Gravity, Gauge Theories and Quantum Cosmology

476 Pages·1986·8.003 MB·English

by  Jayant V. Narlikar, T. Padmanabhan (auth.)

Gravity, Gauge Theories and Quantum Cosmology
Fundamental Theories of Physics
A New International Book Series on the Fundamental Theories of
Physics: Their Clarification, Development and Application
Editor: ALWYN VAN DER MERWE
University of Denver, U.S.A.
Editorial Advisory Board:
ASIM BARUT, University of Colorado, U.S.A.
HERMANN BONDI, Natural Environment Research Council, U.K.
BRIAN D. JOSEPHSON, University of Cambridge, U.K.
CLIVE KILMISTER, University of London, U. K.
GUNTER LUDWIG, Philipps-Universitiit, Marburg, F.R.G.
NATHAN ROSEN, Israel Institute of Technology, Israel
MENDEL SACHS, State University of New York at Buffalo, U.S.A.
ABDUS SALAM, International Centre for Theoretical Physics, Trieste, Italy
HANS-JURGEN TREDER, Zentralinstitut far Astrophysik der Akademie der
Wissenschaften, G.D.R.
Gravity, Gauge Theories
and Quantum Cosmology
by
v.
Jayant NarIikar
and
T. Padmanabhan
Tata Institute of Fundamental Research,
Bombay, India
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht I Boston I Lancaster I Tokyo
Library of Congress Cataloging-in-Publication Data
Narlikar, Jayant Vishnu, 1938-
Gravity, gauge theories, and quantum cosmology.
(Fundamental theories of physics)
Includes bibliographies and index.
1. Quantum gravity. 2. Cosmology. 3. Gauge fields (Physics) 4. Quantum
field theory. 5. General relativity (Physics) I. Padmanabhan, T. (Thanu), 1957- II.
Title. III. Series.
QC178.N26 1986 530.1 86-3282
ISBN-13: 978-94-010-8508-3 e-ISBN-13: 978-94-009-4508-1
001: 10.1007/978-94-009-4508-1
Published by D. Reidel Publishing Company,
P.O. Box 17, 3300 AA Dordrecht, Holland
Sold and distributed in the U.S.A. and Canada
by K!J.lwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
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All Rights Reserved
Softcover reprint of the hardcover I st edition 1986
© 1986 by D. Reidel Publishing Company, Dordrecht, Holland
No part of the material protected by this copyright notice may be reproduced or utilized in any form
or by any means, electronic or mechanical including photocopying, recording or by any information
storage and retrieval system, without written permission from the copyright owner
Table of Contents
Preface Xl
Notation xiii
Chapter 1 / Introduction 1
1.1. Historical Background 1
1.2. What This Book is About 4
PART I QUANTUM THEORY
Chapter 2 / Path Integrals 11
2.1. Action in Classical Physics 11
2.2.· Action in Quantum Physics 13
2.3. The Path Integral 17
2.4. The Quadratic Action 21
2.5. The Schrodinger Equation 24
2.6. The Spreading of Wave Packets 27
2.7. The Harmonic Oscillator 29
Notes and References 37
Chapter 3 / En Route to Quantum Field Theory 38
3.1. The Field as a Dynamical System 38
.3.2. Harmonic Oscillator in an External Potential 39
3.3. The Vacuum Persistence Amplitude 42
3.4. Euclidean Time 46
3.5. The Double-Hump Potential 50
3.6. The Instanton Solutions 54
3.7. The Concept of an Effective Action 57
3.8. Quantum Mechanics at Finite Temperature 61
Notes and References 63
v
Table of Contents
VI
Chapter 4 / Quantum Field Theory 64
4.1. Classical Field Theory (General) 64
4.2. Classical Field Theory (Specific Fields) 69
4.3. Quantization of the Scalar Field 76
4.4. Canonical Quantization 80
4.5. Scalar Field with Quartic Self-interaction 88
4.6. Nonperturbative Methods 93
4.7. Quantum Theory in External Fields 99
4.8. Field Theory at Finite Temperature 104
Notes and References 106
Chapter 5 / Gauge Fields 107
5.1. Gauge Invariance - Electromagnetism 107
5.2. Gauge Invariance - Generalized 109
5.3. General Formalism for Gauge Theory 113
5.4. Spontaneous Symmetry Breaking 117
5.5. SSB with an Abelian Gauge Field 121
5.6. SSB with a Nonabelian Gauge Field 123
5.7. The Salam-Weinberg Model 126
5.8. The Coleman-Weinberg Mechanism 131
5.9. The Gauge Field as a Physical System 133
5.10. The Gauge Field Vacuum and Instantons 136
5.11. Solitons - Monopole Solution 142
Notes and References 148
PART II CLASSICAL GENERAL RELATIVITY
Chapter 6 / General Theory of Relativity 151
6.1. The Need for a General Theory of Relativity 151
6.2. Curved Spacetime 155
6.3. Vectors and Tensors 159
6.4. Metric and Geodesics 162
6.5. Parallel Transport 167
6.6. The Curvature Tensor 172
6.7. Physics in Curved Spacetime 177
6.8. Einstein's Field Equations 184
6.9. The Newtonian Approximation 190
6.10. TheA Term 192
6.11 Conformal Transformations 193
Notes and References 194
Chapter 7 / Gravitating Massive Objects 195
7.1. The Schwarzschild Solution 195
Table of Contents Vll
7.2. Experimental Tests of General Relativity 199
7.3. Gravitational Radiation 209
7.4. Geometrodynamics 212
7.5. Gravitational Collapse 215
7.6. Black Holes 223
Notes and References 233
Chapter 8 / Relativistic Cosmology 234
8.1. Cosmological Symmetries 234
8.2. The Friedmann Models 238
8.3. Observational Cosmology 245
8.4. The Early Universe 254
8.5. The Problems of Singularity, Horizon, and Flatness 264
8.6. Anisotropic Cosmologies 268
Notes and References 272
PART III QUANTIZATION IN CURVED SPACETIME
Chapter 9 / Quantum Theory in Curved Spacetime 275
9.1. Quantum Theory in a Curved Background: Why? 275
9.2. General Covariance and the Particle Concept 279
9.3. Field Theory in Robertson-Walker Spacetime 286
9.4. Field Theory in de Sitter Spacetime 293
9.5. Euclideanization and the Thermal Green's Functions 299
9.6. Field Theory in the Black-Hole Spacetime 301
Notes and References 308
Chapter 10/ The Very Early Universe 310
10.1. Symmetry Breaking in the Early Universe 310
10.2. Cosmological Monopoles 314
10.3. Cosmological Inflationary Scenarios 316
10.4. The Guth Inflation 318
10.5. Inflation with the Coleman-Weinberg Potential 322
10.6. Fine-Tunings in the Early Universe 326
Notes and References 328
PART IV QUANTUM COSMOLOGY
Chapter 11 / Approaches to Quantum Cosmology 333
11.1. Introduction 333
11.2. The Linearized Theory 334
11.3. Canonical Quantization 337
viii Table of Contents
11.4. Manifestly Covariant Quantization 345
11.5. Path Integrals in Euclidean Spacetime 350
11.6. Concluding Remarks 353
Notes and References 353
Chapter 12 / Quantum Conformal Fluctuations 355
12.1. Quantum Gravity via Path Integrals 355
12.2. Conformal Fluctuations 359
12.3. QCF of Friedmann Cosmologies 364
12.4. Bianchi Type I Cosmologies 371
12.5. Universes with Arbitrary Distributions of Massive Particles 373
12.6. The Problems of Singularity and Horizons 382
12.7. The Problem of Flatness 386
12.8. Further Developments 389
Notes and References 391
Chapter 13 / Towards a More Complete Theory 392
13.1. Towards a More Complete Theory 392
13.2. The Average Metric 392
13.3. Quantum Fluctuations and Proper Length 395
13.4. Lower Bound to Proper Length 399
13.5. Quantum Stationary Geometries 404
13.6. QSG and the Back Reaction on the Metric 409
13.7. Solutions of Quantum Gravity Equations 412
13.8. Cosmo genesis and Vacuum Instability 418
Notes and References 426
Chapter 14 / Epilogue 427
PART V APPENDICES
Appendix A / Renormalization 433
Appendix B / Basic Group Theory 441
B.1. Definition of a Group 441
B.2. Generators 443
B.3. Representations 445
Appendix C / Differential Geometry 447
C.1. Basic Concepts 447
C.2. Vectors and I-Forms 449
C.3. Lie Derivative and Covariant Derivative 452
C.4. Curvature and Metric 454
Table of Contents ix
Appendix D / Spacetime Symmetries 457
D .1. Displacement of Spacetime 457
D.2. Killing Vectors 459
D.3. Homogeneity 460
D.4. Isotropy 460
Index 463
Preface
For several decades since its inception, Einstein's general theory of relativity
stood somewhat aloof from the rest of physics. Paradoxically, the attributes
which normally boost a physical theory - namely, its perfection as a theoreti
cal framework and the extraordinary intellectual achievement underlying it -
prevented the general theory from being assimilated in the mainstream of
physics. It was as if theoreticians hesitated to tamper with something that is
manifestly so beautiful.
Happily, two developments in the 1970s have narrowed the gap. In 1974
Stephen Hawking arrived at the remarkable result that black holes radiate
after all. And in the second half of the decade, particle physicists discovered
that the only scenario for applying their grand unified theories was offered by
the very early phase in the history of the Big Bang universe. In both cases, it
was necessary to discuss the ideas of quantum field theory in the background
of curved spacetime that is basic to general relativity.
This is, however, only half the total story. If gravity is to be brought into
the general fold of theoretical physics we have to know how to quantize it. To
date this has proved a formidable task although most physicists would agree
that, as in the case of grand unified theories, quantum gravity will have
applications to cosmology, in the very early stages of the Big Bang universe.
In fact, the present picture of the Big Bang universe necessarily forces us to
think of quantum cosmology.
In this book we have highlighted these interdisciplinary problems, ending
with a simplified version of quantum gravity that appears to provide useful
inputs into the problems of the early universe. In particular, the difficulties
associated with the classical models that have recently gained prominence
under the heads of 'singularity', 'particle horizon' and 'flatness' are resolved
in this picture of the quantum universe.
But this is a beginning, not the end of the quest for a general theory of
quantum gravity. Indeed, none of the topics discussed in the various parts of
this book is free from problems yet to be resolved: thus giving a lie to the
belief shared in some circles that the end of physics is almost at hand.
Our discussion of classical general relativity and cosmology is self-
Xl