is quantum ‘reality’?
The ‘holistic’ nature of a wavefunction
The mysterious ‘quantum jumps’
Probability distribution in a wavefunction
Position states
Momentum-space description
Quantum algebra, geometry, and spin
The quantum procedures U and R
The linearity of U and its problems for R
Unitary structure, Hilbert space, Dirac notation
Unitary evolution: Schrödinger and Heisenberg
Quantum ‘observables’
yes/no measurements; projectors
Null measurements; helicity
Spin and spinors
The Riemann sphere of two-state systems
Higher spin: Majorana picture
Spherical harmonics
Relativistic quantum angular momentum
The general isolated quantum object
The entangled quantum world
Quantum mechanics of many-particle systems
Hugeness of many-particle state space
Quantum entanglement; Bell inequalities
Bohm-type EPR experiments
Hardy’s EPR example: almost probability-free
Two mysteries of quantum entanglement
Bosons and fermions
The quantum states of bosons and fermions
Quantum teleportation
Quanglement
Dirac’s electron and antiparticles
Tension between quantum theory and relativity
Why do antiparticles imply quantum Welds?
Energy positivity in quantum mechanics
Diffculties with the relativistic energy formula
The non-invariance of ∂/∂t
Clifford-Dirac square root of wave operator
The Dirac equation
Dirac’s route to the positron
The standard model of particle physics
The origins of modern particle physics
The zigzag picture of the electron
Electroweak interactions; reXection asymmetry
Charge conjugation, parity, and time reversal
The electroweak symmetry group
Strongly interacting particles
‘Coloured quarks’
Beyond the standard model?
Quantum Field theoryQuantum Field theory
Fundamental status of QFT in modern theory
Creation and annihilation operators
Infinite-dimensional algebras
Antiparticles in QFT
Alternative vacua
Interactions: Lagrangians and path integrals
Divergent path integrals: Feynman’s response
Constructing Feynman graphs; the S-matrix
Renormalization
Feynman graphs from Lagrangians
Feynman graphs and the choice of vacuum
The Big Bang and its thermodynamic legacy
Time symmetry in dynamical evolution
Submicroscopic ingredients
Entropy
The robustness of the entropy concept
Derivation of the second law—or not?
Is the whole universe an ‘isolated system’?
The role of the Big Bang
Black holes
Event horizons and spacetime singularities
Black-hole entropy
Cosmology
Conformal diagrams
Our extraordinarily special Big Bang
Speculative theories of the early universe
Early-universe spontaneous symmetry breaking
Cosmic topological defects
Problems for early-universe symmetry breaking
Inflationary cosmology
Are the motivations for inflation valid?
The anthropic principle
The Big Bang’s special nature: an anthropic key?
The Weyl curvature hypothesis
The Hartle-Hawking ‘no-boundary’ proposal
Cosmological parameters: observational status?
The measurement paradox
The conventional ontologies of quantum theory
Unconventional ontologies for quantum theory
The density matrix
Density matrices for spin 1/2: the Bloch sphere
The density matrix in EPR situations
FAPP philosophy of environmental decoherence
Schrödinger’s cat with ‘Copenhagen’ ontology
Can other conventional ontologies resolve the ‘cat’?
Which unconventional ontologies may help?
Gravity’s role in quantum state reduction
Is today’s quantum theory here to stay?
Clues from cosmological time asymmetry
Time-asymmetry in quantum state reduction
Hawking’s black-hole temperature
Black-hole temperature from complex periodicity
Killing vectors, energy flow—and time travel!
Energy outXow from negative-energy orbits
Hawking explosions
A more radical perspective
Schrödinger’s lump
Fundamental conXict with Einstein’s principles
Preferred Schrödinger-Newton states?
FELIX and related proposals
Origin of fluctuations in the early universe
Supersymmetry, supra-dimensionality, and strings
Unexplained parameters
Supersymmetry
The algebra and geometry of supersymmetry
Higher-dimensional spacetime
The original hadronic string theory
Towards a string theory of the world
String motivation for extra spacetime dimensions
String theory as quantum gravity?
String dynamics
Why don’t we see the extra space dimensions?
Should we accept the quantum-stability argument?
Classical instability of extra dimensions
Is string QFT finite?
The magical Calabi-Yau spaces; M-theory
Strings and black-hole entropy
The ‘holographic principle’
The D-brane perspective
The physical status of string theory?
Einstein’s narrower path; loop variables
Canonical quantum gravity
The chiral input to Ashtekar’s variables
The form of Ashtekar’s variables
Loop variables
The mathematics of knots and links
Spin networks
Status of loop quantum gravity?
More radical perspectives; twistor theory
Theories where geometry has discrete elements
Twistors as light rays
Conformal group; compactified Minkowski space
Twistors as higher-dimensional spinors
Basic twistor geometry and coordinates
Geometry of twistors as spinning massless particles
Twistor quantum theory
Twistor description of massless Welds
Twistor sheaf cohomology
Twistors and positive/negative frequency splitting
The non-linear graviton
Twistors and general relativity
Towards a twistor theory of particle physics
The future of twistor theory?
Where lies the road to reality?
Great theories of 20th century physics—and beyond?
Mathematically driven fundamental physics
The role of fashion in physical theory
Can a wrong theory be experimentally refuted?
Whence may we expect our next physical revolution?
What is reality?
The roles of mentality in physical theory
Our long mathematical road to reality
Beauty and miracles
Deep questions answered, deeper questions posed
Epilogue
Bibliography
Index
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