**The Road to Reality**- Written by Roger Penrose
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Synopsis|Excerpt|Table of Contents

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:

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

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:

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

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

Excerpted from **The Road to Reality** by Roger Penrose. Copyright © 2004 by Roger Penrose. Excerpted by permission of Vintage, a division of Random House LLC. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

Acknowledgements

Notation

Prologue

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

2.1 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

3.1 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

4.1 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

5.1 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

6.1 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

7.1 Complex smoothness; holomorphic functions

7.2 Contour integration

7.3 Power series from complex smoothness

7.4 Analytic continuation

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

9.1 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

10.1 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

11.1 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

12.1 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

13.1 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

14.1 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

15.1 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

16.1 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

19.1 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

20.1 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

21.1 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

22.1 The quantum procedures

22.2 The linearity of

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

23.1 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

24.1 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

25.1 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?

26.1 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

27.1 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

28.1 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?

29.1 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?

30.1 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

31.1 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 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?

32.1 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?

33.1 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?

34.1 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

Bibliography

Index

Contents

Roger Penrose

**Roger Penrose** is Emeritus Rouse Ball Professor of Mathematics at the University of Oxford. He has received numerous prizes and awards, most notably the Wolf Foundation Prize in physics, which he shared with Stephen Hawking. He is the author of three previous books, including *The Emperor’s New Mind*. He lives in Oxford, England.

“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.” —

“A remarkable book . . . teeming with delights.” —

“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.” —

“Extremely comprehensive. . . .

“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.”

—

“For physics fans, the high point of the year will undoubtedly be

—

“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.”—

“Penrose’s work is genuinely magnificent, and the most stimulating book I have read in a long time.”—

“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 Road to Reality by Roger Penrose
- January 09, 2007
- Science; Science - Astrophysics & Space Science
- Vintage
- $26.00
- 9780679776314