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penrose r road reality complete guide 2007

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Textbook in PDF format From one of our greatest living scientists, a magnificent book that provides, for the serious lay reader, the most comprehensive and sophisticated account we have yet had of the physical universe and the essentials of its underlying mathematical theory. Since the earliest efforts of the ancient Greeks to find order amid the chaos around us, there has been continual accelerated progress toward understanding the laws that govern our universe. And the particularly important advances made by means of the revolutionary theories of relativity and quantum mechanics have deeply altered our vision of the cosmos and provided us with models of unprecedented accuracy. What Roger Penrose so brilliantly accomplishes in this book is threefold. First, he gives us an overall narrative description of our present understanding of the universe and its physical behaviors–from the unseeable, minuscule movement of the subatomic particle to the journeys of the planets and the stars in the vastness of time and space. Second, he evokes the extraordinary beauty that lies in the mysterious and profound relationships between these physical behaviors and the subtle mathematical ideas that explain and interpret them. Third, Penrose comes to the arresting conclusion–as he explores the compatibility of the two grand classic theories of modern physics–that Einstein’s general theory of relativity stands firm while quantum theory, as presently constituted, still needs refashioning. Along the way, he talks about a wealth of issues, controversies, and phenomena; about the roles of various kinds of numbers in physics, ideas of calculus and modern geometry, visions of infinity, the big bang, black holes, the profound challenge of the second law of thermodynamics, string and M theory, loop quantum gravity, twistors, and educated guesses about science in the near future. In The Road to Reality he has given us a work of enormous scope, intention, and achievement–a complete and essential work of science 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 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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Penrose R. The Road to Reality. A Complete Guide...2007.pdf 89.9 MB

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penrose r road reality complete guide 2007

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