Microlocal Analysis of Quantum Fields on Curved Spacetimes

230 Pages·2019·1.937 MB·English

by  Christian Gérard

C
h
r
is
Lectures in Mathematics t
ia
n
and Physics G Lectures in Mathematics
é
r
a and Physics
r
d
Christian Gérard
Microlocal Analysis of Quantum M Christian Gérard
i
c
Fields on Curved Spacetimes
r
o
l Microlocal Analysis
o
c
a
This book provides a detailed introduction to microlocal analysis l

A
methods in the study of Quantum Field Theory on curved spacetimes.
n
of Quantum Fields on
a
We focus on free fields and the corresponding quasi-free states l
y
and more precisely on Klein–Gordon fields and Dirac fields. The first s
i
s
chapters are devoted to preliminary material on CCR*-algebras,
o
quasi-free states, wave equations on Lorentzian manifolds, microlocal Curved Spacetimes
f
analysis and to the important Hadamard condition, characterizing Q
physically acceptable quantum states on curved spacetimes. In the u
a
later chapters more advanced tools of microlocal analysis, like the
n
global pseudo-differential calculus on non-compact manifolds, are t
u
used to construct and study Hadamard states for Klein–Gordon fields
m
by various methods, in particular by scattering theory and by Wick

rotation arguments. In the last chapter the fermionic theory of free F
i
e
Dirac quantum fields on Lorentzian manifolds is described in some
l
d
detail.
s

o
This monograph is adressed to both mathematicians and mathematical n

physicists. The first will be able to use it as a rigorous exposition of free C
quantum fields on curved spacetimes and as an introduction to some u
r
interesting and physically important problems arising in this domain. v
e
The second may find this text a useful introduction and motivation to
d
the use of more advanced tools of microlocal analysis in this area of
S
research. p
a
c
e
t
i
m
e
s
ISBN 978-3-03719-094-4
www.ems-ph.org
ESI Gérard | Font: Rotis Sans., Times Ten | 4 Farben: Euroscala | RS: 11.8 mm

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 interna-
tional 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
9 8 7 6 5 4 3 2 1
Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 FreeKlein–GordonfieldsonMinkowskispacetime 7
2.1 Minkowskispacetime . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 TheKlein–Gordonequation . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Pre-symplecticspaceoftestfunctions . . . . . . . . . . . . . . . . . 12
2.4 Thecomplexcase . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 FockquantizationonMinkowskispace 15
3.1 BosonicFockspace . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 FockquantizationoftheKlein–Gordonequation . . . . . . . . . . . 17
3.3 Quantumspacetimefields . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Localalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4 CCRalgebrasandquasi-freestates 21
4.1 Vectorspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Bilinearandsesquilinearforms . . . . . . . . . . . . . . . . . . . . 22
4.3 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.4 States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.5 CCRalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.6 Quasi-freestates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.7 Covariancesofquasi-freestates . . . . . . . . . . . . . . . . . . . . 30
4.8 TheGNSrepresentationofquasi-freestates. . . . . . . . . . . . . . 33
4.9 Purequasi-freestates . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.10 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5 FreeKlein–Gordonfieldsoncurvedspacetimes 43
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Lorentzianmanifolds . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Stationaryandstaticspacetimes . . . . . . . . . . . . . . . . . . . . 52
5.4 Globallyhyperbolicspacetimes . . . . . . . . . . . . . . . . . . . . 52
5.5 Klein–GordonequationsonLorentzianmanifolds . . . . . . . . . . 57
5.6 Symplecticspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
vi Contents
6 Quasi-freestatesoncurvedspacetimes 65
6.1 Quasi-freestatesoncurvedspacetimes . . . . . . . . . . . . . . . . 65
6.2 Consequencesofuniquecontinuation . . . . . . . . . . . . . . . . . 68
6.3 Conformaltransformations . . . . . . . . . . . . . . . . . . . . . . 69
7 MicrolocalanalysisofKlein–Gordonequations 71
7.1 Wavefrontsetofdistributions . . . . . . . . . . . . . . . . . . . . . 71
7.2 Operationsondistributions . . . . . . . . . . . . . . . . . . . . . . 74
7.3 Ho¨rmander’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.4 ThedistinguishedparametricesofaKlein–Gordonoperator . . . . . 77
8 Hadamardstates 81
8.1 Theneedforrenormalization . . . . . . . . . . . . . . . . . . . . . 81
8.2 OlddefinitionofHadamardstates . . . . . . . . . . . . . . . . . . . 83
8.3 ThemicrolocaldefinitionofHadamardstates . . . . . . . . . . . . . 85
8.4 ThetheoremsofRadzikowski . . . . . . . . . . . . . . . . . . . . . 86
8.5 TheFeynmaninverseassociatedtoaHadamardstate . . . . . . . . . 88
8.6 Conformaltransformations . . . . . . . . . . . . . . . . . . . . . . 88
8.7 Equivalenceofthetwodefinitions . . . . . . . . . . . . . . . . . . . 88
8.8 ExamplesofHadamardstates . . . . . . . . . . . . . . . . . . . . . 90
8.9 ExistenceofHadamardstates . . . . . . . . . . . . . . . . . . . . . 91
9 Vacuumandthermalstatesonstationaryspacetimes 93
9.1 GroundstatesandKMSstates . . . . . . . . . . . . . . . . . . . . . 93
9.2 Klein–Gordonoperators . . . . . . . . . . . . . . . . . . . . . . . . 97
9.3 TheKlein–Gordonequationonstationaryspacetimes . . . . . . . . 99
9.4 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
9.5 GroundandKMSstatesforP . . . . . . . . . . . . . . . . . . . . . 101
9.6 Hadamardproperty . . . . . . . . . . . . . . . . . . . . . . . . . . 102
10 Pseudodifferentialcalculusonmanifolds 103
10.1 PseudodifferentialcalculusonRn . . . . . . . . . . . . . . . . . . . 103
10.2 Pseudodifferentialoperatorsonamanifold . . . . . . . . . . . . . . 106
10.3 Riemannianmanifoldsofboundedgeometry . . . . . . . . . . . . . 108
10.4 TheShubincalculus . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.5 Time-dependentpseudodifferentialoperators . . . . . . . . . . . . . 113
10.6 Seeley’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.7 Egorov’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
11 ConstructionofHadamardstatesbypseudodifferentialcalculus 115
11.1 HadamardconditiononCauchysurfacecovariances . . . . . . . . . 115
11.2 ModelKlein–Gordonoperators . . . . . . . . . . . . . . . . . . . . 116
11.3 ParametricesfortheCauchyproblem . . . . . . . . . . . . . . . . . 117
11.4 ThepureHadamardstateassociatedtoamicrolocalsplitting . . . . . 123
Contents vii
11.5 SpacetimecovariancesandFeynmaninverses . . . . . . . . . . . . . 123
11.6 Klein–GordonoperatorsonLorentzianmanifoldsofboundedgeom-
etry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
11.7 Conformaltransformations . . . . . . . . . . . . . . . . . . . . . . 126
11.8 Hadamardstatesongeneralspacetimes . . . . . . . . . . . . . . . . 127
12 AnalyticHadamardstatesandWickrotation 129
12.1 Boundaryvaluesofholomorphicfunctions . . . . . . . . . . . . . . 130
12.2 Theanalyticwavefrontset . . . . . . . . . . . . . . . . . . . . . . . 131
12.3 AnalyticHadamardstates . . . . . . . . . . . . . . . . . . . . . . . 133
12.4 TheReeh–SchliederpropertyofanalyticHadamardstates . . . . . . 134
12.5 ExistenceofanalyticHadamardstates. . . . . . . . . . . . . . . . . 135
12.6 Wickrotationonanalyticspacetimes . . . . . . . . . . . . . . . . . 136
12.7 TheCaldero´nprojectors . . . . . . . . . . . . . . . . . . . . . . . . 137
12.8 TheHadamardstateassociatedtoCaldero´nprojectors . . . . . . . . 138
12.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
13 HadamardstatesandcharacteristicCauchyproblem 143
13.1 Klein–Gordonfieldsinsidefuturelightcones . . . . . . . . . . . . . 143
13.2 Theboundarysymplecticspace . . . . . . . . . . . . . . . . . . . . 145
13.3 TheHadamardconditionontheboundary. . . . . . . . . . . . . . . 147
13.4 ConstructionofpureboundaryHadamardstates . . . . . . . . . . . 150
13.5 Asymptoticallyflatspacetimes . . . . . . . . . . . . . . . . . . . . 151
13.6 ThecanonicalsymplecticspaceonI(cid:2) . . . . . . . . . . . . . . . . 154
14 Klein–GordonfieldsonspacetimeswithKillinghorizons 159
14.1 SpacetimeswithbifurcateKillinghorizons . . . . . . . . . . . . . . 160
14.2 Klein–Gordonfields . . . . . . . . . . . . . . . . . . . . . . . . . . 162
14.3 Wickrotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
14.4 Thedoubleˇ-KMSstateinMC[M(cid:2) . . . . . . . . . . . . . . . 164
14.5 TheextendedEuclideanmetricandtheHawkingtemperature . . . . 165
14.6 TheHartle–Hawking–Israelstate . . . . . . . . . . . . . . . . . . . 166
15 Hadamardstatesandscatteringtheory 169
15.1 Klein–Gordonoperatorsonasymptoticallystaticspacetimes . . . . . 169
15.2 Theinandoutvacuumstates . . . . . . . . . . . . . . . . . . . . . 171
15.3 Reductiontoamodelcase . . . . . . . . . . . . . . . . . . . . . . . 173
16 FeynmanpropagatoronasymptoticallyMinkowskispacetimes 177
16.1 Klein–GordonoperatorsonasymptoticallyMinkowskispacetimes. . 178
16.2 TheFeynmaninverseofP . . . . . . . . . . . . . . . . . . . . . . . 178
16.3 ProofofTheorem16.1 . . . . . . . . . . . . . . . . . . . . . . . . . 180
viii Contents
17 Diracfieldsoncurvedspacetimes 187
17.1 CAR(cid:2)-algebrasandquasi-freestates . . . . . . . . . . . . . . . . . 187
17.2 Cliffordalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
17.3 Cliffordrepresentations . . . . . . . . . . . . . . . . . . . . . . . . 189
17.4 Spingroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
17.5 Weylbi-spinors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
17.6 Cliffordandspinorbundles . . . . . . . . . . . . . . . . . . . . . . 194
17.7 Spinstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
17.8 Spinorconnections . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
17.9 Diracoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
17.10 Diracequationongloballyhyperbolicspacetimes . . . . . . . . . . 200
17.11 QuantizationoftheDiracequation . . . . . . . . . . . . . . . . . . 201
17.12 HadamardstatesfortheDiracequation . . . . . . . . . . . . . . . . 202
17.13 Conformaltransformations . . . . . . . . . . . . . . . . . . . . . . 203
17.14 TheWeylequation . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
17.15 RelationshipbetweenDiracandWeylHadamardstates . . . . . . . . 206
Bibliography 209
GeneralIndex 215
IndexofNotations 219
Chapter 1
Introduction
1.1 Introduction
Quantum Field Theory arose from the need to unify Quantum Mechanics with spe-
cial relativity. It is usually formulated on the flat Minkowski spacetime, on which
classicalfieldequations,suchastheKlein–Gordon,DiracorMaxwellequationsare
easily defined. Their quantization rests on the so-called Minkowski vacuum, which
describesastateofthequantumfieldcontainingnoparticles. TheMinkowskivacuum
is alsofundamentalfortheperturbativeornon-perturbativeconstructionofinteract-
ingtheories,correspondingtothequantizationofnon-linearclassicalfieldequations.
Quantum Field Theory on Minkowski spacetime relies heavily on its symmetry
underthe Poincare´ group. This is apparentin the ubiquitousrole of plane waves in
theanalysisofclassicalfieldequations,butmoreimportantlyinthecharacterization
of the Minkowskivacuum as the uniquestate which is invariantunderthe Poincare´
groupandhassomeenergypositivityproperty.
QuantumFieldTheoryoncurvedspacetimesdescribesquantumfieldsinanexter-
nalgravitationalfield,representedbytheLorentzianmetricoftheambientspacetime.
It is used in situations when both the quantum nature of the fields and the effect of
gravitationareimportant,butthequantumnatureofgravitycanbeneglectedinafirst
approximation. Its non-relativistic analog would be for example ordinary Quantum
Mechanics,i.e. theSchro¨dingerequation,inaclassicalexteriorelectromagneticfield.
Its mostimportant areas of application are the study of phenomenaoccurring in
theearlyuniverseandinthevicinityofblackholes,anditsmostcelebratedresultis
the discovery by Hawking that quantum particles are created near the horizon of a
blackhole.
ThesymmetriesoftheMinkowskispacetime,whichplaysuchafundamentalrole,
areabsentin curvedspacetimes,exceptinsomesimplesituations,likestationaryor
staticspacetimes. Therefore,thetraditionalapproachtoquantumfieldtheoryhasto
bemodified: onehasfirsttoperformanalgebraicquantization,whichforfreetheo-
riesamountstointroducinganappropriatephasespace,whichiseitherasymplectic
or an Euclidean space, in the bosonic or fermionic case. From such a phase space
one can construct CCR or CAR (cid:2)-algebras, and actually nets of (cid:2)-algebras, each
associatedtoaregionofspacetime.
The second step consists in singling out, among the many states on these (cid:2)-
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
theHadamardexpansion.