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

The Road to Reality: A Complete Guide to the Laws of the Universe Cover

 

Synopses & Reviews

Publisher Comments:

Since the earliest efforts of the ancient Greeks to find order amid the chaos around us, there has been continual accelerated progress toward understanding the laws that govern our universe. And the particularly important advances made by means of the revolutionary theories of relativity and quantum mechanics have deeply altered our vision of the cosmos and provided us with models of unprecedented accuracy.

What Roger Penrose so brilliantly accomplishes in this book is threefold. First, he gives us an overall narrative description of our present understanding of the universe and its physical behaviors; from the unseeable, minuscule movement of the subatomic particle to the journeys of the planets and the stars in the vastness of time and space.

Second, he evokes the extraordinary beauty that lies in the mysterious and profound relationships between these physical behaviors and the subtle mathematical ideas that explain and interpret them.

Third, Penrose comes to the arresting conclusion (as he explores the compatibility of the two grand classic theories of modern physics) that Einstein's general theory of relativity stands firm while quantum theory, as presently constituted, still needs refashioning.

Along the way, he talks about a wealth of issues, controversies, and phenomena; about the roles of various kinds of numbers in physics, ideas of calculus and modern geometry, visions of infinity, the big bang, black holes, the profound challenge of the second law of thermodynamics, string and M theory, loop quantum gravity, twistors, and educated guesses about science in the near future. In The Road to Reality he has given us a work of enormous scope, intention, and achievement; a complete and essential work of science.

Review:

"At first, this hefty new tome from Oxford physicist Penrose (The Emperor's New Mind) looks suspiciously like a textbook, complete with hundreds of diagrams and pages full of mathematical notation. On a closer reading, however, one discovers that the book is something entirely different and far more remarkable. Unlike a textbook, the purpose of which is purely to impart information, this volume is written to explore the beautiful and elegant connection between mathematics and the physical world. Penrose spends the first third of his book walking us through a seminar in high-level mathematics, but only so he can present modern physics on its own terms, without resorting to analogies or simplifications (as he explains in his preface, 'in modern physics, one cannot avoid facing up to the subtleties of much sophisticated mathematics'). Those who work their way through these initial chapters will find themselves rewarded with a deep and sophisticated tour of the past and present of modern physics. Penrose transcends the constraints of the popular science genre with a unique combination of respect for the complexity of the material and respect for the abilities of his readers. This book sometimes begs comparison with Stephen Hawking's A Brief History of Time, and while Penrose's vibrantly challenging volume deserves similar success, it will also likely lie unfinished on as many bookshelves as Hawking's. For those hardy readers willing to invest their time and mental energies, however, there are few books more deserving of the effort. 390 illus." Publishers Weekly (Copyright Reed Business Information, Inc.)

Review:

"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

Review:

"Penrose's work is genuinely magnificent, and the most stimulating book I have read in a long time." Scotland on Sunday

Review:

"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:

"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." London Times

Review:

"For physics fans, the high point of the year will undoubtedly be The Road to Reality." Guardian

Review:

"[A] comprehensive guide to physics' big picture, and to the thoughts of one of the world's most original thinkers." George Johnson, The New York Times Book Review

Synopsis:

Aimed at the general reader, this guide to the universe provides a comprehensive account of the present understanding of the physical universe, and the essentials of its underlying mathematical theory.

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 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 Cantors diagonal slash

16.5 Puzzles in the foundations of mathematics

16.6 Turing machines and Gödels 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 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 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 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 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 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 equation

24.8 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 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: 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 lawor 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 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 asymmetry

30.3 Time-asymmetry in quantum state reduction

30.4 Hawkings black-hole temperature

30.5 Black-hole temperature from complex periodicity

30.6 Killing vectors, energy flowand time travel!

30.7 Energy outflow from negative-energy orbits

30.8 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 variables

32.1 Canonical quantum gravity

32.2 The chiral input to Ashtekars variables

32.3 The form of Ashtekars 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 physicsand 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

Epilogue

Bibliography

Index

Contents

Product Details

ISBN:
9780679454434
Subtitle:
A Complete Guide to the Laws of the Universe
Publisher:
Knopf
Author:
Penrose, Roger
Subject:
Physics
Subject:
Astrophysics & Space Science
Subject:
Cosmology
Subject:
Mathematical Physics
Subject:
Physical laws
Edition Description:
American
Publication Date:
20050222
Binding:
Hardback
Grade Level:
General/trade
Language:
English
Illustrations:
423 ILLUSTRATIONS IN TEXT
Pages:
1136
Dimensions:
9.32x6.58x2.23 in. 3.39 lbs.

Related Subjects

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» Science and Mathematics » Physics » Popular

The Road to Reality: A Complete Guide to the Laws of the Universe
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Product details 1136 pages Alfred A. Knopf - English 9780679454434 Reviews:
"Publishers Weekly Review" by , "At first, this hefty new tome from Oxford physicist Penrose (The Emperor's New Mind) looks suspiciously like a textbook, complete with hundreds of diagrams and pages full of mathematical notation. On a closer reading, however, one discovers that the book is something entirely different and far more remarkable. Unlike a textbook, the purpose of which is purely to impart information, this volume is written to explore the beautiful and elegant connection between mathematics and the physical world. Penrose spends the first third of his book walking us through a seminar in high-level mathematics, but only so he can present modern physics on its own terms, without resorting to analogies or simplifications (as he explains in his preface, 'in modern physics, one cannot avoid facing up to the subtleties of much sophisticated mathematics'). Those who work their way through these initial chapters will find themselves rewarded with a deep and sophisticated tour of the past and present of modern physics. Penrose transcends the constraints of the popular science genre with a unique combination of respect for the complexity of the material and respect for the abilities of his readers. This book sometimes begs comparison with Stephen Hawking's A Brief History of Time, and while Penrose's vibrantly challenging volume deserves similar success, it will also likely lie unfinished on as many bookshelves as Hawking's. For those hardy readers willing to invest their time and mental energies, however, there are few books more deserving of the effort. 390 illus." Publishers Weekly (Copyright Reed Business Information, Inc.)
"Review" by , "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."
"Review" by , "Penrose's work is genuinely magnificent, and the most stimulating book I have read in a long time."
"Review" by , "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."
"Review" by , "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."
"Review" by , "For physics fans, the high point of the year will undoubtedly be The Road to Reality."
"Review" by , "[A] comprehensive guide to physics' big picture, and to the thoughts of one of the world's most original thinkers."
"Synopsis" by , Aimed at the general reader, this guide to the universe provides a comprehensive account of the present understanding of the physical universe, and the essentials of its underlying mathematical theory.
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