# The Road to Reality: A Complete Guide to the Laws of the Universe.pdf

*书籍描述*

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“A comprehensive guide to physics’ big picture, and to the thoughts of one of the world’s 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 world’s most original thinkers.” —*Tucson Citizen*

“What a joy it is to read a book that doesn't simplify, doesn't dodge the difficult questions, and doesn't always pretend to have answers. . . . Penrose’s 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*

“Penrose’s 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*

**作者简介**

Roger Penrose is Emeritus Rouse Ball Professor of Mathematics at Oxford University. He has received a number of prizes and awards, including 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 *The Emperor's New Mind*, *Shadows of the Mind*, and *The Nature of Space and Time*, which he wrote with Hawking. He has lectured extensively at universities throughout America. He lives in Oxford.

**目录***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 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 anthr...

**文摘****Prologue**

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 fr...

**内容简介**

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.