The Road to Reality: A Complete Guide to the Laws of the Universe by Roger Penro

Description: The Road to Reality by Roger Penrose One of todays most accomplished scientists presents the only comprehensive and comprehensible account of the physics of the universe. 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. FORMAT Paperback LANGUAGE English CONDITION Brand New Publisher Description Nobel Prize-winner Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive—and comprehensible—account of the physics of the universe. A "guide to physics big picture, and to the thoughts of one of the worlds most original thinkers."—The New York TimesFrom 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. Author Biography Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University. He has received a number of prizes and awards, including the 2020 Nobel Prize in Physics for his work on black hole formation, as well as 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 Cycles of Time, The Road to Reality, The Nature of Space and Time, which he wrote with Hawking, and The Emperors New Mind. He has lectured extensively at universities throughout America. He lives in Oxford. Table of Contents 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 Platos 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 Euclids 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 Wessels 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 calculus6.1 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 functions7.2 Contour integration 7.3 Power series from complex smoothness 7.4 Analytic continuation 8 Riemann surfaces and complex mappings8.1 The idea of a Riemann surface8.2 Conformal mappings 8.3 The Riemann sphere 8.4 The genus of a compact Riemann surface 8.5 The Riemann mapping theorem9 Fourier decomposition and hyperfunctions9.1 Fourier series 9.2 Functions on a circle9.3 Frequency splitting on the Riemann sphere9.4 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 quaternions11.4 How to compose rotations11.5 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 manifolds13 Symmetry groups 13.1 Groups of transformations 13.2 Subgroups and simple groups 13.3 Linear transformations and matrices13.4 Determinants and traces 13.5 Eigenvalues and eigenvectors 13.6 Representation theory and Lie algebras 13.7 Tensor representation spaces; reducibility 13.8 Orthogonal groups13.9 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 derivative14.7 What a metric can do for you14.8 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 bundles15.6 Projective spaces 15.7 Non-triviality in a bundle connection 15.8 Bundle curvature 16 The ladder of infinity 16.1 Finite fields16.2 A Wnite or inWnite geometry for physics? 16.3 Different sizes of infinity 16.4 Cantors diagonal slash 16.5 Puzzles in the foundations of mathematics 16.6 Turing machines and Gödels theorem 16.7 Sizes of infinity in physics17 Spacetime 17.1 The spacetime of Aristotelian physics 17.2 Spacetime for Galilean relativity17.3 Newtonian dynamics in spacetime terms 17.4 The principle of equivalence 17.5 Cartans 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 Einsteins 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 Maxwells 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 Einsteins field equation 19.7 Further issues: cosmological constant; Weyl tensor 19.8 Gravitational field energy20 Lagrangians and Hamiltonians20.1 The magical Lagrangian formalism 20.2 The more symmetrical Hamiltonian picture 20.3 Small oscillations20.4 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 Schrödingers equation 21.4 Quantum theorys 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 spinors22.9 The Riemann sphere of two-state systems22.10 Higher spin: Majorana picture22.11 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 experiments23.5 Hardys 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 Diracs 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 equation24.8 Diracs 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 reversal25.5 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: Feynmans 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 universe28.1 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 Bangs 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 Schrödingers cat with Copenhagen ontology 29.8 Can other conventional ontologies resolve the cat? 29.9 Which unconventional ontologies may help? 30 Gravitys role in quantum state reduction 30.1 Is todays quantum theory here to stay?30.2 Clues from cosmological time asymmetry30.3 Time-asymmetry in quantum state reduction 30.4 Hawkings black-hole temperature30.5 Black-hole temperature from complex periodicity 30.6 Killing vectors, energy flow—and time travel!30.7 Energy outflow from negative-energy orbits30.8 Hawking explosions 30.9 A more radical perspective 30.10 Schrödingers lump 30.11 Fundamental conflict with Einsteins 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 dont 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 Einsteins narrower path; loop variables32.1 Canonical quantum gravity 32.2 The chiral input to Ashtekars variables 32.3 The form of Ashtekars variable32.4 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 space33.4 Twistors as higher-dimensional spinors 33.5 Basic twistor geometry and coordinates 33.6 Geometry of twistors as spinning massless particles33.7 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 posedEpilogue Bibliography Index Contents Review "A comprehensive guide to physics big picture, and to the thoughts of one of the worlds most original thinkers."—The New York Times "Simply astounding. . . . Gloriously variegated. . . . Pure delight. . . . It is shocking that so much can be explained so well. . . . Penrose gives us something that has been missing from the public discourse on science lately–a reason to live, something to look forward to." —American Scientist "A remarkable book . . . teeming with delights." —Nature "This is his magnum opus, the culmination of an already stellar career and a comprehensive summary of the current state of physics and cosmology. It should be read by anyone entering the field and referenced by everyone working in it." —The New York Sun"Extremely comprehensive. . . . The Road to Reality unscores the fact that Penrose is one of the worlds most original thinkers." —Tucson Citizen"What a joy it is to read a book that doesnt simplify, doesnt dodge the difficult questions, and doesnt always pretend to have answers. . . . Penroses appetite is heroic, his knowledge encyclopedic, his modesty a reminder that not all physicists claim to be able to explain the world in 250 pages."—The Times (London)"For physics fans, the high point of the year will undoubtedly be The Road to Reality."—The Guardian"A truly remarkable book...Penrose does much to reveal the beauty and subtlety that connects nature and the human imagination, demonstrating that the quest to understand the reality of our physical world, and the extent and limits of our mental capacities, is an awesome, never-ending journey rather than a one-way cul-de-sac."—London Sunday Times"Penroses work is genuinely magnificent, and the most stimulating book I have read in a long time."—Scotland on Sunday"Science needs more people like Penrose, willing and able to point out the flaws in fashionable models from a position of authority and to signpost alternative roads to follow."—The Independent Review Quote Praise forThe Road to Realityby Roger Penrose "A truly remarkable book...Penrose does much to reveal the beauty and subtlety that connects nature and the human imagination, demonstrating that the quest to understand the reality of our physical world, and the extent and limits of our mental capacities, is an awesome, never-ending journey rather than a one-way cul-de-sac." -London Sunday Times "Penroses work is genuinely magnificent, and the most stimulating book I have read in a long time." -Scotland on Sunday "Science needs more people like Penrose, willing and able to point out the flaws in fashionable models from a position of authority and to signpost alternative roads to follow." -The Independent "What a joy it is to read a book that doesnt simplify, doesnt dodge the difficult questions, and doesnt always pretend to have answers...Penroses appetite is heroic, his knowledge encyclopedic, his modesty a reminder that not all physicists claim to be able to explain the world in 250 pages." -London Times "For physics fans, the high point of the year will undoubtedly beThe Road to Reality." -Guardian From the Hardcover edition. Excerpt from Book Prologue Am-tep was the Kings chief craftsman, an artist of consummate skills. It was night, and he lay sleeping on his workshop couch, tired after a handsomely productive evenings work. But his sleep was restless - perhaps from an intangible tension that had seemed to be in the air. Indeed, he was not certain that he was asleep at all when it happened. Daytime had come - quite suddenly - when his bones told him that surely it must still be night. He stood up abruptly. Something was odd. The dawns light could not be in the north; yet the red light shone alarmingly through his broad window that looked out northwards over the sea. He moved to the window and stared out, incredulous in amazement. The Sun had never before risen in the north! In his dazed state, it took him a few moments to realize that this could not possibly be the Sun. It was a distant shaft of a deep fiery red light that beamed vertically upwards from the water into the heavens. As he stood there, a dark cloud became apparent at the head of the beam, giving the whole structure the appearance of a distant giant parasol, glowing evilly, with a smoky flaming staff. The parasols hood began to spread and darken - a daemon from the underworld. The night had been clear, but now the stars disappeared one by one, swallowed up behind this advancing monstrous creature from Hell. Though terror must have been his natural reaction, he did not move, transfixed for several minutes by the scenes perfect symmetry and awesome beauty. But then the terrible cloud began to bend slightly to the east, caught up by the prevailing winds. Perhaps he gained some comfort from this and the spell was momentarily broken. But apprehension at once returned to him as he seemed to sense a strange disturbance in the ground beneath, accompanied by ominous-sounding rumblings of a nature quite unfamiliar to him. He began to wonder what it was that could have caused this fury. Never before had he witnessed a Gods anger of such magnitude. His first reaction was to blame himself for the design on the sacrificial cup that he had just completed - he had worried about it at the time. Had his depiction of the Bull-God not been sufficiently fearsome? Had that god been offended? But the absurdity of this thought soon struck him. The fury he had just witnessed could not have been the result of such a trivial action, and was surely not aimed at him specifically. But he knew that there would be trouble at the Great Palace. The Priest-King would waste no time in attempting to appease this Daemon-God. There would be sacrifices. The traditional offerings of fruits or even animals would not suffice to pacify an anger of this magnitude. The sacrifices would have to be human. Quite suddenly, and to his utter surprise, he was blown backwards across the room by an impulsive blast of air followed by a violent wind. The noise was so extreme that he was momentarily deafened. Many of his beautifully adorned pots were whisked from their shelves and smashed to pieces against the wall behind. As he lay on the floor in a far corner of the room where he had been swept away by the blast, he began to recover his senses, and saw that the room was in turmoil. He was horrified to see one of his favourite great urns shattered to small pieces, and the wonderfully detailed designs, which he had so carefully crafted, reduced to nothing. Am-tep arose unsteadily from the floor and after a while again approached the window, this time with considerable trepidation, to re-examine that terrible scene across the sea. Now he thought he saw a disturbance, illuminated by that far-off furnace, coming towards him. This appeared to be a vast trough in the water, moving rapidly towards the shore, followed by a cliff-like wall of wave. He again became transfixed, watching the approaching wave begin to acquire gigantic proportions. Eventually the disturbance reached the shore and the sea immediately before him drained away, leaving many ships stranded on the newly formed beach. Then the cliff-wave entered the vacated region and struck with a terrible violence. Without exception the ships were shattered, and many nearby houses instantly destroyed. Though the water rose to great heights in the air before him, his own house was spared, for it sat on high ground a good way from the sea. The Great Palace too was spared. But Am-tep feared that worse might come, and he was right - though he knew not how right he was. He did know, however, that no ordinary human sacrifice of a slave could now be sufficient. Something more would be needed to pacify the tempestuous anger of this terrible God. His thoughts turned to his sons and daughters, and to his newly born grandson. Even they might not be safe. Am-tep had been right to fear new human sacrifices. A young girl and a youth of good birth had been soon apprehended and taken to a nearby temple, high on the slopes of a mountain. The ensuing ritual was well under way when yet another catastrophe struck. The ground shook with devastating violence, whence the temple roof fell in, instantly killing all the priests and their intended sacrificial victims. As it happened, they would lie there in mid-ritual - entombed for over three-and-a-half millennia! The devastation was frightful, but not final. Many on the island where Am-tep and his people lived survived the terrible earthquake, though the Great Palace was itself almost totally destroyed. Much would be rebuilt over the years. Even the Palace would recover much of its original splendour, constructed on the ruins of the old. Yet Am-tep had vowed to leave the island. His world had now changed irreparably. In the world he knew, there had been a thousand years of peace, prosperity, and culture where the Earth-Goddess had reigned. Wonderful art had been allowed to flourish. There was much trade with neighbouring lands. The magnificent Great Palace was a huge luxurious labyrinth, a virtual city in itself, adorned by superb frescoes of animals and flowers. There was running water, excellent drainage, and flushed sewers. War was almost unknown and defences unnecessary. Now, Am-tep perceived the Earth-Goddess overthrown by a Being with entirely different values. It was some years before Am-tep actually left the island, accompanied by his surviving family, on a ship rebuilt by his youngest son, who was a skilled carpenter and seaman. Am-teps grandson had developed into an alert child, with an interest in everything in the world around. The voyage took some days, but the weather had been supremely calm. One clear night, Am-tep was explaining to his grandson about the patterns in the stars, when an odd thought overtook him: The patterns of stars had been disturbed not one iota from what they were before the Catastrophe of the emergence of the terrible daemon. Am-tep knew these patterns well, for he had a keen artists eye. Surely, he thought, those tiny candles of light in the sky should have been blown at least a little from their positions by the violence of that night, just as his pots had been smashed and his great urn shattered. The Moon also had kept her face, just as before, and her route across the star-filled heavens had changed not one whit, as far as Am-tep could tell. For many moons after the Catastrophe, the skies had appeared different. There had been darkness and strange clouds, and the Moon and Sun had sometimes worn unusual colours. But this had now passed, and their motions seemed utterly undisturbed. The tiny stars, likewise, had been quite unmoved. If the heavens had shown such little concern for the Catastrophe, having a stature far greater even than that terrible Daemon, Am-tep reasoned, why should the forces controlling the Daemon itself show concern for what the little people on the island had been doing, with their foolish rituals and human sacrifice? He felt embarrassed by his own foolish thoughts at the time, that the daemon might be concerned by the mere patterns on his pots. Yet Am-tep was still troubled by the question why? What deep forces control the behaviour of the world, and why do they sometimes burst forth in violent and seemingly incomprehensible ways? He shared his questions with his grandson, but there were no answers. . . . A century passed by, and then a millennium, and still there were no answers. . . . Amphos the craftsman had lived all his life in the same small town as his father and his father before him, and his fathers father before that. He made his living constructing beautifully decorated gold bracelets, earrings, ceremonial cups, and other fine products of his artistic skills. Such work had been the family trade for some forty generations - a line unbroken since Am-tep had settled there eleven hundred years before. But it was not just artistic skills that had been passed down from generation to generation. Am-teps questions troubled Amphos just as they had troubled Am-tep earlier. The great story of the Catastrophe that destroyed an ancient peaceful civilization had been handed down from father to son. Am-teps perception of the Catastrophe had also survived with his descendants. Amphos, too, understood that the heavens had a magnitude and stature so great as to be quite unconcerned by that terrible event. Nevertheless, the event had had a catastrophic effect on the little people with their cities and their human sacrifices and insignificant religious rituals. Thus, by comparison, the event itself must have been the result of enormous forces quite unconcerned by those trivial actions of human beings. Yet the nature of those forces was as unknown in Amphoss day as it was to Am-tep. Amphos had studied the structure Details ISBN0679776311 Author Roger Penrose Short Title ROAD TO REALITY Language English ISBN-10 0679776311 ISBN-13 9780679776314 Media Book Format Paperback DEWEY 530.1 Year 2007 Affiliation Oxford University Subtitle A Complete Guide to the Laws of the Universe DOI 10.1604/9780679776314 Place of Publication New York Country of Publication United States AU Release Date 2007-01-09 NZ Release Date 2007-01-09 US Release Date 2007-01-09 UK Release Date 2007-01-09 Pages 1136 Publisher Random House USA Inc Publication Date 2007-01-09 Imprint Random House Inc Illustrations ILL. Audience General We've got this At The Nile, if you're looking for it, we've got it. 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