Quantum Fields in Curved Space

350 Pages·1984·15.343 MB·English

by  N. D. Birrell

CAMBRIDGE MONOGRAPHS
ON MATHEMATICAL PHYSICS
General Editors: P.V. Landshoff, W.H. McCrea, D.W. Sciama, S. Weinberg
QUANTUM FIELDS IN
CURVED SPACE

QUANTUM FIELDS IN
CURVED SPACE
N. D. BIRRELL
Logica Pty Ltd, Australia
AND
P. C. W. DAVIES
Professor of Theoretical Physics, University of Newcastle upon Tyne
| CAMBRIDGE
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© Cambridge University Press 1982
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 1982
First paperback edition (with corrections) 1984
Reprinted 1989,1992,1994
A catalogue record for this publication is available from the British Library
Library of Congress catalogue card number: 81-3851
ISBN 978-0-521-23385-9 Hardback
ISBN 978-0-521-27858-4 Paperback
Cambridge University Press has no responsibility for the persistence or
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Contents
Preface vii
Conventions and abbreviations ix
1 Introduction 1
2 Quantum field theory in Minkowski space 10
2.1 Scalar field 10
2.2 Quantization 12
2.3 Energy-momentum 14
2.4 Vacuum energy divergence 15
2.5 Dirac spinor field 17
2.6 Electromagnetic field 19
2.7 Green functions 20
2.8 Path-integral quantization 28
3 Quantum field theory in curved spacetime 36
3.1 Spacetime structure 37
3.2 Scalar field quantization 43
3.3 Meaning of the particle concept: particle detectors 48
3.4 Cosmological particle creation: a simple example 59
3.5 Adiabatic vacuum 62
3.6 Adiabatic expansion of Green functions 73
3.7 Conformal vacuum 79
3.8 Fields of arbitrary spin in curved spacetime 81
4 Flat spacetime examples 89
4.1 Cylindrical two-dimensional spacetime 90
4.2 Use of Green functions 94
4.3 Boundary effects 96
4.4 Moving mirrors 102
4.5 Quantum field theory in Rindler space 109
5 Curved spacetime examples 118
5.1 Robertson-Walker spacetimes 119
5.2 Static Robertson-Walker spacetimes 122
5.3 The Milne universe 124
5.4 De Sitter space 129
5.5 Classification of conformal vacua 138
5.6 Bianchi I spacetimes and perturbation theory 142
vi Contents
6 Stress-tensor renormalization 150
6.1 The fundamental problem 151
6.2 Renormalization in the effective action 159
6.3 Conformal anomalies and the massless case 173
6.4 Computing the renormalized stress-tensor 189
6.5 Other regularization methods 206
6.6 Physical significance of the stress-tensor 214
7 Applications of renormalization techniques 225
7.1 Two-dimensional examples 225
7.2 Robertson- Walker models 232
7.3 Perturbation calculation of the stress-tensor 237
7.4 Cosmological considerations 243
8 Quantum black holes 249
8.1 Particle creation by a collapsing spherical body 250
8.2 Physical aspects of black hole emission 264
8.3 Eternal black holes 275
8.4 Analysis of the stress-tensor 283
8.5 Further developments 287
9 Interacting fields 292
9.1 Calculation of S-matrix elements 292
9.2 Self-interacting scalar field in curved spacetime 301
9.3 Particle production due to interaction 314
9.4 Other effects of interactions 317
References 323
Index 337
Preface
The subject of quantum field theory in curved spacetime, as an approxi-
mation to an as yet inaccessible theory of quantum gravity, has grown
tremendously in importance during the last decade. In this book we have
attempted to collect and unify the vast number of papers that have
contributed to the rapid development of this area. The book also contains
some original material, especially in connection with particle detector
models and adiabatic states.
The treatment is intended to be both pedagogical and archival. We
assume no previous acquaintance with the subject, but the reader should
preferably be familiar with basic quantum field theory at the level of
Bjorken & Drell (1965) and with general relativity at the level of Weinberg
(1972) or Misner, Thorne & Wheeler (1973). The theory is developed from
basics, and many technical expressions are listed for the first time in one
place. The reader's attention is drawn to the list of conventions and
abbreviations on page ix, and the extensive references and bibliography.
In preparing this book we have drawn upon the material of a very large
number of authors. In adapting certain published material (including that
of the authors) we have gratuitously made what we consider to be
corrections, occasionally without explicitly warning the reader that our use
of that material differs from the original publications.
The bulk of the text was written while we worked together at the
Department of Mathematics, Kings College, London. We are greatly
indebted to many colleagues there and elsewhere for assistance. Special
thanks are extended to T.S. Bunch, S.M. Christensen, N.A. Doughty, J.S.
Dowker, M.J. Duff, L.H. Ford, S.A. Fulling, C.J. Isham, G. Kennedy, L.
Parker and R.M. Wald for critical reading of sections of manuscript.
Finally we should like to thank Mrs J. Bunn for typing the manuscript
and the Science Research Council for financial support.
Vll
Preface to the paperback edition
Since the book first went to press, there have been several important
advances in this subject area. The topic of interacting fields in curved space
has been greatly developed, especially in connection with the phenomenon
of symmetry breaking and restoration in the very early universe, where
both high temperatures and spacetime curvature are significant. A
direct consequence of this work has been the formulation of the so-called
inflationary universe scenario, in which the universe undergoes a de Sitter
phase in the very early stages. This work has focussed attention once more
on quantum field theory in de Sitter space, and on the calculation of < <f>2 ).
A comprehensive review of the inflationary scenario is given in The Very
Early Universe, edited by G.W. Gibbons, S.W. Hawking and S.T.C. Siklos
(Cambridge University Press, 1983).
Further results of a technical nature have recently been obtained
concerning a number of the topics considered in this book. Mention should
be made of the work of M.S. Fawcett, who has finally calculated the
quantum stress tensor for a Schwarzschild black hole {Comrmn. Math.
Phys., 81 (1983), 103), and of W.G. Unruh & R.M. Wald, who have clarified
the thermodynamic properties of black holes by appealing to the effects of
accelerated mirrors close to the event horizon {Phys. Rev. D, 25 (1982), 942;
27 (1983), 2271). Interest has also arisen over field theories in higher-
dimensional spacetimes, in which Casimir and other vacuum effects
become important. For a review, see E. Witten, Nucl. Phys. B, 186 (1981),
412. Finally, much further work has been done on the properties of particle
detectors (see, for example, the paper by K.J. Hinton in J. Phys. A: Gen.
Phys., 16 (1983), 1937).
We are grateful to K.J. Hinton, J. Pfautsch, S.D. Unwin and W.R.
Walker for assistance in revising the text.
Note added at 1986 reprinting
We would like to thank Professor H. Minn for providing corrections to
the original printing.
Conventions and abbreviations
Our notation for quantum field theory mainly follows that of Bjorken &
Drell (1965). The sign conventions for the metric and curvature tensors are
( ) in the terminology of Misner, Thome & Wheeler (1973). That is,
the metric signature is (+ ); Ra = dT* -...; R. = R* -
frt i fr lv mv
Formulae can be changed from our notation to the often used Misner,
Thorne & Wheeler (+ + +) conventions by changing the signs of g^,
0 = <TV V , R' , R^, T; but leaving R , K/, R and T,, unchanged.
M V tit aPyS
For the majority of the book we use units in which h = c = G — 1.
The following special symbols and abbreviations are used throughout:
* complex conjugate
t or h.c. Hermitian conjugate
Dirac adjoint
8
OTd Qt ,H partial derivative
V,or;// covariant derivative
Re (Im) real (imaginary) part
tr trace
In natural logarithm
Boltzmann's constant
7 Euler's constant
iA,B] AB-BA
{A,B} AB + BA
|(a , + «,,,„)
M v
approximately equal to
order of magnitude estimate
X, asymptotically approximate to
== defined to be equal to
normal ordering
IX