Microlocal Analysis of Quantum Fields on Curved Spacetimes 9783037190944

Table of contents :
Introduction......Page 10
Content......Page 11
Notation......Page 14
Minkowski spacetime......Page 16
The Klein–Gordon equation......Page 18
Pre-symplectic space of test functions......Page 21
The complex case......Page 23
Bosonic Fock space......Page 24
Fock quantization of the Klein–Gordon equation......Page 26
Quantum spacetime fields......Page 27
Local algebras......Page 28
Vector spaces......Page 30
Bilinear and sesquilinear forms......Page 31
Algebras......Page 32
States......Page 33
CCR algebras......Page 34
Quasi-free states......Page 35
Covariances of quasi-free states......Page 39
The GNS representation of quasi-free states......Page 42
Pure quasi-free states......Page 46
Examples......Page 50
Background......Page 52
Lorentzian manifolds......Page 56
Globally hyperbolic spacetimes......Page 61
Klein–Gordon equations on Lorentzian manifolds......Page 66
Symplectic spaces......Page 70
Quasi-free states on curved spacetimes......Page 74
Consequences of unique continuation......Page 77
Conformal transformations......Page 78
Wavefront set of distributions......Page 80
Operations on distributions......Page 83
Hörmander's theorem......Page 85
The distinguished parametrices of a Klein–Gordon operator......Page 86
The need for renormalization......Page 90
Old definition of Hadamard states......Page 92
The microlocal definition of Hadamard states......Page 94
The theorems of Radzikowski......Page 95
Equivalence of the two definitions......Page 97
Examples of Hadamard states......Page 99
Existence of Hadamard states......Page 100
Ground states and KMS states......Page 102
Klein–Gordon operators......Page 106
The Klein–Gordon equation on stationary spacetimes......Page 108
Reduction......Page 109
Ground and KMS states for P......Page 110
Hadamard property......Page 111
Pseudodifferential calculus on Rn......Page 112
Pseudodifferential operators on a manifold......Page 115
Riemannian manifolds of bounded geometry......Page 117
The Shubin calculus......Page 120
Seeley's theorem......Page 122
Egorov's theorem......Page 123
Hadamard condition on Cauchy surface covariances......Page 124
Model Klein–Gordon operators......Page 125
Parametrices for the Cauchy problem......Page 126
Spacetime covariances and Feynman inverses......Page 132
Klein–Gordon operators on Lorentzian manifolds of bounded geometry......Page 134
Conformal transformations......Page 135
Hadamard states on general spacetimes......Page 136
Analytic Hadamard states and Wick rotation......Page 138
Boundary values of holomorphic functions......Page 139
The analytic wavefront set......Page 140
Analytic Hadamard states......Page 142
The Reeh–Schlieder property of analytic Hadamard states......Page 143
Existence of analytic Hadamard states......Page 144
Wick rotation on analytic spacetimes......Page 145
The Calderón projectors......Page 146
The Hadamard state associated to Calderón projectors......Page 147
Examples......Page 149
Klein–Gordon fields inside future lightcones......Page 152
The boundary symplectic space......Page 154
The Hadamard condition on the boundary......Page 156
Construction of pure boundary Hadamard states......Page 159
Asymptotically flat spacetimes......Page 160
The canonical symplectic space on I-......Page 163
Klein–Gordon fields on spacetimes with Killing horizons......Page 168
Spacetimes with bifurcate Killing horizons......Page 169
Wick rotation......Page 171
The double -KMS state in M+M-......Page 173
The extended Euclidean metric and the Hawking temperature......Page 174
The Hartle–Hawking–Israel state......Page 175
Klein–Gordon operators on asymptotically static spacetimes......Page 178
The in and out vacuum states......Page 180
Reduction to a model case......Page 182
Feynman propagator on asymptotically Minkowski spacetimes......Page 186
The Feynman inverse of P......Page 187
Proof of Theorem 16.1......Page 189
CAR *-algebras and quasi-free states......Page 196
Clifford algebras......Page 197
Clifford representations......Page 198
Weyl bi-spinors......Page 201
Clifford and spinor bundles......Page 203
Spin structures......Page 205
Spinor connections......Page 206
Dirac operators......Page 207
Dirac equation on globally hyperbolic spacetimes......Page 209
Quantization of the Dirac equation......Page 210
Hadamard states for the Dirac equation......Page 211
Conformal transformations......Page 212
The Weyl equation......Page 213
Relationship between Dirac and Weyl Hadamard states......Page 215
Bibliography......Page 218
General Index......Page 224
Index of Notations......Page 228

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Christian Gérard

Lectures in Mathematics and Physics

Lectures in Mathematics and Physics

Christian Gérard

This book provides a detailed introduction to microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes. We focus on free fields and the corresponding quasi-free states and more precisely on Klein–Gordon fields and Dirac fields. The first chapters are devoted to preliminary material on CCR*-algebras, quasi-free states, wave equations on Lorentzian manifolds, microlocal analysis and to the important Hadamard condition, characterizing physically acceptable quantum states on curved spacetimes. In the later chapters more advanced tools of microlocal analysis, like the global pseudo-differential calculus on non-compact manifolds, are used to construct and study Hadamard states for Klein–Gordon fields by various methods, in particular by scattering theory and by Wick rotation arguments. In the last chapter the fermionic theory of free Dirac quantum fields on Lorentzian manifolds is described in some detail. This monograph is adressed to both mathematicians and mathematical physicists. The first will be able to use it as a rigorous exposition of free quantum fields on curved spacetimes and as an introduction to some interesting and physically important problems arising in this domain. The second may find this text a useful introduction and motivation to the use of more advanced tools of microlocal analysis in this area of research.

ISBN 978-3-03719-094-4

www.ems-ph.org

ESI Gérard | Font: Rotis Sans., Times Ten | 4 Farben: Euroscala | RS: 11.8 mm

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Christian Gérard

Microlocal Analysis of Quantum Fields on Curved Spacetimes

ESI Lectures in Mathematics and Physics Editors Christoph Dellago and Ilaria Perugia (University of Vienna, Austria) Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien Austria The Erwin Schrödinger International Institute for Mathematical Phyiscs is a meeting place for leading experts in mathematical physics and mathematics, nurturing the development and exchange of ideas in the international community, particularly stimulating intellectual exchange between scientists from Eastern Europe and the rest of the world. The purpose of the series ESI Lectures in Mathematics and Physics is to make selected texts arising from its research programme better known to a wider community and easily available to a worldwide audience. It publishes lecture notes on courses given by internationally renowned experts on highly active research topics. In order to make the series attractive to graduate students as well as researchers, special emphasis is given to concise and lively presentations with a level and focus appropriate to a student‘s background and at prices commensurate with a student‘s means. Previously published in this series: Arkady L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations Werner Ballmann, Lectures on Kähler Manifolds Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization Recent Developments in Pseudo-Riemannian Geometry, Dmitri V. Alekseevsky and Helga Baum (Eds.) Boltzmann‘s Legacy, Giovanni Gallavotti, Wolfgang L. Reiter and Jakob Yngvason (Eds.) Hans Ringström, The Cauchy Problem in General Relativity_ Emil J. Straube, Lectures on the 2-Sobolev Theory of the ∂ -Neumann Problem Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, Alan L. Carey (Eds.) Erwin Schrödinger – 50 Years After, Wolfgang L. Reiter and Jakob Yngvason (Eds.)

Christian Gérard

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Author: Christian Gérard Département de Mathématiques Bâtiment 307 Faculté des Sciences d’Orsay Université Paris-Sud 91405 Orsay Cedex France E-mail: [email protected]

2000 Mathematics Subject Classification (primary; secondary): 81T13, 35L10, 35L40, 58J40; 81T28, 35L15, 35L45, 58J47, 53C50 Key words: Quantum Field Theory, curved spacetimes, Hadamard states, microlocal analysis, pseudo-differential calculus

ISBN 978-3-03719-094-4 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2019 European Mathematical Society Contact address: European Mathematical Society Publishing House TU Berlin Mathematikgebäude Straße des 17. Juni 136 10623 Berlin Germany

Email: [email protected] Homepage: www.ems-ph.org

Typeset using the authors’ TEX files: le-tex publishing services GmbH, Leipzig, Germany Printing: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 987654321

Contents

1 Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Free Klein–Gordon fields on Minkowski spacetime 2.1 Minkowski spacetime . . . . . . . . . . . . . 2.2 The Klein–Gordon equation . . . . . . . . . . 2.3 Pre-symplectic space of test functions . . . . . 2.4 The complex case . . . . . . . . . . . . . . .

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3 Fock quantization on Minkowski space 3.1 Bosonic Fock space . . . . . . . . . . . . . . . 3.2 Fock quantization of the Klein–Gordon equation 3.3 Quantum spacetime fields . . . . . . . . . . . . 3.4 Local algebras . . . . . . . . . . . . . . . . . .

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4 CCR algebras and quasi-free states 4.1 Vector spaces . . . . . . . . . . . . . . . . . 4.2 Bilinear and sesquilinear forms . . . . . . . 4.3 Algebras . . . . . . . . . . . . . . . . . . . 4.4 States . . . . . . . . . . . . . . . . . . . . . 4.5 CCR algebras . . . . . . . . . . . . . . . . 4.6 Quasi-free states . . . . . . . . . . . . . . . 4.7 Covariances of quasi-free states . . . . . . . 4.8 The GNS representation of quasi-free states . 4.9 Pure quasi-free states . . . . . . . . . . . . 4.10 Examples . . . . . . . . . . . . . . . . . . .

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5 Free Klein–Gordon fields on curved spacetimes 5.1 Background . . . . . . . . . . . . . . . . . . . . 5.2 Lorentzian manifolds . . . . . . . . . . . . . . . 5.3 Stationary and static spacetimes . . . . . . . . . . 5.4 Globally hyperbolic spacetimes . . . . . . . . . . 5.5 Klein–Gordon equations on Lorentzian manifolds 5.6 Symplectic spaces . . . . . . . . . . . . . . . . .

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Quasi-free states on curved spacetimes 65 6.1 Quasi-free states on curved spacetimes . . . . . . . . . . . . . . . . 65 6.2 Consequences of unique continuation . . . . . . . . . . . . . . . . . 68 6.3 Conformal transformations . . . . . . . . . . . . . . . . . . . . . . 69

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Microlocal analysis of Klein–Gordon equations 7.1 Wavefront set of distributions . . . . . . . . . . . . . . . . 7.2 Operations on distributions . . . . . . . . . . . . . . . . . 7.3 H¨ormander’s theorem . . . . . . . . . . . . . . . . . . . . 7.4 The distinguished parametrices of a Klein–Gordon operator

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Hadamard states 8.1 The need for renormalization . . . . . . . . . . . . 8.2 Old definition of Hadamard states . . . . . . . . . . 8.3 The microlocal definition of Hadamard states . . . . 8.4 The theorems of Radzikowski . . . . . . . . . . . . 8.5 The Feynman inverse associated to a Hadamard state 8.6 Conformal transformations . . . . . . . . . . . . . 8.7 Equivalence of the two definitions . . . . . . . . . . 8.8 Examples of Hadamard states . . . . . . . . . . . . 8.9 Existence of Hadamard states . . . . . . . . . . . .

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Vacuum and thermal states on stationary spacetimes 9.1 Ground states and KMS states . . . . . . . . . . . . . 9.2 Klein–Gordon operators . . . . . . . . . . . . . . . . 9.3 The Klein–Gordon equation on stationary spacetimes 9.4 Reduction . . . . . . . . . . . . . . . . . . . . . . . 9.5 Ground and KMS states for P . . . . . . . . . . . . . 9.6 Hadamard property . . . . . . . . . . . . . . . . . .

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11 Construction of Hadamard states by pseudodifferential calculus 11.1 Hadamard condition on Cauchy surface covariances . . . . . 11.2 Model Klein–Gordon operators . . . . . . . . . . . . . . . . 11.3 Parametrices for the Cauchy problem . . . . . . . . . . . . . 11.4 The pure Hadamard state associated to a microlocal splitting .

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10 Pseudodifferential calculus on manifolds 10.1 Pseudodifferential calculus on Rn . . . . . . 10.2 Pseudodifferential operators on a manifold . 10.3 Riemannian manifolds of bounded geometry 10.4 The Shubin calculus . . . . . . . . . . . . . 10.5 Time-dependent pseudodifferential operators 10.6 Seeley’s theorem . . . . . . . . . . . . . . . 10.7 Egorov’s theorem . . . . . . . . . . . . . .

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11.5 Spacetime covariances and Feynman inverses . . . . . . . . . . . . 11.6 Klein–Gordon operators on Lorentzian manifolds of bounded geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7 Conformal transformations . . . . . . . . . . . . . . . . . . . . . 11.8 Hadamard states on general spacetimes . . . . . . . . . . . . . . . 12 Analytic Hadamard states and Wick rotation 12.1 Boundary values of holomorphic functions . . . . . . . . 12.2 The analytic wavefront set . . . . . . . . . . . . . . . . . 12.3 Analytic Hadamard states . . . . . . . . . . . . . . . . . 12.4 The Reeh–Schlieder property of analytic Hadamard states 12.5 Existence of analytic Hadamard states . . . . . . . . . . . 12.6 Wick rotation on analytic spacetimes . . . . . . . . . . . 12.7 The Calder´on projectors . . . . . . . . . . . . . . . . . . 12.8 The Hadamard state associated to Calder´on projectors . . 12.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Hadamard states and characteristic Cauchy problem 13.1 Klein–Gordon fields inside future lightcones . . . 13.2 The boundary symplectic space . . . . . . . . . . 13.3 The Hadamard condition on the boundary . . . . . 13.4 Construction of pure boundary Hadamard states . 13.5 Asymptotically flat spacetimes . . . . . . . . . . 13.6 The canonical symplectic space on I  . . . . . .

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14 Klein–Gordon fields on spacetimes with Killing horizons 14.1 Spacetimes with bifurcate Killing horizons . . . . . . . . . . 14.2 Klein–Gordon fields . . . . . . . . . . . . . . . . . . . . . . 14.3 Wick rotation . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 The double ˇ-KMS state in MC [ M . . . . . . . . . . . 14.5 The extended Euclidean metric and the Hawking temperature 14.6 The Hartle–Hawking–Israel state . . . . . . . . . . . . . . .

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15 Hadamard states and scattering theory 169 15.1 Klein–Gordon operators on asymptotically static spacetimes . . . . . 169 15.2 The in and out vacuum states . . . . . . . . . . . . . . . . . . . . . 171 15.3 Reduction to a model case . . . . . . . . . . . . . . . . . . . . . . . 173 16 Feynman propagator on asymptotically Minkowski spacetimes 177 16.1 Klein–Gordon operators on asymptotically Minkowski spacetimes . . 178 16.2 The Feynman inverse of P . . . . . . . . . . . . . . . . . . . . . . . 178 16.3 Proof of Theorem 16.1 . . . . . . . . . . . . . . . . . . . . . . . . . 180

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17 Dirac fields on curved spacetimes 17.1 CAR -algebras and quasi-free states . . . . . . . . . . 17.2 Clifford algebras . . . . . . . . . . . . . . . . . . . . . 17.3 Clifford representations . . . . . . . . . . . . . . . . . 17.4 Spin groups . . . . . . . . . . . . . . . . . . . . . . . 17.5 Weyl bi-spinors . . . . . . . . . . . . . . . . . . . . . 17.6 Clifford and spinor bundles . . . . . . . . . . . . . . . 17.7 Spin structures . . . . . . . . . . . . . . . . . . . . . . 17.8 Spinor connections . . . . . . . . . . . . . . . . . . . . 17.9 Dirac operators . . . . . . . . . . . . . . . . . . . . . . 17.10 Dirac equation on globally hyperbolic spacetimes . . . 17.11 Quantization of the Dirac equation . . . . . . . . . . . 17.12 Hadamard states for the Dirac equation . . . . . . . . . 17.13 Conformal transformations . . . . . . . . . . . . . . . 17.14 The Weyl equation . . . . . . . . . . . . . . . . . . . . 17.15 Relationship between Dirac and Weyl Hadamard states .

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Bibliography

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General Index

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Index of Notations

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Chapter 1

Introduction

1.1 Introduction Quantum Field Theory arose from the need to unify Quantum Mechanics with special relativity. It is usually formulated on the flat Minkowski spacetime, on which classical field equations, such as the Klein–Gordon, Dirac or Maxwell equations are easily defined. Their quantization rests on the so-called Minkowski vacuum, which describes a state of the quantum field containing no particles. The Minkowski vacuum is also fundamental for the perturbative or non-perturbative construction of interacting theories, corresponding to the quantization of non-linear classical field equations. Quantum Field Theory on Minkowski spacetime relies heavily on its symmetry under the Poincar´e group. This is apparent in the ubiquitous role of plane waves in the analysis of classical field equations, but more importantly in the characterization of the Minkowski vacuum as the unique state which is invariant under the Poincar´e group and has some energy positivity property. Quantum Field Theory on curved spacetimes describes quantum fields in an external gravitational field, represented by the Lorentzian metric of the ambient spacetime. It is used in situations when both the quantum nature of the fields and the effect of gravitation are important, but the quantum nature of gravity can be neglected in a first approximation. Its non-relativistic analog would be for example ordinary Quantum Mechanics, i.e. the Schr¨odinger equation, in a classical exterior electromagnetic field. Its most important areas of application are the study of phenomena occurring in the early universe and in the vicinity of black holes, and its most celebrated result is the discovery by Hawking that quantum particles are created near the horizon of a black hole. The symmetries of the Minkowski spacetime, which play such a fundamental role, are absent in curved spacetimes, except in some simple situations, like stationary or static spacetimes. Therefore, the traditional approach to quantum field theory has to be modified: one has first to perform an algebraic quantization, which for free theories amounts to introducing an appropriate phase space, which is either a symplectic or an Euclidean space, in the bosonic or fermionic case. From such a phase space one can construct CCR or CAR -algebras, and actually nets of -algebras, each associated to a region of spacetime. The second step consists in singling out, among the many states on these algebras, the physically meaningful ones, which should resemble the Minkowski vacuum, at least in the vicinity of any point of the spacetime. This leads to the notion of Hadamard states, which were originally defined by requiring that their two-point functions have a specific asymptotic expansion near the diagonal, called the Hadamard expansion.

2

1 Introduction

A very important progress was made by Radzikowski, [R1, R2], who introduced the characterization of Hadamard states by the wavefront set of their two-point functions. The wavefront set of a distribution is the natural way to describe its singularities in the cotangent space, and lies at the basis of microlocal analysis, a fundamental tool in the analysis of linear and non-linear partial differential equations. Among its avatars in the physics literature are, for example, the geometrical optics in wave propagation and the semi-classical limit in Quantum Mechanics. The introduction of microlocal analysis in quantum field theory on curved spacetimes started a period of rapid progress, non only for free (i.e. linear) quantum fields, but also for the perturbative construction of interacting fields by Brunetti and Fredenhagen [BF]. For free fields it allowed to use several fundamental results of microlocal analysis, like H¨ormander’s propagation of singularities theorem and the classification of parametrices for Klein–Gordon operators by Duistermaat and H¨ormander.

1.2 Content The goal of these lecture notes is to give an exposition of microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes. We will focus on free fields and the corresponding quasi-free states and more precisely on Klein–Gordon fields, obtained by quantization of linear Klein–Gordon equations on Lorentzian manifolds, although the case of Dirac fields will be described in Chapter 17. There exist already several good textbooks or lecture notes on quantum field theory in curved spacetimes. Among them let us mention the book by B¨ar, Ginoux and Pfaeffle [BGP], the lecture notes [BFr] and [BDFY], the more recent book by Rejzner [Re], and the survey by Benini, Dappiagi and Hack [BDH]. There exist also more physics oriented books, like the books by Wald [W2], Fulling [F] and Birrell and Davies [BD]. Several of these texts contain important developments which are not described here, like the perturbative approach to interacting theories, or the use of category theory. In this lecture notes we focus on advanced methods from microlocal analysis, like for example pseudodifferential calculus, which turn out to be very useful in the study and construction of Hadamard states. Pure mathematicians working in partial differential equations are often deterred by the traditional formalism of quantum field theory found in physics textbooks, and by the fact that the construction of interacting theories is, at least for the time being, restricted to perturbative methods. We hope that these lecture notes will convince them that quantum field theory on curved spacetimes is full of interesting and physically important problems, with a nice interplay between algebraic methods, Lorentzian geometry and microlocal methods in partial differential equations. On the other hand, mathematical physicists with a traditional education, which may lack familiarity with more advanced tools of microlocal analysis, can use this text as an introduction and motivation to the use of these methods.

1.2 Content

3

Let us now give a more detailed description of these lecture notes. The reader may also consult the introduction of each chapter for more information. For pedagogical reasons, we have chosen to devote Chapters 2 and 3 to a brief outline of the traditional approach to quantization of Klein–Gordon fields on Minkowski spacetime, but the impatient reader can skip them without trouble. Chapter 4 deals with CCR -algebras and quasi-free states. A reader with a PDE background may find the reading of this chapter a bit tedious. Nevertheless, we think it is worth the effort to get familiar with the notions introduced there. In Chapter 5 we describe well-known concepts and results concerning Lorentzian manifolds and Klein–Gordon equations on them. The most important are the notion of global hyperbolicity, a property of a Lorentzian manifold implying global solvability of the Cauchy problem, and the causal propagator and the various symplectic spaces associated to it. In Chapter 6 we discuss quasi-free states for Klein–Gordon fields on curved spacetimes, which is a concrete application of the abstract formalism in Chapter 4. Of interest are the two possible descriptions of a quasi-free state, either by it spacetime covariances, or by its Cauchy surface covariances, which are both important in practice. Another useful point is the discussion of conformal transformations. Chapter 7 is devoted to the microlocal analysis of Klein–Gordon equations. We collect here various well-known results about wavefront sets, H¨ormander’s propagation of singularities theorem and its related study with Duistermaat of distinguished parametrices for Klein–Gordon operators, which play a fundamental role in quantized Klein–Gordon fields. In Chapter 8 we introduce the modern definition of Hadamard states due to Radzikowski and discuss some of its consequences. We explain the equivalence with the older definition based on Hadamard expansions and the well-known existence result by Fulling, Narcowich and Wald. In Chapter 9 we discuss ground states and thermal states, first in an abstract setting, then for Klein–Gordon operators on stationary spacetimes. Ground states share the symmetries of the background stationary spacetime and are the natural analogs of the Minkowski vacuum. In particular, they are the simplest examples of Hadamard states. Chapter 10 is devoted to an exposition of a global pseudodifferential calculus on non compact manifolds, the Shubin calculus. This calculus is based on the notion of manifolds of bounded geometry and is a natural generalization of the standard uniform calculus on Rn . Its most important properties are the Seeley and Egorov theorems. In Chapter 11 we explain the construction of Hadamard states using the pseudodifferential calculus in Chapter 10. The construction is done, after choosing a Cauchy surface, by a microlocal splitting of the space of Cauchy data obtained from a global construction of parametrices for the Cauchy problem. It can be applied to many spacetimes of physical interest, like the Kerr–Kruskal and Kerr–de Sitter spacetimes. In Chapter 12 we construct analytic Hadamard states by Wick rotation, a wellknown procedure in Minkowski spacetime. Analytic Hadamard states are defined on analytic spacetimes, by replacing the usual C 1 wavefront set by the analytic

4

1 Introduction

wavefront set, which describes the analytic singularities of distributions. Like the Minkowski vacuum, they have the important Reeh–Schlieder property. After Wick rotation, the hyperbolic Klein–Gordon operator becomes an elliptic Laplace operator, and analytic Hadamard states are constructed using a well-known tool from elliptic boundary value problems, namely the Calder´on projector. In Chapter 13 we describe the construction of Hadamard states by the characteristic Cauchy problem. This amounts to replacing the space-like Cauchy surface in Chapter 11 by a past or future lightcone, choosing its interior as the ambient spacetime. From the trace of solutions on this cone one can introduce a boundary symplectic space, and it turns out that it is quite easy to characterize states on this symplectic space which generate a Hadamard state in the interior. Its main application is the conformal wave equation on spacetimes which are asymptotically flat at past or future null infinity. We also describe in this chapter the BMS group of asymptotic symmetries of these spacetimes, and its relationship with Hadamard states. In Chapter 14 we discuss Klein–Gordon fields on spacetimes with Killing horizons. Our aim is to explain a phenomenon loosely related with the Hawking radiation, namely the existence of the Hartle–Hawking–Israel vacuum, on spacetimes having a stationary Killing horizon. The construction and properties of this state follow from the Wick rotation method already used in Chapter 12, the Calder´on projectors playing also an important role. Chapter 15 is devoted to the construction of Hadamard states by scattering theory methods. We consider spacetimes which are asymptotically static at past or future time infinity. In this case one can define the in and out vacuum states, which are states asymptotic to the vacuum state at past or future time infinity. Using the tools from Chapters 10, 11 we prove that these states are Hadamard states. In Chapter 16 we discuss the notion of Feynman inverses. It is known that a Klein– Gordon operator on a globally hyperbolic spacetime admits Feynman parametrices, which are unique modulo smoothing operators and characterized by the wavefront set of its distributional kernels. One can ask if one can also define a unique, canonical true inverse, having the correct wavefront set. We give a positive answer to this question on spacetimes which are asymptotically Minkowski. Chapter 17 is devoted to the quantization of the Dirac equation and to the definition of Hadamard states for Dirac quantum fields. The Dirac equation on a curved spacetime describes an electron-positron field which is a fermionic field, and the CCR -algebra for the Klein–Gordon field has to be replaced by a CAR -algebra. Apart from this difference, the theory for fermionic fields is quite parallel to the bosonic case. We also describe the quantization of the Weyl equation, which originally was thought to describe massless neutrinos.

1.2.1 Acknowledgments. The results described in Chapters 11, 12, 15, and part of those in Chapters 10 and 13, originate from common work with Michal Wrochna, over a period of several years. I learned a lot of what I know about quantum field theory from my long collaboration with Jan Derezinski, and several parts of these lecture notes, like Chapters 4

1.3 Notation

5

and 5 borrow a lot from our common book [DG]. I take the occasion here to express my gratitude to him. Finally, I also greatly profited from discussions with members of the AQFT community. Among them I would like to especially thank Claudio Dappiagi, Valter Moretti, Nicola Pinamonti, Igor Khavkine, Klaus Fredenhagen, Detlev Bucholz, Wojciech Dybalski, Kasia Rejzner, Dorothea Bahns, Rainer Verch, Stefan Hollands and Ko Sanders.

1.3 Notation We now collect some notation that we will use. 1 We set hi D .1 C 2 / 2 for  2 R. We write A b B if A is relatively compact in B.  If X; Y are sets and f W X ! Y we write f W X  ! Y if f is bijective. If X; Y are equipped with topologies, we write f W X ! Y if the map is continuous, and  ! Y if it is a homeomorphism. f WX 

1.3.1 Scale of abstract Sobolev spaces. Let H a real or complex Hilbert space and A a selfadjoint operator on H. We write A > 0 if A  0 and Ker A D f0g. If A > 0 and s 2 R, we equip Dom As with the scalar product .ujv/s D s .A ujAs v/ and the norm kAs uk. We denote by As H the completion of Dom As for this norm, which is a (real or complex) Hilbert space.

Chapter 2

Free Klein–Gordon fields on Minkowski spacetime Almost all textbooks on quantum field theory start with the quantization of the free (i.e. linear) Klein–Gordon and Dirac equations on Minkowski spacetime. The traditional exposition rests on the so-called frequency splitting, which amounts to splitting the space of solutions of, say, the Klein–Gordon equation into two subspaces, corresponding to solutions having positive/negative energy, or equivalently whose Fourier transforms are supported in the upper/lower mass hyperboloid. One then proceeds with the introduction of Fock spaces and the definition of quantized Klein–Gordon or Dirac fields using creation/annihilation operators. Since it relies on the use of the Fourier transformation, this method does not carry over to Klein–Gordon fields on curved spacetimes. More fundamentally, it has the drawback of mixing two different steps in the quantization of the Klein–Gordon equation. The first, purely algebraic step consists in using the symplectic nature of the Klein–Gordon equation to introduce an appropriate CCR -algebra. The second step consists in choosing a state on this algebra, which on the Minkowski spacetime is the vacuum state. Nevertheless it is useful to keep in mind the Minkowski spacetime as an important example. This chapter is devoted to the classical theory of the Klein–Gordon equation on Minkowski spacetime, i.e. to its symplectic structure. Its Fock quantization will be described in Chapter 3.

2.1 Minkowski spacetime In the sequel we will use notation introduced later in Section 4.1. The elements of Rn D Rt  Rdx will be denoted by x D .t; x/, those of the dual n 0 .R / by  D .; k/.

2.1.1 The Minkowski spacetime. Definition 2.1. The Minkowski spacetime R1;d is R1Cd equipped with the bilinear form  2 Ls .R1Cd ; .R1Cd /0 / given by x x D t 2 C x2 :

(2.1)

8

2 Free Klein–Gordon fields on Minkowski spacetime

Definition 2.2. (1) A vector x 2 R1;d is time-like if x x < 0, null if x x D 0, causal if x x  0, and space-like if x x > 0. (2) C˙ D fx 2 R1;d W x x < 0; ˙t > 0g, resp. C ˙ D fx 2 R1;d W x x  0; ˙t  0g are called the open, resp. closed future/past (solid) lightcones. (3) N D fx 2 R1;d W x  x D 0g, resp. N˙ D N \ f˙t  0g are called the null cone resp. future/past null cones. There is a similar classification of vector subspaces of R1;d . Definition 2.3. A linear subspace V of R1;d is time-like if it contains both spacelike and time-like vectors, null if it is tangent to the null cone N and space-like if it contains only space-like vectors. Definition 2.4. (1) If K  R1;d , I˙ .K/ D K C C˙ , resp. J˙ .K/ is called the time-like, resp. causal future/past of K, and J.K/ J .K/ the causal shadow of K.

D K C C ˙ , D JC .K/ [

(2) Two sets K1 , K2 are called causally disjoint if K1 \J.K2/ D ; or, equivalently, if J.K1/ \ K2 D ;. (3) A function f on Rn is called space-compact, resp. future/past space-compact, if supp f  J.K/, resp. supp f  J˙ .K/ for some compact set K b Rn . The 1 spaces of smooth such functions will be denoted by Csc1 .Rn /, resp. Csc;˙ .Rn /.

2.1.2 The Lorentz and Poincar´e groups. Definition 2.5. (1) The pseudo-Euclidean group O.R1Cd ; / is denoted by O.1; d / and is called the Lorentz group. (2) SO.1; d / is the subgroup of L 2 O.1; d / with det L D 1. (3) If L 2 O.1; d / one has L.JC / D JC or L.JC / D J . In the first case L is called orthochronous and in the second anti-orthochronous. (4) The subgroup of orthochronous elements of SO.1; d / is denoted by SO " .1; d / and called the restricted Lorentz group. Definition 2.6. The .restricted/ Poincar´e group is the set P .1; d / D Rn SO " .1; d / equipped with the product .a2 ; L2 /  .a1 ; L1 / D .a2 C L2 a1 ; L2 L1 /: The Poincar´e group acts on Rn by ƒx D Lx C a for ƒ D .a; L/ 2 P .1; d /.

2.2 The Klein–Gordon equation

9

2.2 The Klein–Gordon equation We recall that the differential operator P D  C m2 D @2t 

d X

@2x i C m2 ;

i D1

for m  0 is called the Klein–Gordon operator. 1 We set .k/ D .k 2 C m2 / 2 and denote by  D .Dx / the ´ Fourier multiplier defined by F .u/.k/ D .k/u.k/, where F u.k/ D .2/d=2 eikxu.x/d x is the (unitary) Fourier transform. Note that  C m2 D @2t C  2 . The Klein–Gordon equation   C m2  D 0

(2.2)

is the simplest relativistic field equation. Its quantization describes a scalar bosonic field of mass m. The wave equation (m D 0) is a particular case of the Klein–Gordon equation. Note that since  C m2 preserves real functions, the Klein–Gordon equation has real solutions, which are associated to neutral fields, corresponding to neutral particles, while the complex solutions are associated to charged fields, corresponding to charged particles. It will be more convenient later to consider complex solutions, but in this chapter we will, as is usual in the physics literature, consider mainly real solutions. The case of complex solutions will be briefly discussed in Section 2.4. We refer the reader to Chapter 4 for a general discussion of the real vs complex formalism in a more abstract framework. We are interested in the space of its smooth, space-compact, real solutions denoted by Solsc;R .KG/. Solsc;R .KG/ is invariant under the Poincar´e group if we set ˛ƒ .x/ D .ƒ1 x/;

ƒ 2 P .1; d /:

(2.3)

2.2.1 The Cauchy problem. If  2 C 1 .Rn / and t 2 R we set .t/.x/ D

.t; x/ 2 C 1 .Rd /. Any solution  2 Solsc;R .KG/ is determined by its Cauchy data on the Cauchy surface †s D ft D sg Rd , defined by the map   .s/  (2.4) %s   D D f 2 C01 .Rd I R2 /: @t .s/ The unique solution in Solsc;R .KG/ of the Cauchy problem  . C m2 / D 0; %s  D f;

(2.5)

is denoted by  D Us f and given by .t/ D cos..t  s//f0 C  1 sin..t  s//f1 ;

 f D

f0 f1

 :

(2.6)

10

2 Free Klein–Gordon fields on Minkowski spacetime

The map Us is called the Cauchy evolution operator. The following proposition expresses the important causality property of Us . Proposition 2.7. One has supp Us f  J.fsg  supp f /:

2.2.2 Advanced and retarded inverses. Let us now consider the inhomogeneous Klein–Gordon equation . C m2 /u D v;

(2.7)

where for simplicity v 2 C01 .Rn /. Since there are plenty of homogeneous solutions, it is necessary to supplement (2.7) by support conditions to obtain unique solutions, by requiring that  vanishes for large negative or positive times. Theorem 2.8. (1) There exist unique solutions uret=adv D Gret=adv v 2 Csc1˙ .Rn / of (2.7). Setting Gret=adv .t/ D ˙ .˙t/ 1 sin.t/; (2.8) where .t/ D ½Œ0;C1Œ .t/ is the Heaviside function, one has ˆ Gret=adv .t  s/v.s; /dsI Gret=adv v.t; / D

(2.9)

R

(2) one has supp Gret=adv v  J˙ .supp v/. The operators Gret=adv are called the retarded/advanced inverses of P . Let us equip C01 .Rn / with the scalar product ˆ .ujv/Rn D uvdx; (2.10) Rn

and C01 .Rd I C2 / with the scalar product ˆ   .f jg/Rd D f 1 g1 C f 0 g0 d x:

(2.11)

Rd

It follows from (2.8) that

 Gret=adv D Gadv=ret ;

where A denotes the formal adjoint of A with respect to the scalar product .j/Rn . The operator G D Gret  Gadv (2.12) is called in the physics literature the Pauli–Jordan or commutator function, or also the causal propagator. Note that G D G  ;

supp Gv  J.supp v/;

(2.13)

11

2.2 The Klein–Gordon equation

ˆ

and

 1 sin..t  s//v.s; /ds:

Gv.t; / D

(2.14)

R

There is an important relationship between G and Us . Namely, if we denote by %s W D 0 .Rd I R2 / ! D 0 .Rn / the formal adjoint of %s W C01 .Rn / ! C01 .Rd I R2 / with respect to the scalar products (2.10) and (2.11), then: %s f .t; x/ D ıs .t/ ˝ f0 .x/  ıs0 .t/ ˝ f1 .x/;

f 2 C 1 .Rd I R2 /:

(2.15)

The following lemma follows from (2.6), (2.8) by a direct computation. Lemma 2.9. One has  for D

0

½

½ 0

Us f D G  ı %s ı f;  .

f 2 C01 .Rd I R2 /;

2.2.3 Symplectic structure. It is well-known that the Klein–Gordon equation is a Hamiltonian equation. Indeed let us equip C01 .Rd I R2 / with the symplectic form: ˆ   f1 g0  f0 g1 d x: (2.16) f  g D Rd

If we identify bilinear forms on C01 .Rd I R2 / with linear operators using the scalar product .j/Rd , we have f  g D .f j g/Rd ; where the operator is defined in Lemma 2.9. If we introduce the classical Hamiltonian ˆ  2  1  f Ef D f1 C f0  2 f0 d x 2 Rd and define A 2 L.C01 .Rd I R2 // by f  Ag D f Eg; f; g 2 C01 .Rd I R2 /; 

we obtain that AD

0  2

½ 0

(2.17)

 :

Setting f .t/ D %t U0 f for f 2 C01 .Rd I C2 / we have, by an easy computation f .t/ D etA f;

(2.18)

which shows that f 7! f .t/ is the symplectic flow generated by the classical Hamiltonian E and the symplectic form . In particular, if fi .t/ D etA fi , i D 1; 2, f1 .t/  f2 .t/ is independent on t.

12

2 Free Klein–Gordon fields on Minkowski spacetime

Equivalently, we can equip Solsc;R .KG/ with the symplectic form 1  2 D %t 1  %t 2 ;

(2.19)

where the right-hand side is independent on t. Fixing the reference Cauchy surface †0 Rd , we obtain the following proposition: Proposition 2.10. The Cauchy data map on †0 %0 W .Solsc;R .KG/; / ! .C01 .Rd I R2 /; /; is symplectic, with %1 0 D U0 , where the Cauchy evolution operator Us was introduced in Subsection 2.2.1. This leads to another interpretation of (2.18): the space Solsc;R .KG/ is invariant under the group of time translations s .; x/ D .  s; x/; and s is symplectic on .Solsc;R .KG/; /. Then (2.18) can be rewritten as sA %0 ı s ı %1 0 De ;

s 2 R:

2.3 Pre-symplectic space of test functions By Proposition 2.10, .Solsc;R .KG/; / is a symplectic space. It is easy to see that ˛ƒ defined in (2.3) is symplectic if ƒ is orthochronous, for example using Theorem 2.12 below. If ƒ is anti-orthochronous, ˛ƒ is anti-symplectic, i.e. transforms into  . Identifying .Solsc;R .KG/; / with .C01 .Rd I R2 /; / using %0 is convenient for concrete computations, but destroys Poincar´e invariance, since one fixes the Cauchy surface †0 . It would be useful to have another isomorphic symplectic space which is Poincar´e invariant and at the same time easier to understand than Solsc;R .KG/. It turns out that one can use the space of test functions C01 .Rn I R/, which is a fundamental step in formulating the notion of locality for quantum fields. Proposition 2.11. Consider the map G W C01 .Rn I R/ ! Csc1 .Rn /. Then: .1/ RanG D Solsc;R .KG/; .2/ Ker G D P C01 .Rn I R/: Moreover, we have

.3/ .%0 G/ ı ı .%0 G/ D G:

2.3 Pre-symplectic space of test functions

13

Proof. (1) By P ı G D 0 and Theorem 2.8 (2), we see that RanG  Solsc;R .KG/. Conversely let  2 Solsc;R .KG/. If fs D %s , then by Lemma 2.9 we obtain that ´  D G ı %s ı fs for s 2 R. Hence, if  2 C01 .R/ with .s/ds D 1 we obtain that ˆ D .s/dx D Gv; R ´ for v D  R %s ı fs ds 2 C01 .Rn /. (2) Since G ı P D 0 we have P C01 .Rn I R/  Ker G. Conversely let v 2 1 C0 .Rn I R/ with Gv D 0. Then for uret=adv D Gret=adv v we have uret D uadv D u, u 2 C01 .Rn / by Theorem 2.8 (2) and v D P u since P ı Gret=adv D ½. (3) We have, using (2.14)   ´ 1  ´  sin.s/u.s/ds ; %0 Gu D cos.s/u.s/ds 

hence

ı .%0 G/u D  and

´

´

cos.s/u.s/ds  1 sin.s/u.s/ds

 ;

.%0 G/ f D G%0 f ´ D   1 sin..t  s//.ı0 .s/ ˝ f0  ı00 .s/ ˝ f1 /ds D  1 sin.t/f0 C cos.t/f1 ;

which yields .%0 G/ ı ı .%0 G/u ´ ´ D  1 sin.t/ cos.s/u.s/ds C  1 cos.t/ sin.s/u.s/ds ´ D  1 sin..t  s//u.s/ds D Gu: This completes the proof of the proposition.



One can summarize Propositions 2.10 and 2.11 in the following theorem: Theorem 2.12. (1) The following spaces are symplectic spaces:   1 n C0 .R I R/ ; .jG/Rn ; .Solsc;R .KG/; /; .C01 .Rd I R2 /; /: P C01 .Rn I R/ (2) The following maps are symplectomorphisms:  1 n  %0 C0 .R I R/ G n !.Solsc;R .KG/; / !.C01 .Rd I R2 /; /: ; .jG/ R 1 n P C0 .R I R/ The first and last of these equivalent symplectic spaces are the most useful for the quantization of the Klein–Gordon equation.

14

2 Free Klein–Gordon fields on Minkowski spacetime

2.4 The complex case Let us now discuss the space Solsc;C .KG/ of complex space-compact solutions. We refer to Section 4.2 for notation and terminology. It is more natural to use the map   .s/ %s  D (2.20) i1 @t .s/ as Cauchy data map and to equip the space C01 .Rd I C2 / of Cauchy data with the Hermitian form ˆ    f qg D f 1 g0 C f0 g1 d x: (2.21) Rd

The space Solsc;C .KG/ is similarly equipped with the form  1 q2 D %t  1 q%t 2 ; which is independent on t. The Cauchy evolution operator becomes U0 f .t/ D cos.t/f0 C i 1 sin.t/f1 :

(2.22)

We have then the following analog of Theorem 2.12: Theorem 2.13. (1) The following spaces are Hermitian spaces:   1 n C0 .R I C/ ; .jiG/Rn ; .Solsc;C .KG/; q/; .C01 .Rd I C2 /; q/: P C01 .Rn I C/ (2) The following maps are unitary:   1 n %0 C0 .R I C/ G ; .jiG/Rn !.Solsc;C .KG/; q/ !.C01 .Rd I C2 /; q/: 1 n P C0 .R I C/

Chapter 3

Fock quantization on Minkowski space We describe in this chapter the Fock quantization of the Klein–Gordon equation on Minkowski spacetime. We recall the definition of the bosonic Fock space over a one-particle space and of the creation/annihilation operators, which are ubiquitous notions in quantum field theory. For example, it is common in the physics oriented literature to specify a state for the Klein–Gordon field by defining first some creation/annihilation operators. We will see in Chapter 4 that this is nothing else than choosing a particular K¨ahler structure on a certain symplectic space. In this approach the quantum Klein–Gordon fields are defined as linear operators on the Fock space, so one has to pay attention to domain questions. These technical problems disappear if one uses a more abstract point of view and introduces the appropriate CCR -algebra, as will be done in Chapter 4. Fock spaces will reappear as the (Gelfand–Naimark–Segal) GNS Hilbert spaces associated to a pure quasi-free state on this algebra. Apart from this fact, they can be forgotten.

3.1 Bosonic Fock space 3.1.1 Bosonic Fock space. Let h be a complex Hilbert space whose unit vectors describe the states of a quantum particle. If this particle is bosonic, then the states of a system of n such particles are described by unit vectors in the symmetric tensor power ˝ns h, where we take the tensor products in the Hilbert space sense, i.e. complete the algebraic tensor products for the natural Hilbert norm. A system of an arbitrary number of particles is described by the bosonic Fock space 1 M ˝ns h; (3.1) s .h/ D nD0

where the direct sum is again taken in the Hilbert space sense and ˝0s h D C by definition. We recall that the symmetrized tensor product is defined by ‰1 ˝s ‰2 D ‚s .‰1 ˝ ‰2 /; where ‚s .u1 ˝    ˝ un / D

1 X u.1/ ˝    ˝ u.n/ : nŠ 2Sn

The vector vac D .1; 0; : : : / is called the vacuum and describe a state with no particles at all. A useful observable on s .h/ is the number operator N , which counts the

16

3 Fock quantization on Minkowski space

number of particles, defined by N j˝ns h D n½: The operator N is an example of a second quantized operator, namely N D d .½/, where n X ½˝j 1 ˝ a ˝ ½˝nj ; d .a/j˝ns h D j D1

for a a linear operator on h.

3.1.2 Creation/annihilation operators. Since s .h/ describes an arbitrary number of particles, it is useful to have operators that create or annihilate particles. One defines the creation/annihilation operators by p a .h/‰n D n C 1h ˝s ‰n ; p a.h/‰n D n.hj ˝ ½˝n1 ‰n ; ‰n 2 ˝ns .h/; h 2 h; where one sets .hju D .hju/ for u 2 h. It is easy to see that a./ .h/ are well 1 defined on Dom N 2 and that .‰1 ja .h/‰2 / D .a.h/‰1 j‰2 /, i.e. a.h/  a .h/ on 1 Dom N 2 . Moreover h 3 h 7! a .h/; resp. a.h/ is C-linear, resp. anti-linear,

(3.2)

1

and as quadratic forms on Dom N 2 one has Œa.h1 /; a.h2/ D Œa .h1 /; a .h2 / D 0; Œa.h1/; a .h2 / D .h1 jh2 /½;

(3.3)

h1 ; h2 2 h;

where ŒA; B D AB  BA, which a version of the canonical commutation relations, abbreviated CCR in the sequel.

3.1.3 Field and Weyl operators. One then introduces the field operators in the Fock representation 1 F .h/ D p .a.h/ C a .h//; 2

h 2 h;

(3.4) 1

which can be easily shown to be essentially selfadjoint on Dom N 2 . One has F .h1 C h2 / D F .h1 / C F .h2 /;

1

 2 R; hi 2 h; on Dom N 2 ;

(3.5)

i.e. h 7! F .h/ is R-linear, and the Heisenberg form of the CCR are satisfied as 1 quadratic forms on Dom N 2 ŒF .h1 /; F .h2 / D ih1  h2 ½:

(3.6)

3.2 Fock quantization of the Klein–Gordon equation

17

for h1  h2 D Im.h1 jh2 /:

(3.7)

Denoting again by F .h/ the selfadjoint closure of F .h/, one can then define the Weyl operators (3.8) WF .h/ D eiF .h/ ; which are unitary and satisfy the Weyl form of the CCR WF .h1 /WF .h2 / D eih1 h2 WF .h1 C h2 /: If hR denotes the real form of h, i.e. h as a real vector space, then .hR ; / is a real symplectic space. Moreover i, considered as an element of L.hR /, belongs to Sp.hR ; / and one has  D ı i D Re.j/  0:

3.1.4 K¨ahler structures. In general, a triple .X ; ; j/, where .X ; / is a real symplectic space and j 2 L.X / satisfies j2 D ½ and ı j 2 Ls .X ; X 0 /, is called a pseudo-K¨ahler structure on X . If ı j  0, it is called a K¨ahler structure. The anti-involution j is called a K¨ahler anti-involution. We will come back to this notion in Section 4.1. Given a K¨ahler structure on X , one can turn X into a complex preHilbert space by equipping it with the complex structure j and the scalar product: .x1 jx2 /F D x1  jx2 C ix1  x2 :

(3.9)

If we choose as one-particle Hilbert space the completion of X for .j/F , we can construct the Fock representation by the map

X 3 x 7! F .x/ which satisfies (3.5), (3.6).

3.2 Fock quantization of the Klein–Gordon equation From the above discussion we see that the first step in the construction of quantum Klein–Gordon fields is to fix a K¨ahler anti-involution on one of the equivalent symplectic spaces in Theorem 2.12, the most convenient one being .C01 .Rd I R2 /; /.

3.2.1 The K¨ahler structure. There are plenty of choices of K¨ahler anti-involutions. The most natural one is obtained as follows: let us denote by h the completion of C01 .Rd I C/ with respect to the scalar product .h1 jh2 /F D .h1 j 1 h2 /Rd :

18

3 Fock quantization on Minkowski space 1

If m > 0, this space is the (complex) Sobolev space H  2 .Rd / and if m D 0 the 1 complex´homogeneous Sobolev space HP  2 .Rd /, except when d D 1, since the 1 integral R jkj d k diverges at k D 0. This is an example of the so-called infrared problem for massless fields in two spacetime dimensions. To avoid a somewhat lengthy digression, we will assume that m > 0 if d D 1. Let us introduce the map V W C01 .Rd I R2 / 3 f 7! f0  if1 2 h:

(3.10)

An easy computation shows that: Im.Vf jVg/F D f  g; i ı V D V ı j;

for j D



0  1  0

 ;

eit  ı V D V ı etA : In other words, j is a K¨ahler anti-involution on C01 .Rd I R2 / and the associated oneparticle Hilbert space is unitarily equivalent to h. Moreover, after identification by V , the symplectic group fetA gt 2R becomes the unitary group feit  gt 2R with positive generator . This positivity is the distinctive feature of the Fock representation.

3.3 Quantum spacetime fields Let us set

ˆ ˆF .u/ D

R

F .eit  u.t; //dt; u 2 C01 .Rn I R/;

(3.11)

1

the integral being for example norm convergent in B.Dom N 2 ; s .h//. We obtain from (2.14) and (3.7) that ŒˆF .u/; ˆF .v/ D i.ujGv/Rn ½;

(3.12)

and ˆF .P u/ D 0. Setting formally ˆF .u/ D

ˆ Rn

ˆF .x/u.x/dx;

we obtain the spacetime fields ˆF .x/, which satisfy ŒˆF .x/; ˆF .x 0 / D iG.x  x 0 /½; . C m2 /ˆF .x/ D 0:

x; x 0 2 Rn ;

(3.13)

19

3.4 Local algebras

3.3.1 The vacuum state. Let us denote by CCRpol .KG/ the -algebra generated

by the ˆF .u/; u 2 C01 .Rn I R/, see Subsections 4.3.1 and 4.5.1 below for a precise definition. The vacuum vector vac 2 s .h/ induces a state !vac on CCRpol .KG/, called the Fock vacuum state, by ! ! N N Y Y !vac ˆF .ui / D vac j ˆF .ui / vac i D1

i D1

s .h/

Clearly, !vac induces linear maps ˝n C01 .Rn I R/ 3 u1 ˝    ˝ uN 7! !vac

N Y

! ˆF .ui / 2 C;

i D1

which are continuous for the topology of C01 .Rn I R/, and hence one can write !vac

N Y

! ˆF .ui / D

i D1

ˆ RN n

!N .x1 ; : : : ; xN /

N Y

ui .xi /dx1 : : : dxN ;

i D1

where the distributions !N 2 D 0 .RN n / are called in physics the N -point functions. Among them the most important one is the 2-point function !2 , which equals ˆ 1 i.t t 0 /.k/Cik.xx0/ 0 n e d k: (3.14) !2 .x; x / D .2/ Rd 2.k/ If we write similarly the distributional kernel of G, we obtain by (2.14) ˆ 1 0 0 n G.x; x / D .2/ sin..t  t 0 /.k//eik.xx / d k: Rd .k/

(3.15)

The fact that !2 .x; x 0 / and G.x  x 0 / depend only on x  x 0 reflects the invariance of the vacuum state !vac under space and time translations.

3.4 Local algebras We recall that a double cone is a subset O D IC .fx1 g/ \ I .fx2 g/;

x1 ; x2 2 Rn with x2 2 JC .x1 /:

We denote by A.O/ the norm closure of Vect.feiˆF .u/ W supp u  Og/ in B. s .h//. From (2.13) and (3.12) it follows that ŒA.O1 /; A.O2/ D f0g; if O1 ; O2 are causally disjoint:

20

3 Fock quantization on Minkowski space

We obtain a representation of the Poincar´e group P .1; d / by -automorphisms of CCRpol .KG/ by setting ˛ƒ ˆF .x/ D ˆF .ƒ1 x/ for ƒ 2 P .1; d /. From the invariance of the vacuum state under translations, we obtain that ˛.a;½/ .A/ D U.a/AU.a/1 for A 2 CCRpol .KG/, where Rn 3 a 7! U.a/ is a strongly continuous unitary group on s .h/. We have ˛ƒ .A.O// D A.LO C a/, for ƒ D .a; L/ 2 P .1; d /.

3.4.1 The Reeh–Schlieder property. One might expect that the closed subspace generated by the vectors A vac for A 2 A.O/ depends on O, since it describes excitations of the vacuum vac localized in O. This is not the case, and actually the following Reeh–Schlieder property holds: Proposition 3.1. For any double cone O the space fA vac W A 2 A.O/g is dense in s .h/. Proof. Let u 2 s .h/ such that .ujA vac / D 0 for all A 2 A.O/. If O1 b O is a smaller double cone and A 2 A.O1 /, the function f W Rn 3 x 7! .ujU.x/A vac / has a holomorphic extension F to Rn C iCC , i.e. f .x/ D F .x C iCC 0/, as distributional boundary values, see Section 12.1. Since U.x/AU  .x/ 2 A.O/, we have f .x/ D 0 for x close to 0, hence by the edge of the wedge theorem, see Subsection 12.1.2, F D 0 and f D 0 on Rn . Vectors of the form U.x/A vac for x 2 Rn ; A 2 A.O1 / are dense in s .h/, hence u D 0. 

Chapter 4

CCR algebras and quasi-free states In this chapter we collect various well-known results on the CCR -algebras associated to a symplectic space and on quasi-free states. We will often work with complex symplectic spaces, which will be convenient later on when one considers Klein–Gordon fields. We follow the presentation in [DG, Section 17.1] and [GW1, Chapter 2].

4.1 Vector spaces In this section we collect some useful notation, following [DG, Section 1.2].

4.1.1 Real vector spaces. Real vector spaces will be usually denoted by X . The complexification of a real vector space X will be denoted by CX D fx1 C ix2 W x1 ; x2 2 X g. 4.1.2 Complex vector spaces. Complex vector spaces will be usually denoted

by Y . If Y is a complex vector space, its real form, i.e. Y , regarded as a vector space over R, will be denoted by YR . Conversely, a real vector space X equipped with an anti-involution j (also called a complex structure), i.e. j 2 L.X / with j2 D ½ can be equipped with the structure of a complex space by setting . C i/x D x C jx;

x 2 X ;  C i 2 C:

If Y is a complex vector space, we denote by Y the conjugate vector space of Y , i.e. Y D YR as a real vector space, equipped with the complex structure j, if j 2 L.YR / is the complex structure of Y . The identity map ½ W Y ! Y will be denoted by y 7! y, i.e. y equals y, but considered as an element of Y . The map ½ W Y ! Y is anti-linear.

4.1.3 Duals and antiduals. Let X be a real vector space. Its dual will be denoted by X 0 . Let Y be a complex vector space. Its dual will be denoted by Y 0 , and its anti-dual, 0 i.e. the space of C-anti-linear forms on Y , by Y  . By definition, Y  D Y . Note that 0 we have a C-linear identification Y 0 Y defined as follows: if y 2 Y and w 2 Y 0 , then wy D wy:

22

4 CCR algebras and quasi-free states 0

This identifies w 2 Y 0 with an element of Y . Similarly, we have a C-linear identifi cation Y Y  .

4.1.4 Linear operators. If Xi , i D 1; 2, are real or complex vector spaces and

a 2 L.X1 ; X2 /, we denote by a0 2 L.X20 ; X10 / or ta its transpose. If Yi , i D 1; 2 are complex vector spaces we denote by a 2 L.Y2 ; Y1 / its adjoint, and by a 2 L.Y 1 ; Y 2 / its conjugate, defined by a y 1 D ay1 . With the above identifications we have a D a0 D a0 . If Xi , i D 1; 2 are real vector spaces and a 2 L.X1 ; X2 /, we denote by aC 2 L.CX1 ; CX2 / its complexification.

4.2 Bilinear and sesquilinear forms If X is a real or complex vector space, a bilinear form on X is given by an operator a 2 L.X ; X 0 /, its action on a couple .x1 ; x2 / is denoted by x1 ax2 . We denote by Ls=a .X ; X 0 / the symmetric/antisymmetric forms on X . A form a is non-degenerate if Ker a D f0g. Similarly, if Y is a complex vector space, a sesquilinear form on Y is given by an operator a 2 L.Y ; Y  /, and its action on a couple .y1 ; y2 / will be denoted by .y1 jay2 / or y 1 ay2 ;

(4.1)

0

the last notation being a reminder that Y  Y . We denote by Lh=a .Y ; Y  / the Hermitian/anti-Hermitian forms on Y . Non-degenerate forms are defined as in the real case. If X is a real vector space and a 2 L.X ; X 0 /, we denote by aC 2 L.CX ; CX  / its sesquilinear extension.

4.2.1 Real symplectic spaces. An antisymmetric form 2 L.X ; X 0 / is called a pre-symplectic form. A non-degenerate pre-symplectic form is called symplectic and a couple .X ; / where is (pre) symplectic a (real) .pre/symplectic space. If .X ; / is symplectic, the symplectic group Sp.X ; / is the set of invertible r 2 L.X / such that r 0 r D equipped with the usual product. The Lie algebra sp.X ; / is the set of a 2 L.X / such that a0 D  a, equipped with the commutator.

4.2.2 Pseudo-Euclidean spaces. A pair .X ; / with  2 L.X ; X 0 / non-degenerate and symmetric is called a pseudo-Euclidean space. If  > 0, it is called an Euclidean space. The orthogonal group O.X ; / is the set of invertible r 2 L.X / such that r 0 r D , equipped with the usual product. The Lie algebra o.X ; / is the set of a 2 L.X / such that a0  D a, equipped with the commutator.

4.3 Algebras

23

4.2.3 Hermitian spaces. A space .Y ; q/ with q Hermitian is called a pre-Hermi-

tian space. If q is non-degenerate, .Y ; q/ is called a Hermitian space. If q > 0 it is called a pre-Hilbert space. The .pseudo/-unitary group U.Y ; q/ is the set of invertible u 2 L.Y / such that u qu D q equipped with the usual product.

4.2.4 Complex symplectic spaces. An anti-Hermitian form on Y is called a (complex) pre-symplectic form. One sets then q D i 2 Lh .Y ; Y  / called the charge. One identifies in this way complex (pre-)symplectic spaces with (pre-)Hermitian spaces. The complex structure on Y is sometimes called the charge complex structure and will often be denoted by j to avoid confusion with the imaginary unit i 2 C. 4.2.5 Charge reversal. Definition 4.1. Let .Y ; q/ a pre-Hermitian space. A map  2 L.YR / is called a charge reversal if 2 D ½ or 2 D ½ and y 1 qy2 D y 2 qy1

y1 ; y2 2 Y :

Note that a charge reversal is anti-linear.

4.2.6 Pseudo-K¨ahler structures. Let .Y ; q/ be a Hermitian space whose complex structure is denoted by j 2 L.YR /. Note that .YR ; Im q/ is a real symplectic space with j 2 Sp.YR ; Im q/ and j2 D ½. The converse construction is as follows: a real symplectic space .X ; / with a map j 2 L.X / such that j2 D ½; j 2 Sp.X ; /; is called a pseudo-K¨ahler space. If in addition  D j is positive definite, it is called a K¨ahler space. We set now Y D .X ; j/; which is a complex vector space, whose elements are logically denoted by y. If .X ; ; j/ is a pseudo-K¨ahler space we can set y 1 qy2 D y1  jy2 C iy1  y2 ;

y1 ; y2 2 Y ;



and check that q 2 Lh .Y ; Y / is non-degenerate.

4.3 Algebras A unital algebra over C equipped with an anti-linear involution A 7! A such that .AB/ D B  A is called a -algebra. A -algebra which is complete for a norm such that kAk D kA k and kABk  kAkkBk is called a Banach -algebra. If moreover kA Ak D kAk2 , it is called a C  -algebra.

24

4 CCR algebras and quasi-free states

4.3.1 Algebras defined by generators and relations. In physics many algebras are defined by specifying a set of generators and the relations they satisfy. Let us recall the corresponding rigorous definition. Let A be a set, called the set of generators, and Cc .AI K/ be the vector space of functions A ! K with finite support (usually K D C). Denoting the indicator function ½fag simply by a, we see that every element of Cc .AI K/ can be written as P  a2B a a, with B  A finite, a 2 K. Thus Cc .AI K/ can be seen as the vector space of finite linear combinations of elements of A. We set A.A; K/ D ˝Cc .AI K/; where ˝E is the tensor algebra over the K-vector space E. Usually one writes a1    an instead of a1 ˝    ˝ an for ai 2 A. Let now R  A.A; K/ (the set of ‘relations’). We denote by I.R/ the two-sided ideal of A.AI K/ generated by R. Then the quotient A.A; K/=I.R/ is called the unital algebra with generators A and relations R D 0; R 2 R.

4.3.2 -algebras defined by generators and relations. Assume that K D C and let i W A ! A some fixed involution. A typical example is obtained as follows: denote by A another copy of A and by A 3 a 7! a 2 A the identity. Then A t A has a canonical involution i mapping a to a (and hence a to a). One then defines the anti-linear involution  on A.A; K/ by .a1    an / D i an    i a1 ;

½ D ½:

If R is invariant under , then I.R/ is also a -ideal, and A.A; K/=I.R/ is called the unital -algebra with generators A and relations R D 0; R 2 R. In this case one usually defines the involution  by adding to R the elements a  i a, for a 2 A, i.e. by adding the definition of  on the generators to the set of relations.

4.4 States A state on a -algebra A is a linear map ! W A ! C which is normalized, i.e. !.½/ D 1, and positive, i.e. !.A A/  0 for A 2 A. The set of states on A is a convex set. Its extreme points are called pure states. Note that if A  B.H/ for some Hilbert space H, a state ! on A given by !.A/ D . jA / for some unit vector may not be pure.

4.4.1 The GNS (Gelfand–Naimark–Segal) construction. If ! is a state on A, one can perform the so-called GNS construction, which we now recall. Let us

25

4.5 CCR algebras

equip A with the scalar product .AjB/! D !.A B/: From the Cauchy–Schwarz inequality one obtains that I D fA 2 A W !.A A/ D 0g is a -ideal of A. We denote by H! the completion of A=I for kk! and by ŒA 2 H! the image of A 2 A. The fact that I is a -ideal implies that for A 2 A the map ! .A/ W H! 3 ŒB 7! ŒAB 2 H! is well defined and defines a linear operator with D! D fŒB W B 2 Ag as invariant domain. If ! D Œ½, then !.A/ D . ! j! .A/ ! /! :

(4.2)

The triple .H! ; ! ; ! / is called the GNS triple associated to !. It provides a Hilbert space H! , a representation ! of A by densely defined operators on H! and a unit vector ! such that (4.2) holds. Vectors in H! are physically interpreted as local excitations of the ground state ! . If A is a C  -algebra, then one can show that ! .A/ 2 B.H! / with k! .A/k  kAk.

4.5 CCR algebras In this section we recall the definition of various -algebras related to the canonical commutation relations.

4.5.1 Polynomial CCR -algebra. Definition 4.2. Let .X ; / be a real pre-symplectic space. The polynomial CCR -algebra over .X ; /, denoted by CCRpol .X ; /, is the unital complex -algebra generated by elements .x/, x 2 X , with relations .x1 C x2 / D .x1 / C .x2 /;

  .x/ D .x/;

.x1 /.x2 /  .x2 /.x1 / D ix1  x2 ½;

x1 ; x2 ; x 2 X ;  2 R: (4.3)

The elements .x/ are called real or selfadjoint fields.

4.5.2 Weyl CCR algebra. One problem with CCRpol .X ; / is that its elements cannot be faithfully represented as bounded operators on a Hilbert space. To cure this problem one uses Weyl operators, which lead to the Weyl CCR -algebra.

26

4 CCR algebras and quasi-free states

Definition 4.3. The algebraic Weyl CCR -algebra over .X ; /, denoted CCRWeyl .X ; /, is the -algebra generated by the elements W .x/, x 2 X , with relations W .0/ D ½; W .x/ D W .x/; x; x1 ; x2 2 X : (4.4) i W .x1 /W .x2 / D e 2 x1 x2 W .x1 C x2 /; The elements W .x/ are called Weyl operators. An advantage of CCRWeyl .X ; / is that it can be equipped with a unique C  -norm see e.g. [DG, Definition 8.60]. Its completion for this norm is called the Weyl CCR C*-algebra over .X ; /, and is still denoted by CCRWeyl .X ; /. We will mostly work with CCRpol .X ; /, but it is sometimes important to work with the C  -algebra CCRWeyl .X ; /, for example in the discussion of pure states, see Section 4.9 below. Of course, the formal relation between the two approaches is x 2 X;

W .x/ D ei.x/ ;

which does not make sense a priori, but from which mathematically correct statements can be deduced.

4.5.3 Charged CCR algebra. Let .Y ; q/ a pre-Hermitian space. As explained above, we denote the complex structure on Y by j. The CCR algebra CCRpol .YR ; Im q/ can be generated instead of the selfadjoint fields .y/ by the charged fields: 1 .y/ D p ..y/ C i.jy//; 2



1 .y/ D p ..y/  i.jy//; 2

y 2 Y : (4.5)

From (4.3) we see that they satisfy the relations 

.y1 C y2 / D

.y1 / C  .y2 /;

.y1 C y2 / D

.y1 / C 

Π.y1 /; .y2/ D ΠΠ.y1 /;







.y1 /;

.y2 / D y 1  qy2 ½; .y/ D



.y/;



.y2 /;

y1 ; y2 2 Y ;  2 C;

.y2 / D 0;

(4.6)

y1 ; y2 2 Y ;

y 2 Y:

Note the similarity with the CCR in (3.3) expressed in terms of creation/annihilation operators, the difference being the fact that q is not necessarily positive. The CCR algebra CCRpol .YR ; Im q/ will be denoted by CCRpol .Y ; q/.

4.6 Quasi-free states In this section we discuss states on CCRpol .X ; / or (equivalently) on CCRWeyl .X ; / which are natural for free theories, the so-called quasi-free states. We start by discussing general states on CCRWeyl .X ; /.

4.6 Quasi-free states

27

4.6.1 States on CCRWeyl .X ;  /. Let .X ; / be a real pre-symplectic space and

! a state on CCRWeyl .X ; /. The function:

X 3 x 7! !.W .x// D G.x/

(4.7)

is called the characteristic function of the state !, and is an analog of the Fourier transform of a probability measure. There is also an analog of Bochner’s theorem: Proposition 4.4. A map G W X ! C is the characteristic function of a state on CCRWeyl .X ; / iff for any n 2 N and any xi 2 X , the n  n matrix i h i G.xj  xi /e 2 xi xj 1i;j n

is positive. P Proof. H) For x1 ; : : : ; xn 2 X , 1 ; : : : ; n 2 C set A D njD1 j W .xj /. Such A are dense in CCRWeyl .X ; /. One computes A A using the CCR and obtains A A D

n X

i

i j W .xj  xi /e 2 xi xj ;

(4.8)

i;j D1

from which H) follows. (H One defines ! using (4.7), and (4.8) shows that ! is positive.



4.6.2 Quasi-free states on CCRWeyl .X ;  /. Definition 4.5. Let .X ; / be a real pre-symplectic space. (1) A state ! on CCRWeyl .X ; / is a quasi-free state if there exists  2 Ls .X ; X 0 / such that   1 ! W .x/ D e 2 xx ; x 2 X : (4.9) (2) The form  is called the covariance of the quasi-free state !. Quasi-free states are generalizations of Gaussian measures. In fact, let us assume that X D Rn and D 0. CCRpol .Rn ; 0/ is simply the algebra of complex polynomials on .Rn /0 if we identify .x/ with the function  7! x . If we consider the Gaussian measure on .Rn /0 with covariance  1

1

d D .2/n=2 .det / 2 e 2  then

ˆ

1

1 

eix d ./ D e 2 xx ;

d ;

28

4 CCR algebras and quasi-free states

which is (4.7). Note also that if xi 2 Rn , then ˆ 2nC1 Y

xi  d ./ D 0;

1

ˆ Y 2n

X

xi  d ./ D

n Y

x.2j 1/  x.2j / ;

2Pair2n j D1

1

which should be compared with Definition 4.8 below. We recall that Pair2m denotes the set of pairings, i.e. the set of partitions of f1; : : : ; 2mg into pairs. Any pairing can be written as fi1 ; j1 g;    ; fim ; jm g for ik < jk and ik < ikC1 , hence can be uniquely identified with a permutation 2 S2m such that .2k  1/ D ik , .2k/ D jk . It will be useful later on to collect some properties of the GNS triple associated to a quasi-free state ! on CCRWeyl .X ; /, see Subsection 4.4.1. For ease of notation, we omit the subscript !. Lemma 4.6. Let us set W .x/ D .W .x// 2 U.H/ for x 2 X . Then: (1) the one-parameter group fW .tx/gt 2R is a strongly continuous unitary group on H; Q (2) let  .x/ be its selfadjoint generator. Then 2 Dom. niD1  .xi // for n 2 N, xi 2 X . Proof. (1) It suffices to prove the continuity of t 7! W .tx/u for u 2 H at t D 0. By density and linearity, we can assume that u D W .y/ , y 2 X . Then ku  W .tx/uk2 D . jW .y/.½  W .tx//.½  W .tx//W .y/ /; and using the CCR (4.4) we have W .y/.½  W .tx//.½  W .tx//W .y/ D 2½  W .tx/eitxy  W .tx/eitxy : Therefore ku  W .tx/uk2 D !.2½  W .tx/eitxy  W .tx/eitxy / 1 2 xxCitxy

D 2  e 2 t

1 2 xxitxy

 e 2 t

;

which tends to 0 when t ! 0. (2) By [DG, Theorem 8.29], it suffices to check that if Xfin  X is a finitedimensional subspace, then Xfin 3 x 7! . jW .x/ / belongs to the Schwartz class S .Xfin / of rapidly decaying smooth functions. This is obvious by (4.9). 

29

4.6 Quasi-free states

Proposition 4.7. (1) One has Dom  .x1 / \ Dom  .x2 /  Dom  .x1 C x2 /;  .x1 C x2 / D  .x1 / C  .x2 / on Dom  .x1 / \ Dom  .x2 /; Œ .x1 /;  .x2 / D ix1  x2 ½ as quadratic forms on Dom  .x1 / \ Dom  .x2 /: (2) One has i . j .x1 / .x2 / / D x1 x2 C x1  x2 ; 2

x1 ; x2 2 X :

(4.10)

(3) One has . j .x1 /     .x2m1 / / D 0; X . j .x1 /     .x2m / / D

(4.11) m Y

. j .x.2j 1/ / .x.2j / /:

2Pair2m j D1

(4.12) Proof. (1) follows from [DG, Theorem 8.25]. 1 2

(2) We have . jW .tx/ / D e 2 t x x , which when differentiated twice with respect to t at t D 0 gives . j2 .x/ / D x x. We then apply (1), i.e. linearity and the CCR to obtain (4.10). (3) in . j .x1 /     .xn / / is the coefficient of t1    tn in the power series expansion of !.W .t1 x1 /    W .tn xn //. One then uses the CCR and (4.9) to compute this function. Details can be found e.g. in [DG, Proposition 17.8]. 

4.6.3 Quasi-free states on CCRpol .X ;  /. From Proposition 4.7 one sees that a quasi-free state ! on CCRWeyl .X ; / induces a state !Q on CCRpol .X ; / by setting ! ! n n Y Y !Q .xi / D j  .xi / : i D1

i D1

Indeed, !Q is well defined on CCRpol .X ; / since it vanishes on elements of the ideal I.R/ for R introduced in (4.3), by Proposition 4.7 (1). This leads to the following definition of quasi-free states on CCRpol .X ; /.

30

4 CCR algebras and quasi-free states

Definition 4.8. (1) A state ! on CCRpol .X ; / is quasi-free if for any m 2 N and xi 2 X one has   ! .x1 /    .x2m1 / D 0; (4.13) m X Y     ! .x1 /    .x2m / D ! .x.2j 1//.x.2j / : (4.14) 2Pair2m j D1

(2) The symmetric form  2 Ls .X ; X 0 / defined by i !..x1 /.x2 // D x1 x2 C x1  x2 2 is called the covariance of the state !.

(4.15)

4.7 Covariances of quasi-free states Proposition 4.9. Let  2 Ls .X ; X 0 /. Then the following are equivalent: (1) there exists a quasi-free state ! on CCRWeyl=pol .X ; / with covariance . (2) C C 2i C  0 on CX ; where C ; C 2 L .CX ; .CX / / are the sesquilinear extensions of ; . 1

1

(3)   0 and jx1  x2 j  2.x1 x1 / 2 .x2 x2 / 2 ; x1 ; x2 2 X . Proof. .1/ H) .2/ If  is the covariance of a state ! on CCRWeyl .X ; / one introduces complex fields  .w/ D  .x1 / C i .x2 /, w D x1 C ix2 2 CX with domain Dom  .x1 / \ Dom  .x2 /. By Proposition 4.7, . .w/ j .w/ / is well defined, positive, and equals w .C C 2i C /w. The same argument, with  ./ replaced by ./, gives the proof for CCRpol .X ; /. .2/ H) .1/ Let us fix x1 ; : : : ; xn 2 X and set bij D xi  xj C 2i xi  xj . Then, for 1 ; : : : ; n 2 C, n P i;j D1

i bij j D wC w C 2i w!C w; w D

n P i D1

i xi 2 CX :

  By (2), the matrix bij is positive. The pointwise product of two positive matrices   is positive, see e.g. [DG, Proposition 17.6], which implies that ebij is positive, and     1 1 i hence e 2 xi xi ebij e 2 xj xj is positive. This matrix equals G.xj  xi /e 2 xi xj 1 with G.x/ D e 2 xx . By Proposition 4.4,  is the covariance of a quasi-free state on Weyl CCR .X ; /. By the discussion following Subsection 4.6.3, it is also the covariance of a quasi-free state on CCRpol .X ; /. The proof of .2/ ” .3/ is an exercise in linear algebra.  We will identify in the sequel the two states on CCRWeyl .X ; / and CCRpol .X ; / having the same covariance .

4.7 Covariances of quasi-free states

31

4.7.1 Quasi-free states on CCRpol .Y; q/. Let now .Y ; q/ a pre-Hermitian

space. Recall that if X D YR and D Im q, then .X ; / is a real pre-symplectic space, and by definition CCRpol .Y ; q/ D CCRpol .X ; /. The complex structure j of Y belongs to Sp.X ; / and also to sp.X ; / since j2 D ½. It follows that fej g 2S1 is a one-parameter group of symplectic transformations. Therefore, one can define a group f˛ g 2S1 of automorphisms of CCRpol .X ; / by ˛ .x/ D .ej x/: (4.16) The gauge transformations ˛ are global gauge transformations, which should not be confused with the local gauge transformations arising for example in electromagnetism. Definition 4.10. A quasi-free state ! on CCRpol .X ; / is called gauge invariant if !.˛ .A// D !.A/;

A 2 CCRpol .X ; /; 2 S1 :

The following lemma follows immediately from Definition 4.10. Lemma 4.11. A quasi-free state ! on CCRpol .X ; / with covariance  is gauge invariant iff j 2 O.X ; / iff j 2 o.X ; /. One can then define O 2 Lh .Y ; Y  / by y 1  y O 2 D y1 y2  iy1 jy2 ;

y1 ; y2 2 Y :

(4.17)

It is then natural to consider the action of ! on products of the charged fields .y/;  .y/ introduced in (4.5). Note that by the CCR (4.6), ! is completely determined by its action on elements AD

n Y



i D1

.yi /

m Y

.yj0 /:

(4.18)

j D1

Proposition 4.12. A quasi-free state ! on CCRpol .Y ; q/ is gauge invariant iff 0 1 n m Y Y  !@ .yi / .yj0 /A D 0; if n ¤ m; (4.19) 0 !@

i D1

j D1

n Y

n Y

i D1



.yi /

j D1

1 .yj0 /A

D

n X Y

!.



0 .yi / .y.i / //:

(4.20)

2Sn i D1

Proof. Using that ˛ .  .y// D ej  .y/, we obtain that if A is as in (4.18) ˛ .A/ D ej.nm/ A, which implies (4.19). The proof of (4.20) is a routine computation, using (4.5) and Definition 4.8. 

32

4 CCR algebras and quasi-free states

Definition 4.13. The sesquilinear forms ˙ 2 Lh .Y ; Y  / defined by !. .y1 / !.





.y2 // D y 1 C y2 ;

.y2 / .y1 // D y 1  y2 ;

y1 ; y2 2 Y ;

(4.21)

are called the complex covariances of the quasi-free state !. Note that since Œ .y1 /;  .y2 / D y 1qy2 ½, we have C   D q. Therefore ! is completely determined by either C or  . Nevertheless, it is more convenient to consider the pair ˙ when discussing properties of !.  is usually called the charge density associated to !. Introducing the selfadjoint fields .y/, we obtain that     i 1 C !..y1 /.y2 // D Re y 1    q y2 C Im.y 1 qy2 /: (4.22) 2 2 It follows that the real and complex covariances of a gauge invariant quasi-free state are connected by the relations   1 1 ˙  D Re  q ; ˙ D O ˙ q; (4.23) 2 2 where O is defined in (4.17). In this situation we will call  the real covariance of the state !, to distinguish it from the complex covariances ˙ . It is easy to characterize the complex covariances of a gauge invariant quasi-free state. Proposition 4.14. Let ˙ 2 Lh .Y ; Y  /. Then the following are equivalent: (1) ˙ are the covariances of a gauge invariant quasi-free state on CCRpol .Y ; q/; (2) ˙  0 and C   D q. Proof. The implication .1/ H) .2/ is immediate using the CCR and the fact that .y/  .y/ and  .y/ .y/ are positive. Let us now prove that .2/ H) .1/. We recall that X D YR . Let  be the real covariance of a gauge invariant quasi-free state. For x 2 X let z D p1 .x  ijx/; z D p1 .x C ijx/. We know that j 2 2 2 O.X ; / \ o.X ; /, which after a standard computation yields O .zjC z/ D x x  ix jx D x  x; O .zjC z/ D x x  ix jx D x  x:

(4.24)

Similarly, using that j 2 Sp.X ; / \ sp.X ; / we obtain .zj C z/ D x  x  ix  jx D ix qx; .zj C z/ D x  x C ix  jx D ix qx:

(4.25)

By Proposition 4.9 (2) we have C C 2i C  0, which implies that O ˙ 12 q D ˙  0.  The fact that C   D q follows from (4.23).

4.8 The GNS representation of quasi-free states

33

4.7.2 Complexification of a quasi-free state. Let .X ; / be a real pre-symplectic space. We equip CX with q D i C , obtaining a pre-Hermitian space. The canonical complex conjugation on CX is a charge reversal on .CX ; q/. Clearly ..CX /R ; Im q/ is isomorphic to .X ˚ X ; ˚ / as real pre-symplectic spaces. If ! is a quasi-free state on CCRpol .X ; / with covariance , then we can consider the quasi-free state !Q on CCRpol ..CX /R; Im q/ with covariance Re C . It is easy to see that !Q is gauge invariant with covariances ˙ equal to 1 ˙ D C ˙ q: 2

(4.26)

Moreover, !Q is invariant under charge reversal. Therefore, by complexifying a quasi-free state ! on a real pre-symplectic space .X ; /, we obtain a gauge invariant quasi-free state !Q on A.CX ; C /. It follows that, possibly after complexifying the real pre-symplectic space .X ; /, one can always restrict the discussion to gauge invariant quasi-free states. Remark 4.15. Let .Y ; q/ pre-Hermitian and ! a quasi-free state on CCRpol .Y ; q/. Assume that ! is not gauge invariant. This means that the complex structure j of Y is irrelevant for the analysis of ! and hence can be forgotten. Therefore, we consider ! simply as a quasi-free state on the real pre-symplectic space .X ; / D .YR ; Im q/. If we want to recover a gauge invariant quasi-free state, we consider the state !Q on CCRpol .CX ; i C /.

4.8 The GNS representation of quasi-free states Let us now discuss the GNS representation of a quasi-free state on CCRpol .X ; /. We will assume for simplicity that its real covariance  is non degenerate, i.e. Ker  D f0g. From Proposition 4.9 (3) we see that Ker   Ker , hence in particular Ker  D f0g if is symplectic. 1 Let X cpl the completion of X for .x  x/ 2 , which is a real Hilbert space. The cpl cpl extension is bounded on X , but may be degenerate on X cpl . Moreover, ! induces a unique quasi-free state ! cpl on CCRWeyl .X cpl ; cpl /. To simplify notation, we forget the superscripts cpl in this section and assume that 1 X is complete for .x x/ 2 . The GNS representation was first constructed by Manuceau and Verbeure [MV] in the case where is non-degenerate. Its extension to the general case was given by Kay and Wald [KW, Appendix A], where it was called a one-particle Hilbert space structure. Another equivalent representation if is non-degenerate is called the Araki–Woods representation, see [AW]. An important fact in this context is the following result, due to Leyland, Roberts and Testard [LRT, Theorem 1.3.2], about dense subspaces of a Fock space s .h/. Another proof can be deduced from the results in [DG, Section 17.3].

34

4 CCR algebras and quasi-free states

Theorem 4.16. Let h a complex Hilbert space and X  h a real vector subspace. Then the space VectfWF .x/ vac W x 2 X g is dense in s .h/ iff CX is dense in h. Note that if we denote by X ? , resp. X perp the space orthogonal to X with respect to the scalar product .j/h , resp. Re.j/h , we have .iX /perp D iX perp , X ? D X perp \ iX perp and iX perp is also the space orthogonal to X with respect to the symplectic form

D Im .j/h . Therefore, an equivalent condition in Theorem 4.16 is that X perp \ iX perp D f0g.

4.8.1 K¨ahler structures. Proposition 4.17. Let  be the real covariance of a quasi-free state on CCRpol .X ; / 1 such that  is non-degenerate and X is complete for .x x/ 2 . Then if dim Ker is even or infinite, there exists an anti-involution j on X such that .; j/ is K¨ahler. Proof. By Proposition 4.9 (3), there exists a bounded anti-symmetric operator c 2 La .X / with kck  1 such that

D 2c: (4.27) ?   We have of course Ker c D Ker and we set Xsing D Ker c, Xreg D Xsing . Since c is anti-symmetric, it preserves Xreg and Xsing . If creg is the restriction of c to Xreg then one can perform its polar decomposition creg D jreg jcjreg , and using the antisymmetry of creg one obtain that j2reg D ½, jreg 2 O.Xreg ; / and Œjcreg j; jreg  D 0, see e.g. [DG, Proposition 2.84]. Since dim Xsing is even or infinite, we can choose an arbitrary anti-involution  jsing 2 O.Xsing; /. Then j D jreg ˚ jsing has the required properties.

4.8.2 The GNS representation. Let us equip X with a complex structure j as in Proposition 4.17, and with the scalar product .x1 jx2 /KW D x1 x2  ix1 jx2 : (4.28) The completion of X for this scalar product is denoted by XKW in the sequel. We set hKW D XKW ˚ ½Rnf1g .jcj/XKW ; where c is as in (4.27) and 1

1

KW .x/ D F ..1 C jcj/ 2 x ˚ .1  jcj/ 2 x/; x 2 X : acting on s .hKW /. Proposition 4.18. The triple .HKW ; KW ; KW /, defined by

HKW D s .hKW /;

KW .x/ D KW .x/;

is the GNS triple of the quasi-free state !.

x 2 X;

KW D vac ;

(4.29)

4.8 The GNS representation of quasi-free states

35

Proof. Using (4.28) we check by standard computations that ŒKW .x1 /; KW .x2 / D iIm .x1 jx2 /hKW D ix1  x2 ; . vac jKW .x1 /KW .x2 / vac / D x1 x2 C 2i x1  x2 : Using the CCR on s .hKW /, we then check that !.A/ D . vac j.A/ vac / for all A 2 CCRpol .X ; /. It remains to prove that KW .CCRWeyl .X ; // KW is dense in HKW , i.e. by The1 1 orem 4.16, that CRX is dense in hKW for Rx D .1 C jcj/ 2 x ˚ .1  jcj/ 2 x. This follows easily from the fact that the complex structure on hKW is j ˚ j.  If is non-degenerate, another equivalent version of the GNS representation is given by the Araki–Woods representation: one equips X with the complex structure j D cjcj1 given in Proposition 4.17, and with the scalar product .x1 jx2 /AW D x1  jx2 C ix1  x2 :

(4.30)

The completion of X for this scalar product is denoted by XAW and equals to jcj XKW , with the notation introduced in Subsection 1.3.1. One sets then  21

%D

1  jcj jcj

as a (possibly unbounded) operator on XAW . From (4.27), (4.30) we obtain that 1 .xj%x/AW D x x, hence X  Dom % 2 . The Araki–Woods representation is then obtained by setting hAW D XAW ˚ ½Rnf1g .jcj/XAW ; and defining the left Araki–Woods representation 1

1

AW;l .x/ D F ..1 C %/ 2 x ˚ % 2 x/;

x 2 X:

(4.31)

Setting

HAW D s .hAW /;

AW;l .x/ D AW;l .x/;

x 2 X;

AW D vac ;

one can show by the same arguments that .HAW ; AW;l ; AW / is an equivalent GNS representation for !.

4.8.3 Doubling procedure. Let us assume that ½f1g .jcj/ D 0, i.e. hAW D XAW ˚ X AW . One defines the right Araki–Woods representation

1 1 AW;r .x/ D F % 2 x ˚ .1 C %/ 2 x ; x 2 X ; which satisfies ŒAW;r .x1 /; AW;r .x2 / D ix1  x2 ;

ŒAW;l .x1 /; AW;r .x2 / D 0;

x1 ; x2 2 X :

36

4 CCR algebras and quasi-free states

One can now combine the left and right Araki–Woods representations by doubling the phase space. This doubling procedure is due to Kay [Ky2]. One sets .Xd ; d / D .X ; / ˚ .X ;  /; d .xd / D AW;l .x/ C AW;r .x 0 /;

xd D .x; x 0 / 2 Xd ;

and the vacuum vector vac induces a quasi-free state !d on CCRpol .Xd ; d / by !d ..x1;d /.x2;d // D . vac jd .x1;d /d .x2;d / vac /HAW : This state is a pure state, see Section 4.9. If we embed CCRpol .X ; / into CCRpol .Xd ; d / by the map X 3 x 7! .x; 0/ 2 Xd , then the restriction of !d to CCRpol .X ; / equals !.

4.8.4 Charged versions. Let us now describe the complex versions of the above constructions. Let ˙ be the complex covariances of a quasi-free state on CCRpol .Y ; q/. Assume that C C  is non-degenerate, which is the case if q is non-degenerate, and that Y is complete for the scalar product C C  . Then there exists d 2 Lh .Y / with kd k  1 such that q D .C C  /d: (4.32) 1 C  Setting X D YR , we have  D 2 Re. C  / and D Im q which implies that the operator c in Proposition 4.17 equals id and hence that jreg D isgn.d /. Since Ker D Ker q, its (real) dimension is even or infinite. Assuming for simplicity that Ker d D f0g we can rewrite .j/KW as 2.y1jy2 /KW D y 1 .C C  /½RC .d /y2 C y 2 .C C  /½R .d /y1 :

(4.33)

Similarly, we can rewrite .j/AW as .y1 jy2 /AW D y 1 q ½RC .d /y2  y 2 q ½R .d /y1 : Finally, let us discuss the doubling procedure in the charged case. We start from a Hermitian space .Y ; q/ and consider .Yd ; qd / D .Y ˚ Y ; q ˚ q/: Let us denote by ˙ d the complex covariances of the doubled state !d . One can show, see e.g. [G2, Section 5.4] that ˙ ˙ d D ˙qd ı cd ;

where cdC

D

% 2 .% C 1/ 2 sgn.d / .% C 1/½RC .d /  %½R .d / 1 1 % 2 .% C 1/ 2 sgn.d / %½RC .d / C .% C 1/½R .d /

D

% 2 .% C 1/ 2 sgn.d / %½RC .d / C .% C 1/½R .d / 1 1 % 2 .% C 1/ 2 sgn.d / .% C 1/½RC .d /  %½R .d /

1

1

cd

1

1

! ; ! ; (4.34)

4.9 Pure quasi-free states

37

j . One can check that cd˙ are a pair of where d is defined above and % D 1jd jd j complementary projections, which is related to the fact that !d is a pure state.

4.9 Pure quasi-free states Let us now discuss pure quasi-free states, which are often called vacuum states in physics. We will always assume that .X ; / is pre-symplectic, and the covariance  is non-degenerate. A basic result, see e.g. [BR, Theorem 2.3.19], says that a state ! on a C  -algebra A is pure iff its GNS representation .H! ; ! / is irreducible, i.e. iff H! does not contain non-trivial closed subspaces, invariant under ! .A/. To be able to apply this result, we will say that a quasi-free state ! on CCRpol .X ; / is pure if it is pure as a state on CCRWeyl .X ; /.

4.9.1 Pure quasi-free states on CCRpol .X ;  /. We use the notation X cpl ;

cpl ; ! cpl introduced in Section 4.8.

Proposition 4.19. A quasi-free state on CCRpol .X ; / with covariance  is pure iff .2cpl ; cpl / is K¨ahler, i.e. there exists an anti-involution jcpl 2 Sp.X cpl ; cpl / such that cpl jcpl D 2cpl . Note that this implies that cpl is non-degenerate on X cpl . Equivalent characterizations of pure quasi-free states are given in [MV, Proposition 12] or [KW, Lemma A.2]. Proof. Let us set A.cpl/ D CCRWeyl .X .cpl/ ; .cpl/ / and let .H.cpl/ ;  .cpl/ ; .cpl/ / be the GNS triple for ! .cpl/ . Using that X is dense in X cpl for , we first obtain that H D Hcpl , D cpl and  cpl jA D . We then claim that .A/ is strongly dense in  cpl .Acpl /. Indeed, if AD

N X



  cpl  cpl W xi 2  cpl Acpl

1

PN cpl and xi;n 2 X with xi ! xi for , we obtain that An D 1 i .W .xi;n // is PN bounded by 1 ji j and that An ! A strongly on the dense subspace .A/ , and hence on H. From this fact we see that a closed subspace K  H is invariant under .A/ iff it is invariant under  cpl .Acpl/, hence ! is pure iff ! cpl is pure. The statement of the proposition is now proved for example in [DG, Theorem 17.13].  There is an alternative characterization of pure quasi-free states, due to Kay and Wald [KW, eq. (3.34)] which is sometimes very useful.

38

4 CCR algebras and quasi-free states

Proposition 4.20. A quasi-free state on CCRpol .X ; / with covariance  is pure iff 1 jx  x1 j2 ; x1 ¤0 4 x1 x1

x x D sup

x 2 X:

(4.35)

Proof. It is easy to see that (4.35) on X is equivalent to (4.35) on X cpl , so we can assume that X is complete for . Note also that from Proposition 4.9 (3) x x is an upper bound of the rhs in (4.35). If ! is pure we have 2 D j by Proposition 4.19, hence x jx D 12 x x, which implies (4.35). Let us now prove the converse implication. Let c 2 La .X / with kck  1 and D 2c, as in the beginning of Section 4.8. Note that Ker c D f0g by (4.35). Performing the polar decomposition of c, see e.g. [DG, Proposition 2.84], we can write c D ujcj D jcju, where u 2 O.Y ; / and u2 D ½. Let us check that jcj D ½, which will prove that ! is pure. If jcj ¤ ½, then there exist ı 2 Œ0; 1Œ and x ¤ 0 with x D ½Œ0;ı .jcj/x, and hence, by Cauchy– Schwarz ˇ ˇ ˇ ˇ ˇx  x1 ˇ D 2ˇjcjx ux1 ˇ  2.jcjx jcjx/ 21 .ux1 ux1 / 21 D 2ı.x x/ 21 .x1 x1 / 21 ; which contradicts (4.35).



4.9.2 Pure quasi-free states on CCRpol .Y; q/. Let us now translate the above results in the case of Hermitian spaces. Note that Proposition 4.14 (2) implies that Ker.C C  / D f0g, hence kyk2! D y  C y C y   y is a Hilbert norm on Y . Denoting by Y cpl the completion of Y for k  k! , the Hermitian forms q; ˙ extend uniquely to q cpl ; ˙;cpl on Y cpl , and ! extends uniquely to a state ! cpl on CCRpol .Y cpl ; q cpl /. As in the real case, q cpl may be degenerate on Y cpl . If Y1  Y cpl with Y  Y1 densely for kk! , then we also obtain unique objects ˙ q1 ; ˙ 1 ; !1 that extend q;  ; !. The next proposition is the version of Proposition 4.19 in the charged case. Proposition 4.21. A gauge invariant quasi-free state ! is pure on CCRpol .Y ; q/ iff there exists Y1  Y cpl with Y  Y1 densely for kk! and projections c1˙ 2 L.Y1 / such that ˙ c1C C c1 D ½; ˙ (4.36) 1 D ˙q1 ı c1 : Moreover (4.36) implies that c1˙ q1 c1 D 0. Note that (4.36) implies that q1 is non-degenerate on Y1 . ˙ Proof. Since ˙ 1 D 1 , we obtain from (4.36) that q1 c1˙ D c1˙ q1 D c1˙ q1 c1˙ D c1˙ q1 .c1C C c1 /; which proves the second claim of the proposition.

4.9 Pure quasi-free states

39

(1) Let us now prove the first claim of the proposition if Y is complete for k  k! , in which case Y1 D Y . Recall that j is the complex structure on Y . The real presymplectic space .X ; / for X D YR , D Im q is then complete for the norm 1 .x x/ 2 and  D Re.˙ 12 q/ D 12 Re.C C  /. By Proposition 4.19, ! is pure iff there exists an anti-involution j1 with 2 D .Im q/j1 or equivalently 2j1 D .Im q/. Since ! is gauge invariant, we have j 2 sp.X ; Im q/ \ o.X ; / (see Lemma 4.11), hence 2j1 j D .Im q/j D j Im q D 2jj1 D 2jj1 ; so Œj; j1  D 0, i.e. j1 is C-linear on Y . Since we know that j1 2 Sp.X ; Im q/ this implies that j1 2 U.Y ; q/. Moreover, since  D Re.C  12 q/, we have Re.2C  q/ D .Im q/j1 , which using that j1 is C- linear and C ; q are sesquilinear yields 2C  q D qjj1 . We now set  D jj1 so that  2 D ½ and  2 U.Y ; q/, C D 1 .q.½ C //. Setting now c ˙ D 12 .½ C /, we see that c ˙ are projections with 2 c C C c  D ½, ˙ D ˙qc ˙ . From   q D q we obtain that c ˙ qc  D 0, which completes the proof of H). To prove the implication (H, assume that (4.36) holds for Y1 D Y and set j1 D j.c C  c  / so that j1 2 U.Y ; q/  Sp.X ; Im q/ is an anti-involution. We have 2C  q D q.c C  c  / D qjj1 D iqj1 hence 2 D 2Re.C  q/ D .Im q/j1 . (2) Let us now prove the proposition in the general case. We use the notation in the proof of Proposition 4.19 and set additionally A1 D CCRWeyl .Y1;R ; Re q1 /, and .H1 ; 1 ; 1 / the associated GNS triple. The same argument as in the proof of Proposition 4.19 shows then that ! is pure iff !1 is pure iff ! cpl is pure. Now the proof of H) follows from (1) by taking Y1 D Y cpl . Conversely, if (4.36) holds for some space Y1 , then an easy computation shows that as identities on L.Y1 ; Y1 /, one has ˙ ˙ ˙  ˙ c1˙ ˙ 1 c1 D 1 ; c1 1 c1 D 0;

hence c1˙ are bounded for kk! . Therefore, they extend to projections on Y cpl satisfying (4.36). This implies that ! cpl is pure, hence ! is pure.  Finally let us prove the analog of Proposition 4.20 in the charged case. Proposition 4.22. A gauge invariant quasi-free state ! with complex covariances ˙ is pure iff y  .C C  /y D

jy  qy1 j2 ; C  y1 2Y ;y1 ¤0 y 1  . C  /y1 sup

8y 2 Y :

(4.37)

Proof. Let us set as before .X ; / D .YR ; Im q/ and let  be the real covariance of !. By Proposition 4.20 ! is pure iff y  y1 D

1 jy  Im qy1 j2 ; sup 4 y1 ¤0 y1  y1

y 2 Y:

Since  D 12 Re.C C  / and q is sesquilinear, this is equivalent to (4.37).



40

4 CCR algebras and quasi-free states

4.9.3 The GNS representation of pure quasi-free states. The GNS representation of a pure quasi-free state is particularly simple, being a Fock representation. In fact with the notations in Section 4.8 we have jcj D 1 and D 2j. Set i .x1 jx2 /F D x1 x2 C x1  x2 ; (4.38) 2 and XF D .X ; j; .j/F / as a complex Hilbert space. Then the GNS representation of ! is .HF ; F ; F /, with HF D s .XF /;

F .x/ D F .x/;

F D vac :

Let us rephrase this in the complex case, where .Y ; q/ is a Hermitian space and ! a gauge invariant quasi-free state with complex covariances ˙ . We have, by (4.23), 2 D Re.C C  /; D Im.C   / D Re..C   /i/:

(4.39)

which yields by an easy computation as in (4.33) 2.y1 jy2 /F D y 1 C y2 C y 2  y1 :

(4.40)

Recall that the Hilbert space Y cpl was introduced in Subsection 4.9.2. We set j1 D j.c C  c  /, and YF D .Y cpl ; j1 ; .j/F /, which is a complex Hilbert space. The GNS representation of ! is .HF ; F ; F / with

HF D s .YF /;

F



.y/ D aF .c Cy/ C aF .c  y/;

F D vac :

Note that the sesquilinear forms ˙ extend continuously to YF (as R-bilinear forms).

4.9.4 The Reeh–Schlieder property for quasi-free states. Let ! be a pure quasi-free state on CCRpol .X ; /. If X1  X is a (real) vector subspace, then by Theorem 4.16 we know that VectfWF .x/ F W x 2 X1 g is dense in the GNS Hilbert space HF iff CX1 is dense in the Hilbert space XF introduced above. It is convenient to have a version of this result in the complex case. We fix a space .Y ; q/ and a pure gauge invariant quasi-free state ! on CCRpol .Y ; q/, with complex covariances ˙ . Let us denote by j the charge complex structure of Y . Proposition 4.23. Let Y1  Y be a complex vector subspace of Y . VectfWF .y/ F W y 2 Y1 g is dense in the GNS Hilbert space HF iff y C y1 D y  y1 D 0 8 y1 2 Y1 H) y D 0; for y 2 Y cpl :

Then (4.41)

Proof. By (4.38) and Theorem 4.16, VectfWF .y/ F W y 2 Y1 g is dense in HF iff y y1 D y  y1 D 0 8 y1 2 Y1 H) y D 0:

(4.42)

Next we use (4.39) and the fact that jY1 D Y1 to obtain that (4.42) is equivalent to (4.41). 

4.10 Examples

41

4.10 Examples 4.10.1 The vacuum state for real Klein–Gordon fields. We can take as real symplectic space .X ; / either the space .C01 .Rd I R2 /; / with defined in C 1 .Rn IR/

(2.16), or the space . P C0 1 .Rn IR/ ; .jG/Rn /. 0 If we take the first version we obtain from (3.10) that 1 1 .f0 jg0 /L2 .Rd / C .f1 j 1 g1 /L2 .Rd / ; 2 2

f; g 2 .C01 .Rd I R2 /; /: (4.43) In the second version, we obtain from (3.14) and (3.15) that ˆ uv D u.x/.x; x 0/v.x 0 /dxdx 0 ; u; v 2 C01 .Rn I R/ f g D

Rn Rn

where 0

.x; x / D .2/

n

ˆ Rd

1 0 cos..t  t 0 /.k//eik.xx / d k: 2.k/

(4.44)

4.10.2 The vacuum state for complex Klein–Gordon fields. It is more convenient to consider complex solutions of the Klein–Gordon equation. We take as Hermitian space .Y ; q/ either .C01 .Rd I C2 /; q/ with q defined in (2.21), or   C01 .Rn IC/ ; .jiG/Rn , see Theorem 2.13. P C 1 .Rn IC/ 0

In the first case the complex covariances ˙ of the vacuum state !vac are given by   1  ˙½ ˙ ; (4.45)  D 1 2 ˙½  where we identify sesquilinear forms with operators using the scalar product on L2 .Rd I C2 /. The projections c ˙ in Proposition 4.21 equal   1 ½ ˙1 : (4.46) c˙ D ½ 2 ˙ Note that

U0 .t/c ˙f D e˙it .f0 ˙  1 f1 /;

so c ˙ are the projections on the spaces of Cauchy data of solutions with positive/ negative energy. If we take the second version and denote by ƒ˙ D .%0 ı G/ ˙ .%0 ı G/ the corresponding complex covariances, their distributional kernels are given by ˆ 1 ˙i.t t 0 /.k/Cik.xx0/ ƒ˙ .x; x 0 / D .2/n d k: (4.47) e 2.k/

42

4 CCR algebras and quasi-free states

4.10.3 Vacuum and KMS states for abstract Klein–Gordon equations.

Let us fix a complex Hilbert space h and  2 > 0 a selfadjoint operator on h. Let us consider the following abstract Klein–Gordon equation: @2t .t/ C  2 .t/ D 0;

 W R ! h:

(4.48)

The main example is the Klein–Gordon equation on an ultra-static spacetime M D R  S , where .S; h/ is a complete Riemannian manifold and M is equipped with the Lorentzian metric g D dt 2 C hij .x/d xi d xj . We take then h D L2 .†; d Volh / and  2 D h C m2 , where h is the Laplace–Beltrami operator on .†; h/. We take as Hermitian space 1

1

Y D   2 h ˚  2 h;

f qf D .f1 jf0 /h C .f0 jf1 /h :

The vacuum state !vac is now defined by the complex covariances ˙ in (4.45), where we again identify sesquilinear forms and operators using the scalar product on h ˚ h. Another natural quasi-free state is the KMS state !ˇ at temperature ˇ 1 , given by the covariances   1 th.ˇ=2/ ˙½ ˙ ; (4.49) ˇ D ˙½  1 th.ˇ=2/ 2 which is not a pure state. !vac resp. !ˇ , is a ground state, resp. a ˇ-KMS state for the dynamics frs gs2R defined by rs ./ D . C s/, for  solution of (4.48). We refer the reader to Section 9.1 for a general discussion of KMS states.

Chapter 5

Free Klein–Gordon fields on curved spacetimes In this chapter we describe some well-known results about Klein–Gordon equations on Lorentzian manifolds. An important notion is the causal structure obtained from a Lorentzian metric, which leads to the notion of globally hyperbolic spacetimes, originally introduced by Leray [Le]. Globally hyperbolic spacetimes are Lorentzian manifolds which admit a Cauchy surface, i.e. a hypersurface intersected only once by each inextensible causal curve. On a globally hyperbolic spacetime M , one can pose and globally solve the Cauchy problem for the Klein–Gordon operators P associated to the metric g. Equivalently one can uniquely solve the inhomogeneous Klein–Gordon equation with support conditions, i.e. introduce the retarded/advanced inverses Gret=adv for P . The causal propagator G D Gret  Gadv is anti-symmetric and hence can be used to equip the space of test functions on the spacetime M with the structure of a presymplectic space, see Lichnerowicz [Li1] and Dimock [Di1]. If one fixes a Cauchy surface †, one can equivalently use the symplectic space of Cauchy data on †, i.e. of pairs of compactly supported smooth functions on †. This is particularly important for the construction of states for quantized Klein–Gordon fields, see Chapter 6.

5.1 Background We now collect some background material on vector bundles and connections on them. Most of it will be used only in Chapter 17 and can be skipped in first reading.

5.1.1 Fiber bundles. Let E; M be two smooth manifolds and  W E ! M

surjective with De  surjective for each e 2 E. The set Ex D  1 .fxg/ is called the  ! M is a fiber bundle with fiber over x 2 M . Let F be another smooth manifold. E  typical fiber F if there exists an open covering fUi gi 2I of M such that for each Ui  there exists i W  1 .Ui /  ! Ui  F such that M ı i D  on  1 .Ui /: 

The maps i are called local trivializations of the bundle E  ! M . The collection  ! M . For Ui ; Uj with f.Ui ; i /gi 2I is called a bundle atlas for the bundle E  Uij D Ui \ Uj ¤ ;, we have i ı j1 .x; f / D .x; tij .x/.f //;

44

5 Free Klein–Gordon fields on curved spacetimes

where the maps tij W Uij ! Aut.F / are called transition maps. One has ti i .x/ D Id;

ti k .x/ D tij .x/ ı tj k .x/;

x 2 Ui \ Uj \ Uk :

(5.1)



A fiber bundle E  ! M can be reconstructed from a covering fUi gi 2I of M and from a set of transition maps satisfying (5.1). 0



5.1.2 Morphisms of bundles. If E  ! M and E 0 ! M are two fiber bundles

with typical fibers F and F 0 , a smooth map  W E ! E 0 is a bundle morphism if  0 ı  D . If tij , resp. tij0 are the transition maps of E, resp E 0 , then there exists i W Ui ! Hom.F; F 0 / such that tij0 ı j D i ı tij on Uij :

(5.2)



! M is trivial if there exists a bundle isomorphism  W E ! A fiber bundle E  M  F . By (5.2), this is the case iff there exists i W Ui ! Aut.F / such that tij D 1 i ı j on Uij :

(5.3) 

5.1.3 Sections of a bundle. A (smooth) section of a bundle E  ! M is a smooth 

!M map f W M ! E such that  ı f D Id. The space of smooth sections of E  will be denoted (somewhat improperly) by C 1 .M I E/. 

5.1.4 Fiber bundles with structure group G . Let E  ! M a fiber bundle

and G a group with an injective morphism  W G ! Aut.F /, where F is the typical  fiber of E. One says that E  ! M has G as structure group and one writes G !  ! M if for all compatible i; j one has E tij .x/ D .gij .x//; with gij W Uij ! G: The maps gij satisfy of course (5.1).

5.1.5 Principal bundles. There is a canonical injective morphism  W G ! 

Aut.G/ given by left multiplication. A bundle P  ! M with G as fiber and structure group for the above action is called a G-principal bundle. Its transition maps are given by maps gij W Uij ! G  Aut.G/: 

! M is a G-principal bundle if there is a right action of Equivalently, a bundle P  G on P , which preserves the fibers and acts freely and transitively on the fibers. It is known that a principal bundle is trivial iff it has a global section.

45

5.1 Background 

5.1.6 Vector bundles. Let K D R or C. A bundle E  ! M with typical fiber

Kn is called a vector bundle of rank n if Ex is an n-dimensional vector space over K for each x 2 M and the maps i;x D F ı i jEx W Ex ! Kn ;

x 2 Ui

are K-linear. If tij are the transition functions of E one has tij W Uij ! GLn .K/. If each fiber Ex is oriented and the maps i;x W Ex ! Kn are orientation preserving,  ! M is said to be oriented, and in this case the transition maps the vector bundle E  tij .x/ take values in GLC n .K/. 

If E  ! M is a vector bundle, we denote by C 1 .M I E/, resp. C01 .M I E/, the space of smooth resp. smooth compactly supported, sections of E. Similarly, one denotes by D 0 .M I E/, E 0 .M I E/ the space of distributional, resp., compactly supported distributional sections of E. If .M; g/ is a spacetime, one denotes by Csc1 .M I E/ the space of smooth spacecompact sections of E, see Subsection 5.2.6 for terminology.

5.1.7 Tangent and cotangent bundles. If M is a smooth manifold of dimen

sion n, its tangent bundle TM  ! M is the vector bundle with fiber Rn and transition maps Uij 3 x 7! Dx ij 2 GLn .R/; where f.Ui ; i /gi 2I is an atlas of M and ij D i ı 1 j . Likewise its cotangent 

! M is the vector bundle with fiber Rn and transition maps bundle T  M  Uij 3 x 7! .t Dx ij /1 2 GLn .R/: We denote by ^p .M / the bundle of p-forms on M and set ^.M / D

n M

^p .M /:

pD0

M is orientable if ^n .M / admits a non-zero global section. If this is the case the transition maps tij of TM can be chosen so that det tij > 0 on Uij . 

5.1.8 Metric vector bundles. A vector bundle E  ! M is a metric vector bundle (of signature .q; p/) if each fiber Ex is equipped with a non-degenerate scalar product hx and i;x W .Ex ; hx / ! Rq;p is orthogonal for x 2 Ui ; where Rq;p is RqCp with the canonical scalar product 

Pq

2 i D1 xi

C

PpCq i DqC1

xi2 .

46

5 Free Klein–Gordon fields on curved spacetimes 

5.1.9 Dual vector bundle. Let E  ! M a vector bundle of rank n. The dual 

bundle E 0  ! M is defined by the fibers Ex0 D .Ex /0 and the transition maps .tij1 /0 . 

5.1.10 Bundle of frames. Let E  ! M a vector bundle of rank n. We can 

! M and defined as associate to it the bundle of frames of E, denoted by Fr.E/  follows: one sets G Fr.Ex /; Fr.E/ D x2M

where Fr.V / is the set of ordered bases (i.e. frames) of the vector space V , i.e. of linear isomorphisms F W Kn ! Ex . The transition functions of Fr.E/ are Tij .x/ W GLn .K/ 2 A 7! tij .x/ ı A 2 GLn .K/;

x 2 Uij ; 

where tij W Uij ! GLn .K/ are the transition functions of E. The bundle Fr.E/  ! M is a GLn .K/-principal bundle. 

5.1.11 The bundle End.E /. Let E  ! M a vector bundle of rank n. One 

defines the vector bundle End.E/  ! M with fibers End.E/x D End.Ex / and transition maps A ! tij .x/ ı A ı tij1 .x/, x 2 M , A 2 End.Kn /. 

5.1.12 The bundle E1  E2 . Let Ei  ! Mi be vector bundles of rank ni , 

i D 1; 2. One can form the vector bundle E1  E2  ! M1  M2 , with fibers E1;x1 ˝ E2;x2 over .x1 ; x2 /. If fUi;ji gji 2Ii and ti;ji ;ki are coverings and transition maps for 

Ei  ! Mi , then one takes fU1;j1  U2;j2 g.j1 ;j2 /2I1 I2 as covering of M1  M2 and t1;j1 ;k1 ˝ t2;j2 ;k2 as transition maps. 

5.1.13 The bundle End.E; E  /. If E  ! M is a complex vector bundle of 

rank n, the bundle End.E; E  /  ! M is the bundle with fibers End.E; E  /x D  End.Ex ; Ex / and transition maps A ! tij .x/ ı A ı tij .x/, x 2 M , A 2 End.Cn ; Cn /. A vector bundle E equipped with a smooth section  of End.E; E  / such that .x/ is a non-degenerate Hermitian form on Ex for all x 2 M is called a Hermitian vector bundle.

5.1.14 Connections on vector bundles. Let E a complex vector bundle over M . Note that C 1 .M I E/ is a C 1 .M / module. A connection r on E is a bilinear map r W C 1 .M I TM /  C 1 .M I E/ ! C 1 .M I E/

47

5.2 Lorentzian manifolds

such that rX .f '/ D X.f /' C f rX '; rf X ' D f rX ';

f 2 C 1 .M /; X 2 C 1 .M I TM /; ' 2 C 1 .M I E/:

If g is a metric on M , there exists a unique connection on TM , called the Levi-Civita connection, denoted by r g or often simply by r, such that X.X1 gX2 / D rX X1 gX2 C X1 grX X2 ; rX1 X2  rX2 X1 D ŒX1 ; X2 ;

X; X1 ; X2 2 C 1 .M I TM /;

X1 ; X2 2 C 1 .M I TM /:

(5.4)

5.1.15 Stokes formula. Let M be a smooth n-dimensional manifold, †  M a smooth hypersurface, i W † ! M the canonical injection, and i  W ^.M / ! ^.†/ the pullback by i . A vector field X over †, i.e. a smooth section of T† M is said to be transverse to † if Tx M D RXx ˚ Tx † for each x 2 †. One still denotes by X any of its smooth extensions as a section of TM , supported in a neighborhood of † in M . If ! 2 C 1 .M I ^p .M //, then X y ! 2 C 1 .M I ^p1 .M //, where y denotes the interior product, and one sets: iX ! D i  .X y !/ 2 C 1 .†; ^p1 .†//: One uses the same procedure to pullback densities on M to densities on †: if  D j!j for ! 2 C 1 .M I ^n .M // is a smooth density on M , we set iX  D jiX !j which is a smooth density on †. In local coordinates .x 1 ; : : : ; x n /, in which † D fx 1 D 0g, X is transverse to † iff X 1 .0; x 2 ; : : : ; x n / ¤ 0, and if  D f dx 1    dx n , then iX  D f .0; x 2 ; : : : ; x n /jX 1 .0; x 2 ; : : : ; x n /jdx 2    dx n : We will always assume that M is orientable, see Subsection 5.1.7, and fix a smooth, nowhere vanishing n-form !or on M . If U  M is an open set such that @U is a finite union of smooth hypersurfaces, then one orients @U by the .n  1/-form iX !or , where X is an outwards pointing, transverse vector field to @U and i W @U ! M is the canonical injection. We recall Stokes’ formula: ˆ ˆ d! D i  !; ! 2 C 1 .M I ^n1 .M //: (5.5) U

@U

5.2 Lorentzian manifolds A Lorentzian manifold is a pair .M; g/, where M is a smooth n-dimensional manifold and g is a Lorentzian metric on M , i.e. a smooth map M 3 x 7! g.x/, where g.x/ 2

48

5 Free Klein–Gordon fields on curved spacetimes

Ls .Tx M; Tx0 M / has signature .1; n1/. It is customary to write g as g .x/dx  dx or g.x/dx 2 and to denote the inverse metric g 1 .x/ 2 Ls .Tx M 0 ; Tx M / as g  .x/d  d  or g 1 .x/d  2 . Definition 5.1. (1) A vector v 2 Tx M is time-like if v  g.x/v < 0, null if v  g.x/v D 0, causal if vg.x/v  0, and space-like if vg.x/v > 0. (2) Similarly, a vector field v on M is time-like, etc., if v.x/ is time-like, etc., for each x 2 M . (3) The cone of time-like, resp. null vectors in Tx M is denoted by C.x/, resp. N.x/. Definition 5.2. A vector subspace V  Tx M is time-like if it contains both spacelike and time-like vectors, null if it is tangent to the lightcone N.x/, and space-like if it contains only space-like vectors. Lemma 5.3. If V  Tx M is a vector subspace, one denotes by V ? its orthogonal with respect to g.x/. Then V is time-like, resp. null, space-like iff V ? is space-like, resp. null, time-like. We refer to [Fr, Lemma 3.1.1] for the proof. There is a similar terminology for submanifolds N  M . Definition 5.4. A submanifold N  M is time-like resp. space-like, null if Tx N is time-like resp. space-like, null for each x 2 N . Null submanifolds are also called characteristic.

5.2.1 Volume forms and volume densities. The metric g induces a scalar product .j/g on the fibers ^px .M / D ^p Tx0 M , defined by .u1 ^    ^ up jv1 ^    ^ vp /g.x/ D det.ui g 1 .x/vj /

1  p  n:

(5.6)

Assuming that M is orientable, one obtains a unique n-form g 2 C 1 .M I ^n M /, called the volume form, such that . g j g /g.x/ D 1 for all x 2 M and g is positively oriented. The volume density is the 1-density d Volg D j g j: If .x 1 ; : : : ; x n / are local coordinates on M such that dx 1 ^    ^ dx n is positively oriented, then one has: 1

g D jg.x/j 2 dx 1 ^    ^ dx n ; where jg.x/j D det.gij .x//.

1

d Volg D jg.x/j 2 dx 1    dx n ;

(5.7)

5.2 Lorentzian manifolds

49

5.2.2 Distributions on M . We denote by D0 .M /, resp. E 0 .M /, the space of distributions on M , resp. compactly supported distributions, see e.g. [H1, Section 6.3] for definitions. The topological dual of C01 .M /, resp. C 1 .M /, is the space of distribution densities, resp. distribution densities of compact support. One identifies each distribution u with the distribution density ud Volg . Setting ˆ  uv d Volg ; (5.8) .ujv/M D M

leads to the following natural notation .ujv/M D hu d Volg jvi;

for u 2 D 0 .M /; v 2 C01 .M /;

(5.9)

where hji is the duality bracket.

5.2.3 Normal vector field. If †  M is a smooth hypersurface which is not null, there is a unique (up to sign) transverse vector field n, which is normal and normalized, i.e. n.x/g.x/v D 0;

jn.x/g.x/n.x/j D 1;

8v 2 Tx †; x 2 †:

The induced metric on †, h D i  g, is non-degenerate and one has

h D in g ;

iX g D X a  na h ;

(5.10)

if X is a vector field on †. This can be easily checked in local coordinates, using (5.7).

5.2.4 Gauss formula. If X is a vector field on M , then ra X a g D d.X y g /;

(5.11)

where r is the Levi-Civita connection associated to g, hence Stokes’ formula can be rewritten as ˆ ˆ r a Xa d Volg D U

@U

iX d Volg :

(5.12)

To express the right-hand side of (5.12), we fix a vector field l that is transverse to @U and outwards pointing. Let  be a 1-form on M such that Ker  D T @U , normalized such that  l D 1. It follows that if X is a vector field on M we have X D . X /l C R;

where R is tangent to @U:

Since R is tangent to @U , we have i  .Ry d Volg / D 0, hence iX d Volg D . X /il.d Volg /:

50

5 Free Klein–Gordon fields on curved spacetimes

Thus, we obtain the Gauss formula ˆ ˆ ra X a d Volg D U

@U

a X a il d Volg :

(5.13)

Let † be one of the connected components of @U . If † is given by ff D 0g for some function f with df ¤ 0 on †, and if we can complete f near † with coordinates y 1 ; : : : ; y n1 such that df ^ dy 1 ^    ^ dy n1 is direct, with @f pointing outwards, then we take l D @f ,  D df and obtain 1

iX .d Volg / D X a ra f jgj 2 dy 1    dy n1 on †:

(5.14)

5.2.5 Non-characteristic boundaries. If † is non-characteristic, we can take l D n, the outwards pointing normal vector field to †. Since in d Volg D d Volh we obtain iX d Volg D na X a d Volh on †: (5.15) 5.2.6 Causal structures. We now recall some notions related to the causal structure on M induced by the metric g. All the objects below are of course unchanged under a conformal transformation g ! c 2 g of the metric, where c 2 C 1 .M / is a strictly positive function. Definition 5.5. (1) A Lorentzian manifold is time orientable if it carries a continuous time-like vector field v. Given such a vector field, one denotes by C˙ .x/ the connected component of C.x/ such that ˙v.x/ 2 C˙ .x/. (2) The vectors in C˙ .x/ are called future/past directed, and one uses the same terminology for time-like vector fields. Such a continuous choice of C˙ .x/ is called a time orientation. (3) A time oriented Lorentzian manifold is called a spacetime. In the sequel, we will always assume that the Lorentzian manifold M is orientable, see Subsection (5.1.6), and by spacetime we will always mean an orientable spacetime. Definition 5.6. Let .M; g/ be a spacetime and  W I 3 s 7! x.s/ 2 M a piecewise C 1 curve. (1)  is time-like, resp. null, space-like, future/past directed if all its tangent vectors x 0 .s/; s 2 I are so. (2)  is inextensible if no piecewise C 1 reparametrization of  can be continuously extended beyond its endpoints. Definition 5.7. (1) The time-like resp. causal future/past of x 2 M , denoted by I˙ .x/, resp. J˙ .x/, is the set of points belonging to time-like, resp. causal future/past directed curves  starting at x.

5.2 Lorentzian manifolds

(2) For K  M one sets I˙ .K/ D

S x2K

I˙ .x/, J˙ .K/ D

S x2K

51

J˙ .x/.

(3) The time-like, resp. causal shadow of K  M is I.K/ D IC .K/ [ I .K/, resp. J.K/ D JC .K/ [ J .K/. (4) Two sets K1 ; K2 are causally disjoint if J.K1/ \ K2 D ;, or, equivalently if J.K2/ \ K1 D ;. (5) A closed set A  M is space-compact, resp. future/past space-compact if A  J.K/, resp. A  J˙ .K/ for some compact set K b M . (6) A closed set A  M is time-compact, resp. future/past time compact if A \ J.K/, resp. A \ J .K/ is compact for each compact set K b M .

K

a space-compact set

a time-compact set Fig. 1

Note that if U  M is an open subset of the spacetime .M; g/, then .U; g/ is a spacetime as well. In this case if K  U , we use the notation J˙U .K/, resp. J˙M .K/ for the future/past causal shadows of K in U resp. in M . One says that U  M is causally compatible if J˙U .x/ D J˙M .x/ \ U for each x 2 U . This is equivalent to the property that a causal curve in M between two points x; x 0 2 U is entirely contained in U . The same terminology is used for an isometric embedding i W .M 0 ; g 0 / ! .M; g/. An example of a non-causally compatible domain U in Minkowski spacetime is given in Fig. 2 below. JCM .x/ \ U

U

JCU .x/

x

U Fig. 2

x

52

5 Free Klein–Gordon fields on curved spacetimes

5.3 Stationary and static spacetimes 5.3.1 Killing vector fields. Let X a smooth vector field on M whose flow s 7! X .s/ is complete. X is called a Killing vector field for .M; g/ if X .s/ are isometries of .M; g/, i.e. X .s/ .g/ D g for s 2 R. Equivalently, X should satisfy Killing’s equation ra Xb C rb Xa D 0; where r is the Levi-Civita connection for g.

5.3.2 Stationary spacetimes. Definition 5.8. The spacetime .M; g/ is stationary if it admits a complete, time-like future directed Killing vector field X . The standard model of a stationary spacetime is as follows: let .S; h/ be a Riemannian manifold , N 2 C 1 .S / with N > 0, and wi dx i be a smooth 1-form on S . Let M D Rt  Sx and g D N 2 .x/dt 2 C .dx i C w i .x/dt/hij .x/.dx j C w j .x/dt/: Then .M; g/ is stationary with Killing vector field @t if N 2 .x/ > wi .x/hij .x/wj .x/, x 2 S. It is known, see e.g. [S2, Proposition 3.1], that a stationary spacetime which is also globally hyperbolic (see Section 5.4) is isometric to such a model.

5.3.3 Static spacetimes. A stationary spacetime .M; g/ with Killing vector field X is called static if there exists a smooth hypersurface S which is everywhere gorthogonal to X . The standard model of a static spacetime is the one above for wi dx i D 0. A static, globally hyperbolic spacetime is isometric to the standard model iff one can choose S to be a Cauchy surface, see [S2, Proposition 3.2]. An ultra-static space time is a spacetime M D R  S with the Lorentzian metric g D dt 2 C h.x/dx 2 , where .S; h/ is a Riemannian manifold. It is known that .M; g/ is globally hyperbolic iff .S; h/ is complete, see [S, Theorem 3.1], [Ky1, Proposition 5.2].

5.4 Globally hyperbolic spacetimes Definition 5.9. A Cauchy surface S is a closed set S  M which is intersected exactly once by each inextensible time-like curve. Definition 5.10. A spacetime .M; g/ is globally hyperbolic if the following conditions hold:

5.4 Globally hyperbolic spacetimes

53

(1) JC .x/ \ J .x 0 / is compact for all x; x 0 2 M , (2) M is causal, i.e. there are no closed causal curves in M . The original definition of global hyperbolicity required the stronger condition of strong causality, see e.g. [BGP, Definition 1.3.8], [W1, Section 8.3]. The fact that the two definitions are equivalent is due to Bernal and Sanchez [BS3]. Here are three elementary examples of non-globally hyperbolic spacetimes: (1) M D R1;1 nfx0 g: JC .x/ \ J .x 0 / may not be compact; (2) M D Rt  0; 1Œ x: JC .x/ \ J .x 0 / may not be compact; (3) M D S1t  Rx : J˙ .x/ D M . x0 JC .x/ \ J .x 0 /

x0 JC .x/ \ J .x 0 /

x

x0

x (1) R

1;1

x

nfx0 g

(2) Rt  0; 1Πx

(3) S1t  Rx

Fig. 3

Later on we will need the following result, which is proved in [BGP, Lemma A.5.7]. Lemma 5.11. Let .M; g/ be globally hyperbolic and K1 ; K2 b M be compact. Then JC .K1 / \ J .K2 / is compact. The following theorem is also due to Bernal and Sanchez [BS1, BS2]. It extends an earlier result of Geroch [Ge]. Theorem 5.12. The following conditions are equivalent: (1) .M; g/ is globally hyperbolic. (2) M admits a Cauchy surface S . (3) There exists an isometric diffeomorphism:  W .M; g/ ! .R  †; ˇ.t; x/dt 2 C ht .x/d x2 /; where † is a smooth .n  1/-dimensional manifold, ˇ > 0 is a smooth function on R  †, t 7! ht .x/d x2 is a smooth family of Riemannian metrics on †, and fT g  † is a smooth space-like Cauchy surface in R  † for each T 2 R.

54

5 Free Klein–Gordon fields on curved spacetimes

5.4.1 Orthogonal decompositions of the metric. An isometry  W M ! R  † such that g D  .ˇdt 2 C ht d x2 / as in Theorem 5.12 is called an orthogonal decomposition. Orthogonal decompositions are very useful to analyze Klein–Gordon equations on .M; g/. The decomposition in Theorem 5.12 is related to the notion of temporal functions. Definition 5.13. A smooth function t W M ! R is called a temporal function if its gradient rt D g 1 dt is everywhere time-like and past directed. It is called a Cauchy temporal function if, in addition, its level sets t 1 .T / are Cauchy surfaces for all T 2 t.M /. Clearly, if  W M ! R  † is the diffeomorphism in Theorem 5.12 (3), then t D R ı  is a Cauchy temporal function. Now let t be a Cauchy temporal function. Without loss of generality we can assume that t.M / D R and set † D t 1 .f0g/, which is a smooth, space-like Cauchy surface. We equip M with an auxiliary complete Riemannian metric hO and set v D krtk1 O rt; h

which is a complete, time-like vector field. Since † is a Cauchy surface, its integral curve through x 2 M intersects † at a unique point .x/ 2 †, and we set  W M 3 x 7! .t.x/; .x// 2 R  †; which is a smooth diffeomorphism. If we set †s D t 1 .fsg/, then Tx †s is orthogonal to Rv.x/, hence is space-like by Lemma 5.3. The image of Tx †s , resp. Rv.x/, under Dx  is f0g  Ty †, resp. R  f0g. Therefore, the metric .1 / g is of the form ˇdt 2 C ht , with ˇ and t 7! ht as in Theorem 5.12. It is known, see [BS4, Theorem 1.2], that for any smooth, space-like Cauchy surface †, there exists a Cauchy temporal function t W M ! R such that † D t 1 .f0g/. Therefore, any smooth space-like Cauchy surface † can be chosen in Theorem 5.12 (3), and the isometry  is completely determined by fixing † and a Cauchy temporal function t with † D t 1 .f0g/.

5.4.2 Neighborhoods of a space-like Cauchy surface. Lemma 5.14. Let †  M be a smooth, space-like Cauchy surface. Then the open neighborhoods V of † such that V  M is causally compatible form a basis of neighborhoods of † in M . Proof. We can assume that M D R  † with metric ˇdt 2 C ht d x2 and identify † with f0g  †. We can also assume that ˇ D 1 by a conformal transformation. If U is a neighborhood of †, we can find a strictly positive function r 2 C 1 .†/ such that

5.4 Globally hyperbolic spacetimes

55

for V D f.t; x/ W jtj < r.x/g one has .i/ V  U; .ii/

1 h .x/ 4 0

 ht .x/  4h0 .x/; 8 .t; x/ 2 V;

.iii/ rr.x/h0 .x/rr.x/ 

1 ; 16

(5.16)

8 x 2 †:

In fact, it suffices to fix an open covering S P fUi gi 2N of † and intervals fIi gi 2N such that I  U  U and choose r D i i i 2N i 2N i i , where fi gi 2N is a partition of unity of † subordinate to fUi gi 2N and the i are chosen small enough. Let now  W Œ1; 1 3 s 7! x.s/ be a future directed causal curve in .M; g/ with x.0/; x.1/ 2 V . Since † is a Cauchy surface, we can assume, modulo a reparametrization of  , that ˙t.s/  0 for ˙s 2 Œ0; 1. By (5.16) (ii), we have t 0 .s/ 

1 0 1 .x .s/h0 .x.s//x0.s// 2 for s 2 Œ1; 1: 2

If f .s/ D t.s/  r.x.s// for s 2 Œ0; 1, then we deduce from (5.16) (iii) and the Cauchy–Schwarz inequality that f 0 .s/ > 0 as long as s 2 Œ0; 1 and f .s/ < 0. Since x.1/ 2 V , we have f .1/ < 0, hence f .s/ < 0 for s 2 Œ0; 1, i.e. x.s/ 2 V for s 2 Œ0; 1. For s 2 Œ1; 0 we use the same argument for f .s/ D t.s/ C r.x.s//. 

5.4.3 Gaussian normal coordinates. If †  M is a smooth space-like Cauchy surface, there is another orthogonal decomposition of the metric using Gaussian normal coordinates to †. It does not depend on the choice of a Cauchy temporal function having † as one of its level sets, but Gaussian normal coordinates exist only in a neighborhood of † in M . Let n 2 T† M be the future directed unit normal vector field to †, so that ny is g-orthogonal to Ty †, future directed, and satisfies g ny g.y/ny D 1. We denote by expx for x 2 M the exponential map at x for the metric g. Proposition 5.15. Let †  M be a smooth space-like Cauchy surface. Then (1) there exist neighborhoods U of f0g  † in R  † and V of † in M such that V  M is causally compatible and  W U 3 .t; x/ 7! expgx .tnx / 2 V is a diffeomorphismI (2) one has  g D dt 2 Cht .x/d x2, where ht is a t-dependent Riemannian metric on † over U . Proof. The map  is clearly a local diffeomorphism. The existence of U; V as in (1) is shown in [O, Proposition 26, Chap. 7] , and V can be chosen to be causally compatible in M by Lemma 5.14.

56

5 Free Klein–Gordon fields on curved spacetimes

Let us explain the proof of (2), following [W1, Section 3.3]. Using local coordinates x i , 1  i  n  1 on † near a point y 2 † we obtain by means of  local coordinates t; x i near a point x 2 V . Let T D @t ; Xi D @x i be the associated coordinate vector fields. Recall that if r is the Levi-Civita connection, then T b rb T a D 0; T

b

rb Xia



Xib rb T a

(5.17) D ŒT; Xi  D 0: a

(5.18)

(5.17) is the geodesic equation, and the Lie bracket ŒT; Xi  vanishes since T; Xi are coordinate vector fields. Denoting by X D X a one of the vector fields Xi , we compute: T b rb .Ta X a / D X a T b rb Ta C Ta T b rb X a D Ta T b rb X a ; using (5.17) and ra gbc D 0. Next, 1 Ta T b rb X a D X b Ta rb T a D X b rb .T a Ta /; 2 by (5.18) and the Leibniz rule for r. Finally, since T a Ta D 1 on † and T b rb .T a Ta / D 0, we have T a Ta D 1 everywhere, which implies that T b rb .Ta X a / D 0. Since Ta X a D 0 on †, we obtain Ta X a D 0, T a Ta D 1 everywhere. This implies (2). 

5.4.4 Spaces of distributions on globally hyperbolic spacetimes. We now recall some useful spaces of distributions on M , characterized by their support properties. We refer the reader to [S1, Chapter 4] for a complete discussion. Definition 5.16. A distribution u 2 D 0 .M / is space, .time/, future/past compact if its support is space, .time/, future/past compact. The spaces of such distributions are 0 0 0 denoted by Dsc .M /, Dtc0 .M /, Dsc;˙ .M /, Dtc;˙ .M /. Similarly, one defines the space, 1 1 1 1 .M /. of smooth functions Csc .M /, Ctc .M /, Csc;˙ .M /, Ctc;˙ The most useful space is Csc1 .M /; the other spaces appear naturally when discussing properties of the retarded/advanced inverses for Klein–Gordon operators, see Section 5.5 below. It is proved in [S1, Theorem 3.1] that a closed set A  M is future/past time compact iff there exists a Cauchy surface † in M such that A  J˙ .†/. Now let us describe the topologies of these spaces. If B  M is closed, we denote by C 1 .B/, resp. D 0 .B/, the smooth functions, resp. distributions with support in B, equipped with the C 1 .M /, resp. D 0 .M / topology. The topologies of the above

57

5.5 Klein–Gordon equations on Lorentzian manifolds

spaces are defined as the following inductive limits: S S 0 .M / D KbM D 0 .J.K//; .i/ Csc1 .M / D KbM C 1 .J.K//; Dsc S S 1 0 .M / D KbM C 1 .J .K//; Dsc;C .M / D KbM D 0 .J .K//; .ii/ Csc;C S S 1 0 .M / D KbM C 1 .JC .K//; Dsc; .M / D KbM D 0 .JC .K//; .iii/ Csc; S S 1 0 .M / D † M C 1 .J .†//; Dtc;C .M / D † M D 0 .J.†//; .iv/ Ctc;C S S 1 0 .M / D † M C 1 .JC .†//; Dtc; .M / D † M D 0 .JC.†//; .v/ Ctc; S .vi/ Ctc1 .M / D †1 ;†2 M C 1 .JC .†1 / \ J .†2 //; S .vii/ Dtc0 .M / D †1 ;†2 M D 0 .JC .†1 / \ J .†2 //: (5.19) In (i), (ii), and (iii) the set of compact subsets K b M is equipped with the order relation K1  K2 if K1  K2 ; in (iv), resp. (v) the set of Cauchy surfaces †  M is equipped with the order relation †  †0 if J .†/  J .†0 /, resp. JC .†/  JC .†0 /; and finally, in (vi) and (vii) the set of pairs of Cauchy surfaces .†1 ; †2 / is equipped with the order relation .†1 ; †2 /  .†01 ; †02 / if JC .†1 / \ J .†2 /  JC .†01 / \ J .†02 /. The various duality relations between these spaces are as follows, see [S1, Theorem 4.3]. Proposition 5.17. One has 0 Dsc .M / D Ctc1 .M /0 ;

Dtc0 .M / D Csc1 .M /0 ;

0 1 Dsc;˙ .M / D Ctc; .M /0 ;

0 1 Dtc;˙ .M / D Csc; .M /0 ;

and all the spaces above are reflexive.

5.5 Klein–Gordon equations on Lorentzian manifolds 5.5.1 Klein–Gordon operator. Let us fix a smooth real 1-form A D A .x/dx  on M and a real function V 2 C 1 .M I R/. A Klein–Gordon operator on .M; g/ is a differential operator P D .r   iqA .x//.r  iqA .x// C V .x/; 1

1

(5.20)

where r  D jgj 2 .x/r jgj 2 .x/g  .x/, A .x/ D g  .x/A .x/, and q 2 R. The quantization of the Klein–Gordon equation P  D 0 for  2 C 1 .M I C/ describes a charged bosonic field of charge q in the external electro-magnetic potential A .x/dx  . If A .x/dx  D 0, then P D g C V .x/, where g D r  r is the d’Alembertian. A typical example of V is V D Scalg C m2 , where Scalg is the n2 ; m D 0 yields the conformal wave scalar curvature on .M; g/, which for  D 4.n1/ operator.

58

5 Free Klein–Gordon fields on curved spacetimes

Recall that we defined the scalar product ˆ uvd Volg ; .ujv/M D M

on

C01 .M /.

Clearly, P is formally selfadjoint with respect to .j/M . Actually, every differential operator of the form P D g C R.x; @x /;

where R.x; @x / is a first-order differential operator on M such that P is formally selfadjoint with respect to .j/M , is of the form (5.20). We are interested in the Klein–Gordon equation P  D 0; and we will always consider its complex solutions in D 0 .M / or C 1 .M /.

5.5.2 Conserved currents. Let us set raA D ra  iqAa ;

r aA D r a  iqAa

and introduce on M the 1-form

We have

Ja .u1 ; u2 / D raA u1 u2  u1 raA u2 ; u1 ; u2 2 C 1 .M /:

(5.21)

r aA Ja .u1 ; u2 / D u1 P u2 C P u1 u2 :

(5.22)

It follows that if ui 2 C 1 .M / with P ui 2 C01 .M / and U  M is an open set with @U a finite union of non-characteristic hypersurfaces, we obtain from Subsection 5.2.4 the Green formula ˆ ˆ    a A  u1 P u2  P u1 u2 d Volg D n ra u1 u2  u1 na raA u2 d Volh ; (5.23) U

@U

where h is the induced metric on @U . To have a satisfactory global theory of Klein–Gordon equations on M , we need to make some assumptions on its causal structure. It turns out that if .M; g/ is globally hyperbolic the theory is particularly nice and complete.

5.5.3 Advanced and retarded inverses. The following extension of Theorem 2.8 is originally due to Leray [Le]. A proof can be found in [BGP, Theorem 3.3.1]. Theorem 5.18. Let .M; g/ be globally hyperbolic and let P be a Klein–Gordon 0 operator on M . Then for v 2 E 0 .M / there exist unique solutions uret=adv 2 Dsc;˙ .M / of the equation P uret=adv D v:

5.5 Klein–Gordon equations on Lorentzian manifolds

59

One has uret=adv D Gret=adv v, where .i/ Gret=adv W E 0 .M / ! D 0 .M /; Gret=adv W C01 .M / ! C 1 .M / continuouslyI .ii/ P ı Gret=adv D Gret=adv ı P D ½I

(5.24)

.iii/ supp Gret=adv v  J˙ .supp v/: Using the continuity and support properties of Gret=adv and the topologies of the spaces introduced in Definition 5.16, one easily obtains the following corollary. Corollary 5.19. The maps Gret=adv extend continuously as follows 1 1 Gret=adv W Csc;˙ .M / ! Csc;˙ .M /;

0 0 Dsc;˙ .M / ! Dsc;˙ .M /;

1 1 .M / ! Ctc;˙ .M /; Gret=adv W Ctc;˙

0 0 Dtc;˙ .M / ! Dtc;˙ .M /

The operator G D Gret  Gadv

(5.25)

is called in physics the Pauli–Jordan function or causal propagator. Using that P D  P  and the uniqueness of Gret=adv , we obtain that Gret=adv D Gadv=ret on C01 .M /, hence (5.26) G D G  ; supp Gv  J.supp v/:

5.5.4 The Cauchy problem. We now discuss the Cauchy problem for P . Let † be a smooth, space-like Cauchy surface in M , n the future unit normal to †, see a A Subsection 5.4.3, and @A n D n ra . As in Section 2.4, we define the Cauchy data map %† by:   †  %†  D ;  2 C 1 .M /: (5.27) i1 @A n † The proof of the following result can be found in [BGP, Theorem 3.2.11]. Theorem 5.20. The Cauchy problem 

P  D 0; %†  D f;

(5.28)

 f0 2 C01 .†I C2 /. has a unique solution  D U† f 2 C .M / for each f D f1 Moreover the map U† W C01 .†I C2 / ! C 1 .M / is continuous and 

1

supp U† f  J.supp f0 \ supp f1 /:

60

5 Free Klein–Gordon fields on curved spacetimes

Let us recall a well-known relation between the Cauchy evolution operator U† and G. We first introduce some notation. Since %† W C01 .M / ! C01 .†I C2 / we obtain by duality the map %† W D 0 .†I C2 / ! D 0 .M /;

(5.29)

where in (5.29) we identify the space C01 .M /0 (resp. C01 .†/0 ), of distribution densities on M (resp. on †), with D 0 .M / (resp. D 0 .†/) using the density d Volg (resp. d Volh ). A concrete expression of %† is %† f D f0 ˝ ı† C i1 f1 ˝ nrı† ; where the distribution ı† is defined by ˆ u d Volh ; hı† d Volg ; ui D †

We also set q† D



0

½

½



0

(5.30)

u 2 C01 .M /:

2 L.C01 .†I C2 //:

(5.31)

Proposition 5.21. Set G† D i1 q† . Then U† D .%† G/ G† ;

on C01 .†I C2 /:

Proof. We apply Green’s formula (5.23) to u2 D u D U† f , u1 D Gadv=ret v, v 2 C01 .M / and U D J˙ .†/. This yields ˆ

ˆ vu d Volg D JC .†/

ˆ

ˆ

†

vu d Volg D J .†/

†



  Gadv vna raA u C na raA Gadv vu d Volh ;

  Gret vna raA u  Gret vna raA u d Volh :

Adding the two equations above, we get, since J.†/ D M , ˆ ˆ vud Volg D  na Ja .Gv; u/d Volh : M

†

By the definition of %† and the fact that G D G  we obtain the proposition.



From Proposition 5.21 and Corollary 5.19 we obtain the following continuous extensions of U† : 0 U† W E 0 .†I C2 / ! Dsc .M /;

D 0 .†I C2 / ! D 0 .M /:

(5.32)

5.6 Symplectic spaces

61

5.6 Symplectic spaces 5.6.1 Symplectic space of Cauchy data. We equip C01 .†I C2 / with the Hermitian form gq† f D

ˆ †

  g 1 f0 C g 0 f1 d Volh :

(5.33)

Abusing the notation, we have gq† f D .gjq† f /† ; for

ˆ



.gjf /† D †

 g0 f0 C g 1 f1 d Volh ;

(5.34)

and the operator q† is defined in (5.31). Clearly, .C01 .†I C2 /; q† / is a Hermitian space, see Subsection 4.2.4.

5.6.2 Symplectic space of solutions. Let us denote by Solsc .P / the space of smooth complex space-compact solutions of the Klein–Gordon equation P  D 0. Proposition 5.22. (1) The Hermitian form q on Solsc .P / defined by: ˆ  1 q2 D i na JaA .1 ; 2 /d Volh

(5.35)

†

is independent on the choice of the space-like Cauchy surface † and .Solsc .P /; q/ is a Hermitian space. (2) If † is a space-like Cauchy surface, the map %† W .C01 .†I C2 ; q† / ! .Solsc .P /; q/ is unitary with inverse U† . Proof. If 1 ; 2 2 Solsc .P /, then by (5.22) we have raA J a .1 ; 2 / D 0. If †; †0 are two space-like Cauchy surfaces with †0  JC .†/, we apply the Gauss formula to U D Int.JC .†/ \ J .†0 // and obtain that ˆ ˆ a A n Ja .1 ; 2 /d Volh D na JaA .1 ; 2 /d Volh : †

†0

In the general case we pick another Cauchy surface †00  JC .†/\JC .†0 / and apply the same argument to obtain (1). Statement (2) follows immediately. 

62

5 Free Klein–Gordon fields on curved spacetimes

5.6.3 Pre-symplectic space of test functions. Theorem 5.23. (1) The sequence P

G

P

0 ! C01 .M / ! C01 .M / ! Csc1 .M / ! Csc1 .M / ! 0 is an exact complex. (2) Let † be a space-like Cauchy surface. Then one has .%† G/ G† .%† G/ D G on C01 .M /: (3) The map

 GW

C01 .M / ; . jiG /M P C01 .M /

 ! .Solsc .P /; q/

is unitary. Proof. (1) The above sequence is clearly a complex since G ı P D 0 and P ı G D 0 on C01 .M /. Let us check that it is exact. Let u 2 C01 .M / with P u D 0. Since u 2 Csc1 .M / we have u D Gret 0 D 0 by Theorem 5.18, which proves exactness at the first C01 .M /. Let u 2 C01 .M / with Gu D 0. We have v D Gret u D Gadv u 2 C01 .M / since supp v  JC .supp u/ \ J .supp u/ is compact by Lemma 5.11. Then u D P v, and so u 2 P C01 .M /, which proves exactness at the second C01 .M /. Let  2 Csc1 .M / with P  D 0, i.e.  2 Solsc .P /. We can find cutoff functions 1 .M / such that C C  D 1 on supp , see Fig. 4 below. We have ˙ 2 Csc;˙ supp   J.K/ and supp ˙  J˙ .K˙ / for K; K˙ compact. Since rC D r on supp  we have supp  \ supp r˙  J.K/ \ JC .KC / \ J .K / which is compact by Lemma 5.11. We set ˙ D ˙  and v D P C D P  , which belongs 1 to C01 .M /, by the compactness of supp  \ supp r˙ . Since ˙ 2 Csc;˙ .M / we have ˙ D ˙Gret=adv v hence  D Gv, which proves exactness at the first Csc1 .M /. 1 Let v 2 Csc1 .M / and ˙ 2 Csc;˙ .M / such that C C  D 1 on supp v. From 1 .M / to Theorem 5.18 (iii) we see that Gret=adv can be extended as a map from Csc; 1 1 Csc;˙ .M /. We set then u D Gret  v C Gadv C v and P u D v, u 2 Csc .M / which proves exactness at the second Csc1 .M /. (2) From U† %† D ½ on Solsc .P /, U† D .%† G/ G† on C01 .†I C2 / and Solsc .P / D GC01 .M / we obtain (2). (3) The map G and the Hermitian form . jiG /M are well defined on C 1 .M /

C01 .M / P C01 .M /

since G ı P D P ı G D 0. By (1), the map G W P C0 1 .M / ! Solsc .P / is bijective, 0 and by (2) and the definition of q in (5.35), it is unitary.  Let us summarize the above discussion.

5.6 Symplectic spaces

63

supp C

supp  †

supp  Fig. 4

Theorem 5.24. The maps  1 

† C0 .M / G ; . jiG /M ! .Solsc .P /; q/ ! .C01 .†I C2 /; q† / 1 P C0 .M / are isomorphisms of Hermitian spaces. As in the Minkowski case, the first and last Hermitian spaces are the most useful.

5.6.4 Time-slice property. We end this section with a remark which is related to the time-slice axiom see e.g. [BGP, Theorem 4.5.1]. Proposition 5.25. Let † a space-like Cauchy surface and V  M a neighborhood of † such that V  M is causally compatible. Then the maps  1 

† C0 .V / G ; . jiG /M ! .Solsc .P /; q/ ! .C01 .†I C2 /; q† / 1 P C0 .V / are isomorphisms of Hermitian spaces. Proof. The space .V; g/ is globally hyperbolic. Let P jV be the restriction of P to V . Since V  M is causally compatible, the causal propagator for P jV equals GjV . If C 1 .V / Œu 2 P C0 1 .V / , then GjV u D .Gu/jV . Applying this remark and Theorem 5.24 for 0 V we obtain the proposition. 

Chapter 6

Quasi-free states on curved spacetimes We saw in Chapter 5 that to a Klein–Gordon operator P on a globally hyperbolic   C 1 .M / spacetime .M; g/ one can associate the Hermitian space P C0 1 .M / ; . jiG /M . Fol0 lowing Chapter 4, one can then consider the associated CCR -algebra and quasi-free states on it. The complex covariances of a quasi-free state induce sesquilinear forms on C01 .M / and it is natural to assume their continuity for the topology of C01 .M /, which allows to introduce their distributional kernels. By Proposition 5.25 one can equivalently use the Hermitian space .C01.†; C2 /; q† / if † is a space-like Cauchy surface. The associated covariances are called Cauchy surface covariances and are very useful for the concrete construction of states.

6.1 Quasi-free states on curved spacetimes Definition 6.1. We denote by CCR.P / the -algebra CCRpol .Y ; q/, see Subsection 4.5.3, for   1 C0 .M / ; . jiG /M : .Y ; q/ D P C01 .M /

6.1.1 Space-time covariances. We will identify distribution densities on M , resp. M  M with distributions using the density d Volg , resp. d Volg  d Volg . Let ! be a gauge invariant quasi-free state on CCR.P /. Its complex covariC 1 .M / ances are sesquilinear forms on P C0 1 .M / , or equivalently sesquilinear forms ƒ˙ on 0 C01 .M / such that uƒ˙ P v D P uƒ˙ v D 0;

u; v 2 C01 .M /;

or in more compact notation ƒ˙ ı P D P  ı ƒ˙ , where P  is the formal adjoint of P defined in Subsection 4.1.4. It is natural to require that ƒ˙ W C01 .M / ! D 0 .M / are continuous, which we will always assume in the sequel. By the Schwartz kernel theorem, ƒ˙ have distributional kernels, still denoted by ƒ˙ 2 D 0 .M  M /, defined by uƒ˙ v D .ƒ˙ ju ˝ v/M M ;

u; v 2 C01 .M /:

(6.1)

Definition 6.2. The maps ƒ˙ W C01 .M / ! D 0 .M / are called the spacetime covariances of !.

66

6 Quasi-free states on curved spacetimes

By Proposition 4.14 we have: Proposition 6.3. Two maps ƒ˙ W C01 .M / ! D 0 .M / are the spacetime covariances of a gauge invariant quasi-free state ! iff .i/ ƒ˙ W C01 .M / ! D 0 .M / are linear and continuous; .ii/ .ujƒ˙ u/M  0; u 2 C01 .M /; .iii/ ƒC  ƒ D iG; .iv/ P ı ƒ˙ D ƒ˙ ı P D 0:

6.1.2 Cauchy surface covariances. Let †  M a space-like Cauchy surface. We again identify distributions on † with distribution densities using the volume form d Volh , where h is the induced Riemannian metric on †. By Theorem 5.24, we can use equivalently the symplectic space .C01 .†I C2 /; q† / to describe CCR.P /. Therefore a quasi-free state ! as above can equivalently be 1 2 defined by a pair ˙ † of sesquilinear forms on C0 .†I C /, or equivalently linear maps 1 2 0 2 ˙ 1 0 ˙ † W C0 .†I C / ! D .†I C /. We will see later that ƒ W C0 .M / ! D .M / is ˙ 1 2 0 2 linear and continuous iff  W C0 .†I C / ! D .†I C / is linear and continuous. Definition 6.4. The maps ˙ † are called the Cauchy surface covariances of the state !. We recall that the scalar product .j/† on C01 .†I C2 / was defined in (5.34). 1 2 0 2 Proposition 6.5. Two maps ˙ † W C0 .†I C / ! D .†I C / are the Cauchy surface covariances of a gauge invariant quasi-free state ! iff 1 2 0 2 .i/ ˙ † W C0 .†I C / ! D .†I C / are linear and continuous; 1 2 .ii/ .f j˙ † f /†  0; f 2 C0 .†I C /;  .iii/ C †  † D q† :

We recall that q† is defined in (5.31) and that G† D i1 q† . Let us now look at the relationship between ƒ˙ and ˙ †. Proposition 6.6. (1) Let ˙ † be Cauchy surface covariances of a quasi-free state !. Then ƒ˙ D .%† G/ ˙ † .%† G/ are the spacetime covariances of !. (2) let ƒ˙ be the spacetime covariances of a quasi-free state !. Then   ˙   ˙ † D .%† G† / ƒ .%† G† /:

are the Cauchy surface covariances of !.

6.1 Quasi-free states on curved spacetimes

67

1 0 ˙ 1 2 Proof. (1) Since %† ˙ † %† G W C0 .M / ! Dtc .M / and † W C .†I C / ! 0 2 ˙ 1 0 D .†I C / are continuous, we see that ƒ W C0 .M / ! D .M / is continuous, by Corollary 5.19. The rest of the conditions in Proposition 6.3 follow from the equalities P ı G D G ı P D 0 and the fact that   1 C0 .M / %† G W ! .C01 .†I C2 /; q† / ; . jiG / M P C01 .M /

is unitary. 1 2 0 2 (2) The fact that ˙ † W C0 .†I C / ! D .†I C / is continuous uses properties of ˙ the wavefront set of ƒ deduced from the equalities P ı ƒ˙ D ƒ˙ ı P D 0 and will be explained later on in Chapter 7, see Subsection 7.2.9. Item (ii) in Proposition 6.5 follows from item (ii) in Proposition 6.3. To check item (iii) in Proposition 6.5, we write      C †  † D .%† G† / iG.%† G† / D G† %† iG%† G† D q† ;

since %† .%† G/ G† D ½, by Proposition 5.21. Therefore ˙ † are the Cauchy surface covariances of a quasi-free state !1 . To check that !1 D !, we use (1) and the fact that %† .%† G/ G† D ½ to conclude that ƒ˙ are the spacetime covariances of !1 ,  and hence !1 D !.

6.1.3 The case of real fields. For comparison with the literature, let us briefly explain the framework for real Klein–Gordon fields. Let P be a real Klein–Gordon operator, i.e. such that P u D P u. Clearly, Gret=adv and hence G are also real operators. Consider the real symplectic space  1  C0 .M I R/ .X ; / D ; .jG/M ; P C01 .M I R/ and denote by CCRR .P / the -algebra CCRpol .X ; /. The real covariance of a quasifree state ! is a (continuous) bilinear form H on C01 .M I R/, i.e. a continuous map H W C01 .M I R/ ! D 0 .M I R/. It satisfies H ı P D P ı H D 0. The two-point function !2 of !, defined by ˆ !2 .x; x 0 /u.x/v.x 0 /d Volg  d Volg D !..u/.v// M M

is equal by (4.15) to i !2 D H C G; 2 and we denote by i !2C D HC C GC W C01 .M / ! D 0 .M / 2 its sesquilinear extension.

68

6 Quasi-free states on curved spacetimes

Let us formulate the version of Proposition 6.3 in the real case, which follows from Proposition 4.9. Proposition 6.7. A map !2 W C01 .M I R/ ! D 0 .M I R/ is the two-point function of a quasi-free state for the real Klein–Gordon operator P iff .i/ !2C W C01 .M / ! D 0 .M / is continuous; .ii/ .uj!2C u/M  0;

u 2 C01 .M /;

.iii/ !2C t !2C D iGC :

6.2 Consequences of unique continuation Next let us examine some consequences on CCR.P / of unique continuation results for the Klein–Gordon operator P . We first introduce some terminology taken from [KW, Chapter 2]. Definition 6.8. Let O  M be an open set. The domain of determinacy D .O/ is the largest open set U  M such that P  D 0, jO D 0 implies jU D 0 for all  2 D 0 .M /. From the existence and uniqueness for the Cauchy problem, see Theorem 5.20, one sees that if † is a Cauchy surface in M , the interior of the domain of dependence D.† \ O/, defined as the set fx 2 M W J.x/ \ †  Og, is included in D .O/. Also, if O ? D fx 2 M W x \ J.O/ D ;g is the causal complement of O, then D .O/ \ O ? D ;. From uniqueness results for the Cauchy problem, see e.g. [H4, Section 28.4], one can get some geometric information on D .O/. In particular, it was shown by Strohmaier in [St] that the envelope of O, see [St, Section 2.4] for the precise definition, is always included in D .O/, provided the operator P is locally analytic in time. This condition means that near any point x0 2 M , there exists local coordinates .t; x/ such that @t is time-like and the coefficients of P (and hence the metric g) are locally analytic in t. Following Definition 6.1 we set

Y .O/ D

C01 .O/ ; for O  M open. P .C01 .O//

Proposition 6.9. Let ! be a quasi-free state on CCR.P / with spacetime covariances ƒ˙ and O  M be open. Then Y .O/ is dense in Y .D .O// for the scalar product ƒC C ƒ . Proof. Let Y cpl be the completion of Y for ƒC C ƒ and A? the orthogonal complement of A  Y cpl . For u 2 Y cpl we set wu˙ .f / D uƒ˙ f;

f 2 C01 .M /:

6.3 Conformal transformations

69

Since ƒ˙  0, the Cauchy–Schwarz inequality yields 1

1

jwu˙ .f /j  .uƒ˙ u/ 2 .f ƒ˙ f / 2 ; which implies that wu˙ 2 D 0 .M /. Moreover since ƒ˙ P D 0 we have P wu˙ D 0. If u 2 Y .0/? we have wu˙ D 0 in O hence wu˙ D 0 in D .O/ hence u 2 Y .D .O//? .  Note that the density result in Proposition 6.9 is valid for any quasi-free state !. It is hence different from the Reeh–Schlieder property, see Section 12.4, which is a property of a given state ! and asserts that Y .O/ is dense in Y .O 0 / for any open sets O; O 0  M .

6.3 Conformal transformations If .M; g/ is globally hyperbolic and c 2 C 1 .M / with c.x/ > 0, then .M; g/ Q for gQ D c 2 g is also globally hyperbolic, with the same Cauchy surfaces as .M; g/. It is easy to see from (5.4) that the Levi-Civita connection rQ for gQ is given by:   rQ X Y D rX Y C c 1 .X dc/Y C .Y dc/X  X gY rc : (6.2) If P is a Klein–Gordon operator on .M; g/ and W W L2 .M; d VolgQ / 3 uQ 7! c n=21 uQ 2 L2 .M; d Volg / then

PQ D W  P W D c n=21P c n=21

n2 is a Klein–Gordon operator on .M; g/. Q In particular, if P D g C 4.n1/ Scalg is the conformal wave operator for g, then PQ is the conformal wave operator for g, Q see e.g. [W1, App. D]. Denoting with tildas the objects associated with g, Q PQ , we have:

Gret=adv D W GQ ret=adv W  ;

Q : G D W GW

(6.3)

6.3.1 Conformal transformations of phase spaces. Let us denote by MQ the manifold M equipped with the density d VolgQ D c n d Volg . If †  M is a space-like Cauchy surface, then nQ D c 1 n, hQ D c 2 h. From (6.2) we obtain that Q r A D W 1 r A W . Let us set   1n=2 f0 c 2 C01 .†I C2 /: U W C01 .†I C2 / 3 f 7! Uf D c n=2 f1 The next proposition follows by easy computations.

70

6 Quasi-free states on curved spacetimes

Proposition 6.10. The following diagram is commutative, with all arrows unitary:



C01 .M / ; . jiG P C01 .M /

? ?  yW

Q/ C01 .M ; . jiGQ Q/ PQ C 1 .M 0

/M



 /MQ

G

%†

Q G

%Q †

! .Solsc .P /; q/ ! .C01 .†I C2 /; q† / ? ? ? 1 ? yW yU ! .Solsc .PQ /; q/ Q ! .C01 .†I C2 /; qQ † /

6.3.2 Conformal transformations of quasi-free states. Let ƒ˙ be the spacetime covariances of a quasi-free state ! for P . From (6.3) and Proposition 6.3 we obtain that Q ˙ D c 1n=2 ƒ˙ c 1n=2 (6.4) ƒ are the spacetime covariances of a quasi-free state !Q for PQ . Q the manifold † equipped with the volume element d Vol Q . Let us denote by † h Then  n=2  c fQ0 Q C2 / U  fQ D ; fQ 2 C01 .†I c n=21 fQ1 and

 1 ˙ 1 ; Q ˙ † D .U / † U

Q˙ Q if ˙ † , resp. † are the Cauchy surface covariances of !, resp. !.

Chapter 7

Microlocal analysis of Klein–Gordon equations The use of microlocal analysis in quantum field theory on curved spacetimes started with the fundamental papers of Radzikowski [R1, R2], who gave a definition of the Hadamard states by means of the wavefront set of their two-point functions, instead of their singularity structure, see e.g. Section 8.2. The work of Radzikowski relied on the analysis by Duistermaat and H¨ormander [DH] of distinguished parametrices for Klein–Gordon operators, which was actually motivated by the desire to understand the notion of ‘Feynman propagators’ on curved spacetimes. On Minkowski spacetime the interplay of microlocal analysis and quantum field theory is much older, see for example the proceedings [P]. In this chapter we first recall basic facts on wavefront sets of distributions on manifolds. We then describe the result of [DH] on distinguished parametrices and some related results due to Junker [J1].

7.1 Wavefront set of distributions We recall the well-known definition of the wavefront set of a distribution u 2 D 0 .M / for M a smooth manifold. We equip M with a smooth density, for which one usually takes d Volg if .M; g/ is a spacetime. We use the notation .j/M in (5.9) for the duality bracket between D 0 .M / and C01 .M /. Let o  T  M be the zero section. The points in T  M n o will be denoted by X D .x; /, x 2 M ,  2 Tx M n f0g. We recall that  T  M n o is conic if .x; / 2 ) .x; / 2 for all  > 0. The cosphere bundle S  M is the quotient of T  M n o by the relation X1 X2 if x1 D x2 and 1 D 2 for some  > 0. A conic set can be seen as a set in S  M and it is called closed if it is closed in S  M in the quotient topology. Definition 7.1. Let  Rn an open set. A point .x0 ; 0 / 2 T  n o does not belong to the wavefront set WFu of u 2 D 0 . / if there exist  2 C01 . / with .x0 / D 1 and a conic neighborhood of 0 , such that jF .u/./j  CN hiN ; 8N 2 N;  2 : One can show that the wavefront set transforms covariantly under diffeomor phisms, i.e. if W 1  ! 2 is a diffeomorphism, then WF.



u2 / D



.WF.u2 //; 8u2 2 D 0 . 2 /:

(7.1)

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7 Microlocal analysis of Klein–Gordon equations

Another useful equivalent definition of WFu is as follows. We set vY .x/ D ei.xy/ ;

Y D .y; / 2 T  ;

x 2 Rn ;   1:

(7.2)

Lemma 7.2. Let  Rn be an open set, .x0 ; 0 / 2 T  n o and u 2 D 0 . /. Then .x0 ; 0 / 62 WFu iff there exist  2 C01 . / with .x0 / ¤ 0 and a neighborhood W of .x0 ; 0 / in T  such that j.vY ju/ j  CN N ;

Y 2 W;   1; N 2 N:

From (7.1) we see that Definition 7.1 immediately extends to distributions on manifolds. Definition 7.3. A point X0 D .x0 ; 0 / 2 T  M no does not belong to the wavefront set WFu of u 2 D 0 .M / if there exist a neighborhood U of x0 and a chart diffeomorphism  WU  ! B.0; 1/ such that .1 / X0 62 WF.1 / uj . The wavefront set WFu is a closed conic subset of T  M n o with M WFu D singsupp u, the singular support of u. From Definition 7.3 we obtain immediately the covariance property of the wavefront set under diffeomorphisms. Proposition 7.4. Let M1 ; M2 be two smooth manifolds and  W M1 ! M2 a diffeomorphism. Then WF. u2 / D  .WF.u2 // 8 u2 2 D 0 .M2 /: The following well-known result, see e.g. [SVW, Theorem 2.8], [H1, Theorem 8.4.8] allows to estimate the wavefront set of distributions defined as partial limits of holomorphic functions. It is usually expressed in terms of the analytic wavefront set, see Section 12.2. Proposition 7.5. Let I  R be an open interval, S a smooth manifold and let F W I ˙ i 0; ıŒ 3 z 7! F .z/ 2 D 0 .S / be a holomorphic function with values in D 0 .S /. Assume that f .t; / D lim!0C F .t ˙ i; / exists in D 0 .I  S /. Then WF.f /  f.t;  / W t 2 I; ˙ > 0g  T  S: Proof. We only prove the C case, and we can assume that S D  Rn . We write t D x 0 , x D .x 0 ; x 0 / for x 0 2 S and Y D .Y 0 ; Y 0 / for Y 0 2 T  I , Y 0 2 T  S . With the notation in (7.2) we have vY .x/ D vY 0 .x 0 /vY 0 .x 0 /. By Lemma 7.2, we need to show that .vY jf /I S 2 O.hi1 /; uniformly for Y 2 W; (7.3)

7.1 Wavefront set of distributions

73

where 0 2 C01 .I /, 0 2 C01 .S /, .x/ D 0 .x 0 /0 .x 0 / and W b fY 2 T  I  S W 0 < 0g is relatively compact. Arguing as in the proof of [H1, Theorem 3.1.14], we first obtain that if K b S , there exist N0 2 N and a semi-norm k  kk of C01 .K/, such that j.vjF .z; //S j  C j Im zjN0 kvkk ; 8v 2 C01 .K/; z 2 I C i 0; ıŒ : For v D 0 vY 0 we obtain: j.0 vY 0 jF .z; //S j  C jImzjN0 hik ;

k 2 N; uniformly for Y 0 2 W 0 b T  S: (7.4) Let 1 2 C01 .   ı; ıŒ / with 1 D 1 in jsj  ı=2 and Q 0 .t C is/ D

N X

@jt 0 .t/

j D0

We have

Q 0 2 C01 .C/;

.is/j 1 .s/: jŠ

Q 0R D 0 ; and @z Q 0 2 O.j Im zjN /;

and Q 0 is called an (N -th order) almost analytic extension of 0 . Let us set 

' Y 0 .z/ D e 2 .zx

0 /2 i.zx 0 / 0

;

which is holomorphic in C and equals vY 0 on R. We apply Stokes formula ˆ

‰ @z g.z/d z ^ dz D



g.z/dz

(7.5)

@

to gY .z/ D ' Y 0 .z/Q 0 .z/.0 vY 0 jF .z; //S , D fIm z > 0g. The right-hand side in (7.5) equals ˆ lim .0 vY 0 jF .t C i; //S ' Y 0 .t C i/Q 0 .t C i/dt D .vY jf /I S : !0

R

0

Q .z/, we obtain using also (7.4) that the Since @z gY .z/ D ' Y 0 .z/.0 vY 0 jF .z; //S @@z integrand in the lhs is bounded by C j Im zjN N0 ecj Im zj hik , uniformly for Y 2 W b f0 < 0g, z 2 supp Q 0 . Therefore the integral in the left-hand side is bounded by C hikCN0 N . Since N was arbitrary, we obtain (7.3). 

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7 Microlocal analysis of Klein–Gordon equations

7.2 Operations on distributions We refer the reader to [H1, Chap. 8].

7.2.1 Operations on conic sets. We first introduce some notation. If  T  M n o is conic, we set

 D f.x; / W .x; / 2 g; and if 1 ; 2  T  M n o are conic, we set 1 C 2 D f.x; 1 C 2 / W .x; i / 2 i g: Let Mi , i D 1; 2 be two manifolds, oi the zero section of T  Mi , M D M1  M2 , and let  T  M n o be a conic set. The elements of T  M n o will be denoted by .x1 ; 1 ; x2 ; 2 /, which allows to consider as a relation between T  M2 and T  M1 , still denoted by . Clearly maps conic sets into conic sets. We set 0 D f.x1 ; 1 ; x2 ; 2 / W .x1 ; 1 ; x2 ; 2 / 2 g  T  .M1  M2 / n o; Exch. / D 1  .T  M2  T  M1 / n o; M1

M2

D f.x1 ; 1 / W 9 x2 such that .x1 ; 1 ; x2 ; 0/ 2 g D .o2 /  T  M1 no1 ; D f.x2 ; 2 / W 9 x1 such that .x1 ; 0; x2 ; 2 / 2 g D 1 .o1 /  T  M2 no2 :

7.2.2 Distribution kernels. If Mi ; i D 1; 2, are smooth manifolds equipped

with smooth densities di and K W C01 .M2 / ! D 0 .M1 / is continuous, we will still denote by K 2 D 0 .M1  M2 / its distribution kernel. Such a kernel is properly supported if the projection 2 W supp K ! M2 is proper. If this is the case, then K W C01 .M2 / ! E 0 .M1 /.

7.2.3 Complex conjugation and adjoints. If u 2 D0 .M /, then WF.u/ D WF.u/:

(7.6)

Similarly, if K W C01 .M2 / ! D 0 .M1 / is continuous and K  W C01 .M1 / ! D 0 .M2 / is its adjoint with respect to some smooth densities d1 ; d2 then: WF.K  /0 D Exch.WF.K/0 /:

(7.7)

7.2.4 Pullback and restriction to submanifolds. Under a condition on WFu it is possible to extend the pullback  u to general smooth maps  W M1 ! M2 . Indeed, let us set  u D u ı  for u 2 C 1 .M2 / and N D f..x1 /; 2 / 2 T  M2 n o2 W tD.x1 /2 D 0g:

7.2 Operations on distributions

75

Then there is a unique extension of the pullback  to distributions u 2 D 0 .M2 / such that N \ WF.u/ D ;; (7.8) and one has

WF. u/   WF.u/:

(7.9)

In particular, if S  M is a smooth submanifold and i W S ! M is the canonical injection, the set Ni is denoted by N  S and called the conormal bundle to S . One has: N  S D f.x; / 2 T  M W x 2 S; jTx S D 0g: The restriction uS D i  u of u 2 D 0 .M / is then well defined if

and one has

WFu \ N  S D ;;

(7.10)

WF.uS /  i  WFu:

(7.11)

7.2.5 Tensor products. If ui 2 D0.Mi / then W F .u1 ˝ u2 / .W F .u1 /  W F .u2 // [ .supp u1  f0g/  W F .u2 / [ W F .u1 /  .supp u2  f0g/ .W F .u1 /  W F .u2 // [ o1  W F .u2 / [ W F .u1 /  o2 :

7.2.6 Products. The map C01 .M /2 3 .u1 ; u2 / 7! u1 u2 uniquely extends to

distributions u1 ; u2 2 D 0 .M / such that:

.WFu1 C WFu2 / \ o D ;;

(7.12)

and one has WF.u1 u2 /  WFu1 [ WFu2 [ .WFu1 C WFu2 /:

7.2.7 Kernels. If K 2 D0 .M1  M2 /, then the map K W C01 .M2 / ! D0 .M1 / uniquely extends to distributions such that u 2 E 0 .M2 /; and one has:

WF.Ku/  0

0 WF.u/ \ WF.K/M D ;; 2

M1WF.K/

0

[ .WF.K/0 .WFu//; 

(7.13) (7.14)



where we interpret WF.K/ as a relation in T M1  T M2 . Quite often one has 0 M1WF.K/ D ;, and (7.14) simplifies to WF.Ku/  WF.K/0 .WFu/;

(7.15)

76

7 Microlocal analysis of Klein–Gordon equations

which justifies the use of WF.K/0 instead of WF.K/. Note for example that WF.Id/0 is equal to the diagonal  D f.X; X / W X 2 T  M n og

(7.16)

which is the relation associated to Id W T  M ! T  M . Similarly, if P is a (properly supported) pseudodifferential operator (see Chapter 10) one has: WF.P /0  ; hence W F .P u/  WF.u/; u 2 D 0 .M /:

(7.17)

7.2.8 Composition of kernels. Finally, let K1 2 D0 .M1 M2 /, K2 2 D0 .M2 

M3 /, where K2 is properly supported. Then K1 ı K2 is well defined if WF.K1 /0M2 \

M2WF.K2 /

0

D ;;

(7.18)

and then WF.K1 ıK2 /0  .WF.K1 /0 ıWF.K2 /0 / [. M1WF.K1 /0 o3 / [ . o1 WF.K2 /0M3 /: (7.19) Again, it often happens that MiWF.Ki /0 and WF.Ki /0Mi C1 are empty. Then (7.18) is automatic and (7.19) simplifies to the beautiful formula: WF.K1 ı K2 /0  WF.K1 /0 ı WF.K2 /0 :

(7.20)

7.2.9 Proof of Proposition 6.6. We end this section by completing the proof of (2) in Proposition 6.6. Consider the map %† G† W C01 .†I C2 / ! D 0 .M /. It is clearly continuous and introducing local coordinates .t; x/ near x0 2 † such that † D ft D 0g we see that WF.%† G† /0  f.X; Y / 2 T  M  T  † W X D i  Y g; where i W † ! M is the canonical embedding. From P ı ƒ˙ D ƒ˙ ı P D 0 we obtain (see the proof of Lemma 7.9) that WF.ƒ˙ /0  N  N . Since † is space-like and hence non-null, we have N \ N  † D ;, which using Subsection 7.2.8 shows that ƒ˙ ı %† G† W C01 .†I C2 / ! D 0 .M / is well defined and continuous. The same argument shows that .%† G† / ı ƒ˙ ı %† G† W C01 .†I C2 / ! D 0 .†I C2 / is well defined and continuous. 

7.3 H¨ormander’s theorem We now state the famous result of H¨ormander on propagation of singularities, see e.g. [H3, Theorem 26.1.1] or [H4, Theorem 3.2.1]. To this end we need some notions from pseudodifferential calculus, which will be recalled later on in Chapter 8.

7.4 The distinguished parametrices of a Klein–Gordon operator

77

The space of (classical) pseudodifferential operators of order m on a manifold X is denoted by ‰ m .X /. If P 2 ‰ m .X /, its principal symbol p D pr .P / is a smooth function on T  X , homogeneous of degree m in . Its characteristic manifold is Char.P / D p1 .f0g/ n o; where o is the zero section in T  X . P is said of real principal type if p is real valued with dp ¤ 0 on Char.P /, which is then a smooth, conic hypersurface in T  M , invariant under the flow of the Hamiltonian vector field Hp . The integral curves of Hp in Char.P / are traditionally called bicharacteristic curves for P . Note also that a Klein–Gordon operator P on a Lorentzian manifold .M; g/ is of real principal type with principal symbol p.x; / D  g 1 .x/. A submanifold S  M is non-characteristic for P iff Char.P / \ N  S D ;. Theorem 7.6. Let X be a smooth manifold and P 2 ‰ m .X / a properly supported pseudodifferential operator. Then for u 2 D 0 .X / one has: (1) WF.u/ nWF.P u/  Char.P / .microlocal ellipticity/. (2) If P is of real principal type, then WF.u/ n WF.P u/ is invariant under the flow of Hp .propagation of singularities/.

7.4 The distinguished parametrices of a Klein–Gordon operator We will recall some deep results of Duistermaat and H¨ormander [DH] on distinguished parametrices of P . These results played a very important role in the work of Radzikowski [R1]. Let us first introduce some notation. Recall that C˙ .x/  Tx M are the cones of future/past time-like vectors. We denote by C˙ .x/  Tx M the dual cones C˙ .x/ D f 2 Tx M W   v > 0; 8v 2 C˙ .x/; v ¤ 0g: We write  B 0 if  2 CC .x/ . In this section P will be a Klein–Gordon operator on .M; g/. We recall that its principal symbol is

pr .P /.x; / D p.x; / D  g 1 .x/: Duistermaat and H¨ormander introduce in [DH] the pseudo-convexity condition of M with respect to P , which says that for any compact set K b M there exists a compact K 0 b M such that the projection on M of any bicharacteristic curve for P with endpoints in K is entirely contained in K 0 . Since projections on M of bicharacteristic curves are null geodesics, and hence causal curves, the pseudo-convexity of M follows easily from global hyperbolicity, using Lemma 5.11.

78

7 Microlocal analysis of Klein–Gordon equations

The characteristic manifold Char.P / will be denoted by N ; it splits into the upper/lower energy shells

N D N C [ N  ; N ˙ D N \ f˙ B 0g:

(7.21)

Recall that X D .x; / denote the points in T  M n o. We write X1 X2 if X1 ; X2 2 N and X1 ; X2 lie on the same integral curve of Hp . For X1 X2 , we write X1 > X2 , resp. X1 < X2 if x1 2 JC .x2 /, resp. x1 2 J .x2 / and x1 ¤ x2 and we write X1  X2 , resp. X1 X2 if X1 comes strictly after, resp. before X2 with respect to the natural parameter on the integral curve of Hp through X1 and X2 . Finally, we set

C D f.X1 ; X2 / 2 N  N W X1 X2 g; and we introduce the following subsets of C :

C ˙ D C \ .N ˙  N ˙ /; Cret D f.X1 ; X2 / 2 C W X1 > X2 g; Cadv D f.X1 ; X2 / 2 C W X1 < X2 g;

(7.22)

CF D f.X1 ; X2 / 2 C W X1 X2 g; CF D f.X1 ; X2 / 2 C W X1  X2 g: Note that

Cret [ Cadv D CF [ CF D C n : Using an orthogonal decomposition of the metric g, one easily obtains that

CF D .Cret \ C C / [ .Cadv \ C  /; CF D .Cret \ C  / [ .Cadv \ C C /:

(7.23)

7.4.1 Parametrices. Definition 7.7. A continuous map GQ W C01 .M / ! D 0 .M / is a left, resp. right parametrix of P if GQ ı P D ½ C R; resp. P ı GQ D ½ C R0 ; where R, resp. R0 has a smooth kernel. If GQ is both a left and a right parametrix, it is called a parametrix of P . Parametrices play in microlocal analysis the role played by pseudo-inverses in Fredholm theory.

79

7.4 The distinguished parametrices of a Klein–Gordon operator

7.4.2 Distinguished parametrices. We now state a theorem of Duistermaat and H¨ormander [DH, Theorem 6.5.3]. Theorem 7.8. For ] D ret; adv; F; F there exists a parametrix GQ ] of P such that WF.GQ ] /0 D  [ C] :

(7.24)

Q 0   [ C] equals GQ ] modulo a Any other left or right parametrix GQ with WF.G/ smooth kernel. The parametrices in Theorem 7.8 are called distinguished parametrices. Those with WF.GQ 0 /   [ Cret=adv are called retarded/advanced parametrices, while those with WF.GQ 0 /   [ CF=F are called Feynman/anti-Feynman parametrices. Note that the closed conic subsets of T  .M  M / n o that can be equal to Q 0 for some parametrix GQ of P were also completely characterized in [DH, WF.G/ Theorems 6.5.6, 6.5.8]. They can be very different from the sets in Theorem 7.8. Lemma 7.9. The retarded/advanced inverses Gret=adv introduced in Subsection 5.5.3 are advanced/retarded parametrices. Proof. We note first that since P is a differential operator, P ˝ ½ and ½ ˝ P are pseudodifferential operators on M  M . Let now GQ be a parametrix of P . We apply (7.17) and Theorem 7.6 (1) to P ˝ ½ or ½ ˝ P , using the fact that P ı GQ  ½ and GQ ı P  ½ have smooth kernels, and obtain that Q 0  .N  N / [ :   WF.G/ Let us assume now that there exists .X1 ; X2 / 2 WF.Gret /0 with .X1 ; X2 / 62  [ Cret . If X1 X2 , then necessarily x1 62 JC .x2 /, hence .x1 ; x2 / 62 supp Gret , which is a contradiction. If X1 6 X2 , then necessarily X1 ; X2 2 N . If B.X / denotes the bicharacteristic curve through X , then B.X1 /  fX2 g \  D ;. We can apply then Theorem 7.6 (2) to P ˝ ½, using that P ı Gret  ½ has a smooth kernel, to obtain that B.X1 /  fX2 g  WF.Gret /0 . In particular, WF.Gret /0 contains .X3 ; X2 / with x3 62 JC .x2 /, which is a contradiction. The proof for Gadv is similar.  By Lemma 7.9, there are canonical advanced/retarded parametrices, namely the advanced/retarded inverses. No such canonical choice exists of Feynman/antiFeynman inverses, at least on general spacetimes .M; g/, a fact already noted by Duistermaat and H¨ormander. This fact is related to the absence of a canonical choice of Hadamard states for P , see Chapter 8 below. We will come back to this question in Chapter 16. We end this section with a proposition about the wavefront set of differences of distinguished parametrices, due to Junker, see [J1, Theorem 2.29].

80

7 Microlocal analysis of Klein–Gordon equations

Proposition 7.10. One has: .1/ WF.GQ ret  GQ adv /0 D C ; .2/ WF.GQ F  GQ FN /0 D C ; .3/ WF.GQ F  GQ ret /0 D C  ; .4/ WF.GQ F  GQ adv /0 D C C : Proof. We will apply the following observation: let S be any of the differences in Proposition 7.10. Since PS; SP 2 C 1 .M  M /, applying Theorem 7.6 to P ˝ ½ and ½ ˝ P we obtain that .i/ WFS 0  N  N ; .ii/ .X1 ; X2 / 2 .N  N / n WFS 0 ) B.X1 /  B.X2 / \ WFS 0 D ;;

(7.25)

where we recall that B.X / is the bicharacteristic curve through X . In the sequel we set N D .N  N / \ . Let us prove assertion (1). Since WF.GQ ret /0 nN and WF.GQ adv /0 nN are disjoint, we obtain that     WF.GQ ret  GQ adv /0 nN D WF.GQ ret /0 n N [ WF.GQ adv /0 n N D C nN : (7.26) Next, (7.25) (i) implies that WF.GQ ret  GQ adv /0  N  N , and (7.25) (ii) combined with (7.26) implies that N  WF.GQ ret  GQ adv /0 . This completes the proof of (1). The proof of (2) is similar. Now let us prove (3). Since WF.Gret /0 \ f.X1 ; X2 / 2 N  N W X1 < X2 g D ;, we have: WF.GQ F  GQ ret /0 \ f.X1 ; X2 / 2 N  N W X1 < X2 g D WFGQ F0 \ f.X1 ; X2 / 2 N  N W X1 < X2 g

(7.27)

D CF \ Cadv D Cadv \ C  ; where in the last step we used (7.23). Applying then (7.25) we obtain (3). The proof of (4) is similar. 

Chapter 8

Hadamard states The main problem one encounters when considering quantum Klein–Gordon fields on a curved spacetime is that there is no notion of a vacuum state. Unless the spacetime is stationary, see Chapter 9, there is no one-parameter group of Killing isometries that can be used to define a vacuum state. One is forced to find a more general class of physically acceptable states, which should be those for which the renormalized stress-energy tensor Tab ./.x/, see Section 8.1, can be rigorously defined. Alternatively one can require that the short distance behavior of their two-point functions, expressed for example in normal coordinates at any point x 2 M , should mimic the one of the vacuum state on Minkowski spacetime. These states are called Hadamard states and play a fundamental role in quantum field theory on curved spacetimes. In this chapter we describe the characterization of Hadamard states due to Radzikowski, [R1, R2], relying on the wavefront set of their two-point functions and various existence and uniqueness theorems for Hadamard states. The microlocal definition of Hadamard states is very convenient and natural for applications.

8.1 The need for renormalization Let us now consider a non-linear Klein–Gordon equation like g .x/ C m2 .x/ C  n .x/ D 0;

(8.1)

or a Klein–Gordon equation coupled to another classical field equation, like the Einstein–Klein–Gordon system: ( R .g/  12 R.g/g D T ./; (8.2) g  C m2  D 0: Here Tab ./ is the stress-energy tensor of , defined as 1 Tab ./ D ra rb   gab .r c rc  C m2  2 /; 2

(8.3)

for a real solution . For complex solutions the stress-energy tensor is defined as Tab ./ D ra rb  C rb ra   gab .r c rc  C m2 /:

(8.4)

82

8 Hadamard states

Note that if  2 C 1 .M / solves the Klein–Gordon equation g  C V .x/ D 0; then one has the identity r a Tab ./ D .V  m2 /.rb  C rb /;

(8.5)

(this vanishes if V D m2 ), which is the basic ingredient of energy estimates for Klein–Gordon equations. To quantize such classical equations, one would like to define expressions like distributions.  n .x/, or Tab ./.x/ as operator-valued ´ ´ n It is hopeless to define M  .x/u.x/d Volg or M Tab ./.x/u.x/d Volg for u 2 C01 .M / as elements of an abstract -algebra. Instead one can hope that given a state ! for the free Klein–Gordon field, those expressions may have a meaning as unbounded operators on the GNS Hilbert space H! . More precisely one can try to proceed as follows: Let ! .u/ for u 2 C01 .M /, be the image of the abstract field .u/ under the map ! of the GNS triple .H! ; ´! ; ! /, and let ! .x/ be the operator-valued distribution on M defined by ! .u/ D M ! .x/u.x/d Volg . Then one can try to define ! .x/! .x 0 /; !2 .x/ D lim 0 x !x

i.e. !2 .x/ will be the trace on the diagonal  D fx D x 0 g of the operator valued distribution ! .x/! .x 0 / on M  M . If this is possible, then one would expect that . ! j!2 .x/ ! /H! will be a well-defined (scalar) distribution on M . In the Minkowski case this means that the two-point function !2 .x; x 0 / has a well-defined trace on . This is clearly impossible, since by (3.14) ˆ !2 .x; x/ D .k/1d k D 1; Rd

an example of ultraviolet divergence. Note also that one has   WF.!2 / D f .x; /; .x 0 ;  0 / W .x; / 2 N C ; .x 0 ;  0 / 2 N  ; .x; / .x 0 ;  0 /g; (8.6) so trying to define !2 j by the arguments of Section 7.2 does not work either.

8.1.1 The Wick ordering. The solution to this problem for the vacuum state on Minkowski is well-known, and called the Wick ordering: it consists in setting W.x/.x 0 /W D .x/.x 0 /  !2 .x; x 0 /½:

(8.7)

If ! is any quasi-free state, then W! .x/! .x 0 /W is clearly well defined as an operatorvalued distribution on M  M . If ! D !vac , let us try to define the operator-valued

83

8.2 Old definition of Hadamard states

distribution W!2 vac .x/W as the trace on  of W!vac .x/!vac .x 0 /W. To this end, we consider the distribution W! .x/! .x 0 /W  W! .y/! .y 0 /W D ! .x/! .x 0 /! .y/! .y 0 /  ! .x/! .x 0 /!2 .y; y 0 /  ! .y/! .y 0 /!2 .x; x 0 / C !2 .x; x 0 /!2 .y; y 0 /½: Using the fact that ! is quasi-free, see Proposition 4.7, we obtain that   ! W! .x/! .x 0 /W  W! .y/! .y 0 / D !2 .x; y/!2 .x 0 ; y 0 / C !2 .x; y 0 /!2 .x 0 ; y/: The right-hand side above has a well-defined trace on fx D x 0 ; y D y 0 g, which equals 2!2 .x; y/2 . Note that !2 .x; y/2 is well defined as an element of D 0 .M  M /, since if is the right-hand side in (8.6) we have . C / \ o D ;. Summarizing we have shown that the vector ˆ W 2 .x/Wu.x/d Volg ! ; u 2 C01 .M / M

is well defined as an element of H! for u 2 C01 .M / (since its norm in H! is finite). Using the same argument one can show that the (unbounded) operator ´ 2 W .x/Wu.x/d Volg is well defined with domain M n nY o D D Vect ! .ui / ! W ui 2 C01 .M /; n 2 N : i D1

8.2 Old definition of Hadamard states The Wick ordering is well understood for the Klein–Gordon field on Minkowski spacetime. The search for a natural class of vacuum states for Klein–Gordon fields on more general globally hyperbolic spacetimes led physicists to introduce the notion of Hadamard states. Originally, Hadamard states were defined by specifying the singularity of their two-point functions !2 .x; x 0 / D !..x/.x 0 // for pair of points .x; x 0 / 2 M  M near the diagonal, see e.g. [KW, Section 3.3]. We will follow here the exposition of Radzikowski in [R1, Chapter 5], see also the PhD thesis of Viet Dang [D, Sections 5.2, 5.3]. Let us first consider the Minkowski case and set Q.x/ D x x for x 2 Rn . We first claim that Q.x C iy/ 2 Cn   1; 0;

x 2 Rn ; y 2 C;

(8.8)

where we recall from Section 2.1 that C D CC [ C  Rn is the cone of time-like vectors. Indeed we have Q.x C iy/ D x x  y y C 2ix y:

84

8 Hadamard states

If Im Q.x Ciy/ D 0 and y 2 C , then x is space-like by Lemma 5.3, hence Re Q.x C iy/ > 0, which proves our claim. Moreover, if b CC is a closed cone and K b Rn is compact, then there exist ı > 0 and R > 0 such that jQ.x C iy/j  ıjyj2 ; Writing

8x 2 K; y 2 \ fjyj  Rg:

(8.9)

jQ.x C iy/j2 D .x x  y y/2 C 4.x y/2 ;

we see that (8.9) is clearly satisfied for x 2 K; x  x  0 and y 2 , since y  y  cjyj2 for y 2 . If x  x < 0, x 2 K, then from Lemma 5.3 we obtain that jx  yj  cjyj for y 2 . This implies (8.9). In the sequel we take the determination of log z which is defined in Cn   1; 0. It follows from (8.8), (8.9) that Q1 .z/; log Q.z/ are holomorphic functions of moderate growth in Rn C iCC , see Section 12.1, hence the boundary values .Q1 /C .x/ D Q1 .x C iCC 0/;

.log Q/C .x/ D log Q.x C iCC 0/

(8.10)

are well defined as distributions on Rn . The limit in (8.10) can be taken in particular along any vector y 2 CC , see Subsection 12.1.2, which implies that the distributions .Q1/C and .log Q/C are invariant under the action of the restricted Lorentz group SO " .1; d /. Now let .M; g/ be a spacetime. There exists a neighborhood U of the zero section in TM such that the map: exp W U 3 .x; v/ 7! .x; expgx .v// 2 M  M is a diffeomorphism onto its range, with V D exp.U / being a neighborhood of the diagonal  in M  M . Clearly, such sets V form a basis of neighborhoods of . Let us also fix a smooth map R W M 3 x 7! R.x/ 2 L.Tx M; Rn / such that R.x/ W .Tx M; g.x// ! .Rn ; / is pseudo-orthogonal and maps the future lightcone CC .x/ into CC , i.e. preserves the time orientation. One can then define the map F W V 3 .x; x 0 / 7! R.x 0 / ı .expgx0 /1 .x/ 2 Rn ; (8.11) which has a surjective differential. Note that Q ı F .x; x 0 / equals the (signed) square geodesic distance .x; x 0 / between x and x 0 . Since NF D ;, we can by Subsection 7.2.4 define the pullbacks of .Q1 /C and .log Q/C by F . 1 /C D F  ..Q1 /C /;

and .log /C D F  ..log Q/C / 2 D 0 .V /:

From the invariance of .Q1 /C and .log Q/C under SO " .1; d /, we deduce that . 1 /C and .log /C are independent of the choice of R.x/. One defines also the van Vleck–Morette determinant 1

1

.x; x 0 / WD  det.r˛ rˇ 0 .x; x 0 //jgj 2 .x/jgj 2 .x 0 /:

8.3 The microlocal definition of Hadamard states

85

Definition 8.1. Let P be a real Klein–Gordon operator. A quasi-free state ! on CCRR .P / is a Hadamard state if there exist a neighborhood V of the diagonal in M  M as above and functions v; w 2 C 1 .V /, such that !2C .x; x 0 / D

1 1  2 .x; x 0 /. 1/C .x; x 0 / .2/2 C v.x; x 0 /.log /C .x; x 0 / C w.x; x 0 /

(8.12) on V:

Note that the function v.x; x 0 / is not arbitrary, since Px !2 D Px 0 !2 D 0. One has v.x; x 0 /

1 X

vi .x; x 0 / .x; x 0 /i ;

i D0 0

where vi .x; x / are the so-called Hadamard coefficients and the symbol means that v

n X

vi i 2 O.j jnC1 /; 8n 2 N

i D0

together with all derivatives.

8.3 The microlocal definition of Hadamard states The situation was radically simplified by Radzikowski, who in [R1] introduced the definition of a Hadamard state via the wavefront set of its two-point function. Let us first introduce the original definition, which deals with real fields, see Subsection 6.1.3.

8.3.1 Hadamard condition for real fields. We use the notation for real Klein– Gordon fields recalled in Subsection 6.1.3. Definition 8.2. Let ! be a quasi-free state on CCRR .P /, with real covariance H . Then ! is a Hadamard state if WF.!2C /0 D f.X; X 0/ 2 T  M  T  M W X; X 0 2 N C ; X X 0 g:

(8.13)

8.3.2 The Hadamard condition for complex fields. As already explained in Chapter 4, it is much more convenient to work with complex fields and gauge invariant states, i.e. in the framework of Chapter 6. In this case the following definition was introduced in [GW1]. Definition 8.3. Let ! be a .gauge invariant/ quasi-free state, with spacetime covariances ƒ˙ W C01 .M / ! D 0 .M /. Then ! is a Hadamard state if WF.ƒ˙ /0 D f.X; X 0 / 2 T  M  T  M W X; X 0 2 N ˙ ; X X 0 g:

86

8 Hadamard states

8.4 The theorems of Radzikowski We now prove the theorems of Radzikowski [R1, R2] on the microlocal characterization of Hadamard states. We will use the formalism of complex fields, in which case Theorem 8.5 is due to Wrochna [W1]. Let us first introduce a list of conditions. Definition 8.4. A pair of continuous maps ƒ˙ W C01 .M / ! D 0 .M / satisfy .Herm/ .Pos/ .CCR/

if ƒ˙  ƒ˙ D 0 modulo C 1 I if ƒ˙  0 modulo C 1 I if ƒC  ƒ D iG modulo C 1 I

.KG/

if P ƒ˙ D ƒ˙ P D 0 modulo C 1 I

.Had/

if WF.ƒ˙ /0 D f.X; X 0 / 2 T  M  T  M W X; X 0 2 N ˙ ; X X 0 gI

.genHad/ .genHadloc/ .Feynm/

if WF.ƒ˙ /0  fX W ˙ B 0g  fX W ˙ B 0gI if WF.ƒ˙ /0 \   f.X; X / W ˙ B 0gI if i1 ƒC C Gadv ; i1 ƒ C Gret are Feynman parametrices of P:

Theorem 8.5. The following conditions are equivalent: (1) ƒ˙ satisfy (Had), (KG), (CCR); (2) ƒ˙ satisfy (genHad), (KG), (CCR); (3) ƒ˙ satisfy (Feynm). Proof. (1)H)(2) is obvious. Let us prove the implication (2)H)(3). Let GQ F be a Feynman parametrix of P . If S ˙ D i.GQ F  Gadv=ret / we have WF.S ˙ /0  C ˙ , by Proposition 7.10 and WF.ƒ˙ /0  N ˙  N ˙ by (genHad) and Theorem 7.6. Hence, WF.ƒ˙  S ˙ /0  N ˙  N ˙ and WF.ƒC  S C /0 \ WF.ƒ  S  /0 D ;: On the other hand, by (CCR) we obtain .ƒC  S C /  .ƒ  S  / D .ƒC  ƒ /  .S C  S  / D iG  iG D 0: Therefore, S ˙  ƒ˙ has a smooth kernel, which implies (3). Finally we prove that (3)H)(1). (KG) and (CCR) are immediate and (Had) follows from Proposition 7.10.  Since the spacetime covariances ƒ˙ of a Hadamard state satisfy (CCR), (KG) and (Had), we immediately obtain the following corollary, which says that these covariances are unique, modulo smooth kernels.

8.4 The theorems of Radzikowski

87

Corollary 8.6. Let ƒ˙ i , i D 1; 2 be the spacetime covariances of two Hadamard ˙ states !i . Then ƒ1  ƒ˙ 2 have smooth kernels. Another important result is the following theorem, due to Duistermaat and H¨ormander [DH, Theorem 6.6.2] in a more general context. The proof we give follows from the existence of Hadamard states, see Section 8.7. Theorem 8.7. (Feynm) implies (Pos). Proof. We know from Theorem 8.13 that Hadamard states for P exist. Let ƒ˙ 1 be the spacetime covariances of a Hadamard state for P , which satisfy (Had), (KG) and (CCR), hence (Feynm). If ƒ˙ satisfy also (Feynm), then ƒ˙  ƒ˙ 1 have smooth ˙ ˙  kernels. Since ƒ1  0, ƒ satisfy (Pos). Finally we prove a variant of a result of Radzikowski [R2] called there a ‘localto-global theorem’. Proposition 8.8. (Pos) and (genHadloc) imply (genHad). The proof follows immediately from Lemma 8.9 below. Lemma 8.9. Let K 2 D 0 .M  M / such that K  0 modulo a smooth kernel. Then for X 2 T  M n o we have .X; X / 62 WF.K/0 ) .X1 ; X /; .X; X2/ 62 WF.K/0 ; 8Xi 2 T  M n o: Proof. We may assume that K  0 and that M D  Rn . Let vY be defined in (7.2). We see that .X1 ; X2 / 62 WF.K/0 iff there exists i 2 C01 .M / with i .xi / ¤ 0 and neighborhoods Wi b T  M of Xi such that .1 vY1 jK2 vY2 /M 2 O.hi1 /; uniformly for Yi 2 Wi : Note also that since K W C01 .M / ! D 0 .M / is continuous, we have j.vY jKvY /M j  C hiN0 uniformly for Y 2 W b T  M; for some N0 depending on ; W . By the Cauchy–Schwarz inequality, we obtain 1

1

2 2 j.1 vY1 jK2 vY2 /M j  .1 vY1 jK1 vY1 /M .1 vY2 jK2 vY2 /M ;

which yields the lemma.



88

8 Hadamard states

8.5 The Feynman inverse associated to a Hadamard state Let ! a Hadamard state with spacetime covariances ƒ˙ . Then GF D i1 ƒC C Gadv D i1 ƒ C Gret

(8.14)

is a Feynman inverse of P , i.e. one has P GF D GF P D ½;

WF.GF /0 D  [ CF :

The operator GF will be called the Feynman inverse associated to !.

8.6 Conformal transformations We use the notation in Section 6.3. Let ! be a quasi-free state for P and !Q the associated quasi-free state for PQ obtained from (6.4), where we recall that PQ D c n=21 P c n=21 and gQ D c 2 g. Clearly, !Q is Hadamard iff ! is Hadamard.

8.7 Equivalence of the two definitions In this section we prove the equivalence of Definition 8.1 and Definition 8.2, following [R1]. Theorem 8.10. A quasi-free state ! for a real Klein–Gordon operator P satisfies Definition 8.1 iff it satisfies Definition 8.2. Proof. Let ƒ˙ the complex covariances of the complexification of the state !2 , see Subsection 4.7.2. By (4.26) we have ƒC D !2C ;

ƒ D !2C  iGC Dt !2C ;

since !2C t !2C D iGC , see Proposition 6.7. Note that if K W C01 .M / ! D 0 .M / we have WF.tK/0 D WF.K/0 . Assume that !2C satisfies (8.12). By Proposition 8.11, !2C satisfies (genHadloc), hence (genHad) by Proposition 8.8. By the above remark, ƒ˙ satisfy (genHad), and of course (CCR) and (KG). By Theorem 8.5, we obtain that i 1 !2C C Gadv is a Feynman parametrix for P , hence !2C satisfies (8.13), again by Theorem 8.5. Conversely, if !2C satisfies (8.13), then by the same argument i 1 !2C C Gadv is a Feynman parametrix for P , hence satisfies (8.12) by the above discussion and the uniqueness of Feynman parametrices modulo smooth kernels. 

8.7 Equivalence of the two definitions

89

Proposition 8.11. Let !2C 2 D 0 .V / a distribution as in Definition 8.1. Then WF.!2C /0  N C  N C :

(8.15)

The proof below shows that actually WF.!2C /0  C C , where C C is defined in (7.22). Proof. We first estimate the wavefront set of .Q1 /C and .log Q/C . If x0 x0 ¤ 0, then near x0 we have .Q1 /C .x/ D Q1 .x/ and .log Q/C .x/ D log jQ.x/j C i , where D 0 if x0 x0 > 0, and D ˙ if x0 2 C˙ . In particular, .Q1 /C and .log Q/C are smooth in fx x ¤ 0g. If x0 x0 D 0 and x0 ¤ 0, then Q.x0 C iy/ D yy C 2ix0 y. It follows that Q1 .x C iy/ and log Q.x C iy/ are holomorphic in Ux0 C i x0 where Ux0  Rn is a small neighborhood of x0 and x0 D fy 2 Rn W ˙x0 y > 0g for x0 2 N˙ . Finally, we saw in Section 8.2 that Q1 .x C iy/ and log Q.x C iy/ are holomorphic in U0 C i 0 , where U0  Rn is a small neighborhood of 0 and 0 D CC , and that Q1 and log Q are of moderate growth in Ux0 C iK, where Ux0 is a small neighborhood of x0 and K b x0 is any relatively compact cone. Note that the cone x0 always contains CC . From Section 12.2 we obtain the estimate [ WF..Q1 /C /; WF..log Q/C /  x0  xı0 ; x0 2N

where the polar cone ı of a cone  Rn is the set ı D f 2 .Rn /0 n o W x   0; 8x 2 g:

(8.16)

It follows that WF..Q1/C /; WF..log Q/C /  f.x; ˙x/ W x 2 N˙ ; x ¤ 0;  > 0g [ f.0; / W   1  D 0; 0 > 0g;

(8.17)

where 0 D  e0 , e0 D .1; : : : ; 0/. Let now u D .Q1 /C or .log Q/C 2 D 0 .Rn /, and let F W V ! Rn be the map in (8.11). By Subsection 7.2.4, we have   WF.F  u/0  f .x;t Dx F /; .x 0 ; tDx 0 F / W .F .x; x 0 /; / 2 WFug: (8.18) Note that we can forget the isometry R.x 0 / in the definition of F if we introduce the orthonormal frame ei .x/ D R1 .x/ei , where .e1 ; : : : ; en / is the canonical basis of Rn . Let us first estimate WF.F  u/0 away from the diagonal x D x 0 . We obtain from (8.17) that the right-hand side in (8.18) is included in   ˚ .x; tDx F v/; .x 0 ; tDx 0 F v/ W v D F .x; x 0 / 2 N; v 2 NC :

90

8 Hadamard states

Since .x; x 0 / D F .x; x 0 /F .x; x 0 /, we have Dx .x; x 0 / D 2Dx F .x; x 0 /F .x; x 0 /; Dx .x; x 0 / D 2Dx 0 F .x; x 0 /F .x; x 0 /; hence the set above equals   ˚ .x; Dx /; .x 0; Dx 0 / W v D F .x; x 0 / 2 N; v 2 NC :

(8.19)

By the Gauss lemma, the radial geodesic between x 0 and x is normal to the hypersurface .; x 0 / D C st, which implies that the vectors rx .x; x 0 /; rx 0 .x; x 0 / are tangent to the (null) geodesic between x 0 and x, and future pointing. This implies that the set in (8.19) is included in N C  N C (actually in C C ). Let us now estimate WF.F  u/0 above the diagonal x D x 0 . If we work in normal coordinates at x, we have Dx F D ½, Dx 0 F D ½ at .x; x/ hence above the diagonal we have also WF.F  u/0  f.X; X / W X 2 N C g. In conclusion we have shown that WF.. 1/C /0 and WF..log /C/0 are included in the right-hand side of (8.15). This implies the same estimate for WF.!2 /0 . 

8.8 Examples of Hadamard states Let us consider one of the simplest examples of globally hyperbolic spacetimes, namely ultra-static spacetimes, see Section 5.3. We assume that .S; h/ is complete. More examples will be given in Chapter 9. The associated Klein–Gordon operator P D g C m2 for m > 0 is @2t C  2 ; where  2 D h C m2 is essentially selfadjoint on C01 .S /. By Subsection 4.10.3, we can construct the vacuum state !vac for P , whose spacetime covariances are given by the analog of (4.47):  ˆ  1 ˙it  ˙ .ujƒvac v/ D u.t; /j e v.t; / dt; u; v 2 C01 .R  S /; 2 R S ´ where .ujv/S D S uv d Volh and ut ./ D u.t; /. One can similarly express the Feynman inverse associated to !vac , which equals ˆ GF u.t; / D GF .t  t 0 /u.t 0 ; /dt 0 ; R

with

  GF .t/ D .2i/1 eit  .t/ C eit  .t/ :

(8.20)

8.9 Existence of Hadamard states

91

Theorem 8.12. The vacuum state !vac is a pure Hadamard state. Proof. We saw in Subsection 4.10.3 that !vac is a pure state. It suffices then to 2 2 verify (genHad). Since m > 0, we see that ƒ˙ vac W L .R  S / ! L .R  S / ˙ 0 ˙ 0 0 have distributional kernels. We have ƒvac .t; t; x; x / D F .t  t ; x; x / for F ˙ u D .2/1e˙it  u, u 2 C01 .S /. By Subsection 7.2.4, it suffices to show that WF.F ˙ /0  f˙ > 0g  T  S  T  S . But this follows from Proposition 7.5, since if we set G ˙ .z/u D .2/1 e˙iz u, u 2 C01 .S /, functional calculus shows that G ˙ .z; / is holomorphic in f˙ Im z > 0g with values in D 0 .S S / with F ˙ .t; / D G ˙ .t ˙i0; /. 

8.9 Existence of Hadamard states In this section we prove the important result of Fulling, Narcowich and Wald [FNW], about existence of Hadamard states. Theorem 8.13. Let P be a Klein–Gordon operator on a globally hyperbolic spacetime .M; g/. Then there exists a pure Hadamard state for P . Proof. By Theorem 5.12 we can assume that M D R  † and g D ˇ.t; x/dt 2 C ht .x/d x2 , where † is a Cauchy surface of .M; g/. We fix an ultra-static metric gus D dt 2 C h.x/d x2 and an interpolating metric gint D  .t/gus C C .t/g, with cutoff functions ˙ such that gint D gus in ft  T C 1g, gint D g in ft  T  1g. We set Pus D gus C m2 , m > 0, and fix a Klein–Gordon operator Pint for gint such that Pint D Pus in ft  T C 1g, Pint D P in ft  T  1g. For †˙T D f˙T g  †, we denote by ˙ T;vac the Cauchy surface covariances on †T of the vacuum state !us for Pus . By Proposition 6.5, ˙ T;vac are also the Cauchy surface covariances of a pure state !int for Pint . Since Pus D Pint on a causally compatible neighborhood V of †T , we have Gvac D Gint on V  V . Therefore, the spacetime covariances of !int and !us , given in Proposition 6.6, coincide on V  V . Since !us is a Hadamard state, the spacetime covariances ƒ˙ int of !int satisfy (Had) over V  V , hence everywhere by propagation of singularities, see e.g. (7.25). Let now ˙ T;int be the Cauchy surface covariances of !int on †T . Again by Proposition 6.5, they are the Cauchy surface covariances of a pure state ! for P . By the same argument as above ! is a Hadamard state. 

Chapter 9

Vacuum and thermal states on stationary spacetimes In this chapter we introduce the notions of vacuum and thermal states for Klein– Gordon fields on stationary spacetimes, see [Ky1, S2]. These states are important examples of Hadamard states, the vacuum state giving in particular a preferred pure Hadamard state on a stationary spacetime.

9.1 Ground states and KMS states It is convenient to introduce these notions first in an abstract framework. We work in the complex framework (to which the real one can be reduced). Thus, let .Y ; q/ be a Hermitian space and frs gs2R be a unitary group on .Y ; q/, i.e. such that rs qrs D q for s 2 R. It follows that frs gs2R induces a group fs gs2R of -automorphisms of CCRpol .Y ; q/ defined by s . ./ .y// D ./ .rs y/. We recall the definitions, see e.g. [S2, Definitions 2.3, 2.4 ], of ground states and KMS states for fs gs2R . We set Dˇ D R C i 0; ˇŒ for ˇ > 0, D1 D R C i 0; C1Œ. Let ! be a state on CCRpol .Y ; q/ which is invariant under fs gs2R i.e. !.A/ D !.s .A// for s 2 R; A 2 CCRpol .Y ; q/. Assume moreover that the function R 3 s 7! !.A s B/ 2 C is continuous for all A; B 2 CCRpol .Y ; q/:

(9.1)

It follows that if .H! ; ! ; ! / is the GNS triple for !, see Subsection 4.4.1, there exists a selfadjoint operator H on H! such that ! .s .A// D eisH ! .A/eisH ;

H ! D 0:

Definition 9.1. A state ! is a non-degenerate ground state for frs gs2R if ! is invariant under fs gs2R , (9.1) holds, and moreover H  0;

Ker H D C ! :

(9.2)

Let us assume in addition that ! is gauge-invariant and quasi-free and let ˙ be its complex covariances. Since !. ./ .y// D 0, we know that ! . ./ .y// ! is orthogonal to ! . It follows then from (9.2) and the spectral theorem that for all y1 ; y2 2 Y there exists a function Fy˙1 ;y2 holomorphic in D1 , bounded and continuous in D 1 , such that FyC1 ;y2 .s/ D y 1 C rs y2 ; Fy1 ;y2 .s/ D rs y 1 C y2 ; (9.3) lim!C1 sups2R jFy˙1 ;y2 .s C i /j D 0:

94

9 Vacuum and thermal states on stationary spacetimes

Definition 9.2. A state ! is a KMS state at temperature T D ˇ 1 if for all A1 ; A2 2 CCR.Y ; q/ there exists a function FA1 ;A2 holomorphic in Dˇ , bounded and continuous in D ˇ , such that FA1 ;A2 .s/ D !.A1 s .A2 //; FA1 ;A2 .s C iˇ/ D !.s .A2 /A1 /;

s 2 R:

If ! is gauge-invariant and quasi-free, taking A1 D .y1 /, A2 D  .y2 /, we obtain as above that for all y1 ; y2 2 Y there exists a function Fy1 ;y2 holomorphic in Dˇ , bounded and continuous in D ˇ , such that Fy1 ;y2 .s/ D y 1 C rs y2 ;

Fy1 ;y2 .s C iˇ/ D y 1  rs y2 :

(9.4)

9.1.1 Positivity of the energy. We now prove an important result, due to Kay and Wald [KW, Section 6.2], which relates the existence of ground or KMS states to the positivity of the classical energy associated to frs gs2R . Theorem 9.3. Let .Y ; q; frs gs2R / be as above and ! be a quasi-free non-degenerate ground state or a quasi-free KMS state. Assume moreover that Y is equipped with a vector space topology for which ˙ ; q are continuous and such that @s rs y D ibrs y, for all y 2 Y for some b 2 L.Y /. Then the classical energy associated to frs gs2R E D qb is positive. Proof. Since q D C  is non-degenerate, .j/! D C C is a Hilbertian scalar product on Y and we denote by Y cpl the completion of Y with respect to .j/! . We still denote by ˙ ; q the bounded extensions of ˙ ; q to Y cpl . The state ! is s invariant, which implies that rs ˙ rs D ˙ . Moreover by Definitions 9.1 and 9.2, the map R 3 s 7! y 1 ˙ rs y2 2 C is continuous for y1 ; y2 2 Y . It follows that frs gs2R extends to a weakly, hence strongly continuous unitary cpl group feisb gs2R on Y cpl , with b cpl selfadjoint on Y cpl . We have b cpl jY D b and Y is a core for b cpl by Nelson’s invariant domain theorem. cpl We first check that (9.3), (9.4) extend to yi 2 Y cpl with rs replaced by eisb . Let cpl cpl y1 ; y2 2 Y , yi;n 2 Y with yi;n ! yi in Y , and let Fn D Fy1;n ;y2;n . Note that cpl cpl Fn .t/ ! y 1 C eit b y2 and Fn .t C iˇ/ ! y 1  eit b y2 uniformly on R. It follows from the three-lines theorem that supz2Dˇ jFn .z/  Fm .z/j  sups2R[RCiˇ jFn .s/  Fm .s/j;

ˇ 0, the function Y 3 y 7! y Ey being the classical energy. The energy space Yen is the completion of Y for the scalar product .y1 jy2 /en D y 1 Ey2 and is a complex Hilbert space. Let rs D eisb be a strongly continuous unitary group on Yen with selfadjoint generator b. We assume that rs W Y ! Y , Y  Dom b, Ker b D f0g, and y 1 Ey2 D y 1 qby2 ;

y1 ; y2 2 Y :

(9.5)

96

9 Vacuum and thermal states on stationary spacetimes

The meaning of (9.5) is that frs gs2R is the symplectic evolution group associated to the classical energy y Ey and the symplectic form D i1 q. One introduces then the dynamical Hilbert space 1

Ydyn D jbj 2 Yen ; see [DG, Subsection 18.2.1], with the scalar product .y1 jy2 /dyn D .y1jjbj1 y2 /en . The group frs gs2R extends obviously as a unitary group on Ydyn whose generator will be still denoted by b. From (9.5) we obtain that y 1 qy2 D .y1 jsgn.b/y2/Ydyn

(9.6)

so q is a bounded sesquilinear form on Ydyn, but in general not on Yen , unless 0 62

.b/. Definition 9.4. The ground state !1 is defined by the covariances y 1 ˙ 1 y2 D .y1 j½R˙ .b/y2 /dyn :

(9.7)

Definition 9.5. The ˇ-KMS state !ˇ is defined by the covariances y 1 C y D y 1 q.1  eˇb /1 y2 ; ˇ 2 y 1  y D y 1 q.eˇb  1/1y2 : ˇ 2

(9.8)

˙ 9.1.3 Infrared problem. The covariances ˙ 1 and ˇ are a priori not defined

on Y if 0 2 .b/. This is usually called an infrared problem. However, if 1

Y  Ydyn \ jbj 2 Ydyn

(9.9)

˙ then using that .1  e /1 behaves like 1 near  D 0, we see that ˙ 1 and ˇ are well defined on Y , and hence !1 and !ˇ are well defined quasi-free states on CCRpol .Y ; q/. Note that (9.9) is equivalent to

y Ejbj1 y < 1;

y Eb 2 y < 1; 8y 2 Y ;

which follows from y Eb 2 y < 1; 8y 2 Y ; since Y  Yen .

(9.10)

9.2 Klein–Gordon operators

97

9.1.4 Pure invariant states. Let .Y ; q/ be a Hermitian space with a unitary group frs gs2R . Assume that rs D eisb on Y and that the classical energy E D qb is positive definite on Y . Then any pure state invariant under the induced group fs gs2R is actually equal to the ground state !1 . As in Theorem 9.3, by rs D eisb on Y we mean that Y is equipped with a vector space topology for which q is continuous and such that @s rs y D ibrs y, for all y 2 Y for some b 2 L.Y /. The classical energy E D qb 2 Lh .Y ; Y  / is thus well defined. Proposition 9.6. Let ! a quasi-free state on CCR.Y ; q/ such that its covariances ˙ are continuous in the topology of Y . Assume that ! is pure and invariant under the induced group fs gs2R , and that E is positive definite on Y . Then ! D !1 . Proof. As in the proof of Theorem 9.3, we obtain that frs gs2R extends as a strongly continuous unitary group on the completion Y cpl of Y for .j/! , whose generator b cpl has Y as a core. Since ! is pure, we deduce from Proposition 4.21 that there exist projections c ˙ 2 B.Y cpl /, selfadjoint for .j/! , with c C C c  D ½, ˙ D ˙qc ˙ . From the invariance of ! we see that Œc ˙ ; b cpl  D 0. Next we compute for y 2 Y : .yj.c C  c  /b cpl y/! D y qby D y Ey: Since Y is a core for b cpl , this implies, by the uniqueness of the polar decomposition of b cpl , that c C  c  D sgn.b cpl /, i.e. c ˙ D ½R˙ .b cpl /. From this fact we deduce that Y cpl is the dynamical Hilbert space Y dyn introduced  in Subsection 9.1.2, and hence ! D !1 .

9.2 Klein–Gordon operators Let us now go back to a concrete situation and consider a globally hyperbolic spacetime .M; g/ with a complete Killing vector field X . For the moment we do not assume X to be time-like. Assume that there exists a space-like Cauchy surface † transverse to X . If n is the future directed normal vector field to †, we have X D N n C w on †;

(9.11)

where N 2 C 1 .†I R/ is called the lapse function and w i is a smooth vector field on † called the shift vector field. We can identify M with Rt  †y by the map  W R  † 3 .t; y/ 7! where

t

t .y/

2 M;

is the flow of X . We have

 g D N 2 .y/dt 2 C hij .y/.dy i C w i .y/dt/.dy j C w j .y/dy j /;

@  X D : @t (9.12)

98

9 Vacuum and thermal states on stationary spacetimes

It follows that X is time-like at y iff N 2 .y/ > w i .y/hij .y/w j .y/;

(9.13)

and space-like at y iff

N 2 .y/ < w i .y/hij .y/w j .y/; where h is the induced metric on †. We fix a Klein–Gordon operator on .M; g/ of the form P D g C V;

(9.14)

V 2 C 1 .M I R/ with X V D 0:

(9.15)

The flow f s gs2R of X induces then a unitary group frs gs2R on the Hermitian spaces C 1 .M / . P C0 1 .M / ; . jiG /M /, .Solsc .P /; q/, defined as: 0

rs Œu D Œu ı

s ;

u 2 C01 .M /;

rs  D  ı

s;

 2 Solsc .P /:

9.2.1 A non-existence result. The next proposition, due to Kay and Wald [KW, Section 6.2], shows that the fact that X is everywhere time-like on †, i.e. that .M; g/ is stationary, is a necessary condition for the existence of a ground or KMS state for X. Proposition 9.7. Let .M; g/ a globally hyperbolic spacetime with a complete Killing vector field X and let P D g C V , where V 2 C 1 .M I R/ with X V D 0. Let fs gs2R be the group of -automorphisms of CCR.P / induced by X . Assume that there exists a Cauchy surface † such that X is transverse to † and space-like at some y0 2 †. Then there exists no KMS state nor non-degenerate ground state on CCR.P / for fs gs2R . Proof. We identify M with R  †, the metric g being then as in (9.12). We choose .Y ; q/ D .Solsc .P /; q/ with q defined in (5.35) and rs .t; y/ D .t C s; y/. We identify .Solsc .P /; q/ with .C01 .†/I C2 ; q† / for q† defined in (5.31) using %† and denote still by frs gs2R the image of rs on .C01 .†/I C2 ; q† /. A standard computation shows that for f 2 C01 .†I C2 /, @s rs f D iNH rs f , where H is defined in (9.20). The associated energy E D qH is given by (9.21) below. For y0 2 † we introduce local coordinates on † near y0 , fix  2 C01 .U / for U a small neighborhood of y0 in †, and set f0 .y/ D ei0 y .y/, f1 D iN 1 wf0 for  1 and 0 2 Ty0 †. Then we have ˆ 1 f Ef  D 2 2 .y/.0h1 .y/0 N 2 .y/.0w.y//2 /jhj 2 dy CO./: (9.16) †

is space-like at y0 , then N 2 .y0 / < w i .y0 /hij .y0 /w j .y0 /, and so there If X D exists a neighborhood U of y0 in † such that @ @t

0 h1 .y/0  N 2 .y/.0 w.y//2 < 0;

y 2 U; for 0 D h.y0 /w.y0 /:

By (9.16) we obtain that f Ef  < 0 for  1. This is a contradiction by Theorem 9.3. 

9.3 The Klein–Gordon equation on stationary spacetimes

99

9.3 The Klein–Gordon equation on stationary spacetimes We assume now that the Killing vector field is everywhere time-like and consider a Klein–Gordon operator P D g C V . We will assume that V is preserved by the Killing field X and is strictly positive: X V D 0;

V > 0:

Remark 9.8. Of course, the condition X V D 0 is necessary for P to be invariant under the flow of X . The condition V > 0 is used in Section 9.5 to ensure that the covariances of the vacuum and thermal state are well defined on C01 .†I C2 /, i.e. to avoid a possible infrared problem. If V takes large negative values the conserved energy E defined in (9.21) may not be positive. In this case it seems impossible to construct vacuum or KMS states. The Klein–Gordon operator P takes the form

with

P D .@t C w  /N 2 .@t C w/ C h0 ;

(9.17)

h0 D r  h1 r C V; w D w i @y i ;

(9.18)

where in (9.17) and (9.18) the adjoints are computed with respect to the scalar product ˆ 1 .ujv/ D uvN jhj 2 dtdy: R† 1 We denote by HQ the Hilbert space L2 .†; N jhj 2 dy/. Let us point out a useful operator inequality which follows from (9.13).

Lemma 9.9. One has h0  w  N 2 w C V on C01 .†/; for the scalar product of HQ : Proof. Let X be a real vector space, k 2 Ls .X ; X 0 / be strictly positive, and c 2 X . Then for  D kc 2 X 0 and  2 CX 0 we have .  hjci /k 1 .  hjci / D  k 1   2Re.hjci k 1 / C jhjcij2  k 1  D  k 1   .2  c kc/jhjcij2 ; whence For u 2

k 1  jcihcj  .1  c kcijcihcj: C01 .†/ 

.uj.h0  w N

(9.19)

we write 2

ˆ w/u/ D †

  1 .@y i u.hij  w i N 2 w j /@y j u C V juj2 N jhj 2 dy:

100

9 Vacuum and thermal states on stationary spacetimes

Applying (9.19) under the integral sign for k D h.y/ and c D N 1 .y/w i .y/, we obtain the lemma.  If %t W Solsc .P / ! C01 .†I C2 / is the Cauchy data map on †t D ftg  † we have, by (9.11) that   .t; / ; %t  D i1 N 1 .@t  w/.t; / and if we identify Solsc .P / with C01 .†I C2 / using the map %0 , we obtain that rs W C01 .†I C2 / ! C01 .†I C2 / is given by rs f D %s U0 f; f 2 C01 .†I C2 /; where  D U0 f is the solution of the Cauchy problem  P  D 0; %0  D f: An easy computation shows that:   ½ iN 1 w 1 ; N @s rs f D iH rs f; H D h0 iw  N 1

f 2 C01 .†I C2 /: (9.20)

The classical energy f Ef D kf1  iN 1 wf0 k2HQ C .f0 jh0 f0 /HQ  .wf0 jN 2 wf0 /HQ ; and the charge

f qf D .f1 jN 1 f0 /HQ C .f0 jN 1 f1 /HQ ;

are both conserved by the evolution e

isH

on

(9.21) (9.22)

C01 .†I C2 /.

9.4 Reduction It is useful to reduce (9.20) to a simpler evolution equation. To this end one introduces Q C hQ 0 ; PQ D NPN D .@t C wQ  /.@t  w/ for

hQ 0 D N h0 N;

Setting %Q t Q D



wQ D N 1 wN;

Q / .t; 1 Q / Q .t; i .@t  w/

 ;

wQ  D N w  N 1 : HQ D



½ iwQ hQ 0 iwQ 

(9.23)  ;

(9.24)

101

9.5 Ground and KMS states for P

we have

%t N D Z %Q t on C 1 .M /; N 1 @s  iH D Z 0 .@s  iHQ /Z 1 on C01 .†I C2 /;

where Z D Setting



N 0



0

½

; Z D 0



½ 0

0 N 1

(9.25)

 ;

(9.26)

Q D kf1  iwf f  Ef Q 0 k2HQ C .f0 jhQ 0 f0 /HQ  .wf Q 0 jwf Q 0 /H ; f  qf Q D .f1 jf0 /HQ C .f0 jf1 /HQ ;

we have

Q Z  EZ D E;

Z  qZ D qQ on C01 .†I C2 /:

(9.27)

9.5 Ground and KMS states for P From Lemma 9.9 we obtain that hQ 0  wQ  wQ  V N 2 ;

(9.28)

which using that V > 0 implies that EQ > 0 on C01 .†I C2 /. We can apply the abstract constructions in Subsection 9.1.2 provided we check (9.10). To check this condition we note that bg D f is equivalent to .hQ 0  wQ  w/g Q 0 D f1  iwQ  f0 ;

g1  iwg Q 0 D f0 :

By Lemma 9.9, hQ 0  wQ  wQ  N V N on C01 .†/. Let hQ be the Friedrichs extension Q acting on the Hilbert space HQ . By the Kato–Heinz theorem, we have of hQ 0  wQ  w, 1 1 1 hQ  .N V N /1, hence C01 .†/  Dom.N V N / 2  Dom hQ  2 . 1 2 1 For f 2 C0 .†I C /, we can express g D b f as g0 D hQ 1 .f1  iwQ  f0 /;

g1 D g0 C iwQ hQ 1 .f1  iwQ  f0 /;

noting that f1  iwQ  f0 2 C01 .†/. We have .f jb 2 f /en D .gjg/en D kf0 k2HQ C .f1  iwQ  f0 jhQ 1 .f1  iwQ  f0 //HQ < 1; since f1  iwQ  f0 2 C01 .†/  Dom hQ  2 . Therefore, (9.10) is satisfied and one can define ground and thermal states !Q ˇ , ˇ 2 0; 1 for PQ , whose covariances, denoted by Q ˙ are introduced in Definitions 9.4 and 9.5. ˇ It is now easy to define the vacuum and thermal states for P , since by (9.27) Q ! .C01 .†I C2 /; q/ is unitary. Z W .C01 .†I C2 /; q/ 1

102

9 Vacuum and thermal states on stationary spacetimes

Definition 9.10. The ground state !1 associated to the Killing vector field X is the quasi-free state on CCRpol .C01 .†I C2 /; q/ defined by the covariances 1  Q / 1 Z 1: ˙ 1 D .Z

(9.29)

The state !1 is a pure state. Definition 9.11. The ˇ-KMS state !fi associated to the Killing vector field X is the quasi-free state on CCRpol .jbjC01 .†I C2 /; q/ defined by the covariances 1  Q / 1 Z 1 : ˙ ˇ D .Z

(9.30)

The state !ˇ is not a pure state. Remark 9.12. If the shift vector field w vanishes, then the spacetime .M; g/ is static and the reduction in Section 9.4 produces an abstract Klein–Gordon operator PQ of ˙ the form considered in Subsection 4.10.3. The formulas giving ˙ 1 and ˇ simplify greatly using (4.46), (4.49).

9.6 Hadamard property In this section we prove that !ˇ , ˇ 2 0; C1 are Hadamard states, a result due to Sahlmann and Verch [SV1]. Theorem 9.13. The states !ˇ with ˇ 2 0; C1 are Hadamard states. 2 D 0 .M M / be the spacetime covariances of !ˇ for 0 < ˇ  1. In Proof. Let ƒ˙ ˇ .t ; t ; y1 ; y2 / D Tˇ˙ .t1  t2 ; y1 ; y2 /, the Killing time coordinates .t; y/ we have ƒ˙ ˇ 1 2 with Tˇ˙ 2 D 0 .R  †  †/, since !ˇ is t invariant. From the ground state or KMS condition, it follows that there exist Fˇ˙ W R ˙ i 0; ˇŒ! D 0 .†  †/ holomorphic such that Tˇ˙ .t; y1 ; y2 / D Fˇ˙ .t ˙ i0; y1 ; y2 /. By Proposition 7.5, we obtain that WF.Tˇ˙ /  f˙ > 0g: Applying then the results on the pullback of distributions in Subsection 7.2.4 we see that 0 WF.ƒ˙ ˇ /  f˙1 > 0g  f˙2 > 0g: Since WF.ƒ˙ /0  N  N , this implies that WF.ƒ˙ /0  N C  N C , using that ˇ ˇ XD

@ @t

is future directed time-like.



Chapter 10

Pseudodifferential calculus on manifolds In this chapter we describe various versions of pseudodifferential calculus on manifolds. The pseudodifferential calculus is a standard tool in microlocal analysis, but it is also useful for the global analysis of partial differential equations on smooth manifolds. Of particular interest to us is the Shubin calculus, which is a global calculus on non compact manifolds relying on the notion of bounded geometry. Its two important properties are the Seeley and Egorov theorems. In applications to quantum field theory the manifold is taken to be a Cauchy surface † in a spacetime .M; g/. It turns out that the Cauchy surface covariances of pure Hadamard states can be constructed as pseudodifferential operators on †. This will be treated in detail in Chapter 11.

10.1 Pseudodifferential calculus on Rn We now recall standard facts about the uniform pseudodifferential calculus on Rn . We refer the reader to [H3, Section 18.1] or [Sh1, Chap. 4] for details.

10.1.1 Symbol classes. Let U  Rn an open set. We denote by S m .T  U /,

m 2 R the symbol class defined by:

a 2 S m .T  U / if j@˛x @ a.x; /j  C˛ˇ .himjˇ j /; 8 ˛; ˇ 2 Nd ; .x; / 2 T  U: (10.1) We denote by Shm .T  U / the subspace of S m .T  U / of symbols homogeneous of degree m in the  variable (outside a neighborhood of the origin) ˇ

a 2 Shm .T  U / if a 2 S m .T  U / and a.x; / D m a.x; /;   1; jj  1: (10.2) If amk 2 S mk .T  U / for k 2 N and a 2 S m .T  U / we write a

1 X

amk

kD0

if rmn1 .a/ D a 

n X

amk 2 S mn1.T  U /; 8 n 2 N:

(10.3)

kD0

.T U / for P k 2 N, then there exists a 2 S m .T  U /, unique modulo If amk 2 S 1  S .T U /, such that a 1 kD0 amk . mk



104

10 Pseudodifferential calculus on manifolds

P We say that a symbol a 2 S m .T  U / is poly-homogeneous if a 1 kD0 amk for mk  amk 2 Sh .T U /. The symbols amk are then clearly unique modulo S 1.T  U /. m The subspace of poly-homogeneous symbols of degree m will be denoted by Sph .T  U /. m  We equip S .T U / with the Fr´echet space topology given by the semi-norms kakm;N WD

jhimCjˇ j @˛x @ aj: ˇ

sup

j˛jCjˇ jN;.x;/2T  U

m m .T  U / is a bit different: we equip Sph .T  U / with the semiThe topology of Sph norms of amk in S mk .T  U / and of rmn1 .a/ in S mn1 .T  U /, for 0  k  n 2 N, where amk and rmn1.a/ are defined in (10.3). We set [ \ 1 m .T  U / WD S.ph/ .T  U /; S 1 .T  U / WD S m .T  U /: S.ph/ m2R

m2R

equipped with the inductive, resp. projective limit topology.

10.1.2 Principal part and characteristic set. The principal part of a 2 S m .T  Rn /, denoted by pr .a/, is the equivalence class of a in S m =S m1 . If m , then a C S m1 has a unique representative in Shm , namely the function a 2 Sph am in (10.3). Therefore, in this case the principal part of a is a function on T  Rn , homogeneous of degree m in . m The characteristic set of a 2 Sph is defined as Char.a/ WD f.x; / 2 T  Rn n o W am .x; / D 0gI

(10.4)

it is clearly conic in the  variable. A symbol a 2 S m .T  Rn / is elliptic if there exist C; R > 0 such that ja.x; /j  C him ;

jj  R:

m .T  Rn / is elliptic iff Char.a/ D ;. Clearly, a 2 Sph m 10.1.3 Pseudodifferential operators on Rn . For a 2 Sph .T  Rn /, we denote

by Op.a/ the Kohn–Nirenberg quantization of a, defined by “ n ei.xy/ a.x; /u.y/dyd; u 2 C01 .Rn /: Op.a/u.x/ D a.x; D/u.x/ WD .2/

10.1.4 Mapping properties. Denote by H s .Rn / the Sobolev space of order s and put H 1 .Rn / D

\ s2R

H s .Rn /;

H 1 .Rn / D

[ s2R

H s .Rn /

10.1 Pseudodifferential calculus on Rn

105

m Then if a 2 Sph .T  Rn / we have the continuous mapping

Op.a/ W H s .Rn / ! H sm .Rn /; hence Op.a/ W H 1 .Rn / ! H 1 .Rn / and Op.a/ W H 1 .Rd / ! H 1 .Rd /. m We denote by ‰ m .Rn / the space Op.Sph .T  Rn // and set ‰ 1 .Rn / D

[

‰ m .Rn /;

‰ 1 .Rn / D

m2R

\

‰ m .Rn /:

m2R

We will often write ‰ m instead of ‰ m .Rn /. We equip ‰ m .Rn / with the Fr´echet m space topology obtained from the topology of Sph .T  Rn /.

10.1.5 Principal symbol. If A D a.x; Dx / 2 ‰ m .Rn /, then the m-homogeneous

function pr .A/ DW am .x; / is called the principal symbol of A.

10.1.6 Composition and adjoint. If we equip ‰ 1 .Rn / with the product and

 involution of L.H 1 .Rn //, then ‰ 1 .Rn / is a graded -algebra with

A 2 ‰ m .Rn /; A1 A2 2 ‰ m1 Cm2 .Rn /; for A 2 ‰ m .Rn /; Ai 2 ‰ mi .Rn /: One has

pr .A / D pr .A/;

pr .A1 A2 / D pr .A1 / pr .A2 /;

pr .ŒA1 ; A2 / D f pr .A1 /; pr .A2 /g; where fa; bg D @ a  @x b  @x a  @ b is the Poisson bracket of a and b. Let s; m 2 R. Then the map m .T  Rn / 3 a 7! Op.a/ 2 B.H s .Rn /; H sm .Rn // Sph

(10.5)

is continuous.

10.1.7 Ellipticity. An operator A 2 ‰ m is elliptic if its principal symbol pr.A/

is elliptic. If A 2 ‰ m is elliptic, then there exists B 2 ‰ m , unique modulo ‰ 1 , such that AB  ½; BA  ½ 2 ‰ 1 . Such an operator B is a parametrix of A in the sense of Definition 7.7. We denote it by A.1/ .

10.1.8 Seeley’s theorem. The uniform pseudodifferential calculus on Rn enjoys

plenty of nice properties. For example, if A 2 ‰ m .Rn /, m > 0 is elliptic, then A with domain Dom A D H m .Rn / is closed as an unbounded operator on L2 .Rn /. If z 2 res.A/, where res.A/  C is the resolvent set of A, the resolvent .A  z/1 belongs to ‰ m .Rn / and its principal symbol equals pr .A/1. If moreover A is symmetric on S .Rn /, then it is selfadjoint on H m .Rn /. If 0 2 res.A/ then As for s 2 R belongs to ‰ ms .Rn / with principal symbol pr .A/s . This last result is an example of Seeley’s theorem.

106

10 Pseudodifferential calculus on manifolds

10.2 Pseudodifferential operators on a manifold The uniform pseudodifferential calculus transforms covariantly under local diffeomorphisms. This means that if Ui  Vi are precompact open sets, D V1 ! V2 is a diffeomorphism and i 2 C01 .Vi / with i D 1 on Ui , for A 2 ‰ m .Rn / one has 1 A



.2 u/ D Bu; 8u 2 D 0 .Rn /;

where B 2 ‰ m .Rn / and

pr .B/.x; / D pr .A/. .x/;t D .x/1 /;

.x; / 2 T  U1 :

(10.6)

This allows to extend the pseudodifferential calculus to smooth manifolds. We follow the exposition in [Sh1, Chap. 1], [H3, Section 18.1].

10.2.1 Pseudodifferential calculus on a manifold. Let M be a smooth, ndimensional manifold. Let U  M be a precompact chart open set and W U ! UQ a chart diffeomorphism, where UQ  Rn is precompact, open. We denote by  W C01 .UQ / ! C01 .U / the map defined by  u.x/ D u ı .x/. Definition 10.1. A linear continuous map A W C01 .M / ! C 1 .M / belongs to ‰ m .M / if the following condition holds: Let U  M be precompact open, W U ! UQ a chart diffeomorphism, 1 ; 2 2 1 C0 .U / and Q i D i ı 1 . Then there exists AQ 2 ‰ m .Rn / such that .

 1

/

1 A2



D Q 1 AQQ 2 :

(10.7)

The elements of ‰ m .M / are called (classical) pseudodifferential operators of order m on M . The subspace of ‰ m .M / of pseudodifferential operators with properly supported kernels is denoted by ‰cm .M /. We set [ 1 m ‰.c/ .M / D ‰.c/ .M /: m2R

We also denote by

R1 .M / D L.E 0 .M /; C 1 .M // the space of smoothing operators, or equivalently of operators with kernels in C 1 .M  M /. If A 2 ‰ m .M / there exists (many) Ac 2 ‰cm .M / such that A  Ac 2 R1 .M /.

10.2 Pseudodifferential operators on a manifold

107

10.2.2 Mapping properties. If A 2 ‰ m .M /, then AW

sm .M /; E 0 .M / ! D 0 .M /; Hcs .M / ! Hloc 1 1 C0 .M / ! C .M / continuously;

while if A 2 ‰cm .M / AW

Hcs .M / ! Hcsm .M /;

E 0 .M / ! E 0 .M /;

s sm Hloc .M / ! Hloc .M /;

D 0 .M / ! D 0 .M /;

C01 .M / ! C01 .M /; C 1 .M / ! C 1 .M /;

s .M /, resp. Hcs .M / are the local, resp. compactly supported Sobolev where Hloc spaces on M .

10.2.3 Principal symbol. From (10.6) and (10.7) it follows that to A 2 ‰ m .M /

one can associate its principal symbol pr .A/ 2 C 1 .T  M n o/, which is homogeneous of degree m in the fiber variable  in Tx M , in fjj  1g. The operator A is called elliptic in ‰ m .M / at X0 2 T  M n o if pr .A/.X0 / ¤ 0. 1 10.2.4 Composition and adjoint. Note that if ‰.c/ .M / D

S

m ‰.c/ .M /, is an algebra, but ‰ .M / is not, since without the proper support then condition, pseudodifferential operators cannot in general be composed. Of course, if M is compact, then ‰ 1 .M / D ‰c1 .M /, so this problem disappears. If we fix a smooth density d on M , then we can define the adjoint A of A 2 1 ‰c .M /. Then ‰c1 .M / is a graded -algebra with

‰c1 .M /

1

m2R

A 2 ‰cm .M /; A1 A2 2 ‰cm1 Cm2 .M /; for A 2 ‰cm .M /; Ai 2 ‰cmi .M /: One has

pr .A / D pr .A/;

pr .A1 A2 / D pr .A1 / pr .A2 /;

pr .ŒA1 ; A2 / D f pr .A1 /; pr .A2 /g; where fa; bg is again the Poisson bracket of a and b.

10.2.5 Ellipticity. For A 2 ‰ m .M / we set Char.A/ D fX 2 T  M n o W pr .A/.X / D 0g; which is a closed, conic subset of T  M n o, called the characteristic set of A. If X0 62 Char.A/ one says that A is elliptic at X0 . If Char.A/ D ;, we say that A is elliptic in ‰ m .M /. An elliptic operator A 2 ‰ m .M / has properly supported parametrices B 2 m ‰c .M /, unique modulo R1 .M / such that AB  ½; ½  AB 2 R1 .M /. Again such a parametrix will be denoted by A.1/ . The essential support essupp.A/ of A 2 ‰ 1 .M / is the closed conic subset of  T X n o defined by X0 62 essupp.A/ if there exists B 2 ‰c1 .M / elliptic at X0 such that A ı B is smoothing.

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10 Pseudodifferential calculus on manifolds

10.2.6 The wavefront set. It is well known that the wavefront set of distributions on M can be characterized by means of pseudodifferential operators. We summarize this type of results in the next proposition. Proposition 10.2. (1) Let u 2 D 0 .M /, X0 2 T  M n o. Then X0 62 WFu iff there exists A 2 ‰c0 .M /, elliptic at X0 such that Au 2 C 1 .M /. (2) Let A 2 ‰ 1 .M /. Then WF.A/0 D f.X; X / W X 2 essupp.A/g: (3) Let K W C01 .M1 / ! D 0 .M2 /. Then .X1 ; X2 / 62 WF.K/0 for Xi 2 T  Mi n oi iff there exists Ai 2 ‰c0 .Mi / elliptic at Xi such that A1 KA2 is smoothing. The above pseudodifferential calculus is sufficient for a large part of microlocal analysis, as long as we study distributions only microlocally, i.e. if near X0 2 T  M n o we identify two distributions u1 and u2 if X0 62 WF.u1  u2 /. However, it is not sufficient for more advanced topics. For example, if M is equipped with a complete Riemannian metric h, the Laplace–Beltrami operator h is elliptic in ‰c2 .M /, with principal symbol   h1 .x/. It is also essentially selfadjoint on C01 .M /. One can show that its resolvent .h C i/1 does not belong to ‰c2 .M /, but only to ‰ 2 .M /. So if M is not compact, one needs an intermediate calculus, lying between ‰c1 .M / and ‰ 1 .M /, which is large enough to be stable under taking resolvents, and small enough to remain a -algebra. There are many possible choices, essentially determined by the behavior of symbols near infinity in M . One of them is Shubin’s calculus, [Sh2], which relies on the notion of bounded geometry. This calculus turns out to be sufficient for constructing Hadamard states on many physically relevant spacetimes, like cosmological spacetimes, Kerr, Kerr–de Sitter, Kerr–Kruskal spacetimes, or cones, double cones and wedges in Minkowski spacetime, see Section 11.7.

10.3 Riemannian manifolds of bounded geometry The notion of a Riemannian manifold .M; g/ of bounded geometry was introduced by Gromov, see e.g. [CG], [Ro]. For our purposes the only use of the metric g is to provide local coordinates near any x 2 M , namely the normal coordinates at x, and to equip the spaces of sections of tensors on M with Euclidean norms. Therefore we will use an alternative definition of bounded geometry, which is easier to check in practice. We denote by ı the flat metric on Rn and by Bn .y; r/  Rn the open ball of center y and radius r. If U  Rn is open, we denote by BT pq .U; ı/ the space of smooth .q; p/ tensors on U , bounded together with all their derivatives on U . We equip BT pq .U; ı/ with its Fr´echet space topology. For q D p D 0 we obtain the space Cb1 .U / of smooth functions bounded together with all their derivatives.

109

10.3 Riemannian manifolds of bounded geometry

Definition 10.3. A Riemannian manifold .M; g/ is of bounded geometry if for each x 2 M , there exist an open neighborhood Ux of x and a smooth diffeomorphism 

x

W Ux ! Bn .0; 1/

with x .x/ D 0, such that if gx D . x1 / g, then (1) the family fgx gx2M is bounded in BT20 .Bn .0; 1/; ı/, (2) there exists c > 0 such that c 1 ı  gx  cı; x 2 M . A family fUx gx2M resp. f x gx2M as above will be called a family of bounded chart neighborhoods, resp. bounded chart diffeomorphisms. One can show, see e.g. [GOW, Theorem 2.4 ] that Definition 10.3 is equivalent to the usual definition, which requires that the injectivity radius r D infx2M rx is strictly positive and that .r g /k Rg is a bounded tensor for all k 2 N, where Rg and r g are the Riemann curvature tensor and Levi-Civita connection associated to g. Here the norm on .q; p/-tensors is the norm inherited from the metric g. The canonical choice of Ux ; x is as follows: one fixes for all x 2 M a linear isometry ex W .Rn ; ı/ ! .Tx M; g.x// and sets 1 x .v/

g .x; r=2/; Ux D BM

D expgx ..r=2/ex v/;

v 2 Bn .0; 1/; g.x/

g .x; r/ is the geodesic ball of center x and radius r and expgx W BTx M .0; rx / where BM ! M the exponential map at x.

10.3.1 Atlases and partitions of unity. It is known (see [Sh2, Lemma 1.2]) that if .M; g/ is of bounded geometry, there exist coverings by bounded chart neighborhoods [ M D Ui ; Ui D Uxi ; xi 2 M; i 2N

T which in addition are uniformly finite, i.e. there exists N 2 N such that i 2I Ui D ; if ]I > N . Setting i D xi , we will call fUi ; i gi 2N a bounded atlas of M . One can associate (see [Sh2, Lemma 1.3]) to a bounded atlas a partition of unity X 1D 2i ; i 2 C01 .Ui / i 2N

such that f. i1 / i gi 2N is a bounded sequence in Cb1 .Bn .0; 1//. Such a partition of unity will be called a bounded partition of unity.

10.3.2 Bounded tensors. We now recall the definition of bounded tensors on a manifold .M; g/ of bounded geometry, see [Sh2]. p

Definition 10.4. Let .M; g/ be of bounded geometry. We denote by BTq .M; g/ the spaces of smooth .q; p/ tensors T on M such that if Tx D . x1 / T , then the family fTx gx2M is bounded in BTqp .Bn .0; 1//. We equip BTqp .M; g/ with its natural Fr´echet space topology.

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10 Pseudodifferential calculus on manifolds

The Fr´echet space topology on BTqp .M; g/ is independent on the choice of the family of bounded chart diffeomorphisms f x gx2M .

10.3.3 Bounded differential operators. For m 2 N we denote by echet space of m-th order differential operators on Bn .0; 1/ Diffm b .Bn .0; 1// the Fr´ with Cb1 .Bn .0; 1// coefficients. We denote by Diffb .M / the space of m-th order differential operators on M such that if Px D . x1 / P , then the family fPx gx2M is bounded in Diffm b .Bn .0; 1//. 10.3.4 Sobolev spaces. Let g be the Laplace–Beltrami operator on .M; g/, defined as the closure of its restriction to C01 .M /. Definition 10.5. For s 2 R we define the Sobolev space H s .M; g/ as H s .M; g/ D hg is=2 L2 .M; d Volg /; with its natural Hilbert space topology. One sets \ H s .M I g/; H 1 .M; g/ D

H 1 .M; g/ D

m2R

[

H s .M; g/;

m2R

equipped with the inductive, resp. projective limit topology. ItP is known (see e.g. [Kr, Section 3.3]) that if fUi ; i gi 2N is a bounded atlas and 1 D i 2i is a subordinate bounded partition of unity, then an equivalent norm on H s .M; g/ is given by X kuk2s D k. i1 / i uk2H s .Bn .0;1// : (10.8) i 2N

10.3.5 Equivalence classes of Riemannian metrics. If g 0 is another Rie-

mannian metric on M , we write g 0 g if g 0 2 BT 02 .M; g/ and .g 0 /1 2 BT 20 .M; g/. One can show, see [GOW, Section 2.5], that then .M; g 0 / is also of bounded geometry, that BT pq .M; g/ D BT pq .M; g 0 / and H s .M; g/ D H s .M; g 0 / as topological vector spaces, and that is an equivalence relation.

10.3.6 Examples. Compact Riemannian manifolds are clearly of bounded geometry, as are compact perturbations of Riemannian manifolds of bounded geometry. Gluing two Riemannian manifolds of bounded geometry along a compact region or taking their cartesian product produces again a Riemannian manifold of bounded geometry. If .K; h/ is of bounded geometry, then the warped product .Rs  K; g/ for g D ds 2 C F 2 .s/h is of bounded geometry if F .s/  c0 > 0; 8s 2 R for some c0 > 0; jF .k/ .s/j  ck F .s/; 8s 2 R; k  1; see [GOW, Proposition 2.13].

10.4 The Shubin calculus

111

10.4 The Shubin calculus We now define the Shubin pseudodifferential calculus, see [Sh2], [Kr], which is a version of the uniform calculus of Section 10.1, adapted to manifolds of bounded geometry. We fix a manifold .M; g/ of bounded geometry.

10.4.1 Symbol classes. Let us first define the symbol classes of Shubin’s calcum .T  Bn .0; 1// was defined in Subsection 10.1.1. lus. Recall that the topology of Sph

 1  Definition 10.6. We denote by BS m ph .T M / the space of all a 2 C .T M / such 1  m  that for each x 2 M , ax D . x / a 2 Sph .T Bn .0; 1// and the family fax gx2M is m m bounded in Sph .T  Bn .0; 1//. We equip BSph .T  M / with the semi-norms

kakm;i;p;˛;ˇ D sup kax km;i;p;˛;ˇ ; x2M m where k  km;i;p;˛;ˇ are the semi-norms defining the topology of Sph .T  Bn .0; 1//. m .T  M / and its Fr´echet space topology is independent of The definition of BSph the choice of the atlas fUx ; x gx2M with the above properties. As usual, we set [ 1 m .T  M / D BSph .T  M /: BSph m2R m .T  M / has a principal part am 2 BShm .T  M / which is homoA symbol a 2 BSph geneous of degree m in the fiber variables. m A symbol a 2 BSph .T  M / is elliptic if there exists C; R > 0 such that

jax .y; /j  C jjm ; 8x 2 M; .y; / 2 T  Bn .0; 1/; m .T  M / means uniform ellipticity. hence ellipticity in BSph

10.4.2 Pseudodifferential operators. Let fUi ; X

M and

i gi 2N

be a bounded atlas of

2i D ½

i 2N

a subordinate bounded partition of unity, see Subsection 10.3.1. Let .

1  i / d Volg

D mi dx;

so that fmi gi 2N is bounded in Cb1 .Bn .0; 1//. We set also Ti W L2 .Ui ; d Volg / ! L2 .Bn .0; 1/; dx/; 1

u 7! mi2 .

1  i / u;

so that Ti W L2 .Ui ; d Volg / ! L2 .Bn .0; 1/; dx/ is unitary.

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10 Pseudodifferential calculus on manifolds

Definition 10.7. Let a 2 BS m .T  M /. We set X i Ti ı Op.Eai / ı Ti i ; Op.a/ D i 2N m m .T  B.0; 1// ! Sph .T  Rn / is an where ai D axi .see Definition 10.6/, and E W Sph extension map.

Such a map Op constructed by means of a bounded atlas and a bounded partition of unity will be called a bounded quantization map. 1 Note that if a 2 BSph .T  M /, then the distributional kernel of Op.a/ is supported in f.x; y/ 2 M  M W d.x; y/  C g; for some C > 0, where d is the geodesic distance on M . In particular, Op.a/ 2 ‰c1 .M /, hence such operators can be composed. However, because of the above support property, Op.Sc1 .T  M // is not stable under composition. 1 To obtain an algebra of operators, it is necessary to add to Op.BSph .T  M // an ideal of smoothing operators, which we introduce below. In the sequel the Sobolev spaces H s .M; g/ will be simply denoted by H s .M /. Definition 10.8. We set

W 1 .M / D

\

B.H m .M /; H m .M //;

m2N

equipped with its natural topology given by the semi-norms kAkm D k.g C 1/m=2 A.g C 1/m=2 kB.L2 .M // : Note that W 1 .M / is strictly included in the ideal R1 .M / of smoothing operators. The next result shows the independence modulo W 1 .M / of Op.BS 1 .T  M // on the above choices of fUi ; i ; E; i g. Proposition 10.9. Let Op0 be another bounded quantization map. Then 1 Op  Op0 W BSph .T  M / ! W 1 .M /:

is continuous. Definition 10.10. We set for m 2 R [ f1g: m ‰bm .M / D Op.BSph .T  M // C W 1 .M /:

Clearly, ‰cm .M /  ‰bm .M /  ‰ m .M /. One can show that ‰bm .M / W H s .M / ! H sm .M /; continuously for s 2 R [ f˙1g:

10.5 Time-dependent pseudodifferential operators

113

10.4.3 Composition and adjoint. To A 2 ‰bm .M / one can associate its principal symbol pr .A/ 2 C 1 .T  M n o/, which is homogeneous of degree m on the fibers. Again, A is elliptic in ‰bm .M / if pr .A/ is elliptic in the sense of Subsection 10.4.1. An elliptic operator A 2 ‰bm .M / has parametrices B 2 ‰bm .M /, unique modulo W 1 .M / such that AB  ½; ½  AB 2 W 1 .M /. As before, such a parametrix will be denoted by A.1/ . If we equip M with the density d Volg , then we can define the adjoint A of A 2 ‰b1 .M /. Then ‰b1 .M / is a graded -algebra with A 2 ‰bm .M /; A1 A2 2 ‰bm1 Cm2 .M /; for A 2 ‰bm .M /; Ai 2 ‰b i .M /: m

We have

pr .A / D pr .A/;

pr .A1 A2 / D pr .A1 / pr .A2 /;

pr .ŒA1 ; A2 / D f pr .A1 /; pr .A2 /g; where fa; bg is the Poisson bracket of a and b. The results on adjoints are still true if d Volg is replaced by an arbitrary smooth, bounded density d on M .

10.5 Time-dependent pseudodifferential operators We also need a time-dependent version of the calculus in Section 10.4, which we will briefly outline, referring to [GOW, Chapter 5] for details. If I  R is an open interval and F is a Fr´echet space whose topology is defined by the semi-norms k  kn , n 2 N, then the space Cb1 .I I F / is also a Fr´echet space, with semi-norms supt 2I k@kt f .t/kn , k; n 2 N. m .T  M //, Cb1 .I I W 1.M // One can define in this way the spaces Cb1 .I I BSph and m Cb1 .I I ‰bm .M // D Op.Cb1 .I I BSph .T  M /// C Cb1 .I I W 1 .M //;

where Op refers of course to quantization in the .x; / variables. An element A of Cb1 .I I ‰bm .M // will be usually denoted by A.t/. All the results in Section 10.4 extend naturally to the time-dependent situation.

10.6 Seeley’s theorem The most important property of the Shubin calculus is its invariance under complex powers, which was shown in [ALNV] and is an extension of a classical result of Seeley [Se]. We consider here the simpler case of real powers, see [GOW, Theorem 5.12]. The Hilbert space L2 .M; d Volg / is denoted simply by L2 .M /.

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10 Pseudodifferential calculus on manifolds

Theorem 10.11. Let a D a.t/ 2 Cb1 .I I ‰bm .M //, be elliptic and symmetric on C01 .I I H 1 .M //. Then a is selfadjoint with domain L2 .I I H m .M //. If a.t/  c ½ for some c > 0 then as .t/ 2 Cb1 .I I ‰bms .M // for all s 2 R and

pr .as /.t/ D . pr .a//s .t/; t 2 I:

10.7 Egorov’s theorem We now state another important property of the Shubin calculus, namely Egorov’s theorem, see [GOW, Section 5.4]. Let us consider an operator .t/ D 1 .t/ C 0 .t/, such that i .t/ 2 Cb1 .I I ‰bi .M //;

i D 0; 1;

1 .t/ is elliptic, symmetric and bounded from below on H 1 .M /:

(10.9)

By Theorem 10.11, 1 .t/ with domain Dom .t/ D H 1 .M / is selfadjoint, hence .t/ with ´ the same domain is closed, with non-empty resolvent set. We denote by t Texp i s . /d the associated propagator, defined by 8   ˆ t   ˆ t @ ˆ ˆ ˆ . /d D i.t/Texp i . /d ; t; s 2 I; Texp i ˆ ˆ @t ˆ s s ˆ     ˆ ˆ < t t @ . /d D iTexp i . /d .s/; t; s 2 I; Texp i ˆ @s  ˆ s s ˆ  ˆ ˆ s ˆ ˆ ˆ . /d D ½; s 2 I: : Texp i

(10.10)

s

The notation Texp comes from the time-ordered exponential, which the standard

is ´t tool to solve (10.10) when .t/ is bounded. The existence of Texp i s . /d is a classic result of Kato, see [SG] for a recent summary. Theorem 10.12. Let a 2 ‰ m .M / and .t/ satisfying (10.9). Then  ˆ t   ˆ s  . /d aTexp i . /d 2 Cb1 .I 2 ; ‰ m .M //: a.t; s/ D Texp i s

Moreover,

t

pr .a/.t; s/ D pr .a/ ı ˆ.s; t/; 

where ˆ.t; s/ W T M n o ! T  M n o is the flow of the time-dependent Hamiltonian

pr ./.t/.

´ t One can show, see [GOW, Lemma 5.14], that Texp i s . /d 2 B.H m .M // for m 2 R [ f˙1g, hence a.t; s/ above is well defined.

Chapter 11

Construction of Hadamard states by pseudodifferential calculus In this chapter we explain the construction in [GW1, GOW] of pure Hadamard states using the global pseudodifferential calculus described in Chapter 10. These Hadamard states are constructed via their Cauchy surface covariances with respect to some fixed Cauchy surface †. It is important to assume that the normal geodesic flow, see Subsection 5.4.3, exists for some uniform time interval. This apparently strong condition can actually be considerably relaxed, since one can perform conformal transformations on the metric. For example the Kerr or Kerr–de Sitter exterior spacetimes and the Kerr–Kruskal spacetime can be treated by this method. An interesting pair of notions that appears in this context is the one of Lorentzian metrics and Cauchy surfaces of bounded geometry, with respect to some reference Riemannian metric. If † and .M; g/ are of bounded geometry, Klein–Gordon operators on .M; g/ can be reduced to a simple model form, which fits into the framework of Chapter 10. It is rather clear that the construction of Hadamard states is intimately related to parametrices for the Cauchy problem on †. Traditionally those parametrices are constructed as Fourier integral operators, using solutions of the eikonal and transport equations. Since we need to control the conditions in Proposition 6.5 on Cauchy surface covariances, like for example positivity, we need a global construction of parametrices, and it turns out that an approach via time-ordered exponentials is more convenient and, we think, more elegant, see Section 11.3. Our construction is also equivalent to a factorization of the Klein–Gordon operator as a product of two first-order pseudodifferential operators, which was already used by Junker [J1, J2], who gave the first construction of the Cauchy covariances of Hadamard states using pseudodifferential calculus. His constructions were however restricted to the case when † is compact.

11.1 Hadamard condition on Cauchy surface covariances The Hadamard condition in Section 8.3 is formulated in terms of the spacetime covariances ƒ˙ . We need a condition in term of the Cauchy surface covariances ˙ † 0 for a space-like Cauchy surface †. We recall that U† W E 0 .†I C2 / ! Dsc .M / is the Cauchy evolution operator for P , see Theorem 5.20.

116

11 Construction of Hadamard states by pseudodifferential calculus

Proposition 11.1. Let c ˙ be linear maps that are continuous from C01 .†I C2 / to C 1 .†I C2 / and from E 0 .†I C2 / to D 0 .†I C2 /, such that for some neighborhood U of † in M we have WF.U† ı c ˙ /0  N ˙  .T  † n o/; over U: Let

ƒ˙ D ˙.%† G/ q† c ˙ .%† G/:

Then

WF.ƒ˙ /0  N ˙  .T  M n o/:

(11.1)

Proof. Clearly ƒ˙ W C01 .M / ! D 0 .M / are continuous. By Proposition 5.21, we have U† D i1 .%† G/ q† , and so ƒ˙ D ˙i1 U† c ˙ .%† G/. We apply Subsection 7.2.8 for M1 D U , M2 D †, M3 D M , K1 D U† c ˙ , K2 D %† G and obtain (11.1), first over U  M , and then over M  M by propagation of singularities, using that P ƒ˙ D 0. 

11.2 Model Klein–Gordon operators We now describe a rather simple class of Klein–Gordon operators to which more complicated ones can be reduced. We fix an .n1/-dimensional Riemannian manifold .†; k0 / of bounded geometry and an open interval I  R with 0 2 I . Let I 3 t 7! ht a time-dependent Riemannian metric on † such that ht 2 Cb1 .I I BT20 .†; k0 // and h1 2 Cb1 .I I BT02 .†; k0 //. t We equip M D I  † with the Lorentzian metric g D dt 2 C ht .x/dx 2 and consider a Klein–Gordon operator P on .M; g/ such that moreover P 2 Diff2b .M; k/ for k D dt 2 C k0 dx 2 . It is easy to see that P is then of the form P D @2t C r.t; x/@t C a.t; x; @x /;

(11.2)

where a.t; x; @x / 2 Cb1 .I I Diffb .†I k0 // such that .i/ pr .a/.t; x; / D  ht .x/; .ii/ a.t; x; @x / D a .t; x; @x /; where the adjoint is defined with respect to the time-dependent scalar product ˆ uv d Volht ; (11.3) .ujv/t D †

and rt D jht j

 21

1 2

@t jht j . The two energy shells for P are 1

N ˙ D f.t; x; ; / W  D ˙. ht .x// 2 ;  ¤ 0g: We set †t D ftg  † in M equipped with the density d Volht .

117

11.3 Parametrices for the Cauchy problem

11.2.1 Cauchy problem. It is usual to rewrite the Klein–Gordon equation .@2t C r.t/@t C a.t//.t/ D 0 as a first-order system 1

i

 @t .t/ D H.t/ .t/;

by setting

where H.t/ D

½ 0 a.t/ ir.t/

 ;

(11.4)

 .t/ D %t : .t/ D 1 i @t .t/ 

We denote by  ˆ t   UH .t; s/ D Texp i H. /d ;

s; t 2 I;

(11.5)

s

the evolution operator associated to H.t/. We equip L2 .†t I C2 / with the timedependent scalar product obtained from (11.3), by setting ˆ    .f jg/t D f 1 g1 C f 0 g0 d Volht : †t

We will use it to define adjoints of linear operators and to identify sesquilinear forms on L2 .†I C2 / with linear operators. For   0 ½  q D ½ 0 we have

 q D UH .s; t/q UH .s; t/;

s; t 2 I;

(11.6)

i.e. the evolution operator UH .t; s/ is symplectic.

11.3 Parametrices for the Cauchy problem 0 Let U0 W E 0 .†I C2 / ! Dsc .M / be the Cauchy evolution operator for P , which solves  P U0 D 0; (11.7) %0 U0 D ½:

We will construct a parametrix UQ 0 for (11.7) such that  P UQ 0 D 0; modulo smoothing errors: %0 UQ 0 D ½;

118

11 Construction of Hadamard states by pseudodifferential calculus

The theory of Fourier integral operators, one of the important topics in microlocal analysis, originated from the construction of parametrices by Lax [La] and Ludwig [Lu] for the Cauchy problem for wave equations (or, more generally, strictly hyperbolic systems). It amounts to looking for UQ 0 as a sum of two oscillatory integrals ˆ ˙ d ei.' .t;x;/y/ a˙ .t; x; /d : .2/ The phase functions ' ˙ .t; x; / are solutions of the eikonal equation ( .@t ' ˙ .t; x; //2  a.t; x; @x ' ˙ .t; x; // D 0; ' ˙ .0; x; / D x ; and the amplitudes a˙ .t; x; / solve a first-order differential equation along the bicharacteristics of P . It is actually simpler and more convenient to use a more operator theoretical approach. Instead, we will try to find time-dependent operators b ˙ .t/ 2 Cb1 .I I ‰b1 .†// such that  ˆ t  U ˙ .t/ D Texp i b ˙ . /d

0

solve the equation P U ˙ .t/ D 0; modulo smoothing errors:

(11.8)

If we try to solve (11.8) exactly, we see that b.t/ should satisfy the Riccati equation i@t b ˙  .b ˙ /2 C a C irb ˙ D 0:

(11.9)

A straightforward computation shows that (11.9) is equivalent to a factorization P D .@t C ib ˙ C r/.@t  ib ˙ /:

(11.10)

Such a factorization was already used by Junker [J1, J2] to construct Hadamard states by pseudodifferential calculus, in the case where the Cauchy surface † is compact.

11.3.1 Solving the Riccati equation. We now explain how to solve (11.9), modulo smoothing errors. The first step consists in reducing the task to the case 1 when a.t/ D a.t; x; @x / is strictly positive, as an operator on L2 .†; jht j 2 dx/. One can find, see [GOW, Proposition 5.11], an operator c1 .t/ 2 Cb1 .I I W 1 .†// and a constant c > 0 such that a.t/ C c1 .t/  c ½, for all 1 t 2 I . One sets then .t/ D .a.t/ C c1 .t// 2 , which by Theorem 10.11 belongs to 1 Cb1 .I I ‰b1 .†//, with principal symbol . ht .x// 2 .

11.3 Parametrices for the Cauchy problem

119

Proposition 11.2. There exists an operator b.t/ 2 Cb1 .I I ‰b1 .†//, unique modulo Cb1 .I I W 1 .†//, such that .i/

b.t/ D .t/ C Cb1 .I I ‰b0 .†//;

.ii/

.b.t/ C b  .t//1 D .2.t// 2 .½ C r1 /.2.t// 2 ; r1 2 Cb1 .I I ‰b1.†//;

.iii/

.b.t/ C b  .t//1  c.t/1 ; for some c > 0;

.iv/

˙ .t/ 2 Cb1 .I I W 1.†//; i@t b ˙ .t/  .b ˙/2 .t/ C a.t/ C ir.t/b ˙ .t/ D r1

1

1

for b C .t/ D b.t/; b  D b  .t/: Proof. We follow the proof in [GOW, Theorem 6.1]. Discarding error terms in 1 .T  †//. Cb1 .I I W 1 .†//, we can assume that .t/ D Op.O /.t/, O .t/ 2 Cb1 .I I BSph We look for b.t/ of the form b.t/ D .t/ C d.t/ for d.t/ D Op.dO /.t/;

0 .T  †//: dO .t/ 2 Cb1 .I I BSph

Since .t/ is elliptic, it admits a parametrix  .1/ .t/ D Op.c/.t/; O

1 c.t/ O 2 Cb1 .I I BSph .†//:

The equation (11.9) becomes, modulo error terms in Cb1 .I I W 1 .†//, d.t/ D

i .1/  .t/@t .t/ C  .1/ .t/r.t/.t/ C F .d /.t/; 2

(11.11)

with F .d /.t/ D

1 .1/ .t/ i@t d.t/ C Œ.t/; d.t/ C ir.t/d.t/  d 2 .t/ :  2

By means of symbolic calculus, we obtain that F .d /.t/ D Op.FQ .dO //.t/ C Cb1 .I I W 1 .†//; with

1 FQ .dO /.t/ D c.t/] O i@t dO .t/ C O .t/]dO .t /  dO .t/]O .t/ C ir.t/]dO .t /  dO .t/]dO .t / ; 2 where the operation ] (sometimes called the Moyal product) is defined by Op.a/Op.b/ D Op.a]b/ modulo BS 1 .†/: The equation (11.11) becomes dO .t/ D dO0 .t/ C FQ .dO /.t/;

(11.12)

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11 Construction of Hadamard states by pseudodifferential calculus

for

i 1 0   O .t/ C c.t/]r.t/]O O  .t// 2 C .T †/ : O dO0 .t/ D .c.t/]@ I I BS t b ph 2 The map FQ has the following property:



j 0 dO1 .t/; dO2 .t/ 2 Cb1 I I BSph .T  †/ ; dO1 .t/  dO2 .t/ 2 Cb1 I I BSph .T  †/

j 1 H) FQ .dO1 /.t/  FQ .dO2 /.t/ 2 Cb1 I I BSph .T  †/ : (11.13) This allows to solve (11.13) symbolically by setting dO1 .t/ D 0; dOn .t/ D dO0 .t/ C FQ .dOn1/.t/; and

dO .t/

X

dOn .t/  dOn1 .t/;

n2N

which is an asymptotic series, since, by (11.13),

n .T  †/ : dOn .t/  dOn1 .t/ 2 Cb1 I I BSph It follows that .t/ C d.t/ solves (11.9) modulo Cb1 .I I W 1 .†//, hence satisfies (i) and (iv) in the proposition. In the rest of the proof .t/ will again denote the square root .t/ D .a.t/ C 1 c1 .t// 2 , which differs from Op.O /.t/ by an error in Cb1 .I I W 1.†//, so that .t/ C d.t/ still solves (11.9) modulo Cb1 .I I W 1 .†//. To satisfy (ii), (iii) we need to further modify .t/ C d.t/ by an error term in Cb1 .I I W 1 .†//, which will not invalidate (i) and (iv). We set s.t/ D .t/ C d.t/ C   .t/ C d  .t/; 1

2 which is selfadjoint, with principal symbol equal to 2.  h1 t .x// . By [GOW, Proposition 5.11], there exist an r1 2 Cb1 .I I W 1 .†// and a constant c > 0 such that c 1 .t/  s.t/ C r1 .t/  c.t/; t 2 I: (11.14)

Now set

1 b.t/ D .t/ C d.t/ C r1 .t/: 2 Property (iii) follows from (11.14) and the Kato–Heinz theorem. To prove property (ii), we write 1 1 b.t/ C b  .t/ D .2/ 2 .t/.½ C rQ1 .t//.2/ 2 .t/;

where rQ1 .t/ 2 Cb1 .I I ‰b1.†//, by Theorem 10.11. Since .½CrQ1 /.t/ is boundedly invertible, we have, again by Theorem 10.11, that .½ C rQ1 /1 .t/ D ½ C r1 .t/; r1 .t/ 2 Cb1 .I I ‰b1.†//; which implies (ii).

11.3 Parametrices for the Cauchy problem

121

We observe then that if b.t/ 2 Cb1 .I I ‰b1 .†// we have .@t b/ .t/ D @t .b  /.t/ C r.t/b  .t/  b  .t/r.t/: Note that the adjoint is computed with respect to the time-dependent scalar product (11.3), so .@t b/ ¤ @t .b  /. This implies that b  .t/ is also a solution of (11.9) modulo Cb1 .I I W 1.†//. The proof of the proposition is complete. 

11.3.2 Parametrices for the Cauchy problem. We can now construct parametrices for the Cauchy problem (11.7). In fact, if u˙ f D .b C  b  /1 .0/. b .0/f0 ˙ f1 /;

f 2 H 1 .†/;

(11.15)

we obtain by an easy computation that UQ 0 f .t/ D U C .t/uC f C U  .t/u f solves



(11.16)

P UQ 0 2 Cb1 .I I W 1 .†//; %0 UQ 0 D ½;

hence is a parametrix for the Cauchy problem (11.7).

11.3.3 Microlocal splitting of Cauchy data. It is easy to see that if u 2 H 1 .†/, then WF.U ˙ ./u/  N ˙ . Therefore, if f 2 Ker u , one has also WF.U0 f /  N ˙ . It turns out that Ker u are complementary spaces, for example in H 1 .†I C2 /, which are moreover orthogonal with respect to q. This is summarized in the next proposition. Proposition 11.3. Let T D



½

½ b C .0/ b  .0/



Then: (1) T

1

C

  21

D .b  b /

(2) T  qT D



1

.b C  b  / 2 .0/:

 .0/

½ 0

b  .0/ b C .0/

0 ½

 :

½ ½

 :

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11 Construction of Hadamard states by pseudodifferential calculus

(3) Let 

C

and c then

˙

D T  ˙ T 1 D



½

D 

0

0 0

 ;





 D

0 0

0



½

.b C  b  /1 b  ˙.b C  b  /1 C C  1 

b .b  b / b ˙b ˙ .b C  b  /1

c C C c  D ½;

.c ˙ /2 D c ˙ ;

.c  / qc ˙ D 0;

 .0/I

Ker u D Ran c ˙ ;

˙.c ˙/ qc ˙  0;

on H 1 .†I C2 /. (4)

WF.U0 c ˙ /0  N ˙  .T  † n o/: 1

1

(5) The map T W L2 .†/ ˚ L2 .†/ ! H 2 .†/ ˚ H  2 .†/ is an isomorphism. Proof. The proof of (1) and (2) is a routine computation, and (3) follows from (2). Note that c ˙ are bounded on H 1 .†I C2 / and H 1 .†I C2 /, since their entries belong to ‰b1 .†/. We have U0 c ˙ D UQ 0 c ˙ modulo C 1 .M  M / and UQ 0 c ˙ D U ˙ ./c ˙ by (3), so WF.U0 c ˙ /0 D WF.U C ./c ˙ /0 . By (11.10), we have P U ˙ ./c ˙ 2 C 1 .M  M / hence WF.U C ./c ˙ /0  N  .T  † n o/. Furthermore, .@t  ib ˙ .t/U C ./c ˙ D 0, but @t  ib ˙ .t/ is not a classical pseudodifferential operator on M . However, one can find ˙ 2 ‰c0 .M /, elliptic near N ˙ , such that ˙ ı .@t  ib ˙ .t// belongs to ‰ 1 .M /. We have ˙ ı .@t  ib ˙ .t//U C./c ˙ D 0 and ˙ ı .@t  ib ˙ .t// is elliptic near N  . Now applying Theorem 7.6 we conclude that WF.U C ./c ˙ /0  N ˙  .T  † n o/, which proves (4). It remains to prove (5). Using the expression of T 1 in (2), we see that the 1 norm kT 1 f kL2 .†IC2 / is equivalent to the norm k.b C  b  / 2 f0 kL2 .†/ C k.b C  1

b  / 2 f1 kL2 .†IC2 / . By (11.14), we have c 1 .0/  b C .0/  b  .0/  c.0/; which by the Kato–Heinz theorem implies that kT 1 f kL2 .†IC2 / is equivalent to 1

1

k.0/ 2 f0 kL2 .†/ C k.0/ 2 f1 kL2 .†/ . By the ellipticity of .0/, this norm is equiva1

1

lent to the norm of H 2 .†/ ˚ H  2 .†/.



Remark 11.4. c ˙ are complementary projections, with Ran c ˙ D Ker u . Moreover, Ran c ˙ are orthogonal for q, with WFU0 c ˙ f  N ˙ for f 2 H 1 .†/. Therefore the pair of projections c ˙ will be called a microlocal splitting of Cauchy data. 1 1 The space H 2 .†/ ˚ H  2 .†/ is the charge space, which appears in the quantization of Klein–Gordon equations. It is more natural in this context than the energy space H 1 .†/ ˚ L2 .†/, which is usually considered in the PDE literature.

123

11.4 The pure Hadamard state associated to a microlocal splitting

11.4 The pure Hadamard state associated to a microlocal splitting It is now straightforward to associate a pure Hadamard state to the pair of projections c ˙ in Proposition 11.3. Theorem 11.5. Let c ˙ be a microlocal splitting and ˙  ˙ 0 D ˙qc :

Then

˙ 0

(11.17)

are the †0 covariances of a pure Hadamard state !b for P .

Proof. We first check the conditions in Proposition 6.5. (i) is obvious and (iii) follows from c C C c  D 1. To check (ii), we note that c ˙ W C01 .†I C2 / ! L2 .†I C2 / since c ˙ W H 1 .†I C2 / ! H 1 .†I C2 /. We have then    C   ˙   ˙ ˙ f j˙ 0 f 0 D ˙ .c C c /f jqc f 0 D ˙ c f jqc f 0  0; by Proposition 11.3 (3). Therefore ˙ 0 are the †0 covariances of a quasi-free state !b for P . If ƒ˙ are the spacetime covariances of !b , we deduce from Proposition 11.1 and Proposition 11.3 (4) that WF.ƒ˙ /0  N ˙  N . Since .ƒ˙ / D ƒ˙ we have WF.ƒ˙ /0  N ˙  N ˙ , hence by Theorem 8.5 !b is a Hadamard state. It remains to prove that !b is pure. To that end, let us first examine the norm k  k! associated to !b , see Subsection 4.9.2. By Proposition 11.3, we have  C  1 C D .T 1 / . C    /2 T 1 D .T 1 / T 1 : 0 C 0 D qT .   /T

Therefore, kf k2! D .f j.C C  /f /L2 .†IC2 / D kT 1 f k2L2 .†IC2 / . By Proposition 11.3 (5), the completion Y cpl of Y D C01 .†I C2 / with respect to the norm k  k! 1 1 equals H 2 .†/ ˚ H  2 .†/. Again by Proposition 11.3 (5), we obtain that c ˙ D T  ˙ T 1 extend by density to projections on Y cpl that satisfy (4.36) in Proposition 4.21. Therefore, !b is a pure state. 

11.5 Spacetime covariances and Feynman inverses We now give more explicit formulas expressing the spacetime covariances ƒ˙ of !b and the Feynman inverse associated to !b , see Section 8.5. It is convenient to formulate these results using the ‘time kernel’ notation: namely, if A W C01 .M I Cp / ! C 1 .M I Cq / we denote by A.t; s/ W C01 .†I Cp / ! C 1 .†I Cq / its operator-valued kernel, defined by ˆ A.t; s/u.s/ds; u 2 C01 .M I Cp /: Au.t/ D R

124

11 Construction of Hadamard states by pseudodifferential calculus

If UH .t; s/ is the propagator introduced in (11.5), we set ˙ UH .t; s/ D UH .t; 0/c ˙UH .0; s/:

The following theorem is shown in [GOW, Theorems 6.8, 7.10]. Theorem 11.6. Let ƒ˙ and GF be the spacetime covariances and the Feynman inverse, respectively, of the state !b constructed in Theorem 11.5. Then ˙ ƒ˙ .t; s/ D ˙0 UH .t; s/1 ;   C  .t; s/ .t  s/  UH .t; s/ .s  t/ 1 ; GF .t; s/ D i1 0 UH

 where i

f0 f1

(11.18)

 D fi and .t/ is the Heaviside function.

Let us conclude this section by stating without proofs some more results taken from [GOW, Chapter 7].

11.5.1 Regular states. Recall that †s D fsg  † for s 2 I and let ˙ s be the Cauchy surface covariances of !b on †s . Then one can show that ˙ ˙ s D ˙qc .s/;

where c ˙ .s/ D T .s/ ˙ T 1 .s/ and T .s/ is defined as in Proposition 11.3, with b ˙ .0/ replaced by b ˙ .s/. A quasi-free state ! for P is called regular if its Cauchy surface covariances ˙ s on †s belong to ‰b1 .†I M2 .C// for some s 2 I . One can show that if ! is regular, 1 then ˙ s on †s 2 ‰b .†I M2 .C// for all s 2 I .

11.5.2 Bogoliubov transformations. It is well known, see e.g. [DG, Theorem 11.20] that if .Y ; q/ is a Hermitian space and if !; !Q are two pure quasi-free states on CCRpol .Y ; q/, then there exists u 2 U.Y ; q/ such that Q ˙ D u ˙ u: Such a map u corresponds to a Bogoliubov transformation. One can show that if ! is a pure, regular Hadamard state for P , with covariances 1 ˙ .†/ such that 0 on †0 ; then there exists a 2 W ˙ 0

D ˙T

1





˙

.0/ U  U T

1

.½ C aa / 2 .0/; with U D a 1

! a : 1 .½ C a a/ 2

11.6 Klein–Gordon operators on Lorentzian manifolds of bounded geometry

125

11.6 Klein–Gordon operators on Lorentzian manifolds of bounded geometry We now introduce a class of spacetimes and associated Klein–Gordon equations whose analysis can be reduced to the model situation in Section 11.2. The results in this section are taken from [GOW, Chapter 3]. We start with some definitions.

11.6.1 Lorentzian manifolds of bounded geometry. Let M a smooth manO is of bounded ifold equipped with a reference Riemannian metric hO such that .M; h/ geometry. Definition 11.7. If g is a Lorentzian metric on M , we say that .M; g/ is of bounded O and g 1 2 BT 2 .M; h/. O geometry if g 2 BT20 .M; h/ 0 Definition 11.8. Let † an .n  1/-dimensional submanifold. An embedding i W † ! M is called of bounded geometry if there exists a family fUx ; x gx2M of bounded chart diffeomorphisms for hO such that if †x D x .i.†/ \ Ux / we have †x D f.v 0 ; vn / 2 Bn .0; 1/ W vn D Fx .v 0 /g; where fFx gx2M is a bounded family in Cb1 .Bn1 .0; 1//. The typical example of an embedding of bounded geometry is as follows: let M D I  S , where I is an open interval and .S; h/ is of bounded geometry, and let hO D dt 2 C h.x/dx 2 . Then the submanifolds ft D F .x/g for F 2 BT 00 .S; h/ are of O bounded geometry in .M; h/. Definition 11.9. A space-like Cauchy surface †  M is of bounded geometry if: O (1) the injection i W † ! M is of bounded geometry for h; (2) if n.y/ for y 2 † is the future directed unit normal to † for g, then O < 1: sup n.y/  h.y/n.y/ y2†

Clearly, the above definitions depend only on the equivalence class of hO for the equivalence relation in Subsection 10.3.5.

11.6.2 Gaussian normal coordinates. The following result is proved in [GOW, Theorem 3.5]. It says that the bounded geometry property of g and † carries over to the Gaussian normal coordinates to †. Theorem 11.10. Let .M; g/ a Lorentzian manifold of bounded geometry and † a Cauchy surface of bounded geometry. Then the following holds:

126

11 Construction of Hadamard states by pseudodifferential calculus

(1) there exists ı > 0 such that the normal geodesic flow to †: W

  ı; ıŒ† ! M .t; y/ 7! expgy .tn.y//

is well defined and is a smooth diffeomorphism onto its range; (2)  g D dt 2 Cht , where fht gt 2 ı;ıŒ is a smooth family of Riemannian metrics on † such that .i/ .†; h0 / is of bounded geometry; .ii/ t 7! ht 2 Cb1 .   ı; ıŒ; BT 02 .†; h0 //; 2 Cb1 .   ı; ıŒ; BT 20 .†; h0 //: .iii/ t 7! h1 t

11.6.3 Klein–Gordon operators on Lorentzian manifolds of bounded geometry. Let .M; g/ a globally hyperbolic spacetime of bounded geometry, with

O We fix a 1-form A dx  2 BT 0 .M; h/ O respect to a reference Riemannian metric h. 1 O and consider the associated Klein–Gordon opand a real function V 2 BT00 .M; h/, O erator P as in Subsection 5.5.1. Note that P 2 Diff2b .M; h/. Let  W   ı; ıŒ† ! M the diffeomorphism in Theorem 11.10 and let us still denote by A dx  , V and P their respective pullbacks by . Then P equals 1

1

1

1

P D jht j 2 .@t iqA0 /jht j 2 .@t iqA0 /jht j 2 .@j iqAj /jht j 2 hjt k .@k iqAk /CV: ´t Setting F .t; x/ D q 0 A0 .s; x/dx, we have eiF .@t  iqA0 /eiF D @t , hence Pred D eiF P eiF is a model Klein–Gordon operator of the form (11.2). If ƒ˙ red are the spacetime covariances of the pure Hadamard state for Pred coniF structed in Theorem 11.5, then ƒ˙ D eiF ƒ˙ are the covariances of a pure red e ˙ Hadamard state for P , on   ı; ı Œ †. Pushing ƒ to M by , we obtain a pure Hadamard state for the original Klein–Gordon operator on M .

11.7 Conformal transformations The conditions in Section 11.6 are rather strong, since they imply in particular that .M; g/ has a Cauchy surface † such that the normal geodesic flow to † exists for some uniform time interval. However it is possible to greatly enlarge the class of Klein–Gordon equations which can reduced to the model case in Section 11.2. Thus, let P D .r   iqA .x//.r  iqA .x// C V .x/

11.8 Hadamard states on general spacetimes

127

be a Klein–Gordon operator on .M; g/, † be a space-like Cauchy surface for .M; g/ O is of bounded geomand hO be a reference Riemannian metric on M such that .M; h/ etry. As in Section 6.3, we consider gQ D c 2 g and PQ D c n=21 P c n=21 . One can check that if O .i/ .M; g/ Q is of bounded geometry for h; .ii/ † is of bounded geometry in .M; g/; Q O A dx  ; c 1 r cdx  2 BT 0 .M; h/; O .iii/ c 2 V 2 BT00 .M; h/; 1 O Therefore, PQ then PQ is Klein–Gordon operator on .M; g/ Q belonging to Diffb .M; h/. can be reduced to the model case, over a causally compatible neighborhood of † in M . The pure Hadamard state for PQ constructed as in Section 11.4 yields by Section 8.6 a pure Hadamard state for P .

11.7.1 Examples. As mentioned in the introduction, the above reduction can be applied for example to the Kerr or Kerr–de Sitter exterior spacetimes and the Kerr– Kruskal spacetime for A D 0, V D m2 . Other examples are cones, double cones and wedges in Minkowski spacetime. We refer the reader to [GOW, Chapter 4] for details.

11.8 Hadamard states on general spacetimes Let us now go back to the general situation, where .M; g/ is a globally hyperbolic spacetime and P a Klein–Gordon operator on .M; g/. Let us fix a space-like Cauchy surface † in .M; g/. We will prove the following theorem, which follows from a 1 construction in [GW1, Section 8.2]. The classes ‰.c/ .†/ were introduced in Section 10.2. Theorem 11.11. Let P a Klein–Gordon operator on the globally hyperbolic spacetime .M; g/ and † a space-like Cauchy surface † in .M; g/. Then: (1) there exists a Hadamard state ! for P whose Cauchy surface covariances ˙ † belong to ‰c1 .†I M2 .C//; (2) the Cauchy surface covariances ˙ † of any Hadamard state ! for P belong to ‰ 1 .†I M2 .C//. Proof. Let us first note that (2) follows from (1). Indeed, let !1 be the Hadamard state in (1) and let ! be another Hadamard state. By Corollary 8.6, ƒ˙  ƒ˙ 1 have ˙   have smooth kernels by Proposition 6.6 (2). Since smooth kernels, hence ˙ † †;1 ˙ 1 ˙ 1 †;1 2 ‰c .†I M2 .C//, we see that † 2 ‰ .†I M2 .C//.

128

11 Construction of Hadamard states by pseudodifferential calculus

It remains to prove (1). By Proposition 5.15, we can assume that M is a neighborhood U of f0g  † in R  † and g D dt 2 C ht .x/dx 2 . Let us fix an atlas fVi ; i gi 2N of † with Vi relatively compact and relatively compact open intervals Ii , i 2 N with 0 2 Ii and Ii  Vi b U . The metrics . i1 / ht can be extended to metrics hQ i t on Rd such that hQ i t 2 1 2 d d Cb1 .RI BT20 .Rd // and hQ 1 i t 2 Cb .RI BT0 .R //, where we equip R with the flat metric ı. This means that for each i 2 N the derivatives in .t; x/ of hQ i t and hQ 1 it are uniformly bounded on R  Rd . The Klein–Gordon operators i ı P ı i1 can similarly be extended as Klein–Gordon operators PQi on R  Rd which belong to Diffb .R1Cd /. P We fix apartitionof unity 1 D i 2N 2i subordinate to the cover fVi gi 2N . Note 0 ½ that if q D ½ 0 , then in view of the expression (5.31) of q† we have q† D

X

i

 i .q/i :

(11.19)

i 2N

Q Let Q ˙ i be the Cauchy surface covariances in Theorem 11.5 for Pi and the Cauchy surface ft D 0g in R  Rd . We set X ˙ D i i .Q ˙ i /i : i 2N  ˙ Q˙ By (11.19), we have C † † D q† . Moreover, †  0, since i  0. Let !U be the associated quasi-free state for P on .U; g/. By Proposition 11.1 and the covariance of the wavefront set under diffeomorphisms, we obtain that !U is a Hadamard state for P on .U; g/. Now we apply the time-slice property Proposition 5.25 and the propagation of singularity theorem to extend !U to a Hadamard state ! for P on .M; g/. Its Cauchy 1 d Q˙ surface covariances on † are of course equal to ˙ † . Since i 2 ‰b .R I M2 .C//, ˙ 1 1 we obtain that † 2 ‰c .†I M2 .C//, by the definition of ‰c .†/. This completes the proof of (1). 

Chapter 12

Analytic Hadamard states and Wick rotation In Minkowski spacetime the Wick rotation consists in the substitution t 7! is. The Minkowski space R1;d becomes the Euclidean space R1Cd and the wave operator  becomes the Laplacian . Being elliptic, the operator  C m2 has a unique inverse GE , given by ˆ GE v.s; / D GE .s  s 0 /v.s 0 ; /ds 0 ; R

with

GE .s/ D .2/1 .es .s/ C es .s//; 1

where we recall that  D .x C m2 / 2 . A remarkable fact is that i1 GE .it/ D GF .t/; where, see (8.20), GF .t/ is the kernel of the Feynman inverse associated to the vacuum state for Cm2 . The Wick rotation or Euclidean approach is particularly important when one tries to construct interacting field theories. It is the basis of the constructive field theory, whose most celebrated achievements are the rigorous constructions of the P .'/2 and '34 theories. We refer the reader to the books of Glimm and Jaffe [GJ] and Simon [Si], or to [DG, Chap. 21], for a detailed exposition. In the Euclidean approach the main goal is the construction of an ‘interacting’ probability measure on a path space, or the construction of its moments, which are called Schwinger functions. The return to the Lorentzian world can be done by ‘reconstruction theorems’, like the Osterwalder-Schrader theorem. This step is actually often forgotten, to such an extent that physicists speaking of quantum field theories often have in mind their Euclidean versions. It is clear that the Wick rotation can be defined if we replace R1;d by an ultra1 static spacetime, see Section 5.3, if we set  D .h C m2 / 2 . Static spacetimes are reduced to ultra-static ones by the procedure explained in Section 9.4 and with some more effort stationary spacetimes can be treated as well, see [G2]. For general spacetimes, its not clear what the Wick rotation should mean, since there is no canonical time coordinate. In this chapter we will explain a result of [GW5], where the Wick rotation is performed in the Gaussian time coordinate near a Cauchy surface of .M; g/. To the elliptic operator obtained by Wick rotation one can associate the so-called Calder´on projectors, which are a standard tool in elliptic boundary value problems. It turns out that it is possible to use the Calder´on projectors to define a pure quasi-free state for a Klein–Gordon operator on .M; g/. This state has the important property of being an analytic Hadamard state. As a consequence, it satisfies the Reeh–Schlieder property.

130

12 Analytic Hadamard states and Wick rotation

12.1 Boundary values of holomorphic functions Let us recall the well-known definition of distributions as boundary values of holomorphic functions.

12.1.1 Notation. We first introduce some notation.  A cone of vertex 0 in Rn , which is convex open and proper, will be simply called a convex open cone. If ; 0 are two cones of vertex 0 in Rn we write 0 b if . 0 \ Sn1 / b . \ Sn1 /.  We recall that ı denotes the polar of , see (8.16). ı is a closed convex cone.  Let  Rn be open and let  Rn be a convex open cone. Then a domain D  Cn is called a tuboid of profile C i if (1) D  C i , (2) for any x0 2 and any subcone 0 b there exists a neighborhood 0 of x0 in and r > 0 such that

0 C ify 2 0 W 0 < jyj  rg  D:  If D  Cn is open, we denote by O .D/ the space of holomorphic functions in D.  We write F 2 Otemp . C i 0/ and say that F is temperate, if F 2 O .D/ for some tuboid D of profile C i and if for any K b , any subcone 0 b , there exist C; r > 0 and N 2 N such that K C ify 2 0 W 0 < jyj  rg  D and jF .x C iy/j  C jyjN ;

x 2 K; y 2 0 ; 0 < jyj  r:

(12.1)

12.1.2 Boundary values of holomorphic functions. If F 2 Otemp . Ci 0/ the limit

lim F .x C iy/ D f .x/ exists in D 0 . /;

 0 3y!0

(12.2)

for any 0 b and is denoted by F .x C i 0/, (see e.g. [Ko, Theorem 3.6]). S ı n If 1 ; : : : ; N are convex open cones such that N 1 i D R , then any u 2 0 D . / can be written as u.x/ D

N X

Fj .x C i j 0/;

(12.3)

j D1

for some Fj 2 Otemp . Ci j 0/. This fact comes from the construction of a so-called decomposition of ı, see e.g. [H1, Theorem 8.4.11]. If n D 1 this is simply the identity ı.x/ D .2i/1 ..x C i0/1  .x  i0/1 /.

12.2 The analytic wavefront set

131

The non-uniqueness of the decomposition (12.3) is described by Martineau’s edge of the wedge theorem, which states that N X

Fj .x C i j 0/ D 0 in D 0 . /

j D1

for Fj 2 Otemp . C i j 0/ iff there exist Hj k 2 Otemp . C i j k 0/, with j k D . j C k /conv (Aconv denotes the convex hull of A) such that X Hj k in C i j ; Hj k D Hkj in j k ; Fj D k

see for example [Ko, Theorem 3.9].

12.1.3 Partial boundary values. One can also obtain distributions as boundary values of partially holomorphic distributions in one variable, as in Proposition 7.5. Let us assume that D I  Y , where I  R is an open interval and Y  Rn1 is open, writing x 2 as .t; y/. We denote by Otemp .I ˙ i0I D 0 .Y // the space of temperate D 0 .Y /-valued holomorphic functions on some tuboid D of profile I ˙ i0. This means that for each K b I there exist r > 0 and N 2 N, such that for each bounded set B  D .Y / there exist CB > 0 such that sup jhu.z; /; './iY j  CB jIm zjN ;

Re z 2 K; ˙Im z > 0; jIm zj  r;

'2B

where h; iY is the duality bracket between D 0 .Y / and D .Y /. Let us set 'z .s/ D .s  z/1 for z 2 C n R. If u 2 D 0 .R  Rn1 / has compact support, then 1 h'z ./; F .; y/iR F .z; y/ D 2i belongs to Otemp .R ˙ i0I D 0 .Rn1 // and u.s; y/ D F .s C i0; y/  F .s  i0; y/; where F .s ˙ i0; y/ D lim!0˙ F .s C i; y/ in D 0 .R  Rn1 /.

12.2 The analytic wavefront set We now recall the definition of the analytic wavefront set of a distribution on Rn originally due to Bros and Iagolnitzer [BI], following [Sj]. We set 

'z .x/ D e 2 .zx/ ; 2

z 2 Cn ; x 2 Rn ;   1:

132

12 Analytic Hadamard states and Wick rotation

Definition 12.1. Let  Rn be an open set. A point .x0 ; 0 / 2 T  n o does not belong to the analytic wavefront set WFa u of u 2 D 0 . / if there exist a cutoff function  2 C01 . / with  D 1 near x0 , a neighborhood W of x0  i0 in Cn , and constants C;  > 0 such that 

2 /

jhuj'z ij  C e 2 ..Im z/

;

z 2 W;   1;

(12.4)

where hji is the duality bracket between D 0 .Rn / and C01 .Rn /. Note that in Definition 12.1 one identifies Rn with .Rn /0 using the quadratic form x x appearing in the definition of 'z . If u 2 E 0 .Rn /, the holomorphic function Cn 3 z 7! T u.z/ D huj'z i is called the F.B.I. transform of u. The C 1 wavefront set WFu can also be characterized by the F.B.I. transform, if one requires instead of (12.4) that 

2

jhuj'z ij  CN e 2 .Im z/ N ;

z 2 W;   1; N 2 N;

(12.5)

see e.g. [De, Corollary 1.4]. The projection of WFa u on Rn is equal to the analytic singular support singsuppa u. The analytic wavefront set is covariant under analytic diffeomorphisms, which allows to extend its definition to distributions on a real analytic manifold M in the usual way. There is an equivalent definition of WFa u based on the representation of a distribution as sum of boundary values of temperate holomorphic functions. The equivalence of the two definitions was shown by Bony [Bo], who also showed the equivalence with a third definition due to H¨ormander, see [H1, Definition 8.4.3]. Definition 12.2. Let u 2 D 0 . / for  Rn open and .x 0 ;  0 / 2  Rn nf0g. Then .x 0 ;  0 / does not belong to WFa u if there exist N 2 N, a neighborhood 0 of x 0 in

, and convex open cones j , 1  j  N , such that u.x/ D

N X

Fj .x C i j 0/ over 0 ;

j D1

for Fj 2 Otemp . 0 C i j 0/, 1  j  N , and Fj holomorphic near x 0 if  0 2 jı . Theorem 7.6 extends to the analytic wavefront set, at least when one considers differential operators. For completeness let us state this extension, see (see [Kw, Theorem 3.3’] or [H5, Theorems 5.1, 7.1]). Theorem 12.3. Let X be a real analytic manifold and P 2 Diffm .X / be an analytic differential operator of order m. Then for u 2 D 0 .X / we have

12.3 Analytic Hadamard states

133

(1) WFa .u/ n WFa .P u/  Char.P / (microlocal ellipticity), (2) If P is of real principal type with @ pm .x; / ¤ 0 on Char.P /, then WFa .u/ n WFa .P u/ is invariant under the flow of Hp (propagation of singularities). The analytic wavefront set of a distribution has deep relations with its support. An example of such a relation is the Kashiwara–Kawai theorem, which we now explain. If F  M is a closed set, the normal set N.F /  T  M n o is the set of .x 0 ;  0 / such that x 0 2 F ,  0 ¤ 0, and there exists a real function f 2 C 2 .M / such that df .x 0 / D  0 or df .x 0 / D  0 and F  fx W f .x/  f .x 0 /g. Note that N.F /   T@F M and N.F / D N  .@F / if @F is a smooth hypersurface. The Kashiwara–Kawai theorem (see e.g. [H2, Theorem 8.5.60 ]) states that N.supp u/  WFa .u/;

8 u 2 D 0 .M /:

(12.6)

We end this section by stating the analog of Proposition 7.5 for the analytic wavefront set, which is proved in [K, Theorem 4.3.10]. Proposition 12.4. Let F 2 Otemp .I ˙ i0I D 0 .Y //. Then WFa .F .t ˙ i0; y//  f  0g:

12.3 Analytic Hadamard states A spacetime .M; g/ is called analytic if M is a real analytic manifold and g is an analytic Lorentzian metric on M . Similarly, a Klein–Gordon operator P as in Subsection 5.5.1 is analytic if .M; g/ and A dx  ; V are analytic. In [SVW] Strohmaier, Verch and Wollenberg introduced the notion of analytic Hadamard states, obtained from Definition 8.3 by replacing the C 1 wavefront set WF by the analytic wavefront set WFa . Definition 12.5. A quasi-free state for P is an analytic Hadamard state if its spacetime covariances ƒ˙ satisfy WFa .ƒ˙ /0  N ˙  N ˙ :

(12.7)

Note that in [SVW] the analytic Hadamard condition is defined also for more general states for P by extending the microlocal spectrum condition of Brunetti, Fredenhagen and K¨ohler [BFK] on the n-point functions to the analytic case. It is quite likely that the results of Section 7.4 on distinguished parametrices for Klein–Gordon operators extend to the analytic setting, although we do not know a published reference. We content ourselves with stating the extension of Corollary 8.6, see [GW5, Proposition 2.8].

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12 Analytic Hadamard states and Wick rotation

Proposition 12.6. Let ƒ˙ i , i D 1; 2 be the spacetime covariances of two analytic ˙ Hadamard states !i . Then ƒ˙ 1  ƒ2 have analytic kernels. C C ˙   Proof. Let R˙ D ƒ˙ 1  ƒ2 . Since ƒ1  ƒ1 D ƒ2  ƒ2 D iG, we have RC D R . On the other hand, from (12.7) we have WFa .R˙ /0  N ˙  N ˙ , hence WFa .RC/0 \ WFa .R /0 D ;. Since R D RC , this implies that WFa .R˙/0 D ;,  and so R˙ have analytic kernels.

12.4 The Reeh–Schlieder property of analytic Hadamard states An important property of analytic Hadamard states, proved in [SVW], is that they satisfy the Reeh–Schlieder property. The Reeh–Schlieder property of a state has important consequences. For example, it allows us to apply the Tomita–Takesaki modular theory to the local von Neumann algebras associated to a bounded region O  M . We start with a lemma, related to a result of Strohmaier, Verch and Wollenberg, see [SVW, Propositions 2.2, 2.6]. Note that the notion of Hilbert space valued distributions, used in [SVW], is not necessary. We first recall some notation. If ƒ˙ are the spacetime covariances of a Hadamard state for P , we denote by cpl Y the completion of Y D C01 .M / with respect to the scalar product .f jg/! D .f jƒC g/M C .f jƒ g/M . Note that ƒ˙ extend as bounded, positive sesquilinear forms on Y cpl , still denoted by ƒ˙ . As in Section 6.2 we set for u 2 Y cpl wu˙ .f / D uƒ˙ f;

f 2 C01 .M /;

and we recall that wu˙ 2 D 0 .M / and 1

1

jwu˙ .f /j  .uƒ˙ u/ 2 .f ƒ˙ f / 2 :

(12.8)

Lemma 12.7. Let X0 D .x0 ; 0 / 2 T  Rn n o. Then for any u 2 Y cpl one has X0 2 WF.a/ .wu˙ / H) .X0 ; X0 / 2 WF.a/ .ƒ˙ /0 : Proof. We can assume that M D Rn . Let X0 D .x0 ; 0 / 2 T  Rn no with .X0 ; X0 / 62 WFa .ƒ˙ /0 . By (12.8), we have for  2 C01 .Rn /

12 jwu˙ .'z /j  C 'z  ƒ˙ 'z ; 'z  ƒ˙ 'z D hƒ˙ j'z ˝ 'z i:

(12.9)

12.5 Existence of analytic Hadamard states

135

 Note that 'z1 ˝ 'z2 D '.z , with the obvious notation. Since .X0 ; X0 / 62 1 ;z2 / ˙ 0 WFa .ƒ / we can, by Definition 12.1, find  equal to 1 near x0 , a neighborhood W of 0 in Cn , and C;  > 0 such that 

hƒ˙ j'z ˝ 'z i  C e 2 ..Im z/

2 C.Im z/2 

/:

By (12.9), this implies that X0 62 WFa .wu˙ /. Using (12.5) one obtains the same result  for the C 1 wavefront set. Theorem 12.8. Let P an analytic Klein–Gordon operator on .M; g/ and ! a pure, analytic Hadamard state for P . Then ! satisfies the Reeh–Schlieder property, i.e. if .H! ; ! ; ! / is the GNS triple of ! and O  M is an open set, the space VectfW! .u/ ! W u 2 C01 .O/g is dense in H! . C01 .O/ . Let P C01 .O/ 1 0, 8f 2 C0 .O/, i.e. supp wu˙  hence N.supp wu˙ /  N.supp wu˙ / 

Proof. We will apply Proposition 4.23 for Y D

C01 .M / P C01 .M /

and Y1 D

u 2 Y cpl such that u  ƒC f D u  ƒ f D M n O. By (12.6), N.supp wu˙ /  WFa .wu˙ /, WFa .ƒ˙ /0 . This contradicts the fact that ! is an analytic Hadamard state, since it is impossible that both .x; / and .x; / belong to N C or to N  . Therefore, @ supp wu˙ D ;, i.e. w ˙ D 0. This implies that u is orthogonal to C01 .M / for .j/! , hence u D 0.  Remark 12.9. Note that much weaker conditions than the Hadamard property of ! are sufficient to ensure that the Reeh–Schlieder property holds: it suffices that if .X; X / 2 WFa .ƒ˙ /0 , then .X; X / 62 WFa .ƒ˙ /0 , where X D .x; / if X D .x; /.

12.5 Existence of analytic Hadamard states The question of the existence of analytic Hadamard states cannot be settled as easily as in the C 1 case. In fact, the deformation argument of Fulling, Narcowich and Wald presented in Section 8.9 relies on cutoff functions, and hence does not apply in the analytic case. Strohmaier, Verch and Wollenberg [SVW, Theorem 6.3] proved that if .M; g/ is stationary, then the vacuum and thermal states associated to the group of Killing isometries are analytic Hadamard states. The following theorem, which essentially settles the existence question, is proved in [GW5] using a general Wick rotation argument. Theorem 12.10. Let .M; g/ be an analytic, globally hyperbolic spacetime having an analytic Cauchy surface. Let P be an analytic Klein–Gordon operator on .M; g/. Then there exists a pure analytic Hadamard state for P .

136

12 Analytic Hadamard states and Wick rotation

12.6 Wick rotation on analytic spacetimes Let .M; g/ be an analytic, globally hyperbolic spacetime and assume that † admits an analytic, space-like Cauchy surface. Let P an analytic Klein–Gordon operator on M . Clearly the diffeomorphism  W U ! V in Proposition 5.15 given by Gaussian normal coordinates to † is analytic. We have  g D dt 2 C h.t; x/dx 2 , where h.t; x/dx 2 is a t-dependent Riemannian metric on †, analytic in .t; x/ on U . One can moreover ensure, after an analytic conformal transformation, that the Riemannian manifold .†; h.0; x/dx 2 / is complete, see [GW5, Section 3.1]. After conjugation by an analytic function of the form eiF , see Subsection 11.6.3, the pullback of P to U can be reduced to a model Klein–Gordon operator P D @2t C r.t; x/@t C a.t; x; @x /; as in Section 11.2.

12.6.1 The Wick rotated operator. The function t 7! r.t; / and the differential operator t 7! a.t; x; @x / extend holomorphically in t in a neighborhood W of f0g  † in C  †. Therefore, there exists a neighborhood V of f0g  † in R  † on which the Wick rotated operator K D @2s  ir.is; x/@s C a.is; x; @x /

(12.10)

obtained from P by the substitution t D is is well defined and analytic in .s; x/ on V . Shrinking V we can assume that V is invariant under the reflection .s; x/ 7! .s; x/. We have pr .K/ D 2 C  h.is; x/, hence after further shrinking V , we can also assume that K is elliptic on V . Note that for the moment K has no realization as an unbounded operator. To fix such a realization, one introduces Dirichlet boundary conditions on the boundary of some open set  V . The natural way to do this is by sesquilinear form arguments. O x/ D .h.is; x/ h.is; x// 21 , which is positive definite, and denote Namely, we set h.s; O x/j 21 dxds/. Similarly, we denote by L2 .†I C2 / the by L2 . / the space L2 . ; jh.s; 1 space L2 .†; jh.0; x/j 2 dxI C2 /. 1 Let H0 . / be the closure of C01 . / with respect to the norm ˆ   1 2 j@s uj2 C @j uhj0k @k u C juj2 jh.0; x/j 2 dxds; kukH 1 ./ D 

and let

Q .v; u/ D .vjKu/L2./ ; Dom Q D C01 . /: One can show, see [GW5, Proposition 3.2], that one can choose close enough to f0g  † so that Q is closeable on C01 . / and its closure Q is sectorial with domain H01 . /, see [Ka, Chapter 6] for terminology. One denotes by K the closed operator associated to Q . One can show that 0 62 .K/ if is close ´ a enough to f0g†. ´This is deduced from the one-dimensional  2 a Poincar´e inequality a j@s uj2 ds  . 2a / a juj2 ds.

12.7 The Calder´on projectors

137

12.7 The Calder´on projectors The Calder´on projectors are a well-known tool in the theory of elliptic boundary value problems. Let us first explain this in an informal way. Let X a smooth manifold and  X an open set with smooth boundary. If F .X /  D 0 .X / is a space of distributions, we denote by F . /  D 0 . / the space of restrictions to of elements in F .X /. So, for example, D 0 . / is the space of extendable distributions on and any u 2 D 0 . / has an extension eu with eu D 0 in X n cl . Now let K be an elliptic, second-order differential operator on X . Let us assume that K has some realization as an unbounded operator, still denoted by K with 0 62

.K/. Set C D and  D X n cl . If u 2 D 0 . ˙ / satisfies Ku D 0 in ˙ , then its trace   u@ ˙  uD 2 D 0 .@ I C2 / @ u@ is well defined, where @ is some fixed transverse vector field to @ . Let n o Z ˙ D f 2 D 0 .@ I C2 / W f D  ˙ u; for some u 2 D 0 . ˙ /; Ku D 0 : Then Z C ; Z  are complementary subspaces in D 0 .@ /. The Calder´on projectors C˙ are the projectors on Z ˙ along Z  . Let us assume for example that X D Rs  S , where .S; h/ is a compact Riemannian manifold, K D @2s h Cm2 and ˙ D R˙ S . Then if u 2 D 0 . ˙ / satisfies Ku D 0 in ˙ we haveu.s;/ D es v./ for v 2 D 0 .S / and  2 D h C m2 . v Further, we have  ˙ u D ˙v and an easy computation shows that   1 ½ ˙ 1 ˙ ; C D ½ 2 ˙ which are exactly the projections c ˙ in (4.46) associated to the vacuum state for the ultra-static spacetime .Rt  S; g/, g D dt 2 C h.x/dx 2 and the Klein–Gordon operator g C m2 . We now define the Calder´on projectors in our concrete situation. We take ˙ D

\ f˙s > 0g, set   u† ; u 2 C 1 . /; u D @s u† and denote by  ˙ the analogous trace operators defined on C 1 . ˙ /. Let also   f0   f D ı.s/ ˝ f0 C ı0 .s/ ˝ f1 ; f D 2 C01 .†/2 ; f1 which is the formal adjoint of  W L2 . / ! L2 .†I C2 /, and   2i@t d.0; y/ ½ ; SD ½ 0 where d.t; y/ D jh.t; x/j1=4 jh.0; x/j1=4.

138

12 Analytic Hadamard states and Wick rotation

Definition 12.11. The Calder´on projectors for K are the operators 1  C˙ D  ˙ K  S:

Note that it is not a priori clear that C˙ are well defined, even as maps from C01 .†/2 to D 0 .†/2 . Despite their name, it is even less clear whether C˙ are projectors on suitable spaces. The first issue is fixed by the following result from [GW5], which is well known if † D @ ˙ is compact. Proposition 12.12. The maps C˙ belong to ‰ 1 .†I M2 .C//. In particular, they are well defined from C01 .†I C2 / to C 1 .†I C2 /.

12.8 The Hadamard state associated to Calder´on projectors  We recall that q D

0

½

½ 0

 .

˙ ˙ Theorem 12.13. Let ˙ Wick D ˙q ı C . Then Wick are the Cauchy surface covariances on † of a pure analytic Hadamard state !Wick for P .

The proof that ˙ Wick are the covariances of a quasi-free state is rather technical. It  relies on various integration by parts formulas and also on the fact that K CK  0. This positivity is a version of reflection positivity in this context. The proof of the purity of !Wick is also quite delicate, since to show that C˙ are projections, one has to give a meaning to C˙ ı C˙ , which seems difficult in this very general situation. One has to use the characterization of quasi-free states in Proposition 4.22 and an approximation argument, see [GW5, Chapter 4]. The essential ingredient for establishing the analytic Hadamard property of !Wick is the following proposition, whose proof is sketched below. Proposition 12.14. WFa .U† C˙ f /  N ˙ ; 8f 2 E 0 .†/2 : Proof. We prove the result for the C case. Let us set 1  v D K  Sf; g D  C v D CC f; u D U† CC f;

where U† is the Cauchy evolution operator for P . Let us assume for simplicity that P is defined and analytic in I  † for I 3 0 an open interval, and that it extends holomorphically in t to .I  iI /  †. This can easily be ensured by a localization

12.8 The Hadamard state associated to Calder´on projectors

139

argument. Writing z D t C is, we denote the holomorphic extension of P by Pz , and hence P by Pt and K by Pis . We set also I r=l D I \ f˙t > 0g; I ˙ D I \ f˙s > 0g; D D I  iI; D C D I  iI C ; D r=l D I r=l  iI: Step 1: we can write v as: v.s; y/ D v r .is C 0; y/  v l .is  0; y/; with v r=l 2 Otemp .D r=l I D 0 .†//. We have Pis v D ı.s/ ˝ h0 .x/ C ı0 .s/ ˝ h1 .x/ on I  †: 1

1 1  , this implies that Pz v r=l D w in D r=l  †, Using that ı.s/ D 2i s C i0 s  i0 where 1 1 ˝ h1 .x/ C r.z; x/; w.z; y/ D ˝ h0 .x/ C 2z 2iz 2 and r.z; x/ 2 O .DI D 0 .†//. Note that w 2 Otemp .D C I D 0 .†//. We now define distributions ur=l .t; x/ on I r=l  † by ur=l .t; x/ D v r=l .t C i0; y/; so that Pt ur=l .t; x/ D Pz v r=l .t C i0; x/ D w.t C i0; x/. In Fig. 5 below we explain the relation between v, v r=l and ur=l , the arrows corresponding to boundary values.

is vl ul

v

†

0

vr ur

t

0 Fig. 5.

Since Pt is hyperbolic with respect to dt, we can extend ur=l as uQ r=l 2 D 0 .I  †/ with Pt uQ r=l .t; x/ D w.t C i0; x/; uQ r=l .t; x/ D ur=l .t; x/ in I r=l : By Proposition 12.4, WFa .w.t C i0; x//  f  0g and WFa ur=l  f  0g over I r=l  †, and so by Theorem 12.3 we know that WFa uQ r=l  f  0g over I  †.

140

12 Analytic Hadamard states and Wick rotation

One can then deduce from Martineau’s edge of the wedge theorem that there exist vQ r=l .z; x/ 2 Otemp .D C I D 0 .†// such that uQ r=l .t; x/ D vQ r=l .t C i0; x/, Pz vQ r=l D w and vQ r=l .z; x/ D v r=l .z; x/ for z 2 D C \ D r=l . Now let v.z; Q x/ D vQ r .z; x/  vQ l .z; x/ 2 Otemp .D C I D 0 .†// and uQ D v.t Q C i0; x/. Q  f  0g, and so WFa .u/ Q  NC We have Pz vQ D 0 hence Pt uQ D 0 and WFa .u/ by microlocal ellipticity. It remains to check that uQ D U† C!C f or, equivalently, that %† u D  C v, which will complete the proof of the proposition. Note that since v.z; Q x/ D vQ r .z; x/  vQ l .z; x/, we have v.s; x/ D v.is; Q x/ for s > 0. If we were allowed to take directly the limit s ! 0C , this would imply that u.0; Q x/ D lims!0C v.is; Q x/ D lims!0C v.0; x/, and similarly i1 @t u.0; Q x/ D lims!0C ps v.0; x/ i.e. %† uQ D  C v D CC f . To justify this computation we use the fact that uQ 2 C 1 .I I D 0 .†//, which in turn follows from the fact that Pt uQ D 0. If ' 2 C01 .†/, then we have hu.t; Q /j'i D Q C i; /j'i in D 0 .I /. Since hu.t; Q /j'i 2 C 1 .I /, we actually have lim!0C hv.t hu.t; Q /j'i D lim!0C hv.t Q C i; /j'i in C 1 .I /, which justifies the above computation. 

12.9 Examples We conclude this chapter by giving some explicit examples of Calder´on projectors and of the quasi-free state they generate in the ultra-static case. We have then P D @2t C  2 ;

1

K D @2s C  2 ; for  D .h C m2 / 2 :

One can realize K as a selfadjoint operators in various ways. Let us list a few examples.

12.9.1 Boundary conditions at infinity. Let K1 the natural selfadjoint realization of K on L2 .R/ ˝ L2 .†/. We saw in Section 12.7 that the associated Calder´on projectors for C D RC  † are   1 ½ ˙ 1 ˙ C1 D ; ½ 2 ˙ and the associated state is the vacuum !vac for P .

12.9.2 Dirichlet boundary conditions. Let now KT be the selfadjoint realization of K on L2 .   T; T Œ / ˝ L2 .†/ with Dirichlet boundary conditions on s D ˙T . We can easily compute KT1 , namely KT1 v D u  r, where ˆ C1

0 0 1 .s  s 0 /e.ss / C .s 0  s/e.ss / v.s 0 /ds 0 ; u.s/ D .2/ 1

12.9 Examples

141

and

1

r.s/ D .2/1 e4T   1 e.2T s/ v C  es v C  es v  C e.sC2T / v  ; ˆ T 0 ˙ e˙s  v.s 0 /ds 0 : v D T

Taking C D 0; T Œ †, the Calder´on projectors are   1 ½ ˙ 1 th.T / CT˙ D : ½ 2 ˙ coth.T /

(12.11)

The associated state is a pure Hadamard state for P . If m D 0 the infrared singularity at  D 0 is smoothed out by the Dirichlet boundary condition. When T ! 1, CT˙ ˙ converge to C1 .

12.9.3 ˇ-periodic boundary conditions. Let Sˇ D   ˇ=2; ˇ=2 Œ with endper

points identified be the circle of length ˇ and Kˇ be the ˇ-periodic realization of K per on L2 .Sˇ / ˝ L2 .†/. The kernel of .Kˇ /1 has the following well-known expression:

1 es C e.sˇ / per ; s 2  0; ˇ Œ; .s/ D Kˇ 2.1  eˇ  / extended to s 2 R by ˇ-periodicity. Let us take C D  0; ˇ=2 Œ. Since @ C D f0g  † [ fˇ=2g  †, we can identify C01 .@ C I C2 / with C01 .†I C2 / ˚ C01.†I C2 / by writing f 2 C01 .@ C I C2 / as f D f .0/ ˚ f .ˇ=2/ for f .i / 2 C01 .†I C2 /. We set

T f .0/ ˚ f .ˇ=2/ D f .ˇ=2/ ˚ f .0/ and denote by d the operator  ˚ . Then an easy computation shows that the Calder´on projectors are:

Cˇ˙





1 0 ½ ˙d1 coth ˇ2 d C T sh1 ˇ2 d 1@ A:



D 2 ˙d coth ˇ d  T sh1 ˇ d ½ 2

2

Since @ C consists of two copies of †, the projections Cˇ˙ are associated to a pure quasi-free state on the doubled phase space .Yd ; qd / obtained from .Y ; q/ D .C01 .†I C2 /; q/, see Subsection 4.8.4. If we restrict this state to CCR.Y ; q/, we obtain the thermal state !ˇ at temperature ˇ 1 for P , see Subsection 4.10.3.

Chapter 13

Hadamard states and characteristic Cauchy problem In this chapter we describe a different construction of Hadamard states which relies on the use of characteristic cones and is due to Moretti [Mo1, Mo2]. The original motivation was to construct a canonical Hadamard state on spacetimes with some asymptotic symmetries. The class of spacetimes considered are those that are asymptotically flat at past (or future) null infinity. After a conformal transformation, the original spacetime .M; g/ can be regarded as the interior of a future light cone I  , called the past null infinity in some larger space time .MQ ; g/, Q where gQ D 2 g in M . Since I  is a null hypersurface, any normal vector field to I  is also tangent to I  , so the trace on I  of a solution  2 Solsc .P / of the Klein–Gordon equation in M consists of a single scalar function. The symplectic form on Solsc .P / induces a boundary symplectic form qI  on a space HI  of scalar functions on I  . One can use this boundary symplectic space as a new phase space and a quasi-free state !I  on CCRpol .HI  ; qI  / induces a quasi-free state ! on CCR.P /. The Hadamard condition for ! is rather easy to characterize in terms of !I  , since the covariances of !I  are simply scalar distributions, and not 2  2 matrices as in the case of a space-like Cauchy surface † considered in Chapter 11. The past null infinity in an asymptotically flat spacetime .M; g/ is traditionally denoted by I  and the metric gQ and conformal factor induce on I  a conformal frame, consisting of a degenerate Riemannian metric hQ on I  and a vector field n. The group of diffeomorphisms of I  leaving the set of conformal frames invariant is called the .Bondi–Metzner–Sachs/ BMS group, which is interpreted as the group of asymptotic symmetries of M at past null infinity. At the end of this chapter we give a short description of these objects. The BMS group GBMS acts on HI  by symplectic transformations, and a natural state on I  should be invariant under the action of GBMS . We will describe the construction of this state due to Moretti [Mo1].

13.1 Klein–Gordon fields inside future lightcones 13.1.1 Future lightcones. Let .M; g/ a globally hyperbolic spacetime and p 2 M a base point. It is known, see [W1, Section 8.1], that on any spacetime M , IC .p/ is open with IC .p/cl D JC .p/cl , @IC .p/ D @JC .p/. Moreover, any causal curve from p to q 2 @IC .p/ must be a null geodesic. Since .M; g/ is globally hyperbolic, JC .p/ is closed, see [BGP, Appendix A.5], hence IC .p/cl D JC .p/.

144

13 Hadamard states and characteristic Cauchy problem

We set

M0 D IC .p/;

C D @IC .p/ n fpg;

(13.1)

so C is the future lightcone from p, with its tip p removed and M0 is the interior of C . The following results on the causal structure of M0 are due to Moretti [Mo1, Theorem 4.1] and [Mo2, Lemma 4.3]. Proposition 13.1. The spacetime .M; g0 / is globally hyperbolic. Moreover JCM0 .K/ D JCM .K/;

JM0 .K/ D JM .K/ \ M0 ; 8K  M0 :

(13.2)

Proposition 13.2. Let K b M0 . Then there exists a neighborhood U of p in M such that no null geodesic starting from K intersects C cl [ U .

13.1.2 Klein–Gordon fields in M0 . Let P D P .x; @x / be a Klein–Gordon operator in M , Gret=adv its retarded/advanced inverses and P0 D P0 .x; @x / the restriction of P to M0 . From Proposition 13.1 we obtain immediately that the retarded/advanced inverses Gret=adv;0 of P0 are the restrictions of Gret=adv to M0 and hence G0 D GM0 M0 ; where G; G0 are the Pauli–Jordan functions for P; P0 .

13.1.3 Null coordinates near C . Clearly, the cone C will in general not be an embedded submanifold of M , due to the possible presence of caustics. Let us introduce some assumptions from [GW2], which avoid this problem and are a version of the notion of asymptotic flatness (with past time infinity). We will come back to this notion in Section 13.5. We assume that there exists a function f 2 C 1 .M / such that .1/ C  f 1 .f0g/;

ra f ¤ 0 on C;

ra f .p/ D 0;

ra rb f .p/ D 2gab .p/;

.2/ the vector field r f is complete on C: a

(13.3) It follows that C is a smooth hypersurface, although C cl is not. Moreover, since C is a null hypersurface, r a f is tangent to C . To construct null coordinates near C , one needs to fix a compact submanifold S  C , of codimension 2 in M , such that r a f is transverse to S . Then S is diffeomorphic to Sn2 and C to R  Sn2 . One can then, see e.g., [GW2, Lemmas 2.5, 2.6], prove the following standard fact: Proposition 13.3. There exist a neighborhood U of C in M and a diffeomorphism  W U ! R  R  Sn2 x 7! .f .x/; s.x/; .x//

13.2 The boundary symplectic space

145

such that .1 / .r a f C / D @s ;

 1   . / g C D 2df ds C hij .s; /d i d j ; (13.4)

where hij .s; /d i d j is a smooth s-dependent Riemannian metric on Sn2 . Moreover, if hij . /d i d j is the standard metric on Sn2 one has 1

1

jhij .s; /j 2 D O.e2s.n2/ /jhij . /j 2 for s 2   1; R; R > 0:

(13.5)

The above diffeomorphism depends only on f satisfying (13.3) and on the choice of the submanifold S . Restricting  to C gives a diffeomorphism C W C ! RSn2 that is rather easy to describe: let us first fix normal coordinates .y 0 ; y/ at p such that in a neighborhood of p, C D f.y 0 /2  jyj2 D 0; y 0 > 0g. If ft gt 2R is the flow of r a f on C , we define s D s.x/ for x 2 C by x D s .x 0 / for a unique x 0 2 S . One sees that t .x 0 / ! p when t ! 1 and one defines y .t .x 0 // 2 Sn2 . .x/ D limt !1 jyj

13.1.4 Change of gauge. One can view the choice of .f; S / as the choice of a gauge. If ! 2 C 1 .M / is such that ! > 0 on C and !.p/ D 1, then f 0 D !f also satisfies (13.3). Let also S 0 be another submanifold transverse to r a f . If 0 W U 0 ! R  R  Sn2 is the corresponding diffeomorphism in Proposition 13.3 one can easily see that D .0C / ı .C /1 W .s; / 7! .s 0 .s; /; /; for some function s 0 .s; / on R  Sn2 . Explicitly, if S 0 is given in the .s; / coordinates by fs D b. /g, one has ˆ s ! 1 . ; /d : (13.6) s 0 .s; / D b. / C 0

The map is quite similar to the so-called supertranslations, see Section 13.5. If h0 .s 0 ; 0 /d 02 is the corresponding metric in (13.4), then h0 d 02 D . / hd 2 .

13.2 The boundary symplectic space Let us consider the symplectic space .Solsc .P0 /; q/. Clearly, any solution 0 2 Solsc .P0 / extends to a solution  2 Solsc .P /, hence its trace on C %C 0 D 0C ;

(13.7)

is well defined. Note that since C is null, a vector field n normal to C is also tangent to C , so @n 0C is determined by 0C .

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13 Hadamard states and characteristic Cauchy problem

We would like to introduce a boundary symplectic space .HC ; qC / of functions on C which will play the role of .C01 .†I C2 /; q/ for a Cauchy surface † in M0 and such that %C W .Solsc .P0 /; q/ ! .HC ; qC / is weakly symplectic, i.e. such that %C qC %C D q. Note that this implies that %C is injective. The map %C is sometimes called a bulk-to-boundary correspondence. The space HC should be small enough to admit interesting quasi-free states, and depend only on C , not on a particular gauge .f; S /. 1 the set of g 2 D 0 .C / such that Let us denote by Hf;S ˆ RSn2

ˇ ˇ ˇ ˛ ˇ ˇ1 ˇ@ @ g.s; /ˇ2 ˇh.s; /ˇ 2 dsd < 1; 8.˛; ˇ/ 2 Nn1 ; s

equipped with its Fr´echet space topology and 1 1 D fg 2 Hf;S Hf;S;R W supp g    1; R; R 2 Rg: 1 depends on .f; S /, but the inductive limit The space Hf;S

HC D

[

1 Hf;S;R

R2R

does not. This can be verified quite easily using (13.6) and the estimates in [GW2, Lemmas 2.7, 2.8]. Proposition 13.4. Set ˆ   1  g 1 qC g2 D i @s g 1 g2  g 1 @s g2 jh.s; /j 2 dsd ; RSn2

g1 ; g2 2 HC : (13.8)

Then: (1) qC is well defined and independent on the choice of the gauge .f; S /, (2) .HC ; qC / is a Hermitian space, (3) %C W .Solsc .P0 /; q/ ! .HC ; qC / is unitary. Proof. qC is clearly well defined on HC . Its independence on the choice of the gauge follows from the discussion of changes of gauge in Subsection 13.1.4. Let us now prove (2). We denote by m. /d 2 the canonical metric on Sn2 and set Ug.s; / D jmj1=4 jhj1=4 g ı .C /1 .s; /: (13.9) We have 1

g 1 qC g2 D i

ˆ RSn2

  1 @s Ug 1 Ug2  Ug 1 @s Ug2 jmj 2 . /dsd :

13.3 The Hadamard condition on the boundary

147

We can integrate by parts in s with no boundary terms since Ug ! 0 when s ! 1 and supp Ug    1; R and obtain that ˆ 1 g 1 qC g2 D 2i1 Ug 1 @s Ug2 jmj 2 . /dsd : (13.10) RSn2

Hence, if g 1 qC g2 D 0 for all g1 2 HC , we have @s Ug2 D 0, and so Ug2 D 0. Now let us prove (3). We first show that %C maps Solsc .P0 / into HC . Let us verify that %C W C01 .M / ! HC continuously: (13.11) This can be easily deduced from [GW2, Lemma 2.8]. Next, if 0 2 Solsc .P0 / with supp 0  J M0 .K/ for some K b M0 , then we can extend 0 as  2 Solsc .P / with supp   J.K/ and supp  \ C  .JC.K/ \ JC .p// [ J .K/ \ JC .p/. The first set is empty by Proposition 13.1, the second is compact by Lemma 5.11. Therefore, %C 0 D %C u for some u 2 C01 .M / hence belongs to HC . We now fix a Cauchy surface † in M0 and pick ´ 1 ; 2 2 Solsc .P0 /, gi D %C i . For Ja .1 ; 2 / as in 5.5.2 we have  1  q2 D † Ja .1 ; 2 /na d Volh . Applying the Gauss formula as in 5.2.4 using the coordinates .f; s; / on C , we obtain that  1  q2 D g 1 qC g2 . To justify the application of the Gauss formula to the non-smooth surface C cl , it suffices to replace C cl in an -neighborhood of p by a piece of a smooth Cauchy surface in M , the contribution of the integral on this part tends then to 0 when  ! 0. 

13.3 The Hadamard condition on the boundary Let !C a quasi-free state on CCRpol .HC ; qC /, with covariances ˙ C . We will call !C a boundary state. From Proposition 13.4 we see that !C induces a state !0 for CCR.P0 / , called the induced bulk state, by setting ˙  ƒ˙ 0 D .%C ı G0 /C .%C ı G0 /:

(13.12)

We would like to give sufficient conditions on ˙ C which ensure that the induced state !0 is a Hadamard state. Recall that we use the density d Volg to identify distributions with distributional 1 densities on M0 . Similarly, we use the density jh.s; /j 2 dsd to identify distributions with distributional densities on C . Changing the gauge .f; S / amounts to multiplying distributions on C by a smooth, non-zero function hence does not change their wavefront set. We will denote by X D .x; / resp. Y D .y; / the elements of T  X n o resp.  T C n o. If necessary, we introduce near C the local coordinates .f; s; / as in Proposition 13.3, which we will denote by .r; s; y/, the dual variables being .%; ; /.

148

13 Hadamard states and characteristic Cauchy problem

Let i  W TC M ! T  C be the pullback by the injection i W C ! M and recall that N C D .i  /1 .o/ is the conormal bundle of C in M , see Subsection 7.2.4. Recall also that N ˙  T  M n o are the two connected components of the characteristic manifold N of p. 

Lemma 13.5. Consider the function F .y; / D r a f .y/ on T  C and denote T  C ˙ D fY 2 T  C W ˙F .Y / > 0g;

T  C 0 D fY 2 T  C W F .Y / D 0g: (13.13)

Then (1) i  W TC M \ N ˙ ! T  C ˙ is bijective. (2) .i  /1 .T  C 0 / \ N D N  C . (3) For Y 2 T  C , X 2 T  M let us write Y X if Y 2 T  C ˙ and .i  /1 .Y / X . Let ; 2 C01 .M / with p 62 supp . Then WF.%C G/0  f.Y; X / W Y X; x 2 M0 g. The sets T  C ˙ , T  C 0 are clearly independent on the choice of f . Proof. Let us use the above coordinates, so that F .Y / D . By Proposition 13.3 we have p.X / D 2% C h.0; y; /; X 2 TC M: (13.14) for h.s; y; / D   h1 .s; y/ and N  C D fr D D  D 0g. The proof of (1) and (2) is then easy. Let us prove (3). Since p 62 supp , the singularity of C at p is harmless. We check that WF.%C /0 D f.Y; X / 2 T  C n o  T  M n o W Y D i  X g and know that WF. G/0  f.X1 ; X2 / W X1 X2 ; x2 2 M0 g, see Proposition 7.10. Then we apply the composition rule in Subsection 7.2.8. 

T C 0

T C C

T C 

Fig. 6

Theorem 13.6. Let ˙ C be the covariances of a boundary state !C . Assume that  W H ! H are continuous and let ˙ C C C  ˙ ƒ˙ 0 D .%C ı G0 / C .%C ı G0 /:

13.3 The Hadamard condition on the boundary

149

Then 0 (1) ƒ˙ 0 2 D .M0  M0 / are the spacetime covariances of a quasi-free state !0 for P0 . (2) Assume that

0   WF.˙  T  C D ;: C/ \ T C

(13.15)

Then the bulk state !0 is a Hadamard state for P0 . Proof. Assertion (1) follows from (13.11). The proof of assertion (2) relies on a idea due to Moretti [Mo2], which allows to ˙ avoid the difficulties caused by the tip p of C . Note first that since ˙ C D C we deduce from (13.15) that 0  ˙ WF.˙ [ T  C 0 /  .T  C ˙ [ T  C 0 /: C /  .T C

(13.16)

0 1 It clearly suffices to estimate WF.ƒ˙ 0 / for  2 C0 .M0 /. Observe first that %C G0  D %C G since G0 D GM0 M0 . By the support property of G, we can pick 2 C01 .M / such that %C G D %C G. By Proposition 13.2, we can split as 0 C 1 , where i 2 C01 .M /, 0 D 1 near p, and no null geodesics from supp  intersect C in supp 0 . Using that WF.G/0  C we obtain that 0 G W D 0 .M0 / ! C01 .M / continuously, hence

%

0 G

W D 0 .M / ! HC

 continuously, by (13.11). Since by assumption ˙ C W HC ! HC is continuous, in the ˙ definition of ƒ0 we can replace %C G by %C 1 G, modulo a smoothing operator on M0 . From Lemma 13.5 we know that 1 G/

WF.%C WF..%C

0

1 G/

 f.Y; X / W Y X; x 2 M0 g;  0

/  f.X; Y / W Y X; x 2 M0 g:

We observe that if Y X for x 2 M0 , then Y 62 T  Y 0 . Indeed, if we assume that Y 2 T  Y 0 and Y D i  X 0 for X 0 X , then necessarily X 0 2 N  C , by Lemma 13.5. Since C is null, N  C is invariant under the Hamiltonian flow of p, hence X 2 N  C and x 2 C , which is a contradiction. This implies that we can find a pseudodifferential operator Q 2 ‰c0 .C / with essential support (see Subsection 10.2.5) disjoint from T  C 0 such that %C

1 G

D Q%C

1 G

modulo a smoothing operator;

 ˙ and hence we can replace ˙ C by Q C Q with  C  T C C; WF.Q ˙ C Q/  T C

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13 Hadamard states and characteristic Cauchy problem

by (13.16). We can then apply twice the rules for composition of kernels in Subsection 7.2.8 and obtain by Lemma 13.5 that 0 ˙ WF.ƒ˙  N ˙; 0 /  N

i.e. condition (genHad) in Definition 8.4 is satisfied. By Theorem 8.5 !0 is a Hadamard  state for P0 .

13.4 Construction of pure boundary Hadamard states It is now rather easy to construct, for each given gauge .f; S /, a boundary state !C 1 which induces a Hadamard state !0 in M0 . We denote L2 .R  Sn2 I jmj 2 d ds/ 2 n2 2 n2 simply by L .R  S / and recall that the map U W HC ! L .R  S / was defined in (13.9). Theorem 13.7. Set  g1 ˙ C g2 D 2.Ug1 j½R˙ .Ds /jDs jUg2 /L2 .RSn2 / : Then the following holds: pol (1) ˙ C are the covariances of a pure quasi-free state !C on CCR .HC ; qC /. (2) !C depends on the choice of f but not of S . (3) !C induces a Hadamard state !0 in P0 . (4) Assume that dimM  4. Then the state !0 is pure. Proof. The fact that ˙ C are the covariances of a quasi-free state is obvious. To prove that !C is pure, we can apply Proposition 4.21. The completion of U HC for 1 2 n2 2 /, on which ½˙ the norm obtained from ˙ R .Ds / are C is equal to jDs j L .R  S complementary projections. This completes the proof of (1). Changing the surface S amounts to replacing s by s 0 D s b. / for some function b on Sn2 , so Ds 0 D Ds , which proves (2). Statement (3) follows from Theorem 13.6 and the fact that in the coordinates .f; / on C , T  C ˙ is given by f˙ > 0g. We refer the reader to [GW2] for details. It remains to explain the proof of (4). The fact that !C is pure does not automatically ensure that !0 is pure. To prove this one has to show, again by Proposition 4.21, 1 that U%C Solsc .P0 / is dense in jDs j 2 L2 .R  Sn2 /. This can be deduced from the solvability of the characteristic Cauchy problem  P0  D 0 in M0 ; (13.17) C D g; in energy spaces, by adapting a method due to H¨ormander [H6]. We refer again the reader to [GW2]. The restriction to n  4 comes from the use of a Hardy-type inequality on the cone C . 

13.5 Asymptotically flat spacetimes

151

13.5 Asymptotically flat spacetimes The above method of constructing a bulk Hadamard state from a boundary state was originally developed by Moretti [Mo1, Mo2] for spacetimes that are asymptotically flat at past (or future) null infinity. In this case it is important to consider only the conformal wave equation and to assume that the spacetime dimension n is equal to 4 (the value of n is important when one takes the trace of some identities between tensors). In this section we would like to explain this notion and its relationship to the previous sections. Our exposition below follows [Mo2], [DMP1] or [W1, Chapter 11], with some slight differences. For example, the conformal factor already incorporates a change of gauge ! 0 D ! such that (13.18) is satisfied. Definition 13.8. A spacetime .M; g/ is asymptotically flat at past null infinity if there exists another spacetime .MQ ; g/ Q such that:   Q (1) M  M is open, I D @M is a smooth hypersurface homeomorphic to R  S2 , (2) there exists 2 C 1 .MQ / with > 0 on M , D 0; d ¤ 0 on I  , Q

(3) gj Q M D 2 jM g and I  \ JCM .M / D ;, ea r eb D 0 on I  , (4) gQ ab r

(5) If i W I  ! MQ is the canonical injection, then ea is complete on I  ; .i/ na D r eb / D 0: ea r .ii/ i  .r

(13.18)

Q / such that conditions (2), (3), (4), (5) hold. Let us denote by M the set of .g; From conditions (2), (3) we see that if .g; Q / and .gQ 0 ; 0 / belong to M, then there 1 Q Q Moreover from exists ! 2 C .M /, ! > 0 such that 0 D ! , gQ 0 D ! 2 g. ea ! D 0 on I  , see Lemma 13.10 below. conditions (4) and (5) it follows that na r

13.5.1 Conformal frames. Let .g; Q / 2 M. The manifold I  is null for gQ

and is naturally equipped with the vector field n, which is tangent to I  and with hQ D g Q I  , which is a degenerate Riemannian metric with kernel spanned by n. Q n/ is called the conformal frame on I  associated to Definition 13.9. The pair .h; .g; Q /. The set of all conformal frames associated to elements of M is denoted by C . The above change of conformal factor ! 0 D ! is called a gauge transforQ ! 1 n/ on the associated Q n/ ! .hQ 0 ; n0 / D .! 2 h; mation and induces the change .h; conformal frames.

152

13 Hadamard states and characteristic Cauchy problem

Q n/ Lemma 13.10. (1) Let .g; Q / 2 M. Then the associated conformal frame .h; satisfies:

Ln hQ D 0; n is complete:

Q Ker h.x/ D Rn.x/; x 2 I  ;

(13.19)

Q n/; .hQ 0 ; n0 / 2 C . Then there exists ! 2 C 1 .I  / with ! > 0 and (2) Let .h; Q ! 1 n/. Ln ! D 0 such that .hQ 0 ; n0 / D .! 2 h; Proof. Let us complete x 0 D with local coordinates x i , 1  i  3, and remove the tildes to simplify notation. Then if b D i  .ri rj /, we have bij D  ij0 D  12 g 0k .@i gj k C@j gi k @k gij / since g 00 D 0 on I  . We compute the Lie derivative Ln hij D nk @k gij C @i nk gkj C gi k @i nk . Using again that g 00 D 0, we see that g 0k gkj D ıj0 D 0. Taking derivatives of this identity we obtain that bij D 12 Ln hij , which proves (1). Let us prove (2). The existence of ! 2 C 1 .I  / with ! > 0 is obvious. To show that Ln ! D 0 we compute

Ln .! 2 h/ D ! 2 Ln h C 2! Ln .!/h; L! 1 n .h/ D ! 1 Ln h C d! 1 ˝ hn C hn ˝ d! 1 ; whence

L! 1 n .! 2 h/ D ! Ln h C 2Ln !h  d ln ! ˝ hn  hn ˝ d ln !: Q n/ and .hQ 0 ; n0 / this implies that Ln ! D 0. Using (1) for .h;



Q n/ be a conformal frame and S; S 0  I  be 13.5.2 Bondi frames. Let now .h; two smooth surfaces transverse to n. Since n is complete, its flow defines a diffeomorphism S 0 S W S ! S 0 ; by identifying points in S and S 0 which are on the same integral curve of n. This Q n/. Moreover, the flow of n defines a diffeodiffeomorphism is independent on .h; morphism n;S

W Ru  S ! I  ; with hQ D hS .y/dy 2 ;

(13.20)

where hS .y/dy 2 is a Riemannian metric on S , independent on u. 1 0 1 n;S .S / D f.u; y/ W u D f .y//g for some f 2 C .S / and

We have

SD

1 n;S 0

n;S .f0g

ı

 S /;

n;S .u; y/

nD.

@ n;S / @u ;

D .u  f .y/; S 0

.

n;S /

S .y//;



.u; y/ 2 Ru  S:

(13.21)

Since I  is diffeomorphic to R  S2 , S is diffeomorphic to S2 . Let mS denote the unique Riemannian metric on S of constant Gaussian curvature equal to 1. By uniqueness, we have mS D .S 0 S / mS 0 .

13.5 Asymptotically flat spacetimes

153

Q n/ is a Bondi frame if for some .and hence Definition 13.11. A conformal frame .h; Q S D mS . for all/ surface S transverse to n one has h Lemma 13.12. The set C of conformal frames contains a unique Bondi frame .hQ B ; nB /. Q n/ 2 C and S transverse to n. After transportation by n;S , all Proof. Let us fix .h; conformal frames are of the form .!S2 hS ; !S1 @u / for some !S 2 C 1 .S /, !S > 0. It is well known that any Riemannian metric on S2 is conformal to the standard metric. This means that there is a unique such !S with !S2 hS D mS .  If we fix a transverse surface S and identify S with S2 we can introduce the socalled Bondi coordinates on I  , .u; ; '/, such that nB D @u and hQ B D d 2 C sin2 d' 2 . The existence of a unique Bondi frame implies the following rigidity result: we saw that there exists a diffeomorphism W I  ! Ru  S2 such that the natural 2 image of C under is the set of pairs .! mS2 ; ! 1 @u / for ! > 0 an arbitrary smooth function on S2 .This implies that if .Mi ; gi / i D 1; 2, are two asymptotically flat spacetimes, there exists a diffeomorphism W I1 ! I2 such that .C1 / D C2 . Another illustration of this rigidity is the fact that the BMS group defined below is independent of the asymptotically flat spacetime .M; g/.

13.5.3 The BMS group. We now recall the definition of the Bondi–Metzner– Sachs group, see e.g. [W1, Chapter 11] or [DMP1]. Its physical interpretation is the group of asymptotic symmetries of .M; g/ near past null infinity. If  W I  ! I  Q n/ by is a diffeomorphism, we let  act on .h; Q  n/: Q n/ D ..1 / h; ˛ .h; Definition 13.13. The BMS group GBMS is the group of diffeomorphisms  W I  ! I  such that ˛ .C /  C . One can associate to  2 GBMS a conformal factor ! by the rule ˛ .hQ B ; nB / D .!2 .1 / hQ B ; !1  nB /;

(13.22)

where .hQ B ; nB / is the Bondi frame. From ˛1 ı ˛2 D ˛1 ı2 we obtain the identity !1 ı2 D .!1 ı 2 /!2 :

(13.23)

It is convenient to describe the action of the BMS group in Bondi coordinates .u; ; '/ on I  associated to the Bondi frame. Let us identify S2 with C by stereographic projection: . ; '/ 7! z D ei' coth. 2 /, so that d 2 C sin2 d' 2 D 4.1 C zz/2 dzd z. Functions on C will be denoted by f .z; z/, to emphasize the fact that they do not need to be holomorphic (nor anti-holomorphic). One can prove that GBMS can

154

13 Hadamard states and characteristic Cauchy problem

be identified with the semi-direct product of SO " .1; 3/ and C 1 .S2 / as follows, see [DMP1]: " 1 Let …  W SL.2; C/! SO .1; 3/ be the covering map with … .½/ D f˙½g. For a ƒ bƒ ƒD… one sets cƒ dƒ Kƒ .z; z/ D

1 C jzj2 jaƒ z C bƒ j2 C jcƒ z C dƒ j2

and one associates to .ƒ; f / 2 SO " .1; 3/  C 1.S2 / the map:  W I  ! I  given in the Bondi coordinates fixed above by the rule .u; z; z/ 7! .u0 ; z 0 ; z 0 /; where u0 D Kƒ .z; z/.u C f .z; z// and z 0 D We have

a ƒ z C bƒ : cƒ z C dƒ

! .z; z/ D Kƒ .z; z/1 :

(13.24) (13.25)

The diffeomorphisms obtained for ƒ D ½ are called supertranslations.

13.6 The canonical symplectic space on I  Assume that .M; g/ and .MQ ; g/ Q (and hence .M; g/) Q are globally hyperbolic and the inclusion i W .M; g/ Q ! .MQ ; g/ Q is causally compatible, see Subsection 5.2.6. Let P D g C 16 Scalg , resp. PQ , be the conformal wave operator on .M; g/, resp. .MQ ; g/. Q By Proposition 6.10, the map .Solsc .P /; q/ 3  7! Q D 1  2 Solsc .PQ ; q/ Q is an injective homomorphism of pseudo-Hermitian spaces, and we can consider Q I  2 C 1 .I  /: v D  Since an element  of the BMS group corresponds to a change ! 0 D ! , we see that the natural action of  2 GBMS on functions on I  is U v D .! v/ ı 1 ;

(13.26)

and by (13.23) GBMS 3  7! U 2 L.C 1 .I  // is a group homomorphism. In analogy with Proposition 13.4, one can now equip suitable subspaces of C 1 .I  /, such as, for example, C01 .I  /, with a canonical Hermitian form. Let .hQ B ; nB / be the Bondi frame and S be transverse to nB .

13.6 The canonical symplectic space on I 

155

Definition 13.14. We set for v1 ; v2 2 C01 .I  / ˆ    v 1 qv2 D i @u w 1 w2  w 1 @u w2 du d VolmS ; RS

where

wDvı

nB ;S :

(13.27)

Proposition 13.15. (1) the Hermitian form q is independent on the choice of the transverse surface S , (2) one has .U / qU D q for  2 GBMS , i.e. GBMS acts as unitary transformations of .C01 .I C /; q/. Proof. Let us first prove (1). If S 0 is another transverse surface and w 0 D v ı then from (13.21) it follows that wj0 .u0 ; y 0 / D wj .u0 C f 0 .y 0 /; S

S 0 .y

0

//;

nB ;S 0 ,

(13.28)

and .S S 0 / mS 0 D mS , which implies (1). To prove (2), we work again with the Bondi frame .hQ B ; nB /, and identify I  with R  S using nB ;S and S with C as in Subsection 13.5.3. The charge q takes the form ˆ   4 v 1 qv2 D i dudzd z: @u w 1 w2  w 1 @u w2 .1 C zz/2 RC We equip R  C with the density 4.1 C jzj2 /2 dudzd z and denote by w 7! V w the action of  2 GBMS obtained from U and the identification (13.27). The operator Du D i1 @u is essentially selfadjoint on C01 .R  C/, and integrating by parts we obtain that v 1 qv2 D 2.w1 jDu w2 /L2 .RC/ : From (13.24) it follows, by an easy computation, that V V D Kƒ ½; V Du V D Du ;

(13.29)

where we consider V as an operator on L2 .R  C/ and Kƒ is the operator of multiplication by Kƒ .z; z/. This implies (2).  There is a considerable freedom in the choice of a symplectic space Y on which q is defined. A natural canonical choice is the space H 1 .I  / defined as the completion of 1 C0 .I  / with respect to the norm kvk21 D kwk2L2 .RS/ C k@u wk2L2 .RS/ ; where as above w D v ı

nB ;S

and R  S is equipped with the density du d VolmS .

156

13 Hadamard states and characteristic Cauchy problem

The operator Du D i1 @u acting on L2 .R  S / is essentially selfadjoint on C01 .R  S / and H 1 .I  / is the inverse image of Dom Du under the map v 7! w D v ı nB ;S . A change of transverse surface S does not change the space H 1 .I  /, but simply equips it with an equivalent norm. The group GBMS acts on .H 1 .I  /; q/ by bounded unitary transformations and q is non-degenerate on H 1 .I  /, since Du is injective.

13.6.1 The canonical quasi-free state on I  . We now describe the con-

struction of a canonical quasi-free state !I  on CCRpol .H 1 .I  /; q/, due to Moretti [Mo1]. Proposition 13.16. Let us set v 1 ƒ˙ v2 D 2.w1 j½R˙ .Du /jDu jw2 /L2 .RS/ ; for wi D vi ı

nB ;S .

vi 2 H 1 .I  /;

Then

˙

(1) ƒ are independent of the choice of the transverse surface S , (2) ƒ˙ are the covariances of a pure, quasi-free state !I  on CCRpol .H 1 .I  /; q/ which is invariant under the action of GBMS . Proof. If S; S 0 are two transverse surfaces and w D v ı nB ;S , w 0 D v ı nB ;S 0 , then w 0 D US 0 S w, where US 0 S is given in (13.28). We check that US 0 S W L2 .R  S / ! L2 .R  S 0 / is unitary with US 0 S Du US0 S D Du . This implies that ƒ˙ are independent of the choice of S . To prove (2), we use the notation in the proof of Proposition 13.15. Let S D 1

 V Kƒ 2 , which is unitary by (13.29). Since Kƒ commutes with Du we have Sƒ Du Sƒ  D Du , hence Sƒ ½R˙ .Du /jDu jSƒ D ½R˙ .Du /jDu j by functional calculus. Using again the fact that Kƒ commutes with Du , this implies that Vƒ ½R˙ .Du /jDu jVƒ D ½R˙ .Du/jDu j, i.e. that U ƒ˙ U D ƒ˙ . 

Moretti proved in [Mo1] that !I  is the unique pure quasi-free state ! on CCRpol .H 1 .I  /; q/ with the following two properties: (1) ! is invariant under GBMS , (2) if fTs gs2R  GBMS is the one-parameter subgroup of translations in u and ˛s D UTs , then ! is a non-degenerate ground state for f˛s gs2R , see Definition 9.1.

13.6.2 Construction of a quasi-free state in M . To obtain quasi-free states for P in M from states on CCRpol .H 1 .I  /; q/, %I  Solsc .P / should be contained in H 1 .I  / for %I   D . 1 /I  . If we introduce coordinates .u; y/ on I  as in Subsection 13.5.2, then it follows Q 1 .fu  CK g/ for any from Definition 13.8 (3) that J M .K/ \ I  is included in n;S  K b M , so the support of %I  for  2 Solsc .P / only extends towards 1 in the u variable.

13.6 The canonical symplectic space on I 

157

If .M; g/ is asymptotically flat with past time infinity, see [Mo2, Appendix A] for a precise definition, then u D 1 corresponds to an actual point i  of MQ , and the situation is essentially the same as the one in Section 13.1, i.e. .M; g/ is modulo a conformal transformation the interior of a smooth, future lightcone. In more complicated situations, like the Schwarzschild spacetime, see [DMP3] or cosmological spacetimes, see [DMP4], it is necessary to prove some decay estimates of %I   and its derivative in u when u ! 1 to ensure that %I  Solsc .P /  H 1 .I  /. The discussion of these estimates is beyond the scope of this survey.

Chapter 14

Klein–Gordon fields on spacetimes with Killing horizons As recalled in the Introduction, one of the most spectacular results of QFT on curved spacetimes is the Hawking effect, discovered by Hawking [Ha]. Hawking considered a Klein–Gordon field in a spacetime describing the formation of a black hole by gravitational collapse of a spherically symmetric star, the spacetime being eventually equal to the Schwarzschild spacetime in the exterior of the black hole horizon. Considering the state which in the past is the vacuum state for the region outside of the star, he gave some heuristic arguments to show that in the far future and far away from the horizon this state is a thermal state at Hawking temperature TH D .2/1 . The first complete justification of the Hawking effect is due to Bachelot [B], who considered the same situation as Hawking. Another derivation of the Hawking effect is due to Fredenhagen and Haag [FH]. They considered the same situation as Hawking and the more general case of a state for the Klein–Gordon field whose two-point function is assumed to be asymptotic to that of the vacuum at spatial infinity and of Hadamard form near the horizon. We discuss in this chapter another phenomenon related to the Hawking radiation, namely, the existence of a ‘vacuum state’ for a Klein–Gordon field on spacetimes with a bifurcate Killing horizon, see Section 14.1 for a precise definition. The existence of such a state is related to the so-called Unruh effect, [U], which we now briefly describe. In the Minkowski spacetime .R1;d ; / one considers a right wedge MC D f.t; x/ W jtj < x1 g, where x1 is a space coordinate. The spacetime .MC ; / is invariant under the Lorentz boosts with generator X D a.x1 @t C t@x1 /; where a > 0 is an arbitrary constant. Although X is not globally time-like in R1;d , it is time-like in MC and its integral curves in MC are worldlines of uniformly accelerated observers, with acceleration equal to a. Since X is time-like in MC , one can construct, for any ˇ > 0, the associated ˇ-KMS state !ˇ for the Klein–Gordon operator  C m2 restricted to MC , see Chapter 9. Unruh proved that if ˇ D .2/a1, then !ˇ is the restriction to MC of the Minkowski vacuum !vac . This result is interpreted as the fact that the Minkowski vacuum state is seen by uniformly accelerated observers with acceleration a as a thermal state at temperature a.2/1. Note that the Killing vector field X vanishes at B D ft D x1 D 0g, which is the intersection of the two null hyperplanes ft D ˙x1 g, whose union is an example of a bifurcate Killing horizon. In spacetimes with a bifurcate Killing horizon, the

160

14 Klein–Gordon fields on spacetimes with Killing horizons

existence of a state analogous to the Minkowski vacuum, called the Hartle–Hawking– Israel state, was conjectured by Hartle and Hawking [HH] and Israel [Is], using formal Wick rotation arguments. We will explain the rigorous construction of the HHI state in [G2], which is based on methods already used in Chapter 12, namely the Calder´on projectors from the theory of elliptic boundary value problems. For static Killing horizons, i.e. when X is orthogonal to some Cauchy surface in the exterior region, the HHI state was already constructed by Sanders in [S3]. The condition that the Killing vector field X generating the horizon is time-like in the exterior region excludes the physically important Kerr spacetime. In fact, applying Proposition 9.7 to the exterior region of the Kerr spacetime, we know that no KMS state for X exists in the exterior region. Much more general non-existence results on the Kerr spacetime were shown by Kay and Wald in [KW]. For example assuming the existence of some solutions of the Klein–Gordon equation exhibiting superradiance, it is shown in [KW] that there exist no X -invariant state which is Hadamard near the horizon. Therefore, it is expected that no HHI state exists in the Kerr spacetime.

14.1 Spacetimes with bifurcate Killing horizons Let .M; g/ be a globally hyperbolic spacetime with a complete Killing vector field X . We assume that B D fx 2 M W X.x/ D 0g is a compact, connected submanifold of codimension 2, called the bifurcation surface. If moreover there exists a smooth, space-like Cauchy surface † containing B , the triple .M; g; X / is called a spacetime with a bifurcate Killing horizon, see [KW, Chapter 2]. If N; w are the lapse function and shift vector field associated to X; † as in Section 9.2, the Cauchy surface † splits as † D † [ B [ †C ; †˙ D fy 2 † W ˙N.y/ > 0g; i.e. X is future/past directed on †˙ . Accordingly one can split M as M D MC [ M [ F [ P ; where the future cone F D I C .B /, the past cone P D I  .B /, and the right/left wedges M˙ D D.†˙ /, are all globally hyperbolic when equipped with g. The boundary of the future cone @F may be a black hole horizon, in which case the past cone @P is the corresponding white hole horizon. The bifurcate Killing horizon is H D @F [ @P ; and the Killing vector field X is tangent to H. In Fig. 7 below the vector field X is represented by arrows.

14.1 Spacetimes with bifurcate Killing horizons

H

H

H

F

M

161

MC

B

†

P

H Fig. 7

14.1.1 The surface gravity. An important quantity associated to the Killing horizon H is its surface gravity, defined by 1  2 D  .r b X a rb Xa /jB ; 2

 > 0:

It is a fundamental fact, see [KW, Chapter 2], that the scalar  is constant on B and actually on the whole horizon H.

14.1.2 Wedge reflection. In concrete situations, like the Schwarzschild or Kerr spacetimes, the metric g is originally defined only on the right wedge MC and first extended to the future cone F by a new choice of coordinates. The regions P , M are constructed as copies of F , MC , with reversed time orientation, glued together along B . This motivates one to assume the existence of a wedge reflection, i.e. an isometric involution R of M [ U [ MC , where U is a neighborhood of B in M , such that R reverses the time orientation, R D Id on B and R X D X . It can be shown, see [S3], that there exists a smooth, space-like Cauchy surface  ! †. The restriction r of R to † is called a weak † with B  † such that R W †  wedge reflection. We have rjB D Id;



r W †˙ ! † :

(14.1)

In the sequel we will fix such a Cauchy surface.

14.1.3 Stationary Killing horizons. The bifurcate Killing horizon H is called stationary, resp. static, if the Killing vector field X is time-like in MC , resp. timelike and orthogonal to † in MC .

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14 Klein–Gordon fields on spacetimes with Killing horizons

14.2 Klein–Gordon fields Let us consider a Klein–Gordon operator P D g C V; where V 2 C 1 .M I R/ has the same invariance properties as g, i.e. X  V D 0, V ı R D V . We also strengthen the condition V > 0 in Section 9.3 to V .x/  m2 ;

x 2 M; m > 0;

i.e. we restrict our attention to massive Klein–Gordon fields. If X is time-like in MC , we can apply Sections 9.3, 9.5 to the Klein–Gordon operator P , on the globally hyperbolic spacetime .MC ; g/, with Cauchy surface †C . We obtain, for each ˇ > 0 the ˇ-KMS state !ˇ acting on CCRpol .C01 .†C I C2 /; q/. Following Section 4.8, we can associate to !ˇ the doubled state !d , which is associated to the doubled Hermitian space .C01 .†C I C2 / ˚ C01 .†C I C2 /; q ˚ q/:

(14.2)

14.3 Wick rotation The key step in the construction of the Hartle–Hawking state is the interpretation of the double ˇ-KMS state !D using the Wick rotation in Killing time coordinates. We will now explain this important step.

14.3.1 The Wick rotated metric. As in Section 9.2, we can identify MC with

R  †C , the metric g taking the form

g D N 2 .y/dt 2 C hij .y/.dy i C w i .y/dt/.dy j C w j .y/dt/; see (9.12). As in Chapter 12, we can perform the Wick rotation, replacing the Killing time coordinate t by is. In this way we obtain from g the complex metric    g eucl D N 2 .y/ds 2 C hij .y/ dy i C iw i .y/ds dy j C iw j .y/ds : If  D .; / 2 CTy M and y 2 †C , then  g eucl .y/ D .N 2 .y/  w.y/h.y/w.y// C h.y/ Ci.w.y/h.y/ C  w.y/h.y//: Since X D @t@ is time-like in MC , we know that N 2 .y/ > w i .y/hij .y/w j .y/, from which we deduce that j Im. g eucl .y//j  c.y/Re. g eucl .y//;

y 2 †C ;

(14.3)

14.3 Wick rotation

163

for some c.y/ > 0. It is convenient to have some uniformity in y in the inequality (14.3), which follows if we require that there exists ı > 0 such that X.y/ C ıw.y/ is time-like for y 2 †:

(14.4)

One can show that it suffices to assume that (14.4) holds away from a compact neighborhood of B in †, i.e. near spatial infinity. From (14.4) we deduce the uniform version of (14.3), namely, there exists c > 0 such that j Im. g eucl .y//j  cRe. g eucl .y//;

y 2 †C :

(14.5)

Another useful fact is that jg eucl j.y/ D j det g eucl .y/j D N 2 .y/jh.y/ > 0 for all 1 y 2 †, so the density d Volg eucl D jg eucl j 2 dsdy is positive.

14.3.2 The Wick rotated operator. The Klein–Gordon operator P takes the form

P D .@t C w  /N 2 .@t C w/ C h0 ;

see (9.17), and becomes after Wick rotation the differential operator P eucl D .@s C iw  /N 2 .@s C iw/ C h0 : One can define the Laplace–Beltrami operator g eucl associated to the complex metric g eucl as in the Riemannian case and one has P eucl D g eucl C V .y/. It also follows from (14.5) that P eucl is an elliptic differential operator. Let us now associate to P eucl some densely defined operator. It is a well-known fact that to describe quantum fields at temperature ˇ 1 by Euclidean methods, the Euclidean time s should belong to the circle Sˇ of length ˇ. Therefore, we set M eucl D Sˇ  †C and consider the sesquilinear form ˆ Qˇ .u; u/ D uP eucl u d Volg eucl ; Dom Qˇ D C01 .M eucl /: M eucl

It follows from (14.5) that Qˇ is sectorial, i.e., j Im Qˇ .u; u/j  cRe Qˇ .u; u/;

u 2 Dom Qˇ ;

and hence closeable. The domain of its closure Qˇcl equals the Sobolev space H 1 .M eucl /, defined as the completion of C01 .M eucl / with respect to the norm ˆ   2 kuk1 D ruRe.g eucl /1 .y/ru C V .y/uu d Volg eucl : M eucl

By the Lax–Milgram theorem, one associates to Qˇcl a boundedly invertible operator 

Pˇeucl W H 1 .M eucl / ! H 1 .M eucl / ; which corresponds to imposing ˇ-periodic boundary conditions for the operator P eucl .

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14 Klein–Gordon fields on spacetimes with Killing horizons

14.3.3 Calder´on projectors. Consider the open set

D 0; ˇ=2Œ †C  M eucl : Note that @ has two connected components f0g  †C and fˇ=2g  †C , both identified with †C . We will use the notation introduced in Section 12.7 for spaces of distributions on . One defines the outer unit normal to @ for the complex metric g eucl as the unique complex vector field  such that .i/ .x/g eucl.x/v D 0; 8v 2 Tx @ ; .ii/ .x/g eucl.x/.x/ D 1; .iii/ Re .x/ is outwards pointing: @ We see that  equals N 1 . @s  iw/ on f0g  †C and its opposite on fˇ=2g  †C . One can then define the trace   u@ u D 2 C 1 .@ I C2 /  ru@

for u 2 C 1 . / with P eucl u D 0 in and the Calder´on projectors cˇ˙ associated to .Pˇeucl ; / as in Section 12.7, see [G2, Section 8.7] for the precise definitions. The important observation now is that the doubled state !d constructed from !ˇ can be expressed in terms of the Calder´on projectors cˇ˙ . In fact one has, see [G2, Proposition 8.8]: Proposition 14.1. The covariances of !d are equal to 1 ˙ ˙ d D ˙Q ı .½ ˚ T / cˇ .½ ˚ T /;   ½ 0 . where T D 0 ½

Q D q ˚ q;

Two comments are in order at this point. First, the Calder´on projectors cˇ˙ are defined on C01 .@ I C2 /, or equivalently on C01 .†C I C2 / ˚ C01 .†C I C2 /, which is exactly the doubled phase space on which the doubled state !d is defined. Second the operator T takes care of the fact that !d is associated to the Hermitian form q ˚ q, see (14.2), and not Q D q ˚ q.

14.4 The double ˇ-KMS state in MC [ M Recall that the wedge reflection R maps MC to M and reverses the time orientation. It is hence easy to obtain from !d a pure quasi-free state !D in MC [ M ,

14.5 The extended Euclidean metric and the Hawking temperature

165

called the double ˇ-KMS state. This provides a first extension of the thermal state !ˇ in MC to a pure state in MC [ M . The Cauchy surface covariances ˙ D of !D are the sesquilinear forms on .C01 .†C I C2 /; q/ ˚ .C01 .† I C2 /; q/ given by  1 ˙  ˙ D D ˙Q ı .½ ˚ r / cˇ .½ ˚ r /;

where r  f .y/ D f .r.y//. Note that 

R† D T r  W .C01 .† I C2 /; q/!.C01 .†C I C2 /; q/: is exactly the unitary map on Cauchy data induced by the wedge reflection R W  C01 .MC /  ! C01 .M /.

14.5 The extended Euclidean metric and the Hawking temperature The constructions carried out up to now are valid for any ˇ > 0. The Euclidean metric g eucl usually degenerates at the bifurcation surface B . In fact, for ! 2 B , let n! the unit normal to B for the induced metric h on †, pointing towards †C . Using n! one can introduce Gaussian normal coordinates .u; !/ on a neighborhood of B in †, with †C corresponding to u > 0. One can then show that in the coordinates .s; u; !/, the Euclidean metric g eucl near u D 0 takes the form  2 u2 ds 2 C du2 C k.!/d! 2 ; modulo higher-order terms depending only on .u2 ; !/, where the Riemannian metric k.!/d! 2 is the restriction of h.y/dy 2 to B , see [G2, App. A]. We recognize in the first two terms the expression of the flat Riemannian metric dX 2 C d Y 2 , if X D u cos.s/, Y D u sin.s/, i.e. if .u; s/ are polar coordinates. Since s 2 Sˇ , we see that if ˇ D .2/ 1, i.e. if ˇ 1 equals the Hawking eucl temperature .2/1 , then g eucl extends across B to a smooth complex metric gext , eucl 2 living on a smooth manifold Mext , which near B is diffeomorphic to R  B . For other values of ˇ, no such smooth extension exists, and g eucl has a conical singularity at B . eucl It is also important to understand the open set ext  Mext corresponding to  eucl M . Its boundary @ ext is obtained by gluing together along B the two connected components f0g  †C and fˇ=2g  †C of @ . Actually, @ ext is diffeomorphic to †. The reason for this is that in coordinates .u; !/, the weak wedge reflection r becomes simply the reflection .u; !/ 7! .u; !/, and †C is identified with † by r.

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14 Klein–Gordon fields on spacetimes with Killing horizons

14.6 The Hartle–Hawking–Israel state eucl eucl One can associate to the extended metric gext a Laplace–Beltrami operator Pext and ˙ consider its Calder´on projectors cext for the open set ext . ˙ to conSince the boundary @ ext is diffeomorphic to †, it is tempting to use cext struct Cauchy surface covariances on †, which, if the required positivity properties are satisfied, will define a quasi-free state on the whole of M . It turns out that this is indeed the case, the resulting state being the sought-for Hartle–Hawking–Israel state. Let us thus summarize the main result of [G2].

Theorem 14.2. There exists a state !HHI for P in .M; g/, called the Hartle–Hawking– Israel state, such that: (1) !HHI is a pure Hadamard state in M ; (2) the restriction of !HHI to MC [ M is the double ˇ-KMS state !D at Hawking temperature TH D .2/1, where  is the surface gravity of the horizon; (3) !HHI is the unique extension of !D such that its spacetime covariances ƒ˙ map C01 .M / into C 1 .M / continuously. In particular, it is the unique Hadamard extension of !D . Proof. Let us now explain some ingredients of the proof of Theorem 14.2, which essentially relies on known results on Calder´on projectors and Sobolev spaces. We res .N /, resp. Hcs .N / denote the local, resp. compactly supported Sobolev call that Hloc spaces on the manifold N . ˙ Let us first check that !HHI is indeed an extension of !D , i.e., that ˙ HHI equal D 1 2 on C0 .† n B I C /. ˙ eucl The Calder´on projectors cext are constructed using the inverse of Pext , which as eucl for P is constructed from a sesquilinear form Qext . Clearly, Qext and Qˇ coincide eucl on C01 .Mext n B /. Near B the topology of the domain of the closure of Qext is 1 eucl eucl .Mext /. Since B is of codimension 2 in Mext , this implies that the topology of Hloc 1 eucl ˙ C0 .Mext n B / is a form core for Qext . This immediately implies that ˙ HHI and D 1 2 coincide on C0 .† n B I C /. From this fact one can also easily deduce that ˙ HHI are indeed the Cauchy surface covariances of a state, i.e., that C  ˙ HHI  0; HHI  HHI D q:

(14.6)

Let us explain this argument: it is known that Calder´on projectors for second-order 1

1 2

elliptic operators, hence in particular c ˙ , are continuous from Hc2 .†/ ˚ Hc 1 2

1 Hloc .†/ ˚ Hloc2 .†/. 1 1 Hc2 .†/ ˚ Hc 2 .†/.

From this we deduce immediately that

˙ HHI

.†/ to

are continuous on

Since B is of codimension 1 in †, we know that the space C01 .†n B I C2 / is dense 1

 21

in Hc2 .†/ ˚ Hc

1 2 ˙ .†/. The restrictions of ˙ HHI to C0 .† n B I C / equal D , and so

14.6 The Hartle–Hawking–Israel state

167

satisfy (14.6), since they are the Cauchy surface covariances of the state !D . By the above density result, this implies that (14.6) holds on C01 .†I C2 /, as claimed. The purity of !HHI follows similarly from the purity of !D . Further, let us explain how to prove that !HHI is a Hadamard state. The restriction of !HHI to MC is a Hadamard state for P , since it is a .2/ 1-KMS state for a timelike, complete Killing vector field. The restriction of !HHI to M is also a Hadamard state for P . This implies that the restriction of !HHI to MC [ M is a Hadamard state. The same is true of the restriction of a reference Hadamard state !ref in M (see Theorem 11.11) to MC [ M . Passing to Cauchy surface covariances on †C [ † , this ˙ implies that if  2 C01 .†˙ /, then  ı .˙ HHI  ref / ı  is a smoothing operator on ˙ †. This implies that ˙ HHI  ref is smoothing, which shows that !HHI is a Hadamard state. ˙ If fact let a be one of the entries of ˙ HHI ref , which is a scalar pseudodifferential m operator belonging to ‰ .†/ for some m 2 R. We know that  ı a ı  is smoothing for any  2 C01 .†nB /. Then its principal symbol pr .a/ vanishes on T  .†nB / hence on T  † by continuity, so a 2 ‰ m1 .†/. Iterating this argument, we obtain that a is smoothing. For the proof of the uniqueness statement (3) we refer the reader to [G2]. 

Chapter 15

Hadamard states and scattering theory In this chapter we study the construction of Hadamard states from scattering data, i.e., from data at future or past time infinity. This construction is related to the construction of Hadamard states from past or future null infinity on asymptotically flat spacetimes, which we reviewed in Chapter 13. The geometric assumption on the spacetime .M; g/ is that it should be asymptotically static, at past or future time infinity, see Section 15.1. Roughly speaking, this means that M should be of the form R  † and g should tend to a standard static metric gout=in , see Subsection 5.3.3, when t ! ˙1. The existence of the out and in vacuum states !out=in for a Klein–Gordon operator P on .M; g/, i.e., of states looking like the Fock vacua for the static Klein–Gordon operators Pout=in on .M; gout=in/ at large positive or negative times, is often taken for granted in the physics literature. We will explain the result of [GW3], which provides a proof of the existence of !out=in and more importantly of their Hadamard property.

15.1 Klein–Gordon operators on asymptotically static spacetimes Let us now introduce a class of spacetimes that are asymptotically static at future and past time infinity and corresponding Klein–Gordon operators We fix an .n  1/dimensional manifold † and set M D Rt  †y , y D .t; y/. We equip M with the Lorentzian metric     (15.1) g D c 2 .y/dt 2 C d yi C b i .y/dt hij .y/ d yj C b j .y/dt ; where c 2 C 1 .M /, h.t; y/d y2 , resp. b.t; y/ is a smooth t-dependent Riemannian metric, resp. vector field on †. If there exist a reference Riemannian metric k.y/d y2 on † and constants c0 ; c1 > 0 such that .t; y/ 2 M; (15.2) then it follows from [CC, Theorem 2.1] that t W M ! R is a Cauchy temporal function for .M; g/, see Definition 5.13, hence in particular .M; g/ is globally hyperbolic. h.t; y/  c1 k.y/;

b.t; y/h.t; y/b.t; y/  c1 ;

c0  c.t; y/  c1 ;

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15 Hadamard states and scattering theory

It is natural to use the framework of bounded geometry and to equip † with a reference Riemannian metric k such that .†; k/ is of bounded geometry. The version of (15.2) is then ( h 2 Cb1 .RI BT20 .†; k//; h1 2 Cb1 .RI BT02 .†; k//; .bg/ (15.3) b 2 Cb1 .RI BT01 .†; k// c; c 1 2 Cb1 .RI BT00 .†; k//: A concrete example of .†; k/ is Rd equipped with the uniform metric.

15.1.1 Asymptotically static spacetimes. Let us consider a Klein–Gordon operator

P D .r   iqA .x//.r  iqA .x// C V .x/

on .M; g/. We now impose conditions on h; b; c; A; V which mean that .M; g/ is asymptotically static at t D ˙1. Let us first introduce a convenient notation. Definition 15.1. Let F be a Fr´echet space whose topology is defined by the seminorms k  kn , n 2 N. For I  R an interval, we denote by S ı .I I F /, ı 2 R, the space of functions I 3 t 7! X.t/ 2 F such that suphtiıCm k@m t X.t/kn < 1; 8 m; n 2 N: t 2I

We introduce two static metrics 2 .y/dt 2 C hout=in .y/d y2 gout=in D cout=in

and time-independent potentials Vout=in and assume the following conditions 8 0  ˙ ˆ ˆ h.y/  hout=in .y/ 2 S .R I BT 2 .†; k//; ˆ ˆ ˆ < b.y/ 2 S 0 .RI BT 1 .†; k//; A.y/ 2 S 0 .RI BT 0 .†; k//; 0 1 .as/ 0  ˙ ˆ c.y/  cout=in .y/ 2 S .R I BT 0 .†; k//; ˆ ˆ ˆ ˆ : V .y/  Vout=in .y/ 2 S  .R˙ I BT 00 .†; k//; for some  > 0, 0 > 1. Here the space S ı .RI BTqp .†; k//, ı 2 R is defined as in Definition 15.1. The above conditions are standard scattering type conditions, with ; 0 measuring the rate of convergence of h; b, etc. to their limits at t D ˙1. The condition 0 > 1 is traditionally called a short-range condition in the scattering theory for Schr¨odinger equations, while  > 0 corresponds to the weaker long-range condition.

171

15.2 The in and out vacuum states

15.2 The in and out vacuum states 15.2.1 The asymptotic Klein–Gordon operators. It follows from condition (as) that when t ! ˙1, P is asymptotic to the Klein–Gordon operator Pout=in D gout=in C Vout=in ; associated to the static metric gout=in . We can introduce the ultra-static metric 2 gout=in D dt 2 C hQ out=in .y/d y gQ out=in D cout=in

and obtain from Section 6.3 that n=21 Q n=21 ; Pout=in cout=in Pout=in D cout=in

where

n2 2 ScalgQout=in C cout=in VQout=in ; PQout=in D gQout=in C 4.n  1/

n2 and VQout=in D Vout=in  4.n1/ Scalgout=in . The ultra-static Klein–Gordon operator PQout=in equals @2t C aQ out=in.y; @y /, and to avoid technical complications coming from infrared problems we will assume that

n2 2 ScalgQout=in C cout=in VQout=in  m2 ; for some m > 0; 4.n  1/

.pos/

which simply means that 1 aQ out=in  m2 > 0 on L2 .†; jhQ out=in j 2 .y/d y/;

2 hout=in . for hQ out=in D cout=in vac , see Subsection 4.10.2, whose It follows that PQout=in admits a vacuum state !Q out=in Cauchy surface covariances are

1 D Q ˙;vac out=in 2



Qout=in ˙½ 1 ˙½ Qout=in



1

2 ; Qout=in D aQ out=in :

vac , whose Cauchy surface By Subsection 6.3.2, Pout=in admit the vacuum state !out=in covariances on †0 D f0g  † are

˙;vac out=in

D

 .Uout=in /1

ı

Q ˙;vac out=in

ı

1 Uout=in ;

U D

1n=2 0 cout=in n=2 0 cout=in

! :

172

15 Hadamard states and scattering theory

15.2.2 The out and in vacuum states. We have seen that †s D fsg  † are Cauchy surfaces for .M; g/. Denoting by %s W Solsc .P / ! C01 .†s I C2 / the Cauchy data map on †s , see (5.27) and by Us f , f 2 C01 .†s I C2 /, the solution of the Cauchy problem on †s we set U .t; s/ D %t Us W C01 .†s I C2 / ! C01 .†t I C2 /: If ! is a quasi-free state for P , with spacetime covariances ƒ˙ , we will denote by ˙ t its Cauchy surface covariances on †t , called the time t covariances of the state !. From Propositions 5.21, 6.6 it easily follows that  ˙ ˙ s D U .t; s/ ı t ı U .t; s/;

s; t 2 R:

(15.4)

We would like to define quasi-free states !out=in for P , called the out/in vacua vac which look like the ‘free’ vacua !out=in when t ! ˙1. Taking (15.4) into account, we see that !out=in should be defined by the time 0 covariances: ˙;vac  ˙ out=in .0/ D lim U .t; 0/ ı out=in ı U .t; 0/; t !˙1

(15.5)

where the limit above is taken as sesquilinear forms on C01 .†0 I C2 /. Of course, the reference time t D 0 is completely arbitrary. The following theorem is the main result of [GW3]. Theorem 15.2. Assume the conditions (bg), (as) and (pos). Then: (1) the limits (15.5) when t ! C1, resp. 1, exist and are the time 0 covariances of a quasi-free state for P denoted by !out , resp. !in , called the out resp. in vacuum state. (2) !out=in are pure Hadamard states. vac 15.2.3 Wave operators. The static vacua !out=in are invariant under time transla-

tions: if Uout=in .t; s/ is the Cauchy evolution operator for Pout=in , then Uout=in .t; s/ D Uout=in .t C T; s C T / and D Uout=in .t; s/ ı ˙;vac ı Uout=in .t; s/: ˙;vac out=in out=in Therefore we can rewrite (15.5) as ˙;vac  ˙ out=in .0/ D lim .Uout=in .0; t/ ı U .t; 0// ı out=in ı .Uout=in .0; t/ ı U .t; 0//: t !˙1

If the exponent  in conditions (as) satisfies  > 1, then one can prove that the strong limits Wout=in D s lim Uout=in .0; t/ ı U .t; 0/ (15.6) t !˙1

15.3 Reduction to a model case

173

exist on some natural energy spaces. The operators Wout=in are called .inverse/ wave operators and (15.5) takes the more familiar form ˙;vac  ˙ out=in .0/ D Wout=in out=in Wout=in ;

which is often found in the physics literature. Note however that the existence of Wout=in requires  > 1, while the existence of !out=in only requires  > 0.

15.3 Reduction to a model case We now give some ideas of the proof of Theorem 15.2. The existence of !out=in , at least in the short-range case  > 1, is not very difficult, using the arguments outlined in Subsection 15.2.3. The Hadamard property is more delicate. For example, the covariances U .t; 0/ ı ˙;vac out=in ı U .t; 0/ in the right-hand side of (15.5) are not Hadamard for P for finite t. vac In fact, the free vacua !out=in are Hadamard states for Pout=in , but not for P . It is only after taking the limit t ! 1 that one obtains a Hadamard state for P . The proof of Theorem 15.2 is done by reduction to a model case, similar to the one considered in Section 11.2. Since we want to use the time coordinate t and not the Gaussian time, we use the orthogonal decomposition associated to t explained in Subsection 5.4.1.

15.3.1 Orthogonal decomposition. One can identify f0g  † with † and use the vector field

v D .rt grt/1 rt D @t C b i .y/@yi

as in Subsection 5.4.1 to construct an orthogonal decomposition of g by the diffeomorphism  W R  † 3 .t; x/ 7! .t; y.t; 0; x// 2 R  †; where y.t; s; / W † ! † is the flow of the time-dependent vector field b i .y/@yi on †. The metric  g takes the form O x/d x2 :  g D cO 2 .t; x/dt 2 C h.t; After a further conformal transformation, the operator PQ D cO 1n=2  P cO 1Cn=2 take on the form, see [GW3, Section 5.2] PQ D @2t C r.t; x/@t C a.t; x; @x /; i.e. is a model Klein–Gordon operator of the type considered in Section 11.2.

174

15 Hadamard states and scattering theory

15.3.2 Properties of the model operator. In the sequel the model operator PQ will be denoted by P for simplicity. Let us first introduce classes of time-dependent pseudodifferential operators on † that are analogs of the classes of time-dependent tensors S ı .RI BTqp .†; k// defined in Subsection 15.1.1. We set m;ı m .I I †/ D Op.S ı .I I BSph .†/// C S ı .I I W 1.†//; ‰td m .†/ and W 1 .†/ are defined in Definitions 10.6, 10.8, and we use Defwhere BSph inition 15.1. One can show that the conditions (bg), (as), (pos) imply the conditions 8 2;ı ˆ .R˙ I †/; ı > 0; a.t; x; @x / D aout=in .x; Dx/ C ‰td ˆ < 0;1ı .td/ .RI †/; r.t/ 2 ‰td ˆ ˆ : aout=in .x; @x / 2 ‰ 2 .†/ elliptic; aout=in.x; Dx / D aout=in.x; Dx /  C1 > 0;

for ı D min.; 0  1/. The asymptotic Klein–Gordon operators are now Pout=in D @2t C aout=in .x; @x /: The decay conditions (td) lead to an improvement of the properties of the generator 1 b.t/ constructed in Section 11.3. Indeed, setting .t/ D a.t; x; @x / 2 and out=in D 1

2 one can show that b.t/ in Proposition 11.2 can be chosen so that aout=in

0;1ı 1;ı b.t/ D .t/ C ‰td .RI˙ †/ D out=in C ‰td .R˙ I †/; 1;1ı .RI †/: i@t b  b 2 C a C irb 2 ‰td

(15.7)

15.3.3 Almost diagonalization. In Chapter 11 the microlocal splitting deduced from a solution b.t/ was used to construct a pure Hadamard state. It is also possible, see [GOW, Chapter 6], to use it to diagonalize the evolution U .t; s/ associated to P , modulo smoothing error terms. Let us set   ½ ½ .b C  b  / 21 .t/; T .t/ D i1 b C b  where we recall that b C .t/ D b.t/, b  .t/ D b  .t/. Then one can check that   b  ½ 1 C   21 T .t/ D i.b  b / .t/: b C ½ We now define

U .t; s/ D T .t/ ı U ad .t; s/ ı T .s/1 ;

t; s 2 R;

(15.8)

175

15.3 Reduction to a model case

which is (at least formally) a two-parameter group. Computing the infinitesimal generator of fU ad .t; s/gt;s2R one obtains   b  C rb 0 ad .t/ C R1 .t/; (15.9) H .t/ D 0 b C C rbC 1;1ı 0;1ı .RI †/ ˝ M.C2 / and rb˙ 2 ‰td .RI †/, i.e. H ad .t/ is where R1 2 ‰td diagonal, modulo the regularizing in space and decaying in time error term R1 .t/. There is a similar well-known exact diagonalization of the Cauchy evolutions Uout=in.t; s/ for Pout=in . If 0 1 1 1 2 p 1   2 1  out=in A 2 Tout=in D .i 2/ @ out=in ; ; out=in D aout=in 1 1 2 2 out=in out=in

then

ad 1 Uout=in.t; s/ D Tout=in ı Uout=in .t; s/ ı Tout=in ;

ad ad and the (time-independent) generator Hout=in of Uout=in .t; s/ equals

 ad Hout=in

D

0 out=in 0 out=in

 :

vac The vacua !out=in are pure states associated to the projections ˙ cout=in

C

D Tout=in ı  ı

1 Tout=in ;

for 

C

 D

½ 0

0 0



;   D ½   C:

Rather straightforward arguments show that the existence of the limits in Theorem 15.2 follows from the existence of ad .t; 0/; (15.10) lim Wout=in .t/ı ˙ ıWout=in .t/1; for Wout=in .t/ D U ad .0; t/Uout=in

t !˙1

for example in B.L2 .†I C2 //. Using the properties of H ad .t/ one can actually prove that s lim Wout=in .t/ ı  ˙ ı Wout=in .t/1 D  ˙ C W 1 .†/ ˝ M.C2 /: t !˙1

(15.11)

This implies not only the existence of the out=in vacuum states, but also their Hadamard property. Indeed, if c ˙ D T .0/ ˙T .0/1 then ˙ .0/ D ˙q ı c ˙ are the Cauchy surface covariances on †0 of the Hadamard state associated to the microlocal .0/ difsplitting obtained from b, see Section 11.4. From (15.11) we obtain that ˙ out=in ˙ fer from  .0/ by a smoothing error, which proves that !out=in are Hadamard states.

Chapter 16

Feynman propagator on asymptotically Minkowski spacetimes We have seen in Section 7.4 that a Klein–Gordon operator P on a globally hyperbolic spacetime .M; g/ possesses four distinguished parametrices, the retarded/ advanced parametrices GQ ret=adv and the Feynman/anti-Feynman parametrices GQ F=F , unique modulo smooth kernels and uniquely characterized by the wavefront set of their distributional kernels. One can ask if there exist true inverses of P , corresponding to the above parametrices and canonically associated to the spacetime .M; g/. By Lemma 7.9, there exists true retarded/advanced inverses of P , namely Gret=adv , see Theorem 5.18, which are uniquely determined by the causal structure of .M; g/. The situation is more complicated for the Feynman/anti-Feynman inverses. Of course, given a Hadamard state ! for P , the Feynman inverse associated to !, see (8.14), has the correct wavefront set, but it depends on the choice of the Hadamard state !, and hence is not canonical. There are some situations where such a canonical Feynman inverse exists. If .M; g/ is stationary with Killing vector field X and P is invariant under X , one can, under the conditions in Chapter 9, construct the vacuum state !vac associated to X and the corresponding Feynman inverse GF is a canonical choice of a Feynman inverse, respecting the symmetries of .M; g/. In the particular case of the Minkowski spacetime R1;d and P D @2t  x C m2 , the Feynman inverse obtained from the vacuum state is equal to the Fourier multiplier by the distribution 1 :  2  .k 2 C m2 / C i0 In this chapter we will describe the results of [GW4, GW6], devoted to this question on spacetimes which are asymptotically Minkowski, and hence have in general no global symmetries, only asymptotic ones. It turns out that it is possible in this case to define a canonical Feynman inverse GF , which is the inverse of P between some appropriate Sobolev type spaces. More concretely, one introduces spaces Y m , XFm for m 2 R, see Section 16.3, where Y m is a space of functions decaying fast enough when t ! ˙1, while the functions in XFm satisfy asymptotic conditions at t D ˙1 which are analogs of the wavefront set condition which characterizes Feynman parametrices. One can show that P W XFm ! Y m is invertible, and that its inverse GF is a Feynman parametrix in the sense of Subsection 7.4.2. Vasy [Va] considered the same problem by working directly on the scalar operator P using microlocal methods. He constructed the Feynman inverse GF between microlocal Sobolev spaces, as the boundary value .P  i0/1 of the resolvent of P .

178

16 Feynman propagator on asymptotically Minkowski spacetimes

16.1 Klein–Gordon operators on asymptotically Minkowski spacetimes In this section we recall the framework considered in [GW4].

16.1.1 Asymptotically Minkowski spacetimes. We consider M D R1Cd equipped with a Lorentzian metric g such that ı .R1Cd /; ı > 1; .aM.i// g .x/   2 Sstd

.aM.ii// .R1Cd ; g/ is globally hyperbolic; 1 .aM.iii// .R1Cd ; g/ has a temporal function tQ with tQ  t 2 Sstd .R1Cd / for  > 0; ı .R1Cd / denotes the class of smooth funcwhere  is the Minkowski metric and Sstd 1 tions f such that, for hxi D .1 C jxj/ 2 ,

@˛x f 2 O.hxiıj˛j /;

˛ 2 N1Cd :

Recall that tQ is called a temporal function if r tQ is a time-like vector field, and is called a Cauchy temporal function if in addition its level sets are Cauchy surfaces for .M; g/. It is shown in [GW4] that if .aM.i// holds, then .aM.ii// is equivalent to the familiar non trapping condition for null geodesics of g, and if .aM.i/; .ii/; .iii// hold, then there exists a Cauchy temporal function tQ such that tQ  t 2 C01 .M /. Replacing t by t  c, tQ by tQ  c for c 1 we can also assume that † D ft D 0g D ftQ D 0g is a Cauchy surface for .M; g/, which can be canonically identified with Rd . In the sequel we will fix such a temporal function tQ.

16.1.2 Klein–Gordon operator. We fix a real function V 2 C 1 .M I R/ such that

ı .aM.iv// V .x/  m2 2 Sstd .R1Cd /; for some m > 0; ı > 1;

and consider the Klein–Gordon operator P D g C V:

16.2 The Feynman inverse of P We now introduce the Hilbert spaces XFm , Y m between which P will be invertible. The spaces Y m are standard spaces of right-hand sides for the Klein–Gordon equations, their essential property being that their elements are L1 in t, with values in some Sobolev spaces of order m. The spaces XFm incorporate the Feynman boundary conditions, which are imposed at t D ˙1.

16.2 The Feynman inverse of P

179

16.2.1 Hilbert spaces. Using the Cauchy temporal function tQ we can identify 1

Q

M with R  † using the flow t of the vector field v D d gtQg 1d dt tQ , and obtain the diffeomorphism (16.1)  W R  † 3 .t; x/ 7! t .x/ 2 M; such that

 g D c 2 .t; x/dt 2 C h.t; x/d x2 :

For m 2 R we denote by H m .Rd / the usual Sobolev spaces on Rd . We set, for 1 <  < 12 C ı 2

Y m D fu 2 D 0 .M / W  u 2 hti L2 .RI H m .Rd //g; with norm kvkY m D k ukL2 .RIH m .Rd // . The exponent  is chosen such that hti L2 .R/  L1 .R/. Similarly we set

X m D fu 2 D 0 .M / W  u 2 C 0 .RI H mC1 .Rd // \ C 1 .RI H m .Rd //; P u 2 Y m g: We equip X m with the norm  where %s u D

kukX m D k%0 ukE m C kP ukY m ;  u†s is the Cauchy data map on †s D tQ1 .fsg/ and E m D i1 @ u n

†s

H mC1 .Rd / ˚ H m .Rd / is the energy space of order m. From the well-posedness of the inhomogeneous Cauchy problem for P one easily deduces that X m is a Hilbert space.

16.2.2 Feynman boundary conditions. Let us set

! p 2 1 ½ ˙  C m x ˙ p : D cfree 2 ˙ x C m2 ½   0 ½ ˙ ˙ Of course, free D ˙q ı cfree for q D ½ 0 are the Cauchy surface covariances on † of the free vacuum state !free associated to Pfree . We set then ˙ XFm D fu 2 X m W lim cfree %t u D 0 in E m g: t !1

It is easy to see that XFm is a closed subspace of X m . The following theorem is proved in [GW6]. Theorem 16.1. Assume .aM/. Then P W XFm ! Y m is boundedly invertible for all m 2 R. Its inverse GF is called the Feynman inverse of P . It satisfies WF.GF /0 D  [ CF : We recall that CF was defined in Section 7.4. In particular GF is a Feynman parametrix for P .

180

16 Feynman propagator on asymptotically Minkowski spacetimes

16.3 Proof of Theorem 16.1 We now give some ideas of the proof of Theorem 16.1. As in Section 15.3, the first step consists in the reduction to a model Klein–Gordon equation, by using successively the diffeomorphism  in Subsection 16.2.1 and the conformal transformation  g ! c 2 .t; x/ g. After this reduction, we work on R1Cd with elements x D .t; x/ equipped with the Lorentzian metric g D dt 2 C hij .t; x/d xi d xj ; where t 7! ht D hij .t; x/d xi d xj is a smooth family of Riemannian metrics on Rd . The Klein–Gordon operator P D g C V takes the form P D @2t C r.t; x/@t C a.t; x; @x /; where

1

(16.2)

1

a.t/ D a.t; x; @x / D jhj 2 @i hij jhj 2 @j C V .t; x/; 1

1

r.t/ D r.t; x/ D jhj 2 @t .jhj 2 /.t; x/: The operator a.t/ is formally selfadjoint for the time-dependent scalar product ˆ 1 uvjht j 2 d x; .ujv/t D †

and P is formally selfadjoint for the scalar product ˆ 1 .ujv/ D uvjht j 2 d xdt: R†

Conditions .aM/ on the original metric g and potential V imply similar asymptotic conditions on a.t; x; @x / and r.t; x/ when t ! ˙1. More precisely, one has 8 2;ı ˆ .R˙ I Rd /; a.t; x; @x / D aout=in .x; @x / C ‰std ˆ ˆ ˆ ˆ < r.t/ 2 ‰ 0;1ı .RI Rd /; std .Hstd/ ˆ 2;0 ˆ aout=in .x; @x / 2 ‰sc .Rd / is elliptic; ˆ ˆ ˆ : aout=in .x; @x / D aout=in.x; @x /  C1 > 0; m;ı .R˙ I Rd / is the class of time-dependent pseudodifferential operators on where ‰std d R associated to symbols m.t; x; k/ such that

@t @˛x @ˇk m.t; x; k/ 2 O..hti C hxi/ı j˛j hkimjˇ j /;

 2 N; ˛; ˇ 2 Nd ; t 2 R˙ :

m;ı Similarly, ‰sc .Rd / is the class of pseudodifferential operators on Rd associated to symbols m.x; k/ such that

@˛x @ˇk m.x; k/ 2 O.hxiıj˛j hkimjˇ j /;

˛; ˇ 2 Nd :

We refer the reader to [GW4, Section 2.3] for more details.

16.3 Proof of Theorem 16.1

181

The Hilbert spaces Y m and X m become

Y m D hti L2 .RI H m .Rd //; X m D fu 2 C 0 .RI H mC1 .Rd // \ C 1 .RI H m .Rd // W P u 2 Y m g; equipped with the norm kuk2X m D k%0 uk2E m C kP uk2Y m ; where %t u D



u.t /



i1 @t u.t /

and the energy space E m is defined in Subsection 16.2.1.

The subspaces XFm become  XFm D fu 2 X m W lim cout %t u D lim cinC %t u D 0 in E m g t !1 t !C1

0

where ˙ D cout=in

1 2

1

1 ˙aout=in 1@ A 1 2 2 ˙aout=in 1

are the projections for the out/in vacuum state !out=in associated to the Klein–Gordon operator @2t C aout=in.x; @x /.

16.3.1 A further reduction. It is convenient to perform a further reduction to 1 1 the case r D 0. Namely, setting R D jh0 j 4 jht j 4 , we see that 1

1

L2 .†; jh0 j 2 d x/ 3 uQ 7! RuQ 2 L2 .†; jht j 2 d x/ is unitary and that where

R1 PR D PQ D @2t C a.t; Q x; @x /; a.t/ Q D rR1 @t R C R1 .@2t R/ C R1 a.t/R

Q x; @x / satisfies also .Hstd/, with the is formally selfadjoint for .j/0 . Clearly, a.t; same asymptotic aout=in .x; @x /. It is also immediate that the Hilbert spaces Y m , X m and XFm introduced in Section 2.2 are invariant under the map u 7! Ru and hence we can assume that r.t; x/ D 0.

16.3.2 Almost diagonalization. One can then perform the same almost diagonalization as in Subsection 15.3.3. The stronger spacetime decay in conditions .Hstd/ givestronger decay  conditions on the off diagonal terms. More precisely, 0 ½ if H.t/ D is the generator of the Cauchy evolution for P and T .t/ is a.t/ 0 as in Subsection 15.3.3 we have T 1 .Dt  H.t//T D Dt  H ad .t/ D P ad ;

182

16 Feynman propagator on asymptotically Minkowski spacetimes

where H ad .t/ is almost diagonal, i.e. ad .t/; H ad .t/ D H d .t/ C V1   C  .t/ 0 d ; H .t/ D 0   .t/

(16.3)

where  ˙ .t/ belong to ‰ 1;0 .RI Rd /, with principal symbols equal to 1 ad ˙.k  h1 .t; x/k/ 2 , and V1 .t/ is an off-diagonal matrix of time-dependent operd ators on R such that ad .hxi C hti/m V1 .t/.hxi C hti/mCı W H p .Rd / ! H p .Rd /

(16.4)

is uniformly bounded in t for all m; p 2 R. Compared with the situation in Section ad 15.3, we obtain extra decay in x and hence compactness properties of V1 . ad We denote by U .t; s/, resp. U .t; s/, for t; s 2 R, the Cauchy evolution generated by H.t/, resp. H ad .t/. Recall from (15.8) that

U .t; s/ D T .t/ ı U ad .t; s/ ı T .s/1 :

(16.5)

Moreover, U .t; s/.ad/ are unitary with respect to the Hermitian scalar product     ½ 0 0 ½ .ad/ .ad/ ad  (16.6) f q g D .f jq g/H0 ; q D ½ 0 ; q D 0 ½ ; 1

where H0 D L2 .Rd ; jh0 j 2 d xI C2 /, which implies the identity H ad .t/ q ad D q ad H ad .t/;

(16.7)

where the adjoint is computed with respect to the scalar product of H0 . The spaces corresponding to Y m , XFm with the scalar operator P replaced by the matrix operator Dt  H ad .t/ are the following:

Y ad;m D hti L2 .R; Hm /; X ad;m D fuad 2 C 0 .RI HmC1 / \ C 1 .RI Hm / W P ad uad 2 Y ad;m g; equipped with the norm kuad k2X ad;m D k%0 uad k2Hm C kP ad uk2Y ad;m ; ad where Hm D H m .Rd / ˚ H m .Rd / and %ad D uad .t/. The subspace XFad;m is t u defined as ad  ad ad m XFad;m D fuad 2 X ad;m W lim  C %ad t u D lim  %t u D 0 in H g; t !1 t !C1

where C D



1 0 0 0



;  D



0 0 0 1

 :

16.3 Proof of Theorem 16.1

183

Note that  ˙ are the spectral projections on R˙ for the Hamiltonian 1 0 1 2 a 0 ad A Hout=in D @ out=in 1 2 0 aout=in ad ad and we will denote by Uout=in .t; s/ the evolution generated by Hout=in .

Proposition 16.2. Assume .aM/. Then the operator P ad W XFad;m ! Y m is Fredholm of index 0. Proof. Set P d D Dt  H d .t/. Then P d W XFad;m ! Y ad;m is boundedly invertible, with inverse GFd given by ˆ t d ad  GF v .t/ D i U d .t; 0/ C U d .0; s/v ad .s/ds 1 ˆ C1 U d .t; 0/  U d .0; s/v ad .s/ds: i t

ad is compact from X ad;m to Y m , It is easy to show, see [GW4, Lemma 3.7], that V1 ad;m ad;m hence also from XF to Y m since XF is closed in X ad;m . 

Now let us prove that P ad W XFad;m ! Y m is injective, and hence boundedly invertible by Proposition 16.2. The proof of Lemma 16.3 below is inspired by the work of Vasy [Va, Proposition 7], which in turn relies on arguments of Isozaki [I] from N -body scattering theory. Lemma 16.3. One has: Ker P ad jX ad;m D f0g for all m 2 R: F

ad uad , from Proof. We first note that if uad 2 Ker P ad jX ad;m , we have uad D GFd V1 0

F

ad which we deduce that uad 2 XFad;m for any m0 , using that V1 is smoothing in x. Therefore, it suffices to prove the lemma for m  1. ´ C1 Let us set  .t/ D jt j ½Œ1;2 .s/s r ds for some 0 < r < 1. Note that supp   fjtj  2 1 g. Let us still denote by  the operator  ˝ ½C2 . Recalling that q ad is defined in (16.6), we compute for u 2 XFad;m : ˆ     ad ad P u .t/jq ad  .t/uad .t/ H0   .t/uad .t/jq ad P ad uad .t/ H0 dt R ˆ     Dt uad .t/jq ad  .t/uad .t/ H0  uad .t/jq ad  .t/Dt uad .t/ H0 dt D R ˆ   ad u .t/jq ad ŒH ad .t/;  .t/uad.t/ H0 dt; C R

184

16 Feynman propagator on asymptotically Minkowski spacetimes

using that H ad .t/q ad D q ad H ad .t/,  .t/ q ad D q ad  .t/ and uad .t/ 2 Dom H ad .t/ since m  1. We have ŒH ad .t/;  .t/ D 0, and since  is compactly supported in t we can integrate by parts in t in the second line and obtain ˆ ˆ    ad ad  P u .t/jq ad  .t/uad.t/ H0 dt   .t/uad.t/jq ad P ad uad .t/ H0 dt R R ˆ   ad ad ad D i u .t/jq @t  .t/u .t/ H0 dt: R

(16.8) Note that we used here that the scalar product in H0 does not depend on t, which is the reason for the reduction to r D 0 in Subsection 16.3.1. Since P ad uad D 0, this yields ˆ   ad (16.9) u .t/jq ad @t  .t/uad .t/ H0 dt D 0: R

We claim that: .i/ k ˙ uad .t/k2H0 2 O.t 1ı /; when t ! 1; .ii/ k ˙ uad .t/k2H0 D c ˙ C O.t 1ı /; when t ! ˙1;

(16.10)

for c ˙ D limt !˙ k ˙ uad .t/k2H0 . The proof of (16.10) is elementary: we have ad 2 O.t ı / in B.H0 / when t ! ˙1, see e.g. [GW1, Section H ad .t/  Hout=in 2.5], which using that ı > 1 and the Cook argument yields  ad uad D limt !˙ Uout=in .0; t/uad.t/ exists in H0 ; Wout=in 

ad .0; t/uad.t/kH0 2 O.t 1ı /: kWout=in uad  Uout=in ad .0; t/ is unitary on H0 , this yields (16.10). We then compute Since Uout=in

ˆ

  ad u .t/jq ad @t  .t/uad.t/ H0 dt R ˆ ˆ @t  .t/k C uad .t/k2H 0 dt  @t  .t/k  uad .t/k2H 0 dt D I C C I  : D R

R

Since @t  .t/ D sgn.t/½Œ1 ;21 .jtj/jtjr , we have, using (16.10): ˆ ˆ j@t  .t/jk ˙ uad .t/k2H 0 dt  C ½Œ1 ;21 .jtj/jtjrıC1dt 2 O. rCı2 /; 0 R

ˆ

R˙

˙ ad

@t  .t/k u

ˆ .t/k2H 0 dt

D

R˙

½Œ1 ;21 .jtj/c Cjtjr dt C O.rCı2 /

D Cc ˙  r1 C O. rCı1 /:

16.3 Proof of Theorem 16.1

185

Using (16.9), this yields C r1 .c C C c  / 2 O. rCı2 /, hence c C D c  D 0, since ı > 1. Therefore by (16.10) we have limt !˙1 kuad .t/kH0 D 0. Since the Cauchy evolution U ad .t; s/ is uniformly bounded in B.H0 / we have uad .0/ D 0, hence u D 0.  The reduction explained at the beginning of Section 16.3 shows that Theorem 16.1 follows from Theorem 16.4. P W XFm ! Y m is boundedly invertible, with inverse GF D 0 T GFad T 1 1 : Moreover, GF is a Feynman inverse of P , i.e. WF.GF /0 D  [ CF :

(16.11)

Proof. It is straightforward using the expression of T to check that 1

1

0 T 2 B.X ad;mC 2 ; X m /; T 1 1 2 B.Y m ; Y ad;mC 2 /; and so GF W Y m ! X m . Since .Dt H.t//T GFad T 1 D T GFad T 1 .Dt H.t// D ½, we obtain that P GF D GF P D ½. We have also %0 T GFad T 1 1 D T GFad T 1 1 v. ad;mC 1

2 ! XFm , hence %F GF D From [GW4, Equation (3.25)] we obtain that 0 T W XF m m 0, i.e. GF W Y ! XF . To prove the second statement, let GQ F D 0 T GFd T 1 1 . We have GFd  GFad D ad R1 D GFd V1 GFad by the resolvent identity. It is shown in [GW1, Lemma 3.7] that 0 0 ad ad;m V1 W X ! Y m is bounded for all m0 > m, hence R1 W Y ad;m ! X ad;m 0 for all m > m, i.e. is smoothing in the x variables. We use then that Dt R1 D ad ad H d .t/R1 C V1 GFad , R1 Dt D R1 H ad .t/ C GFd V1 to gain regularity in 0 1Cd 2 1 1Cd the t variable and obtain that R1 W E .R I C / ! C .R I C2 /. Therefore, Q GF  GF is a smoothing operator. Let also GF;ref be defined as GQ F with U d .t; s/ replaced by U ad .t; s/. From (16.4) it follows that U d .; /  U ad .; /, and hence GF;ref  GQ F have smooth kernels in M  M . Using (16.5), we see that GF;ref is the Feynman inverse associated to a Hadamard state, see Theorems 11.5, 11.6. Therefore, WF.GF;ref /0 D  [ CF , which completes the proof of the theorem. 

Chapter 17

Dirac fields on curved spacetimes In this chapter we will give a brief description of quantized Dirac fields on curved spacetimes. Usually Dirac equations on a Lorentzian manifold are introduced starting from spin structures, see [Di2, Li2] or [LM, Chaps. 1, 2]. Here we use the approach through spinor bundles, with which analysts may be more comfortable. We will follow the exposition by Trautman [T] and refer to [FT] for a comparison between the two approaches. The quantization of Dirac fields on curved spacetimes is due to Dimock [Di2]. The definition of Hadamard states for quantized Dirac fields on globally hyperbolic spacetimes was given by Hollands [Ho1] and Sahlmann and Verch [SV2] and is completely analogous to the Klein–Gordon case. Another nice reference is [S4]. The massless Dirac equation can be written as a pair of uncoupled Weyl equations which were for some time supposed to describe neutrinos and anti-neutrinos. We describe the quantization of the Weyl equation, the corresponding definition of Hadamard states, and the relationship between Hadamard states for Weyl and for Dirac fields.

17.1 CAR -algebras and quasi-free states The fermionic version of Chapter 4, namely CAR - algebras and quasi-free states on them, is quite parallel to the bosonic case. A detailed exposition can be found for example in [DG, Sections 12.5, 17.2]. The complex case, corresponding to charged fermions, is the most important in practice, although the real case corresponding to neutral or Majorana fermions is sometimes also considered. For simplicity we will only consider the complex case. Definition 17.1. Let .Y ; / be a pre-Hilbert space. The CAR -algebra over .X ; /, denoted by CAR.Y ; /, is the unital complex -algebra generated by elements .y/,  .y/, y 2 Y , with the relations 

.y1 C y2 / D

.y1 / C  .y2 /;

.y1 C y2 / D

.y1 / C 

Π.y1 /; .y2/C D ΠΠ.y1 /;





.y1 /;

.y2 /C D y 1  y2 ½; .y/ D



 

.y2 /;

y1 ; y2 2 Y ;  2 C;

.y2 /C D 0; y1 ; y2 2 Y ;

.y/;

where ŒA; BC D AB C BA is the anti-commutator.

(17.1)

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17 Dirac fields on curved spacetimes

Quasi-free states on CAR.Y ; / are defined in a way quite similar to the bosonic case. Definition 17.2. A state ! on CAR.Y ; / is a .gauge invariant/ quasi-free state if Q Q 0 !. niD1  .yi / m j D1 .yj // D 0; if n ¤ m; Q Q P Q !. niD1  .yi / njD1 .yj0 // D 2Sn sgn. / niD1 !.  .yi .y.i / //: A quasi-free state is again characterized by its covariances ˙ 2 Lh .Y ; Y  /, defined by !. .y1 /



.y2 // D y 1 C y2 ;

!.



.y2 / .y1 // D y 1  y2 ;

y1 ; y2 2 Y :

One has the following analog of Proposition 4.14. Proposition 17.3. Let ˙ 2 Lh .Y ; Y  /. Then the following statements are equivalent: (1) ˙ are the covariances of a gauge invariant quasi-free state on CAR.Y ; /; (2) ˙  0 and C C  D . Let us note an important difference with the bosonic case. Since  > 0, one can always consider the completion .Y cpl ; / of .Y ; / and uniquely extend any quasi-free state ! to CAR.Y cpl ; /. This is related to the fact that the -algebra CAR.Y ; / can be equipped with a unique C  -norm, see e.g. [DG, Proposition 12.50]. Therefore, if necessary, one can assume that .Y ; / is a Hilbert space. Let us conclude this section with the characterization of pure quasi-free states, see e.g. [DG, Theorem 17.31]. Proposition 17.4. A quasi-free state ! on CAR.Y ; / is pure iff there exist projections c ˙ 2 L.Y / such that ˙ D  ı c ˙ ;

c C C c  D ½:

Note that c ˙ are bounded selfadjoint projections on .Y ; /.

17.2 Clifford algebras We now collect some standard facts about Clifford algebras. For simplicity, we will only discuss the case of Lorentzian signature. Let X be an n-dimensional real vector space and  2 Lh .X ; X 0 / be a symmetric non-degenerate bilinear form of signature .1; d /.

17.3 Clifford representations

189

Definition 17.5. The Clifford algebra Cliff.X ; / is the abstract real algebra generated by the elements .x/, x 2 X , and the relations .x1 C x2 / D .x1/ C .x2 /; .x1 /.x2/ C .x2/.x1 / D 2x1 x2 ½;

x1 ; x2 2 X ;  2 R:

As a vector space Cliff.X ; / is isomorphic to ^X . Cliff.X ; / has an involutive automorphism ˛ defined by ˛..x// D .x/, which defines a Z2 -grading Cliff.X ; / D Cliff0 .X ; / ˚ Cliff1 .X ; /. The set Cliff0 .X ; / of elements of even degree is a sub-algebra of Cliff.X ; /. The Clifford algebras Cliff.0/ .R1;d / will be simply denoted by Cliff.0/ .1; d /.

17.2.1 Volume element. Let .x1 ; : : : ; xn / be an orthonormal basis of .X ; /, i.e. such that x1  x1 D 1, xi  xi D 1 for 2  i  n. In particular, this fixes an orientation of X . Set  D .x1/    .xn /I  is called the volume element and is independent of the choice of the oriented orthonormal basis .x1 ; : : : ; xn /. One has  ½; if n 2 f0; 1g mod 4; .x/ D .1/nC1.x/; 2 D (17.2) ½; if n 2 f2; 3g mod 4:

17.2.2 Pseudo-Euclidean group. Each r 2 O.X ; / induces an automorphism

rO of Cliff.X ; /, defined by

r..x// O D .rx/;

x 2 X:

The map O.X ; / 3 r 7! rO 2 Aut.Cliff.X ; // is a group morphism. More generally, if r W .X ; / ! R1;d is orthogonal, then it induces an isomorphism rO W Cliff.X ; / ! Cliff.R1;d /.

17.3 Clifford representations Let S a complex vector space. A morphism  W Cliff.X ; / ! L.S / is called a representation of Cliff.X ; / in S . It is called faithful if it is injective. It is called irreducible if ŒB; .A/ D 0 for all A 2 Cliff.X ; / implies B D ½S for  2 C. We set  .x/ D ..x// for x 2 X . Let { 2 f1; ig such that  { D i; if n 2 f0; 1g mod 4; (17.3) 2 D { 2 ½; i.e. { D ½; if n 2 f2; 3g mod 4:

190

17 Dirac fields on curved spacetimes

Proposition 17.6. (1) Assume that n D 2m is even. Then there is a unique up to equivalence, faithful and irreducible representation of Cliff.X ; /, called the Dirac representation in a space S of dimension 2m , whose elements are called Dirac spinors. One has C ˝ .Cliff.X ; // D End.S /. Setting H D {./, we have H 2 D ½ and ŒH; .Cliff0 .X ; // D 0. Setting We=o D f 2 S W H D ˙ g, the representation  restricted to Cliff0 .X ; / splits as the direct sum C ˚  of two irreducible representations on We=o . The elements of We=o are called even/odd Weyl spinors. (2) Assume that n D 2m C 1 is odd. Then there is a unique up to equivalence, faithful and irreducible representation of Cliff0 .X ; /, called the Pauli representation in a space S of dimension 2m , whose elements are called Pauli spinors. Setting ./ D { ½, the representation of Cliff0 .X ; / extends to an irreducible representation  of Cliff.X ; / in S . One has C ˝ .Cliff.X ; // D C ˝ .Cliff0 .X ; // D End.S /. The representations  ı ˛ and  are not equivalent, and none of them is faithful. If n is odd then Cliff0 .X ; / D Cliff0 .X ; / D Cliff1 .X ; /, which is used in (2) of Proposition 17.6 to extend  from Cliff0 .X ; / to Cliff.X ; /. In the sequel  will denote a representation of Cliff.X ; / as in Proposition 17.6, which will be called a spinor representation. We have C ˝ .Cliff.X ; // D End.S /:

(17.4)

17.3.1 Charge conjugations. Let  a spinor representation. Proposition 17.7. (1) Assume that n is even. Then there exists  2 End.SR / antilinear such that  .x/ D  .x/ and  2 D ½ if n 2 f2; 4g mod 8,  2 D ½ if n 2 f0; 6g mod 8. (2) Assume that n is odd. Then there exists  2 End.SR / anti-linear such that  .x/ D .1/.nC1/=2  .x/ and  2 D ½ if n 2 f1; 3g mod 8,  2 D ½ if n 2 f5; 7g mod 8. We refer, e.g. to [DG, Theorem 15.19] for the proof. An anti-linear map  as above is called a charge conjugation, with some abuse of terminology if  2 D ½ (if  2 D ½, then S becomes a quaternionic vector space). Later on we will be only interested in the existence of a true charge conjugation, i.e. with  2 D ½, which is the case iff n 2 f1; 2; 3; 4g mod 8. We have .x/ D .x/ iff n 2 f1; 2; 4g mod 8, .x/ D .x/ iff n D 3 mod 8. If ; Q are two such charge conjugations, then  1 Q 2 Aut.S / (in particular, it is C-linear) and commutes with  .x/ for all x 2 X . Since  is irreducible, we have Q D ,  2 C and from  2 D Q 2 we obtain that  D 1.

191

17.3 Clifford representations

Let us denote by C./ the set of charge conjugations in Proposition 17.7. By the above discussion, we have C./ S1 ; (17.5) or, more pedantically, the group S1 acts freely and transitively on C./.

17.3.2 Positive energy Hermitian forms. Proposition 17.8. Let us equip .X ; / with an orientation and a time orientation, so that .X ; / R1;d . Let  W Cliff.X ; / ! End.S / be a spinor representation. Then there exists a Hermitian form ˇ 2 Lh .S; S  / such that   .x/ˇ D ˇ .x/;

x 2 X ; iˇ .e/ > 0;

for all time-like, future directed e 2 X . Hermitian forms ˇ as above are called positive energy Hermitian forms. Proof. Let us fix a positively oriented orthonormal basis .e0 ; e1 ; : : : ; en / of .X ; / with e0 time-like and future directed. We set 0 D i .e0 /;

j D  .ej /;

1  j  n:

From the j we obtain an irreducible representation of Cliff.Rn /, defined as in Definition 17.5 with  replaced by the Euclidean scalar product on Rn . It is well known that one can equip S with a positive definite scalar product  2 Lh .S; S  / such that j D j for this scalar product. Setting ˇ D i ı 0 , we obtain that j ˇ D ˇj and iˇ0 > 0. Let now e 2 X be time-like future directed. We can assume that ee D 1, and hence there exists r 2 SO " .X ; / such that e D re0 . It is well known that there exists an element U of the restricted spin group Spin" .X ; /, see Section 17.4, such that .rx/ D U.x/U 1 , for x 2 X . Denoting by A the adjoint of A 2 End.S / for the Hermitian form ˇ, one then checks that .rx/ D U  .x/.U  /1 hence U U  D ˙½. Since Spin" .X ; / is connected, we have U U  D ½. Now we have  .e/ D U .x0 /U  , hence  iˇ .e/ > 0. As in Subsection 17.3.1, we denote by B./ the set of positive energy Hermitian forms on S . Then the same argument yields B./ RC ;

(17.6)

with the same meaning that the group RC acts freely and transitively on B./.

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17 Dirac fields on curved spacetimes

17.4 Spin groups The spin group Spin.X ; / is the group Spin.X ; / D f.x1 /    .x2p / W xi xi D ˙1; p 2 Ng  Cliff.X ; /: The restricted spin group Spin" .X ; / is the connected component of ½ in Spin.X ; /. One can show that a D .x1 /    .x2p / belongs to Spin" .X ; / iff the number of indices i , 1  i  p, with xi xi D 1 is even. The spin groups Spin."/ .R1;d / will be simply denoted by Spin."/ .1; d /. If a 2 Spin."/ .X ; /, then a .x/a1 D  .Ad.a/x/;

Ad.a/ 2 SO ."/ .X ; /;

(17.7)

and we have the exact sequence of groups: Ad

1 ! Z2 ! Spin."/ .X ; / ! SO ."/ .X ; / ! 1: Let us fix a spinor representation 0 W Cliff.1; d / ! L.S0 / (recall that S0 is a complex vector space of dimension 2Œn=2 ). We denote 0 ..v// by 0 .v/ for v 2 R1;d and identify Spin" .1; d / with its image in L.S0 /. We fix a positive energy Hermitian form ˇ0 and a charge conjugation 0 on S0 . One can show that Spin" .1; d / is the set of elements a 2 GL.S0 / such that .i/ a ˇ0 a D ˇ0 ; a0 D 0 a; .ii/ a0 .v/a1 D 0 .Ad.a/v/; 8v 2 R1;d :

(17.8)

This characterization of Spin" .1; d / inside GL.S0 / is independent on the choice of ˇ 0 ; 0 .

17.5 Weyl bi-spinors Let us assume that n D 4, and let  W Cliff.X ; / ! End.S / be a spinor representation, so that dimC S D 4. To simplify notation, we denote .A/ simply by A for A 2 Cliff.X ; /. Let  be a charge conjugation as in Proposition 17.7 and let ˇ 2 Lh .S; S  / be a positive energy Hermitian form as in Proposition 17.8. Recall that .x/ D .x/;

 2 D ½;

  .x/ˇ D ˇ.x/;

iˇ.e/ > 0 for e 2 X future directed time-like:

(17.9)

If  is the volume element we have 2 D ½;  ˇ D ˇ, hence H D i satisfies H D ½; H  ˇ D ˇH . We recall that S D We ˚ Wo for We=o D Ker.H ½/. 2

193

17.5 Weyl bi-spinors

Since  D  we have H D H  hence dimC We=o D 2 and 

 W We=o ! W o=e :

(17.10)

We obtain also that ue=o ; ve=o 2 We=o

ue=o ˇve=o D 0; hence



 ˇ D We=o ! Wo=e :

(17.11) (17.12)

Let ˇQ D   ˇ 2 Lh .S; S  /, i.e. Q 2 D v 2 ˇv1 ; vN 1  ˇv

v1 ; v2 2 S:

Q for x 2 X . Moreover, we have From (17.9) we obtain that .x/ ˇQ D ˇ.x/ Q iˇ.e/ D i  ˇ.e/ D   iˇ.e/ < 0; if e 2 X is future directed time-like, using that  and hence   is anti-linear and that Œ; .e/ D 0. Therefore, by (17.6), we have ˇQ D ˛ˇ; ˛ 2 R . Using that  2 D ½ we obtain that ˛ 2 D 1, hence vN 2 ˇv1 D vN 1 ˇv2 ;

vi 2 S:

(17.13)

17.5.1 Weyl bi-spinors. We know that S D We ˚ Wo , but we can use ˇ to obtain a different decomposition. We introduce the space of Weyl spinors: S D We ; and identify linearly S with S ˚ S0 by the map S3

7!

e

˚

o

DW  ˚  2 S ˚ S0 ;

where D e ˚ o with e=o 2 We=o . We have D  ˚ . The space S is canonically equipped with the symplectic form 1  D p .ˇ/1 2 L.S; S0 /: 2 The fact that  is anti-symmetric follows from (17.13), and Ker  D f0g since Ker ˇ D f0g.

17.5.2 Another identification. We can identify X with La .S ; S/ as real vector spaces by

X 3 x 7! ˇ.x/ 2 La .We ; We /:

(17.14)

194

17 Dirac fields on curved spacetimes

This map is injective, since  is faithful, and since both spaces have the same dimension, it is bijective. By complexification we obtain an isomorphism T W CX 3 z 7! ˇ.z/ 2 L.We ; We / We ˝ We0 D S ˝ S:

(17.15)

In the next proposition we still denote by  2 Ls .CX ; .CX /0 / the bilinear extension of . Proposition 17.9. The map 

T W .CX ; /!.S ˝ S;  ˝ / is an isomorphism, i.e.

T 0 ı . ˝ / ı T D :

(17.16)

Proof. Let a.x/ D .x/ 2 L.We ; W e /. Since a.x/2 D x  x ½, we have .det a.x//2 D .x  x/2 , hence det a.x/ D ˙x  x, p where the sign ˙ is independent on x by connectedness. Note also that a.x/ D 2 ı ˇ.x/. Let B D .s1 ; s2 / be a symplectic basis of S with s1 s2 D 1. We denote by B 0 the dual basis of S0 and by B the basis B considered as a basis of S. Computing the determinants of a.x/,  and ˇ.x/ in the above bases, we obtain that 2 det ˇ.x/ D 2 det ˇ.x/ det  D det a.x/ D ˙x x. Since iˇ.e/ > 0 for e 2 X time-like and future directed, we have det ˇ.e/ < 0 so det a.e/ D ee and det a.x/ D x x for all x 2 X . P If Œj k .x/ is the matrix of ˇ.x/ in B 0 ; B, so that T .x/ D j;k j k .x/s j ˝ sk , we check that hT .x/j. ˝ /T .x/i D 2 detŒˇ.x/ D det a.x/ D x x. 

17.6 Clifford and spinor bundles In this section and the next two we will use notions on fiber bundles, recalled in Section 5.1. Let .M; g/ be an orientable and time orientable Lorentzian manifold. After fixing an orientation and a time orientation of M , we can assume that the transition maps oij of TM , see Subsection 5.1.7, take values in SO " .R1;d /. Equivalently, one can view oij as the transition maps of the principal bundle Fr"on .TM / of oriented and time oriented orthonormal frames of TM . Definition 17.10. The Clifford bundle Cliff.M; g/ is the bundle over M with typical fiber Cliff.R1;d / defined by the transition maps oO ij 2 Aut.Cliff.R1;d //, where oij W Uij ! SO " .R1;d / are the transition maps of TM . Note that Cliff.M; g/ is a bundle of algebras.

17.6 Clifford and spinor bundles

195 

Definition 17.11. Let .M; g/ a Lorentzian manifold. A complex vector bundle S  ! M is a spinor bundle over .M; g/ if there exists a morphism  W Cliff.M; g/ ! End.S / of bundles of algebras over M such that for each x 2 M the map x W Cliff.Tx M; gx / ! End.Sx / is a spinor representation. Let us fix a spinor representation 0 W Cliff.1; d / ! L.S0 /, a positive energy Hermitian form ˇ0 and a charge conjugation 0 on S0 as at the end of Section 17.4. 

! M be a spinor bundle over M . Then one can assume that Lemma 17.12. Let S  its transition maps tij W Uij ! GL.S0 / satisfy: tij ı 0 .a/ ı tij1 D 0 .oO ij .a//; a 2 Cliff.1; d / on Uij :

(17.17)

Proof. By Subsections 5.1.2 and 5.1.11, we deduce from the existence of the bundle morphism  that there exist i W Ui ! Hom.Cliff.1; d /; L.S0// such that tij ı j .a/ ı tij1 D i .oO ij .a//; a 2 Cliff.1; d /: By irreducibility of the spinor representation, there exists Vi W Ui ! GL.S0 / such that i .a/ D Vi ı 0 .a/ ı Vi1 ; a 2 Cliff.1; d /: Let us set tQij D Vi1 ı tij ı Vj . We check that tQij satisfy (17.17) and note that changing tij to tQij corresponds by Subsection 5.1.2 to a vector bundle isomorphism. This completes the proof of the lemma. 

17.6.1 The bundles B./ and C./. Let B.0 /, resp. C.0 /, the sets of positive energy Hermitian forms, resp. of charge conjugations, associated to 0 , see Subsections 17.3.1 and 17.3.2. 

! M be a spinor bundle and  W Cliff.M; g/ ! End.S / Definition 17.13. Let S  the associated morphism.  The bundle B./  ! M is the bundle with typical fiber B.0 / and transition maps ˇ 7! tij ˇtij ; ˇ 2 B.0 /: 

! M is the bundle with typical fiber C.0 / and transition maps The bundle C./   7! tij1 tij ;  2 C.0 /:

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17 Dirac fields on curved spacetimes

Note that using that tij1 0 .v/tij D 0 .oij v/ for v 2 R1;d , we obtain that the transition maps above preserve the fibers. By the definition of B./ and C./, we immediately obtain the following proposition. Proposition 17.14. There exist canonical bundle morphisms B./ ! End.S ; S  /; C./ ! End.S ; SN /: From Subsections 17.3.2 and 17.3.1, we see that B./, resp. C./ are principal bundles over M with fiber RC , resp. S1 . Being principal, these bundles are trivial iff they admit a global section. Remark 17.15. Local sections of B./ can be pieced together using a partition of unity on M , since the set B.0 / is convex. Therefore B./ is a trivial bundle.

17.7 Spin structures Next, let us explain the relationship between spin structures and spinor bundles, following [T]. Definition 17.16. A spin structure on M is a Spin" .1; d /-principal bundle 

!M Spin.M /  with a bundle map  W Spin.M / ! Fr"on .TM / such that 8a 2 Spin" .1; d /; q 2 Spin.M / one has .qa/ D .q/Ad.a/:

(17.18)

We recall that a principal bundle admits a right action of its structure group, see Subsection 5.1.5, which is used in (17.18). If sij W Uij ! Spin" .1; d / are the transition maps of Spin.M / and oij W Uij ! SO " .1; d / are the transition maps of Fr"on .TM /, (17.18) means that oij .x/ D Ad.sij /.x/;

x 2 Uij :

Theorem 17.17. Let .M; g/ be an orientable and time-orientable Lorentzian man ! M be a spin structure over .M; g/. Then there exists ifold and let Spin.M /   ! M with canonical global sections ˇ;  of the buna canonical spinor bundle S  dles B./; C./. 

! M is a spinor bundle over Remark 17.18. Conversely, one can show that if S  .M; g/ such that the bundle C./ is trivial, then M admits a spin structure  Spin.M /  ! M . The two constructions are inverse to one another, modulo bundle isomorphisms.

17.8 Spinor connections

197

Proof. Recall that sij W Uij ! Spin" .1; d / are the transition maps of Spin.M /. Let 

S ! M be the vector bundle with typical fiber S0 and transition maps tij D 0 .sij / W Uij ! GL.S0 /: We define the bundle morphism  W Cliff.M; g/ ! End.S / by i D 0 W Ui ! Hom.Cliff.1; d /; L.S0 //; see Subsection 5.1.2. From (17.8) (ii), we obtain that  is indeed a morphism of bundles of algebras, ie that S is a spinor bundle over M . From (17.8) (i) and the definition of tij , we see that the local sections of B./, resp. C./ defined by ˇi .x/ D ˇ0 , resp. i .x/ D 0 for x 2 Ui can be patched together as global sections of B./, resp. C./. This completes the proof of the theorem. 

17.8 Spinor connections Let r be the L Levi-Civita connection on .M; g/. Since Cliff.M; g/ is a vector subbundle of nkD0 ˝k TM , r induces a unique connection r Cl , defined by Cl .Y / D .rX Y /; X; Y 2 C 1 .M I TM /: rX

Since r is metric for g, r Cl is adapted to the algebra structure of Cliff.M; g/, i.e. Cl Cl Cl rX ..Y1/.Y2 // D rX .Y1 /.Y2/ C .Y1 /rX .Y2 /: 

! M be a spinor bundle and let us denote ..X // simply by .X / Let now S  for X a vector field on M . One can show, see [T], that there exists a (non unique) connection r S on S such that S C .Y /rX ; X; Y 2 C 1 .M I TM /;

S ..Y / / D .rX Y / rX

2 C 1 .M I S /:

The following result is shown in [T, Proposition 9]. 

Theorem 17.19. Let S  ! M a spinor bundle. Assume that the bundle C./ is trivial. Then given a section ˇ 2 C 1 .M I B.// and a section  2 C 1 .M I C.//, there exists a unique connection r S on S such that S .i/ rX ..Y / / D .rX Y /

S C .Y /rX ;

S S .ii/ X.. jˇ // D .rX jˇ / C . jˇrX /; S

S

.iii/ rX . / D rX ; for all X; Y 2 C 1 .M I TM / and

2 C 1 .M I S /.

(17.19)

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17 Dirac fields on curved spacetimes 

From Theorem 17.17 we see that if Spin.M /  ! M is a spin structure over M ,  ! M , canonical sections ˇ;  and spin then there exists a canonical spinor bundle S  connection r S .

17.9 Dirac operators In the rest of this chapter we will assume that the hypotheses of Theorem 17.19 are satisfied. One defines a Dirac operator, acting on smooth sections of S as follows: let U  M a chart open set for S and the bundle of frames Fr.TM /. Choose sections e , 1    n of Fr.TM / over U , i.e. .e1 .x/;    en .x// is a ordered basis of Tx M for x 2 M (not necessarily orthogonal). We define = D g  .e /reS ; D = C m.x/ DDD

(17.20)

where r S is the connection on S from Theorem 17.19 and m 2 C 1 .M I End.S // is such that m ˇ D ˇm where ˇ is the section of B./ in Theorem 17.19. Such an operator will be called a Dirac operator.

17.9.1 Characteristic manifold. Denoting by X D .x; / the elements of T  M no, the principal symbol d.x; / of D is the section of C 1 .T  M n oI End.S //, homogeneous of degree 1 in , given by d.x; / D .g 1.x//: From the Clifford relations we obtain that d 2 .x; / D  g 1 .x/ ½:

(17.21)

The characteristic manifold of D is Char.D/ D f.x; / 2 T  M n o W d.x; / is not invertibleg; and by (17.21) we have Char.D/ D f.x; / 2 T  M n o W  g 1 .x/ D 0g D N : As usual, we denote by N ˙ the two connected components of N .

17.9.2 Charge conjugation. Assume that the charge conjugation  satisfies  2 D

½, i.e. that n 2 f1; 2; 3; 4g mod 8 by Proposition 17.7. By (17.19), we have Œ; rXS  D

0. Assuming also that m is real, i.e. Œm;  D 0, we obtain that D D D

if n 2 f1; 2; 4g mod 8;

D D D if n D 3 mod 8 and m D 0:

17.9 Dirac operators

17.9.3 Conserved current. Let J.

1;

1;

1  2 / 2 C .M I T M / by

J.

1;

2 /X

D

1 ˇ.X /

2

2;

2 C 1 .M I S /.

199

Define the 1-form

X 2 C 1 .M I TM /:

The following lemma follows easily from (17.19). Lemma 17.20. We have r  J .

1;

2/

D D

1 ˇ

2

C

1 ˇD

2;

i

2 C 1 .M I S /:

Proposition 17.21. The Dirac operator D is formally selfadjoint on C01 .M I S / with respect to the Hermitian form ˆ  . 1 j 2 /M D (17.22) 1 ˇ 2 d Volg : M

Proof. We apply the identity r J g D´d.J  y g´/, where g is the volume form on .M; g/, and the Stokes formula (5.11) U d! D @U ! to ! D J  y g , U b M an open set with smooth boundary, containing supp i .  

17.9.4 Decomposition of the Dirac operator. Let us assume that n D 4 and that m in (17.20) is scalar, i.e. m.x/ D m.x/½ for m 2 C 1 .M I R/. Section 17.5 provides a section H 2 C 1 .M I End.S // locally defined by H D i.e1 /    .e4 /, where .e1 ; : : : ; e4 / is an oriented orthonormal frame of TM . We have H 2 D ½; H .X / D .X /H; X 2 C 1 .M I TM /: Using (17.19), the fact that r is metric for g, and the Clifford relations, one can prove = D H D. = that r Cl H D 0, which implies that DH 1 Using Pe=o D 2 .1 ˙ H /, we can construct the vector bundles We=o D Pe=o S and identify C 1 .M I S / with C 1 .M I We / ˚ C 1 .M I Wo /. The Dirac operator becomes   =o m D = e=o D .g  .e /re /C 1 .M IWe=o / : (17.23) DD ; with D =e m D By Subsection 17.9.2, there exists a charge conjugation  with  2 D ½ and D D  D,  W We=o  ! W o=e , and we obtain that = e=o D  D = o=e : D

(17.24) 



As in Subsection 17.5.1, we identify S  ! M with S ˚ S0  ! M and a section 1 1  2 C .M I S / with .; / 2 C .M I S / ˚ C 1 .M I S0 /. We can rewrite the Dirac equation = Cm D0 D as ( = C pm  1  D 0; ˇ D 2 (17.25) =  0 ˇ D C pm N 1  D 0: 2

200

17 Dirac fields on curved spacetimes

17.10 Dirac equation on globally hyperbolic spacetimes Assume now that .M; g/ is a globally hyperbolic spacetime. We denote by Solsc .D/ the space of smooth, space compact solutions of the Dirac equation D

D 0:

17.10.1 Retarded/advanced inverses. Since .M; g/ is globally hyperbolic, D admits unique retarded/advanced inverses Gret=adv W C01 .M I S / ! Csc1 .M I S / such that ( DGret=adv D Gret=adv D D ½; supp Gret=adv u  J˙ .supp u/;

u 2 C01 .M I S /;

see eg [DG, Theorem 19.61]. Using the fact that D is formally selfadjoint with respect to .j/M and the uniqueness of Gret=adv we obtain that  D Gadv=ret ; Gret=adv

where the adjoint is computed with respect to .j/M . Therefore, the causal propagator G D Gret  Gadv satisfies

8 DG D GD D 0; ˆ < supp Gu  J.supp u/; ˆ :  G D G:

u 2 C01 .M I S /;

(17.26)

17.10.2 The Cauchy problem. Let †  M be a smooth, space-like Cauchy surface and denote by n its future directed unit normal and by S† the restriction of the spinor bundle S to †, so that † W C 1 .M I S / 3 is surjective. The Cauchy problem  D D 0; † D f;

7!

† 2 C 1 .†I S† /

f 2 C01 .†I S† /;

is globally well-posed, the solution being denoted by D U† f . From [DG, Theorem 19.63], we obtain that ˆ U† f .x/ D  G.x; y/.n.y//f .y/d Volh ; (17.27) †

where h is the Riemannian metric induced by g on †.

17.11 Quantization of the Dirac equation

201

We equip C01 .†I S† / with the Hermitian form .f1 jf2 /† D

ˆ †

f 1 ˇf2 d Volh :

(17.28)

 For g 2 E 0 .†I S† /, we define † g 2 D 0 .M I S / by

ˆ M

 † gˇu d Volg D

ˆ †

gˇ† ud Volh ; u 2 C 1 .†I S† /;

 is the adjoint of † with respect to the scalar products .j/M and .j/† . We i.e. † can rewrite (17.27) as

U† f D .† G/ .n/f;

f 2 C01 .†I S† /:

(17.29)

17.11 Quantization of the Dirac equation For

1;

2 Solsc .D/ we set ˆ  iJ . 1 ; 1  2 D 2

2 /n



d Volh D .†

1 ji.n/†

2 /† :

(17.30)

†

Since r  J . 1 ; 2 / D 0, the right-hand side of (17.28) is independent on the choice of †, and  is a positive definite scalar product on Solsc .D/. Setting ˆ f 1 † f2 D i f 1 ˇ.n/f2d Volh ; †

we obtain that † W .Solsc .D/; / ! .C01 .†I S† /; † / is unitary, with inverse U† . We also get that G W C01 .M I S / ! Solsc .D/ is surjective with kernel DC01 .M I S / and, see e.g. [DG, Theorem 19.65], that  GW

C01 .M I S / ; i.jG/M DC01 .M I S /

 ! .Solsc .D/; /

is unitary. Summarizing, the maps

C01 .M IS / ; i.jG/M DC01 .M IS /

are unitary.



G

†

! .Solsc .D/; / ! .C01 .†I S† /; † /

(17.31)

202

17 Dirac fields on curved spacetimes

17.12 Hadamard states for the Dirac equation We denote by CAR.D/ the -algebra CAR.Y ; / for .Y ; / one of the equivalent pre-Hilbert spaces in (17.31). We use the Hermitian form .j/M in (17.22) to pair C01 .M I S / with D 0 .M I S / and to identify continuous sesquilinear forms on C01 .M I S / with continuous linear maps from C01 .M I S / to D 0 .M I S /. Thus, a quasi-free state ! on CAR.D/ is defined by its spacetime covariances ƒ˙ which satisfy .i/ ƒ˙ W C01 .M I S / ! D 0 .M I S / are linear continuous; .ii/ ƒ˙  0 with respect to .j/M ; .iii/ ƒC C ƒ D iG;

(17.32)

.iv/ D ı ƒ˙ D ƒ˙ ı D D 0: Alternatively, one can define ! by its Cauchy surface covariances ˙ † , which satisfy 1 0 .i/ ˙ † W C0 .†I S† / ! D .†I S† / are linear continuous;

.ii/ ˙ †  0 for .j/† ; .iii/

C †

C

 †

(17.33)

D i.n/:

Using (17.29) one can show as in Proposition 6.6 that ƒ˙ D .† G/ ˙ † .† G/;   ˙  ˙ † D .† .n// ƒ .† .n//:

(17.34)

By the Schwartz kernel theorem, we can identify ƒ˙ with distributional sections in D 0 .M  M I S  S /, still denoted by ƒ˙ . The wavefront set of such sections is defined in the natural way: choosing a local trivialization of S  S , one can assume that S  S is trivial with fiber Mp .C/ for p D rank S , and the wavefront set of a matrix valued distribution is simply the union of the wavefront sets of its entries. We recall that N ˙ are the two connected components of N , see Subsection 17.9.1. Definition 17.22. ! is a Hadamard state if WF.ƒ˙ /  N ˙  N ˙ : The following version of Proposition 11.1 gives a sufficient condition for the Cauchy surface covariances ˙ † to generate a Hadamard state. Its proof is analogous, using (17.34). ˙ ˙  Proposition 17.23. Let ˙ are linear continuous from † D i.n/c , where c 1 1 0 C0 .†I S† / to C .†I S† / and from E .†I S† / to D 0 .†I S† /. Assume that

WF.U† ı c ˙ /0  N ˙  .T  † n o/; over U  †; for some neighborhood U of † in M . Then ! is a Hadamard state.

17.13 Conformal transformations

203

The existence of Hadamard states for Dirac equations on globally hyperbolic spacetimes can be shown by the same deformation argument as in the Klein–Gordon case, see e.g. [Ho1].

17.13 Conformal transformations Let c 2 C 1 .M / with c.x/ > 0 and gQ D c 2 g. If .X Q / are the generators of Cliff.M; g/, Q we have Q .X / D c.X /. eS on S for the metric gQ we need to fix a HerTo define the spinor connection r mitian form ˇQ and a charge conjugation . Q It is natural to choose Q D , but several choices of ˇQ are possible. The choice that we will adopt is ˇQ D c 1 ˇ which has the advantage that if n D 4 the isomorphism T in Proposition 17.9 is unchanged. From Theorem 17.19 we deduce that eS D r S C 1 c 1 .X /.rc/  c 1 X dc ½: r X X 2 =Q is the associated Dirac operator, we have If D = n=21 : =Q D c n=2 Dc D

(17.35)

Equivalently, if we introduce the map W W C01 .M I S / 3 Q 7! c n=21 Q 2 C01 .M I S /; and denote by .j/MQ the Hermitian form (17.22) with ˇ and d Volg replaced by ˇQ and d VolgQ , respectively, we have .

1 jW

Q 2 /M D .W 

1j

Q 2/ Q ; M

W

D c n=2 ;

(17.36)

and (17.35) can be rewritten as: =Q C c 1 m: DQ D W  DW D c n=2 Dc n=21 D D Q . We have then G D W GW Remark 17.24. The choice ˇQ D ˇ is often used in the mathematics literature. It leads to eS D r S C 1 c 1 .X /.rc/  1 c 1 X dc ½; r X X 2 2 = .n1/=2 : =Q D c .nC1/=2 Dc D

204

17 Dirac fields on curved spacetimes

17.13.1 Conformal transformations of phase spaces. Setting U W C01 .†I S† / 3 f 7! Uf D c 1n=2 f 3 C01 .†I S† /; we obtain the following analog of Proposition 6.10. Proposition 17.25. The following diagram is commutative, with all arrows unitary:



C01 .M IS / ; . jiG DC01 .M IS /

/M

Q IS / C01 .M ; . jiGQ 1 Q Q DC .M IS /

/MQ

? ?  yW

0





G

%†

Q G

%Q †

! .Solsc .D/; / ! .C01 .†I S† /; † / ? ? ? 1 ? yW yU Q / ! .Solsc .D/; Q ! .C01 .†I S† /; Q † /

17.13.2 Conformal transformations of quasi-free states. Let ƒ˙ be the spacetime covariances of a quasi-free state ! for D. Then Q ˙ D c 1n=2 ƒ˙ c n=2 ƒ

(17.37)

Q and are the spacetime covariances of a quasi-free state !Q for D,  1 ˙ 1 n=21 Q ˙ D c 1n=2 ˙ ; † D .U / † U †c

Q˙ Q if ˙ † , resp. † are the Cauchy surface covariances of !, resp. !.

17.14 The Weyl equation = D 0 and assume n D 4. AcWe consider now the massless Dirac equation D cording to Subsection 17.9.4, the Dirac equation decouples as two independent Weyl equations ( = D 0; ˇ D (17.38) =  0 ˇ D D 0: Let us set

= W C 1 .M I S / ! C 1 .M I S/: D D ˇ D

Note that D D D by Proposition 17.21.

17.14.1 Characteristic manifold. The characteristic manifold of D is Char.D/ D f.x; / 2 T  M n o W pr .D/.x; / not invertibleg:

17.14 The Weyl equation

It is easy to see that

Char.D/ D N :

205

(17.39)

of Indeed, fix x 2 M and choose a basis .w1 ; w2 / of W   ex . By (17.23), the matrix 0 de .x; / , where d.x; / in the basis .w1 ; w2 ; w1 ; w2 / of Sx equals 0 de .x; / de .x; / 2 M2 .R/. From (17.21) we obtain that de .x; /2 D  g 1 .x/ ½2 , which implies (17.39).

17.14.2 Retarded/advanced inverses. D has the retarded/advanced inverses Gret=adv D Gret=adv ˇ 1 W C01 .M I S/ ! Csc1 .M I S /; and the causal propagator G D Gret  Gadv D Gˇ 1 :  Let us denote by r† W C 1 .M I S / ! C 1 .†I S† / the trace on †, and by r† W   1 1 1 C .†I S† / ! C .M I S/ its adjoint, so that r† D ˇ† ˇ . We also set

.X / D ˇ.X / W C 1 .†; S† / ! C 1 .†I S† /: The Cauchy problem

(

D D 0; r†  D f 2 C01 .†I S† /;

has the unique solution

ˆ

 D U† f D 

G.x; y/ .n.y//f .y/d Volh ; †

or equivalently

U† D .r† G/ .n/:

We see that .Solsc .D/; / is a pre-Hilbert space, and from (17.31) we obtain the unitary maps:   r† C01 .M I S/ G (17.40) ; iG ! .Solsc .D/; / ! .C01 .†I S† /; † /: 1  DC0 .M I S /

17.14.3 Quasi-free states. As before, we denote by CAR.D/ the -algebra CAR.Y ; / for .Y ; / one of the equivalent pre-Hilbert spaces in (17.40). A quasifree state ! on CAR.Y ; / is defined by its spacetime covariances L˙ , which satisfy .i/ L˙ W C01 .M I S/ ! D 0 .M I S / are linear continuous; .ii/ L˙  0; .iii/ LC C L D iG; .iv/ DL˙ D L˙ D D 0:

(17.41)

206

17 Dirac fields on curved spacetimes

˙ Alternatively, one can define ! by its Cauchy surface covariances l† which satisfy: ˙ .i/ l† W C01 .†I S† / ! D 0 .†I S† / are linear continuous; ˙ .ii/ l†  0;

.iii/ One has

C l†

(17.42)

 C l† D i .n/: ˙ L˙ D .r† G/ l† .r† G/; ˙   l† D .r† .n//L˙ .r† .n//:

(17.43)

Here are the identities corresponding to those in Section 17.13, obtained by a conformal transformation gQ D c 2 g: Q D c 1n=2 Gc n=2C1 ; Q D c 1n=2Dc n=21 ; G D Q ˙ D c 1n=2 L˙ c n=2C1 ; lQ˙ D c 1n=2 l ˙ c n=21 : L † †

(17.44)

Definition 17.26. The state ! on CAR.D/ is a Hadamard state if WF.L˙ /0  N ˙  N ˙ : We have the following version of Proposition 17.23. ˙ Proposition 17.27. Let l† D i .n/c ˙, where c ˙ are linear continuous from 1  1  C0 .†I S† / to C .†I S† / and from E 0 .†I S† / to D 0 .†I S† /. Assume that

WF.U† ı c ˙ /0  N ˙  .T  † n o/ over U  †; for some neighborhood U of † in M . Then ! is a Hadamard state.

17.15 Relationship between Dirac and Weyl Hadamard states Finally, let us describe the relationship between Hadamard states for the Weyl and Dirac equations. Proposition 17.28. Let !D be a quasi-free Hadamard state for D with spacetime covariances L˙ . Then   0 L˙ ˇ ƒ˙ D 0 L ˇ = are the spacetime covariances of a quasi-free Hadamard state !D for D.

17.15 Relationship between Dirac and Weyl Hadamard states

207

Proof. We check (17.32). Condition (i) is obvious. We have .LC C L /ˇ D iGˇ D iG on C01 .M I Wo /, hence .LC C L /ˇ D iG D iG on C01 .M I We /, since G D G and  is anti-linear, which proves condition (iii). Condition (iv) is also immediate. To check the positivity condition (ii), we write using (17.13) and the fact that ˇ D ˇ  : . jˇƒ˙ / D .

˙ o jˇL ˇ

o/

.

D .

˙ o jˇL ˇ

o/

C .

D .

˙

o jˇL

ˇ

o/

 e jˇL ˇ

e/

 e jˇL ˇ

e/

C .ˇ



e jL

ˇ

e/

 0;

as needed. It remains to prove the Hadamard condition. The fact that WF.L˙ ˇ/0  N ˙  N ˙ follows from the Hadamard property of !D . This implies that WF.L˙ ˇ/  N   N  since  is anti-linear, and completes the proof that !D= is Hadamard.  The converse of Proposition 17.28 is much easier. Proposition 17.29. Let ƒ˙ be the spacetime covariances of a Hadamard state for ˙ 1 = Then setting ƒ˙ D. o D ƒ jC0 .M IWo / , the maps 1 L˙ D ƒ˙ o ˇ

are the covariances of a Hadamard state for D.

Bibliography [ALNV] B. Ammann, R. Lauter, V. Nistor, and A. Vasy, Complex powers and non-compact manifolds. Comm. in PDE 29 (2004), 671–705. [AW]

H. Araki and E.J. Woods, Representations of the canonical commutation relations describing a non-relativistic infinite free Bose gas. J. Math. Phys. 4 (1963), 637–662.

[B]

A. Bachelot, The Hawking effect. Annales I.H.P. Phys. Theor. 70 (1999), 41–99.

[BFr]

Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations (R. Brunetti and K. Fredenhagen, eds.). Springer Lecture Notes in Physics 786, 2009.

[BGP]

C. B¨ar, N. Ginoux, and F. Pf¨affle, Wave Equations on Lorentzian Manifolds and Quantization. ESI Lectures in Mathematics and Physics, EMS, 2007.

[BSt1]

C. B¨ar and A. Strohmaier, An index theorem for Lorentzian manifolds with compact space-like Cauchy boundary. preprint arXiv:1506.00959 (2015).

[BSt2]

C. B¨ar and A. Strohmaier, A rigorous geometric derivation of the chiral anomaly in curved backgrounds. Comm. Math. Phys. 347 (2016), 703–721.

[BDH]

M. Benini, C. Dappiagi, and T.-P. Hack, Quantum Field Theory on curved backgrounds—a primer. Int. J. of Mod. Physics 28 (2013), 1330023.

[BS1]

A. Bernal and M. Sanchez, On smooth Cauchy hypersurfaces and Gerochs splitting theorem. Comm. Math. Phys. 243 (2003), 461–470.

[BS2]

A. Bernal and M. Sanchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. 257 (2005), 43–50.

[BS3]

A. Bernal and M. Sanchez, Globally hyperbolic spacetimes can be defined as ‘causal’ instead of ‘strongly causal’. Class. Quantum Gravity 24 (2007), 745–749.

[BS4]

A. Bernal and M. Sanchez, Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions. Lett. Math. Phys. 77 (2006), 183–197.

[BD]

N. Birrell and P. Davies, Quantum Fields in Curved Space. Cambridge University Press, 1982.

[Bo]

J.M. Bony, Equivalence des diverses notions de spectre singulier analytique. S´eminaire Goulaouic-Schwartz, 1976–77.

[BR]

O. Bratteli and D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Volume 1. Springer-Verlag, second edition, Berlin, 1987.

[BI]

J. Bros and D. Iagolnitzer, Support essentiel et structure analytique des distributions. S´eminaire Goulaouic-Schwartz, 1975–76.

[BF]

R. Brunetti and K. Fredenhagen, Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds. Comm. Math. Phys. 208 (2000), 623– 661.

[BDFY] Advances in Algebraic Quantum Field Theory (R. Brunetti, C. Dappiagi, K. Fredenhagen, and J. Yngvason, eds.). Mathematical Physics Studies Springer, 2015. [BFK]

R. Brunetti, K. Fredenhagen, and M. K¨ohler, The microlocal spectrum condition and Wick polynomials of free fields on curved spacetimes. Comm. Math. Phys. 180 (1996), 633–652.

210

Bibliography

[CC]

Y. Choquet-Bruhat and Y. Cotsakis, Global hyperbolicity and completeness. J. Geom. and Phys. 43, (2002), 345–350.

[CG]

J. Cheeger and M. Gromov, Bounds on the von Neumann dimension of L2 -cohomology and the Gauss-Bonnet theorem for open manifolds. J. Diff. Geom. 21 (1985), 1–34.

[D]

N.V. Dang, Renormalization of quantum field theory on curved spacetimes, a causal approach. PhD thesis Universit´e Paris VI, arXiv:1312.5674 (2013).

[DMP1] C. Dappiagi, V. Moretti, and N. Pinamonti, Rigorous steps towards holography in asymptotically flat spacetimes. Rev. Math. Phys. 18 (2006), 349–405. [DMP2] C. Dappiagi, V. Moretti, and N. Pinamonti, Hadamard states from light-like hypersurfaces. preprint arXiv:1706.09666 (2017). [DMP3] C. Dappiagi, V. Moretti, and N. Pinamonti, Rigorous construction and Hadamard property of the Unruh state in Schwarzschild spacetime. Adv. Theor. Math. Phys. 15 (2011). 355–447. [DMP4] C. Dappiagi, V. Moretti, and N. Pinamonti, Distinguished quantum states in a class of cosmological spacetimes and their Hadamard property. J. Math. Phys. 50 (2009) 062304. [De]

J.M. Delort, F.B.I. Transformation, Second Microlocalization and Semilinear Caustics. Springer LNM 1522, 1992.

[DG]

J. Derezinski and C. G´erard, Mathematics of Quantization and Quantum Fields. Cambridge Monographs in Mathematical Physics, Cambridge University Press, 2013.

[Di1]

J. Dimock, Algebras of local observables on a manifold. Comm. Math. Phys. 77 (1980), 219–228.

[Di2]

J. Dimock, Dirac quantum fields on a manifold. Trans. Amer. Math Soc. 269 (1982), 133–147.

[DH]

J.J. Duistermaat and L. H¨ormander, Fourier integral operators. II. Acta Math. 128 (1972), 183–269.

[El]

J. Eldering, Persistence of non compact normally hyperbolic invariant manifolds in bounded geometry. PhD thesis Utrecht University, 2012.

[FP]

C. Fewster and M. Pfenning, A quantum weak energy inequality for spin-one fields in curved spacetime. J. Math. Phys. 44 (2003), 4480–4513.

[FH]

K. Fredenhagen and R. Haag, On the derivation of Hawking radiation associated with the formation of a black hole. Comm. Math. Phys. 127 (1990), 273–284.

[Fr]

F. Friedlander, The Wave Equation on a Curved Space-time. Cambridge University Press, Cambridge, 1975.

[FT]

T. Friedrich and A. Trautman, Spin spaces, Lipschitz groups, and spinor bundles. Ann. Global Anal. Geom., 18 (2000) 221–240.

[F]

S.A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time. Cambridge University Press, 1989.

[FNW]

S.A. Fulling, F.J. Narcowich, and R.M. Wald, Singularity structure of the two-point function in quantum field theory in curved spacetime, II. Ann. Phys. 136 (1981), 243–272.

[GHV]

J. Gell-Redman, N. Haber, and A. Vasy, The Feynman propagator on perturbations of Minkowski space. Comm. Math. Phys. 342 (2016), 333–384.

[G1]

C. G´erard, On the Hartle–Hawking–Israel states for spacetimes with static bifurcate Killing horizons. arXiv:1608.06739 (2016).

Bibliography

211

[G2]

C. G´erard, The Hartle–Hawking–Israel state on stationary black hole spacetimes. arXiv:1806:07645 (2018).

[GOW]

C. G´erard, O. Oulghazi, and M. Wrochna: Hadamard states for the Klein–Gordon equation on Lorentzian manifolds of bounded geometry. Comm. Math. Phys. 352 (2017), 519–583.

[GW1]

C. G´erard and M. Wrochna, Construction of Hadamard states by pseudodifferential calculus. Comm. Math. Phys. 325 (2014), 713–755.

[GW2]

C. G´erard and M. Wrochna, Construction of Hadamard states by characteristic Cauchy problem. Anal. PDE 9 (2016), 111–149.

[GW3]

C. G´erard and M. Wrochna, Hadamard property of the in and out states for Klein–Gordon fields on asymptotically static spacetimes. Ann. Henri Poincar´e, 18 (2017), 2715–2756.

[GW4]

C. G´erard and M. Wrochna, The massive Feynman propagator on asymptotically Minkowski spacetimes I. arXiv:1609.00192 (2016), to appear in Amer. Journal of Math.

[GW5]

C. G´erard and M. Wrochna, Analytic Hadamard states, Calder´on projectors and Wick rotation near analytic Cauchy surfaces. Comm. Math. Phys. 366 (2019), 29–65.

[GW6]

C. G´erard and M. Wrochna, The massive Feynman propagator on asymptotically Minkowski spacetimes II. arXiv:1806.05076 (2018), to appear in Int. Math. Res. Notices.

[Ge]

R. Geroch, Domain of dependence. J. Math. Phys. 11 (1970), 437–449.

[GJ]

J. Glimm and A. Jaffe, Quantum Physics. A Functional Integral Point of View. Springer, 1987.

[Ha]

D. H¨afner, Creation of fermions by rotating charged black holes. M´em. SMF 117 (2009).

[HH]

J. Hartle and S. Hawking, Path-integral derivation of black-hole radiance. Phys. Rev. D 13 (1976) 2188–2203.

[H]

S.W. Hawking, Particle creation by black holes. Comm. Math. Phys. 43 (1975), 199–220.

[Ho1]

S. Hollands, The Hadamard condition for Dirac fields and adiabatic states on RobertsonWalker spacetimes. Comm. Math. Phys. 216 (2001), 635–661.

[H1]

L. H¨ormander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis. Springer, Berlin Heidelberg New York, 1985.

[H2]

L. H¨ormander, The Analysis of Linear Partial Differential Operators III. Distribution Theory and Fourier Analysis. Springer, Berlin Heidelberg New York, 1985.

[H3]

L. H¨ormander, The Analysis of Linear Partial Differential Operators IV. Distribution Theory and Fourier Analysis. Springer, Berlin Heidelberg New York, 1985.

[H4]

L. H¨ormander, On the existence and regularity of solutions of linear pseudodifferential equations. Enseign. Math. 17 (1971), 99–163.

[H5]

L. H¨ormander, Uniqueness theorems and wavefront sets for solutions of linear differential equations with analytic coefficients. Comm. Pure Appl. Math. 24 (1971), 671–704.

[H6]

L. H¨ormander, A remark on the characteristic Cauchy problem. J. Funct. Anal. 93 (1990), 270–277.

[I]

Isozaki H., A generalization of the radiation condition of Sommerfeld for N-body Schr¨odinger operators. Duke Math. J. 74 (1994), 557–584.

[Is]

W. Israel, Thermo-field dynamics of black holes. Phys. Lett. 57 (1976), 107–110.

212

Bibliography

[J1]

W. Junker, Hadamard states, adiabatic vacua and the construction of physical states for scalar quantum fields on curved spacetime. Rev. Math. Phys. 8 (1996), 1091–1159.

[J2]

W. Junker, Erratum to “Hadamard states, adiabatic vacua and the construction of physical states . . . ”. Rev. Math. Phys. 207 (2002), 511–517.

[K]

A. Kaneko, Introduction to Hyperfunctions. Mathematics and Its Applications, Kluwer, Dordrecht, 1988.

[Ka]

T. Kato, Perturbation Theory for Linear Operators. Springer Classics in Mathematics 123, 1995.

[Kw]

T. Kawai, Construction of local elementary solutions for linear partial differential operators with real analytic coefficients. Publ. R.I.M.S. Kyoto Univ. 7 (1971), 363–397.

[Ky1]

B. Kay, Linear spin-zero quantum fields in external gravitational and scalar fields, I. Comm. Math. Phys. 62 (1978), 55–70.

[Ky2]

B. Kay, Purification of KMS states. Helv. Phys. Acta 58 (1985), 1030–1040.

[KW]

B.S. Kay and R.M. Wald, Theorems on the uniqueness and thermal properties of stationary, nonsingular, quasifree states on spacetimes with a bifurcate Killing horizon. Phys. Rep. 207 (1991), 49–136.

[Ko]

H. Komatsu, Microlocal analysis in Gevrey classes and in complex domains. In Microlocal Analysis and Applications (L. Cattabriga, L. Rodino, eds.). Lecture Notes Math. 1495, Springer, 1989.

[Kr]

Y. Kordyukov, Lp -theory of elliptic differential operators on manifolds of bounded geometry. Acta Appl. Math. 23 (1991), 223–260.

[LM]

H.B. Lawson and M.L. Michelsohn, Spin Geometry. Princeton University Press, 1989.

[La]

P.D. Lax, Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24 (1957), 627–646.

[Le]

J. Leray, Lectures on Hyperbolic Differential Equations. Princeton I.A.S., 1955.

[LRT]

P. Leyland, J. Roberts, and D. Testard, Duality for quantum free fields. preprint CPT (1978).

[Li1]

A. Lichnerowicz, Propagateurs et commutateurs en relativit´e g´en´erale. Publ. Math. IHES 10 (1961), 5–56.

[Li2]

A. Lichnerowicz, Champs spinoriels et propagateurs en relativit´e g´en´erale. Bull. Soc. Math. France 92 (1964), 11–100.

[Lu]

D. Ludwig, Exact and asymptotic solutions of the Cauchy problem. Comm. Pure. Appl. Math. 13 (1960), 473–508.

[MV]

J. Manuceau and A. Verbeure, Quasi-free states of the CCR-algebra and Bogoliubov transformations. Comm. Math. Phys. 9 (1968), 293–302.

[Mo1]

V. Moretti, Uniqueness theorem for BMS-invariant states of scalar QFT on the null boundary of asymptotically flat spacetimes and bulk-boundary observable algebra correspondence. Comm. Math. Phys. 268 (2006), 727–756.

[Mo2]

V. Moretti, Quantum out-states holographically induced by asymptotic flatness: invariance under spacetime symmetries, energy positivity and Hadamard property. Comm. Math. Phys. 279 (2008), 31–75.

[O]

B. O’Neill, Semi-Riemannian Geometry. With Applications to Relativity. Pure and Applied Mathematics 103, Academic Press, London, 1983.

Bibliography

213

[P]

F. Pham, Hyperfunctions and Theoretical Physics. Springer Lect. Notes in Math. 449, 1973.

[R1]

M. Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved spacetime. Comm. Math. Phys. 179 (1996), 529–553.

[R2]

M. Radzikowski, A local-to-global singularity theorem for quantum field theory on curved spacetime. Comm. Math. Phys. 180 (1996), 1–22.

[Re]

K. Rejzner, Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies, Springer, 2016.

[Ro]

J. Roe, An index theorem on open manifolds I. J. Diff. Geom. 27 (1988), 87–113.

[SV1]

H. Sahlmann and R. Verch, Passivity and microlocal spectrum condition. Comm. Math. Phys. 214 (2000), 705–731.

[SV2]

H. Sahlmann and R. Verch, Microlocal spectrum condition and Hadamard form for vector-valued quantum fields in curved spacetime. Rev. Math. Phys. 13 (2001), 1203– 1246.

[S]

M. Sanchez, On the geometry of static spacetimes. Nonlinear Anal. 63 (2005), 455–463.

[S1]

K. Sanders, A note on space-like and time-like compactness. Class. Quant. Grav. 30 (2013), 115014.

[S2]

K. Sanders, Thermal equilibrium states of a linear scalar quantum field in stationary spacetimes. Int. J. Modern Phys. A 28 (2013), 1330010.

[S3]

K. Sanders, On the construction of Hartle–Hawking–Israel states across a static bifurcate Killing horizon. Letters in Math. Phys. 105 (2015), 575–640.

[S4]

K. Sanders, The locally covariant Dirac field. Rev. Math. Phys. 22 (2010), 381–430.

[Se]

R. Seeley, Complex powers of an elliptic operator. In Singular Integrals. Proc. Symp. Pure Math., Amer. Math. Soc., Providence, RI, 1967, 288–307.

[SG]

J. Schmid and M. Griesemer, Kato’s theorem on the integration of non-autonomous linear evolution equations. Math. Phys. Anal. Geom. 17 (2014) 265–271.

[Sh1]

M. Shubin, Pseudodifferential Operators and Spectral Theory. Springer Series in Soviet Mathematics, Springer, 2001.

[Sh2]

M.A. Shubin, Spectral theory of elliptic operators on non-compact manifolds, Ast´erisque 207 (1992), 37–108.

[Si]

B. Simon, The P ./2 Euclidean (Quantum) Field Theory. Princeton University Press, 1974.

[Sj]

J. Sj¨ostrand, Singularit´es Analytiques Microlocales. Ast´erisque 95 (1982).

[St]

A. Strohmaier, On the local structure of the Klein–Gordon field on curved spacetimes. Lett. Math. Phys. 54 (2000), 249–261.

[SVW]

A. Strohmaier, R. Verch, and M. Wollenberg, Microlocal analysis of quantum fields on curved spacetimes: analytic wavefront sets and Reeh–Schlieder theorems. J. Math. Phys. 43 (2002), 5514–5530.

[SZ]

A. Strohmaier and S. Zelditch, A Gutzwiller trace formula for stationary spacetimes. preprint arXiv:1808.08425 (2018).

[T]

A. Trautman, Connections and the Dirac operators on spinor bundles. J. Geom. and Phys. 58 (2008), 238–252.

214

Bibliography

[Va]

A. Vasy, Essential self-adjointness of the wave operator and the limiting absorption principle on Lorentzian scattering spaces (2017), arXiv:1712.09650.

[U]

W.G. Unruh, Notes on black-hole evaporation. Physical Review D 14 (1976), 870–892.

[W1]

R.M. Wald, General Relativity. University of Chicago Press, 1984.

[W2]

R.M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics, University of Chicago Press, 1994.

General Index Cauchy surface covariances, 66 Cauchy temporal function, 54 causal complement, 68 causal curve, 50 causal future/past, 8, 51 causal propagator, 10, 59, 200 causal shadow, 8, 51 causal spacetime, 53 causal vector, 8, 48 causally compatible, 51 causally disjoint, 8 CCR -algebra, 25 Change of gauge, 145 characteristic function, 27 characteristic manifold, 77, 198, 205 bicharacteristic curves, 77 characteristic submanifold, 48 bifurcate Killing horizon, 160, 161 charge, 23, 100 bifurcation surface, 160 charge complex structure, 23 BMS group, 153 charge conjugation, 190 Bogoliubov transformation, 124 charge density, 32 Bondi frame, 153 charge reversal, 23 bosonic field, 9, 57 charge space, 122 bosonic Fock space, 15 charged CCR -algebra, 26 bounded atlas, 109 charged fields, 26 bounded differential operators, 110 classical energy, 94, 100 bounded embedding, 125 Clifford algebra, 188 bounded geometry, 109, 125 Clifford bundle, 194 bounded tensors, 109 Clifford representation, 189 bulk state, 147, 149 commutator function, 10 bulk-to-boundary correspondence, 146 complex covariances of a state, 32 bundle atlas, 43 conformal factor, 151 bundle morphism, 44 conformal frame, 151 bundle of frames, 46 conformal transformation, 50, 69, 88, 126, 203, 206 Calder´on projector, 137, 138, 140, 164 conformal wave equation, 151 CAR -algebra, 187 conformal wave operator, 57 Cauchy data, 12, 61 conic set, 71, 74 Cauchy evolution operator, 10, 14, 59, conical singularity, 165 60, 117 conjugate map, 22 Cauchy problem, 10, 59, 60, 117 Cauchy surface, 53 conjugate vector space, 21 abstract Sobolev spaces, 5 advanced/retarded inverses, 10, 59, 200, 205 advanced/retarded parametrices, 79 almost analytic extension, 73 analytic Hadamard state, 133 analytic wavefront set, 131 anti-commutator, 187 anti-dual, 21 asymptotic Klein–Gordon operator, 171 asymptotically flat spacetime, 151 asymptotically Minkowski spacetime, 178 asymptotically static spacetime, 169, 170

216

General Index

connection, 46 conormal bundle, 75 conserved current, 58, 199 cosphere bundle, 71 cotangent bundle, 45 covariance of a state, 27, 30 creation/annihilation operators, 16 d’Alembertian, 57 Dirac operator, 198 Dirac representation, 190 Dirac spinors, 190 distinguished parametrices, 79 domain of determinacy, 68 double KMS state, 165 doubling procedure, 35 edge of the wedge theorem, 130 Egorov theorem, 114 elliptic operator, 105, 107, 113 elliptic symbol, 104, 107, 111 energy shells, 78, 116 energy space, 95, 122, 179 Euclidean approach, 129 Euclidean metric, 165 evolution operator, 117 F.B.I. transform, 132 faithful representation, 189 fermionic fields, 187 Feynman inverse, 88, 90, 124, 179 Feynman parametrix, 79 fiber bundle, 43 gauge invariant state, 31, 188 gauge transformation, 31, 151 Gauss formula, 50 Gauss lemma, 90 Gaussian measure, 28 Gaussian normal coordinates, 55, 126 geodesic equation, 56 globally hyperbolic spacetime, 53 GNS construction, 25 Green’s formula, 58 ground state, 25, 93, 96, 102

Hadamard condition, 85, 115, 149 Hadamard state, 85 Hartle–Hawking–Israel state, 166 inextensible curve, 50 infrared problem, 96 K¨ahler structure, 17, 34 Killing vector field, 52 KMS state, 94, 96, 102 Laplace–Beltrami operator, 42, 108, 110, 163, 166 lapse function, 97 Levi-Civita connection, 47 lightcone, 8, 144 Lorentz group, 8 Lorentzian manifold, 48 Majorana fermions, 187 microlocal ellipticity, 77 Minkowski spacetime, 7 Minkowski vacuum, 19 Moyal product, 119 non trapping condition, 178 normal vector field, 49 null cone, 8 null geodesics, 77 orientable manifold, 45 parametrix, 78, 105, 113, 117 Pauli–Jordan function, 10 Poincar´e group, 8 positive energy Hermitian form, 191 principal bundle, 44 pseudo-Euclidean space, 22 pseudo-K¨ahler structure, 23 pseudodifferential operator, 104, 106 pure quasi-free state, 37, 38 pure state, 24 quasi-free state, 27, 30, 31, 188 Reeh–Schlieder property, 20, 40, 135 Riccati equation, 118 section of a bundle, 44

General Index

Seeley’s theorem, 105, 114 shift vector field, 97 Shubin’s calculus, 111 Sobolev spaces, 104, 107, 110 Space-time covariances, 65 spacetime, 50 spin group, 192 spin structure, 196 spinor bundle, 195 spinor connection, 197 spinor representation, 190 static spacetime, 52 stationary spacetime, 52 Stokes formula, 47 stress-energy tensor, 81 strong causality, 53 supertranslations, 154 surface gravity, 161

tangent bundle, 45 temporal function, 54 trivial bundle, 44 unique continuation, 68 Unruh effect, 159 vacuum state, 41 vector bundle, 45 volume density, 48 volume element, 189 volume form, 48 wavefront set, 72 Weyl CCR algebra, 26 Weyl equation, 204 Weyl spinors, 190, 193 Wick ordering, 83 Wick rotation, 136, 162

217

Index of Notations ˙ B 0, 77 ^p .M /, 45 a./ .h/, 16 B./, 191 m BSph .T  M /, 111 p BTq .M; g/, 109

Cadv , 78 Cb1 .I I ‰bm .M //, 113 CF , 78 C˙ , 138 Cret , 78 C./, 191 Csc1 .M /, 56 Ctc1 .M /, 56 C˙ .x/ , 77 CAR.D/, 202 CAR.D/, 205 CCR.P /, 65 CCRpol .HC ; qC /, 147 CCRpol .X ; /, 25 CCRpol .Y ; q/, 26 CCRR .P /, 67 CCRWeyl .X ; /, 26 Char.A/, 107 Char.a/, 104 Char.P /, 77 Cliff.1; d /, 189 Cliff.M; g/, 194 Cliff.X ; /, 189 D, 198 D, 204 = 198 D, D .O/, 68 0 Dsc .M /, 56 Dtc0 .M /, 56 d Volg , 48 Diffb .M /, 110

essupp.A/, 107 , 189 Exch. /, 74 F .x C i 0/, 130 Fr.E/, 46 G, 10, 59, 200 G, 205 GBMS , 153 g eucl , 162 Gret=adv , 10, 59, 200 Gret=adv , 205  ˙ , 137 M1 , 74 M2 , 74 s .h/, 15 .X /, 205 .x/, 189 0 , 74 H ad .t/, 175 HC , 146 Hp , 77 H s .M; g/, 110 I˙ .K/, 8, 51 Ja .u1 ; u2 /, 58 J˙ .K/, 8, 51 J. 1 ; y2 /, 199 ƒ˙ , 66 ˙ † , 66

N ˙ , 78 N , 75 !1 , 96 !ˇ , 96

g , 48 Op.a/, 104

220

Index of Notations

P .1; d /, 8 P eucl , 163 Pout=in , 171 F .h/, 17 .x/, 25 ‰bm .M /, 112 ‰cm .M /, 106 ‰ m .M /, 106 ‰ m .Rn /, 105 .y/, 26, 187 qC , 146 q† , 61 R1;d , 7 † , 59 S, 193 m .T  U /, 103 Sph m S .T  U /, 103

pr .A/, 105 SO " .1; d /, 8 Solscsc .D/, 200 Solscsc;C .KG/, 14 Solscsc .P /, 61 Solscsc;R .KG/, 9 Spin.M /, 196 Spin.X ; /, 192 Spin" .X ; /, 192

Tab ./,

´82 t Texp i s . /d , 114 TM , 45 T  M , 45 U ad .t/, 175 UH .t; s/, 117 U† , 60 U† , 205 .ujv/M , 49 vY , 72

We=o , 190 WF .h/, 17 W 1 .M /, 112 W .x/, 26 WFa u, 132 WFu, 71 X m , 179 XFm , 179 X 0 , 21 Y  , 21 Y , 21 Y m , 179

Christian Gérard

Lectures in Mathematics and Physics

Lectures in Mathematics and Physics

Christian Gérard

This book provides a detailed introduction to microlocal analysis methods in the study of Quantum Field Theory on curved spacetimes. We focus on free fields and the corresponding quasi-free states and more precisely on Klein–Gordon fields and Dirac fields. The first chapters are devoted to preliminary material on CCR*-algebras, quasi-free states, wave equations on Lorentzian manifolds, microlocal analysis and to the important Hadamard condition, characterizing physically acceptable quantum states on curved spacetimes. In the later chapters more advanced tools of microlocal analysis, like the global pseudo-differential calculus on non-compact manifolds, are used to construct and study Hadamard states for Klein–Gordon fields by various methods, in particular by scattering theory and by Wick rotation arguments. In the last chapter the fermionic theory of free Dirac quantum fields on Lorentzian manifolds is described in some detail. This monograph is adressed to both mathematicians and mathematical physicists. The first will be able to use it as a rigorous exposition of free quantum fields on curved spacetimes and as an introduction to some interesting and physically important problems arising in this domain. The second may find this text a useful introduction and motivation to the use of more advanced tools of microlocal analysis in this area of research.

ISBN 978-3-03719-094-4

www.ems-ph.org

ESI Gérard | Font: Rotis Sans., Times Ten | 4 Farben: Euroscala | RS: 11.8 mm

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Microlocal Analysis of Quantum Fields on Curved Spacetimes

Christian Gérard

Microlocal Analysis of Quantum Fields on Curved Spacetimes