Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive and comprehensible account of the physics of the universe. From 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.
|Publisher:||Knopf Doubleday Publishing Group|
|Product dimensions:||6.10(w) x 9.20(h) x 2.10(d)|
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Am-tep was the King’s chief craftsman, an artist of consummate skills. It was night, and he lay sleeping on his workshop couch, tired after a handsomely productive evening’s 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 dawn’s 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 parasol’s 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 scene’s 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 God’s 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-tep’s 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 artist’s 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 father’s 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-tep’s 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-tep’s 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 Amphos’s day as it was to Am-tep.
Amphos had studied the structure of plants, insects and other small animals, and crystalline rocks. His keen eye for observation had served him well in his decorative designs. He took an interest in agriculture and was fascinated by the growth of wheat and other plants from grain. But none of this told him ‘why?’, and he felt unsatisfied. He believed that there was indeed reason underlying Nature’s patterns, but he was in no way equipped to unravel those reasons.
One clear night, Amphos looked up at the heavens, and tried to make out from the patterns of stars the shapes of those heroes and heroines who formed constellations in the sky. To his humble artist’s eye, those shapes made poor resemblances. He could himself have arranged the stars far more convincingly. He puzzled over why the gods had not organized the stars in a more appropriate way? As they were, the arrangements seemed more like scattered grains randomly sowed by a farmer, rather than the deliberate design of a god. Then an odd thought overtook him: Do not seek for reasons in the specific patterns of stars, or of other scattered arrangements of objects; look, instead, for a deeper universal order in the way that things behave.
Amphos reasoned that we find order, after all, not in the patterns that scattered seeds form when they fall to the ground, but in the miraculous way that each of those seeds develops into a living plant having a superb structure, similar in great detail to one another. We would not try to seek the meaning in the precise arrangement of seeds sprinkled on the soil; yet, there must be meaning in the hidden mystery of the inner forces controlling the growth of each seed individually, so that each one follows essentially the same wonderful course. Nature’s laws must indeed have a superbly organized precision for this to be possible.
Amphos became convinced that without precision in the underlying laws, there could be no order in the world, whereas much order is indeed perceived in the way that things behave. Moreover, there must be precision in our ways of thinking about these matters if we are not to be led seriously astray.
It so happened that word had reached Amphos of a sage who lived in another part of the land, and whose beliefs appeared to be in sympathy with those of Amphos. According to this sage, one could not rely on the teachings and traditions of the past. To be certain of one’s beliefs, it was necessary to form precise conclusions by the use of unchallengeable reason. The nature of this precision had to be mathematical – ultimately dependent on the notion of number and its application to geometric forms. Accordingly, it must be number and geometry, not myth and superstition, that governed the behaviour of the world.
As Am-tep had done a century and a millennium before, Amphos took to the sea. He found his way to the city of Croton, where the sage and his brotherhood of 571 wise men and 28 wise women were in search of truth. After some time, Amphos was accepted into the brotherhood. The name of the sage was Pythagoras.
Table of ContentsPreface
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
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?
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
17.1 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
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
26.10 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
27.4 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
27.12 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
31.3 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