com INTRODUCTION TO QUANTUM EFFECTS IN GRAVITY This is the first introductory textbook on quantum field theory in gravitational backgrounds intended for undergraduate and beginning graduate students in the fields of theoretical astrophysics, cosmology, particle physics, and string theory. The book covers the basic (but essential) material of quantization of fields in expanding universe and quantum fluctuations in inflationary spacetime. It also contains a detailed explanation of the Casimir, Unruh, and Hawking effects, and introduces the method of effective action used for calculating the backreaction of quantum systems on a classical external gravitational field. The broad scope of the material covered will provide the reader with a thorough perspective of the subject.
Complicated calculations are avoided in favor of sim- pler ones, which still contain the relevant physical concepts. Every major result is derived from first principles and thoroughly explained. The book is self-contained and assumes only a basic knowledge of general relativity. Exercises with detailed solutions are provided throughout the book.
Mukhanov is Professor of Physics at Ludwig-Maximilians University, Munich. His main result is the theory of inflationary cosmological perturbations. Professor Mukhanov is author of Physical Foundations of Cosmol- ogy (Cambridge University Press, 2005). He also serves on the editorial boards of leading research journals and is Scientific Director of the Journal of Cosmology and Astroparticle Physics (JCAP).
Sergei Winitzki is Research Associate in the Department of Physics at Ludwig- Maximilians University, Munich. His main areas of research include quantum cosmology, the theory of dark energy, the global structure of spacetime, and quantum gravity.com INTRODUCTION TO QUANTUM EFFECTS IN GRAVITY VIATCHESLAV MUKHANOV AND SERGEI WINITZKI Ludwig-Maximilians University, Munich www.com cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.org Information on this title: www. Winitzki 2007 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 2007 Printed in the United Kingdom at the University Press, Cambridge A catalog record for this publication is available from the British Library ISBN 978-0-521-86834-1 hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.com Contents Preface page ix Part I Canonical quantization and particle production 1 1 Overview: a taste of quantum fields 3 1.2 Quantum field and its vacuum state 4 1.3 The vacuum energy 7 1.4 Quantum vacuum fluctuations 8 1.5 Particle interpretation of quantum fields 9 1.6 Quantum field theory in classical backgrounds 9 1.7 Examples of particle creation 10 2 Reminder: classical and quantum theory 13 2.3 Quantization of Hamiltonian systems 19 2.4 Hilbert spaces and Dirac notation 20 2.5 Operators, eigenvalue problem and basis in a Hilbert space 22 2.6 Generalized eigenvectors and basic matrix elements 26 2.7 Evolution in quantum theory 29 3 Driven harmonic oscillator 33 3.1 Quantizing an oscillator 33 3.2 The “in” and “out” states 35 3.3 Matrix elements and Green’s functions 38 v www.com vi Contents 4 From harmonic oscillators to fields 42 4.1 Quantum harmonic oscillators 42 4.2 From oscillators to fields 43 4.3 Quantizing fields in a flat spacetime 45 4.4 The mode expansion 48 4.5 Vacuum energy and vacuum fluctuations 50 4.6 The Schrödinger equation for a quantum field 51 5 Reminder: classical fields 54 5.1 The action functional 54 5.2 Real scalar field and its coupling to the gravity 56 5.3 Gauge invariance and coupling to the electromagnetic field 58 5.4 Action for the gravitational and gauge fields 59 5.5 Energy-momentum tensor 61 6 Quantum fields in expanding universe 64 6.1 Classical scalar field in expanding background 64 6.4 Hilbert space; “a- and b-particles” 69 6.5 Choice of the physical vacuum 71 6.1 The instantaneous lowest-energy state 71 6.2 Ambiguity of the vacuum state 74 6.6 Amplitude of quantum fluctuations 78 6.1 Comparing fluctuations in the vacuum and excited states 80 6.7 An example of particle production 81 7 Quantum fields in the de Sitter universe 85 7.1 De Sitter universe 85 7.1 Bunch–Davies vacuum 90 7.3 Fluctuations in inflationary universe 92 8 Unruh effect 97 8.2 Comoving frame of accelerated observer 100 8.3 Quantum fields in inertial and accelerated frames 103 8.5 Occupation numbers and Unruh temperature 107 www.com Contents vii 9 Hawking effect. Thermodynamics of black holes 109 9.2 Kruskal–Szekeres coordinates 111 9.3 Field quantization and Hawking radiation 115 9.4 Hawking effect in 3+1 dimensions 118 9.2 Thermodynamics of black holes 120 9.1 Laws of black hole thermodynamics 121 10 The Casimir effect 124 10.1 Vacuum energy between plates 124 10.2 Regularization and renormalization 125 Part II Path integrals and vacuum polarization 129 11 Path integrals 131 11.2 Propagator as a path integral 132 11.3 Lagrangian path integrals 137 11.4 Propagators for free particle and harmonic oscillator 138 11.3 Euclidean path integral 142 11.4 Ground state as a path integral 144 12 Effective action 146 12.1 Driven harmonic oscillator (continuation) 146 12.1 Green’s functions and matrix elements 146 12.2 Euclidean Green’s function 148 12.3 Introducing effective action 150 12.4 Calculating effective action for a driven oscillator 152 12.6 The effective action “recipe” 157 12.2 Effective action in external gravitational field 159 12.1 Euclidean action for scalar field 161 12.3 Effective action as a functional determinant 163 12.1 Reformulation of the eigenvalue problem 164 12.3 Heat kernel 167 www.com viii Contents 13 Calculation of heat kernel 170 13.1 Perturbative expansion for the heat kernel 171 13.2 Trace of the heat kernel 176 13.3 The Seeley–DeWitt expansion 178 14 Results from effective action 180 14.1 Renormalization of the effective action 180 14.2 Finite terms in the effective action 183 14.1 EMT from the Polyakov action 185 14.3 Conformal anomaly 187 Appendix 1 Mathematical supplement 193 A1.1 Functionals and distributions (generalized functions) 193 A1.2 Green’s functions, boundary conditions, and contours 202 A1.3 Euler’s gamma function and analytic continuations 206 Appendix 2 Backreaction derived from effective action 212 Appendix 3 Mode expansions cheat sheet 216 Appendix 4 Solutions to exercises 218 Index 272 www.com Preface This book is an expanded and reorganized version of the lecture notes for a course taught at the Ludwig-Maximilians University, Munich, in the spring semester of 2003. The course is an elementary introduction to the basic concepts of quantum field theory in classical backgrounds. A certain level of familiarity with general relativity and quantum mechanics is required, although many of the necessary concepts are introduced in the text.
The audience consisted of advanced undergraduates and beginning graduate students. There were 11 three-hour lectures. Each lecture was accompanied by exercises that were an integral part of the exposition and encapsulated longer but straightforward calculations or illustrative numerical results. Detailed solutions were given for all the exercises.
Exercises marked by an asterisk ∗ are more difficult or cumbersome. The book covers limited but essential material: quantization of free scalar fields; driven and time-dependent harmonic oscillators; mode expansions and Bogolyubov transformations; particle creation by classical backgrounds; quantum scalar fields in de Sitter spacetime and the growth of fluctuations; the Unruh effect; Hawking radiation; the Casimir effect; quantization by path integrals; the energy- momentum tensor for fields; effective action and backreaction; regularization of functional determinants using zeta functions and heat kernels. Topics such as quantization of higher-spin fields or interacting fields in curved spacetime, direct renormalization of the energy-momentum tensor, and the theory of cosmological perturbations are left out. The emphasis of this course is primarily on concepts rather than on compu- tational results.
Most of the required calculations have been simplified to the barest possible minimum that still contains all relevant physics. For instance, only free scalar fields are considered for quantization; background spacetimes are always chosen to be conformally flat; the Casimir effect, the Unruh effect, and the Hawking radiation are computed for massless scalar fields in suitable ix www.com x Preface 1 + 1-dimensional spacetimes. Thus a fairly modest computational effort suffices to explain important conceptual issues such as the nature of vacuum and parti- cles in curved spacetimes, thermal effects of gravitation, and backreaction. This should prepare students for more advanced and technically demanding treatments suggested below.
The authors are grateful to Josef Gaßner and Matthew Parry for discussions and valuable comments on the manuscript. Special thanks are due to Alex Vikman who worked through the text and prompted important revisions, and to Andrei Barvinsky for his assistance in improving the presentation in the last chapter. The entire book was typeset with the excellent LyX and TEX document prepa- ration system on computers running Debian GNU/Linux. We wish to express our gratitude to the creators and maintainers of this outstanding free software.
Suggested literature The following books offer a more extensive coverage of the subject and can be studied as a continuation of this introductory course. Davies, Quantum Fields in Curved Space (Cambridge University Press, 1982). Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge University Press, 1989). Mostepanenko, Vacuum Quantum Effects in Strong Fields (Friedmann Laboratory Publishing, St.com Part I Canonical quantization and particle production www.com 1 Overview: a taste of quantum fields Summary Quantum fields as a set of harmonic oscillators.
Particle interpretation of field theory. Examples of particle production by external fields. We begin with a few elementary observations concerning the vacuum in quantum field theory.1 Classical field A classical field is described by a function x t, where x is a three-dimensional coordinate in space and t is the time. At every point the function x t takes values in some finite-dimensional “configuration space” and can be a scalar, vector, or tensor.
The simplest example is a real scalar field x t whose strength is charac- terized by real numbers. A free massive scalar field satisfies the Klein–Gordon equation 2 3 2 ¨ − + m2 = 0 − + m2 ≡ (1.1) t2 j=1 xj2 which has a unique solution x t for t > t0 provided that the initial conditions ˙ x t0 are specified. x t0 and Formally one can describe a free scalar field as a set of decoupled “harmonic oscillators.” To explain why this is so it is convenient to begin by considering a field x t not in infinite space but in a box of finite volume V , with some boundary conditions imposed on the field . The volume V should be large enough to avoid artifacts induced by the finite size of the box or by physically irrelevant boundary conditions.
For example, one might choose the box as a cube 3 www.com 4 Overview: a taste of quantum fields with sides of length L and volume V = L3 , and impose the periodic boundary conditions, x = 0 y z t = x = L y z t and similarly for y and z. The Fourier decomposition is then 1 x t = √ k teik·x (1.2) V k where the sum goes over three-dimensional wavenumbers k with components 2nx kx = nx = 0 ±1 ±2 L √ and similarly for ky and kz. The normalization factor V in equation (1.2) is chosen to simplify formulae (in principle, one could rescale the modes k by any constant).1), we find that this equation is replaced by an infinite set of decoupled ordinary differential equations: ¨ k + k2 + m2 k = 0 with one equation for each k In other words, each complex function k t satisfies the harmonic oscillator equation with the frequency k ≡ k +m 2 2 where k ≡ k.