QUANTUM

Edited by M. A. Markov Academy of Sciences of the USSR Moscow, USSR and P. C. West King's College London, England

PLENUM PRESS • NEW YORK AND LONDON Library of Congress Cataloging in Publication Data

Seminar on Quantum Gravity (2nd: 1981: Moscow, R.S.F.S.R.) Quantum gravity.

"Proceedings of the Second Seminar on Quantum Gravity, held I October B-15, 1981, in Moscow, USSR"-T.p. verso. Includes bibliographical references and index. l. Quantum gravity-Congresses. 2. Cosmology-Congresses. 3. Black holes (Astronomy)-Congresses. I. Markov, M. A. (Moisei Aleksandrovich), 1908- II. West, P. C. (Peter C.) Ill. Title. QC178.S455 1981 530.1 83-24446 ISBN-13: 978-1-4612-9678-2 e-ISBN-13: 978-1-4613-2701-1 DOl: 10.1007/978-1-4613-2701-1

Proceedings of the second Seminar on Quantum Gravity, held October 13-15, 1981, in Moscow, USSR

©1984 Plenum Press, New York Softcover reprint of the hardcover 1st edition 1984 A Division of Plenum Publishing Corporation 233 Spring Street, New York, N.Y. 10013

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No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical. photocopying, microfilming, recording, or otherwise, without written permission from the Publisher ORGANIZING COMMITTEE

Markov M.A. Chairman

Frolov V.P. Scientific Secretary

Zlobina K.K. Secretary

Berezin V.A.

Ogievetsky V.I.

Tavkhelidze A.N.

Fradkin E.S.

EDITORIAL BOARD

Markov M.A. Chief Editor

Berezin V.A.

Frolov V.P. PREFACE

Three years have passed after the First Moscow Seminar on Quantum Gravity. It is a rather long time interval for the modern theoretical . The talks given at the present Second Seminar which took place in October 13-15, 1981 in Moscow contain the discussion of new results obtained during this period and the problems which arose. More than one hundred Soviet scientists and a number of the foreign guests attended this Seminar, which as the previous one was held by the Nuclear Physics Department of the Academy of Sciences of the USSR and the Institute for Nuclear Research of the Academy of Sciences of the USSR.

The aim of the Seminar was to discuss the most important problems of the modern Quantum Gravity, namely: i) Quantum Gravity: the state of art; ii) Quantum effects in Cosmology; iii) Quantum black-hole physics; iv) the recent development in and v) Quantum Gauge .

The Editorial Board expresses its sincere gratitude to all physicists who have contributed to these Proceedings for their cooperation with the Board in respect of time limitation, accurate and patient fulfilment of all tiresome requirements, set by the Board. The talks in the Proceedings are arranged in sections in accordance with their presentation at the Seminar.

Editorial Board: M.A. Markov V.A. Berezin V.P. Frolov FOREWORD

In recent years it has emerged that the nuclear weak, nuclear strong and electromagnetic are mediated by one particles whose dynamics are controlled by gauge theories within the framework of relativistic quantum field . Further, there is now dramatic experimental confirmation of the of the electromagnetic and nuclear weak forces and there exist theories which unify all of these three forces. Gravity on the other hand, the only other of , is mediated by a spin two particle described by Einstein's theory of . A theme running through many articles in this book is the place of Einstein's theory in relation to the rest of physics. A major obstacle to progress in discovering the relation between the force of gravity and the other three forces of nature is the inconsistency, noticed as long ago as the 1930's, between Einstein's theory of general relativity and as embodied in per• turbative relativistic . One possible solution to this problem, which is discussed in several articles, is to modify Einstein's theory by adding terms involving higher derivatives. Al• so discussed is the way these modifications may help alleviate one of the other problems of general relativity; namely the occurrence of singularities. Gravity is the most complex quantum field and investigation of its structure has often lead to a better understanding of quantum field theory in general. Three articles are devoted to the interplay of and symmetry within the context of gauge theories. One area where both gravity and the other forces of nature are important is in the early . A substantial part of the book is devoted to the processes that occur in this exotic domain. These include a discussion of how particles could have been created, in particular, the development of baryon asymmetry and how the inhomo• genities which have led to galaxies could have emerged. One of the exciting possibilities that is considered in this context is that the universe, underwent a period of exponential expansion. It is examined to what extent this overcomes many of the problems of the standard cosmology. Much work on the interplay between gravity and quantum was stimulated by the discovery that black holes can emit radiation

ix x FOREWORD and . Eight articles are devoted to a discussion of this ef• fect. They include the ways the laws of are modi• fied, the possibility of very small black holes, the role of thermo• dynamics and the effect these objects could have on the evolution of the universe. Finally several authors discuss . This symmetry provides the only known non-trivial extension of general relativity. The extent to which these supergravity theories can provide a con• sistent theory of gravity and quantum mechanics or may unify gravity with the three other forces of nature is examined.

P.c. West CONTENTS

SECTION 1. QUANTUM GRAVITY

The Problems of General Relativity 1 M.A. Markov

The Loss of Quantum due to Virtual Black Holes .. 19 S.W. Hawking

Asymptotic Freedom in Renormalizable Gravity and Supergravity . • • . . • • • . • . • . 29 E.S. Fradkin and A.A. Tseytlin

Pregeometry 47 H. Terazawa

On High-Frequency Background of Gravity. . • . 57 H.-H. v. Borzeskowski

SECTION II. QUANTUM EFFECTS IN COSMOLOGY

Complete Cosmological Theories 71 L.P. Grischuk and Y.B. Zeldovich

Particle Production and Dynamics in the Early Universe. • • 87 J.B. Hartle

Nonsingular Model of the Universe with the Quantum• Gravitational de Sitter Stage and its Observational Consequences •••.•••.• 103 A.A. Starobinsky

Black Holes Evaporation and Big Mass Particle (Maximon, Intermediate ) Creation in Nonstationary Universe ..•••.••.••••.•••..•. 129 V.I. Man'ko and M.A. Markov

xi xii CONTENTS

Origin of the Primordial Inhomogeneities of the Universe 143 D.A. Kompaneets, V.N. Lukash and I.D. Novikov

Quantum Fluctuations in Cosmology and Galaxy Formation ...... 161 G.V. Chibisov and V.F. Mukhanov

Can de Sitter be a Final State of the Contracting Universe? • • • • • • • • 173 V.A. Berezin

A New Inflationary Universe Scenario: A possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and PrimDrdial Monopole Problems • • • 185 A.D. Linde

Self-Consistent Treatment of Vacuum Quantum Effects in Isotropic Cosmology ••••••••. 197 A.A. Grib, S.G. Mamayev and V.M. Mostepanenko

Quantum Field Theories with Spontaneous Symmetry Breaking in External Gravitational Fields of Cosmological Type • • • • • • . • • • • • • • 213 V.G. Lapchinsky, V.I. Nekrasov, V.A. Rubakov and A.V. Veryaskin

Grand Unified Phase Transitions and the Early Universe. 231 V.A. Kuzmin, M.E. Shaposhnikov and 1.1. Tkachev

SECTION III. QUANTUM EFFECTS IN BLACK HOLES

On the Generalized Second-Law of Thermodynamics • • • 249 W.G. Unruh

Towards the Theory of Mini Black Holes with 253 Subplanckian Mass • • • • • • • • • • Y.B. Zeldovich

Spherically-Symmetric Collapse in Quantum Gravity • • • • 267 V.P. Frolov and G.A.. Vilkovisky

Temperature-Dependent G and Thermodynamics • • 291 P.C.W. Davies

Vacuum Polarization Near Black Holes. • • • • • • • • • • 303 V.P. Frolov

The Relativity of Vacuum • • • • • • • • • • • • • • • • 323 N. Sanchez CONTENTS xiii

Black Hole Evaporations and Their Cosmological Consequences. 337 B.J. Carr

SECTION IV. SUPERSYMMETRY

Geometries Inherent to N = 1 Supergravity •.••• 367 A.S. Galperin, V.I. Ogievetsky and E.S. Sokatchev

Currents in Supersymmetric Theories 389 P.C. West

Counterterms in Extended Supergravities ••..•..•• 415 R.E. Kallosh

Divergences and Anomalies in Kaluza-Klein Theories • • . 431 M.J. Duff and D.J. Toms

SECTION V. GAUGE THEORIES. TWISTORS

Gauge Algebra and Quantization • • 463 I.A. Batalin and G.A. Vilkovisky

Renormalization of General Gauge Theories 481 I.V. Tyutin and B.L. Voronov

Twistor theory 503 Z. Perjes

The Group of Paths in and • • • 527 M. B. Mensky

INDEX ...... 547