Microlocal analysis of quantum fields on curved spacetimes 1st edition christian gerard

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Lectures in Mathematics and Physics

Christian Gérard

Microlocal Analysis of Quantum Fields on Curved Spacetimes

ESI Lectures in Mathematics and Physics

Editors

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Erwin Schrödinger International Institute for Mathematical Physics

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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.

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Previously published in this series:

Arkady L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations

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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

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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: christian.gerard@math.u-psud.fr

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.

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© 2019 European Mathematical Society

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9 8 7 6 5 4 3 2 1

2FreeKlein–GordonfieldsonMinkowskispacetime

6Quasi-freestatesoncurvedspacetimes

6.1Quasi-freestatesoncurvedspacetimes

6.2Consequencesofuniquecontinuation

6.3Conformaltransformations

7MicrolocalanalysisofKlein–Gordonequations

7.1Wavefrontsetofdistributions

7.2Operationsondistributions

7.3H¨ormander’stheorem

7.4ThedistinguishedparametricesofaKlein–Gordonoperator

8Hadamardstates

8.1Theneedforrenormalization

8.2OlddefinitionofHadamardstates

8.3ThemicrolocaldefinitionofHadamardstates

8.4ThetheoremsofRadzikowski

8.5TheFeynmaninverseassociatedtoaHadamardstate

8.6Conformaltransformations

8.7Equivalenceofthetwodefinitions

8.8ExamplesofHadamardstates

8.9ExistenceofHadamardstates

9Vacuumandthermalstatesonstationaryspacetimes

9.1GroundstatesandKMSstates

9.2Klein–Gordonoperators

9.3TheKlein–Gordonequationonstationaryspacetimes

9.5GroundandKMSstatesfor P

9.6Hadamardproperty

10.3Riemannianmanifoldsofboundedgeometry

10.5Time-dependentpseudodifferentialoperators

10.6Seeley’stheorem

10.7Egorov’stheorem

11ConstructionofHadamardstatesbypseudodifferentialcalculus

11.3ParametricesfortheCauchyproblem

11.4ThepureHadamardstateassociatedtoamicrolocalsplitting

11.5SpacetimecovariancesandFeynmaninverses

11.6Klein–GordonoperatorsonLorentzianmanifoldsofboundedgeometry ...................................125

11.7Conformaltransformations

11.8Hadamardstatesongeneralspacetimes

12AnalyticHadamardstatesandWickrotation

12.1Boundaryvaluesofholomorphicfunctions

12.2Theanalyticwavefrontset

12.3AnalyticHadamardstates

12.4TheReeh–SchliederpropertyofanalyticHadamardstates

12.5ExistenceofanalyticHadamardstates

12.6Wickrotationonanalyticspacetimes

12.7TheCalder´onprojectors ........................137

12.8TheHadamardstateassociatedtoCalder´onprojectors

12.9Examples

13HadamardstatesandcharacteristicCauchyproblem

13.1Klein–Gordonfieldsinsidefuturelightcones

13.3TheHadamardconditionontheboundary

13.4ConstructionofpureboundaryHadamardstates

13.5Asymptoticallyflatspacetimes

13.6Thecanonicalsymplecticspaceon I

14Klein–GordonfieldsonspacetimeswithKillinghorizons

15Hadamardstatesandscatteringtheory

16FeynmanpropagatoronasymptoticallyMinkowskispacetimes

16.2TheFeynmaninverseof P

viii Contents

17Diracfieldsoncurvedspacetimes

17.1CAR -algebrasandquasi-freestates

17.2Cliffordalgebras ............................188

17.3Cliffordrepresentations

17.4Spingroups

17.5Weylbi-spinors

17.6Cliffordandspinorbundles

17.7Spinstructures

17.8Spinorconnections

17.9Diracoperators

17.10Diracequationongloballyhyperbolicspacetimes

17.11QuantizationoftheDiracequation

17.12HadamardstatesfortheDiracequation

17.13Conformaltransformations

17.14TheWeylequation ...........................204

17.15RelationshipbetweenDiracandWeylHadamardstates

Chapter1 Introduction

1.1Introduction

QuantumFieldTheoryarosefromtheneedtounifyQuantumMechanicswithspecialrelativity.ItisusuallyformulatedontheflatMinkowskispacetime,onwhich classicalfieldequations,suchastheKlein–Gordon,DiracorMaxwellequationsare easilydefined.Theirquantizationrestsontheso-called Minkowskivacuum,which describesastateofthequantumfieldcontainingnoparticles.TheMinkowskivacuum isalsofundamentalfortheperturbativeornon-perturbativeconstructionofinteractingtheories,correspondingtothequantizationofnon-linearclassicalfieldequations.

QuantumFieldTheoryonMinkowskispacetimereliesheavilyonitssymmetry underthePoincar´egroup.Thisisapparentintheubiquitousroleofplanewavesin theanalysisofclassicalfieldequations,butmoreimportantlyinthecharacterization oftheMinkowskivacuumastheuniquestatewhichisinvariantunderthePoincar´e groupandhassome energypositivity property.

QuantumFieldTheoryon curvedspacetimes describesquantumfieldsinanexternalgravitationalfield,representedbytheLorentzianmetricoftheambientspacetime. Itisusedinsituationswhenboththequantumnatureofthefieldsandtheeffectof gravitationareimportant,butthequantumnatureofgravitycanbeneglectedinafirst approximation.Itsnon-relativisticanalogwouldbeforexampleordinaryQuantum Mechanics,i.e.theSchr¨odingerequation,inaclassicalexteriorelectromagneticfield. Itsmostimportantareasofapplicationarethestudyofphenomenaoccurringin theearlyuniverseandinthevicinityofblackholes,anditsmostcelebratedresultis thediscoverybyHawkingthatquantumparticlesarecreatednearthehorizonofa blackhole.

ThesymmetriesoftheMinkowskispacetime,whichplaysuchafundamentalrole, areabsentincurvedspacetimes,exceptinsomesimplesituations,like stationary or static spacetimes.Therefore,thetraditionalapproachtoquantumfieldtheoryhasto bemodified:onehasfirsttoperforman algebraicquantization,whichforfreetheoriesamountstointroducinganappropriate phasespace,whichiseithera symplectic oran Euclidean space,inthe bosonic or fermionic case.Fromsuchaphasespace onecanconstructCCRorCAR -algebras,andactually nets of -algebras,each associatedtoaregionofspacetime.

Thesecondstepconsistsinsinglingout,amongthemanystatesonthesealgebras,thephysicallymeaningfulones,whichshouldresembletheMinkowski vacuum,atleastinthevicinityofanypointofthespacetime.Thisleadstothe notionof Hadamardstates,whichwereoriginallydefinedbyrequiringthattheir two-pointfunctionshaveaspecificasymptoticexpansionnearthediagonal,called the Hadamardexpansion

AveryimportantprogresswasmadebyRadzikowski,[R1, R2],whointroduced thecharacterizationofHadamardstatesbythe wavefrontset oftheirtwo-pointfunctions.Thewavefrontsetofadistributionisthenaturalwaytodescribeitssingularitiesinthecotangentspace,andliesatthebasisof microlocalanalysis,afundamental toolintheanalysisoflinearandnon-linearpartialdifferentialequations.Among itsavatarsinthephysicsliteratureare,forexample,thegeometricalopticsinwave propagationandthesemi-classicallimitinQuantumMechanics.

Theintroductionofmicrolocalanalysisinquantumfieldtheoryoncurvedspacetimesstartedaperiodofrapidprogress,nononlyforfree(i.e.linear)quantumfields, butalsofortheperturbativeconstructionofinteractingfieldsbyBrunettiandFredenhagen[BF].Forfreefieldsitallowedtouseseveralfundamentalresultsofmicrolocal analysis,likeH¨ormander’spropagationofsingularitiestheoremandtheclassification ofparametricesforKlein–GordonoperatorsbyDuistermaatandH¨ormander.

1.2Content

Thegoaloftheselecturenotesistogive anexpositionofmicrolocalanalysismethods inthestudyofQuantumFieldTheoryoncurvedspacetimes.Wewillfocuson free fields andthecorresponding quasi-freestates andmorepreciselyon Klein–Gordon fields,obtainedbyquantizationoflinearKlein–GordonequationsonLorentzianmanifolds,althoughthecaseof Diracfields willbedescribedinChapter 17.

Thereexistalreadyseveralgoodtextbooksorlecturenotesonquantumfieldtheoryincurvedspacetimes.AmongthemletusmentionthebookbyB¨ar,Ginouxand Pfaeffle[BGP],thelecturenotes[BFr ]and[BDFY],themorerecentbookbyRejzner[Re],andthesurveybyBenini,DappiagiandHack[BDH].Thereexistalsomore physicsorientedbooks,likethebooksbyWald[W2],Fulling[F]andBirrelland Davies[BD].Severalofthesetextscontainimportantdevelopmentswhicharenot describedhere,liketheperturbativeapproachtointeractingtheories,ortheuseof categorytheory.

Inthislecturenoteswefocusonadvancedmethodsfrommicrolocalanalysis,like forexample pseudodifferentialcalculus,whichturnouttobeveryusefulinthestudy andconstructionofHadamardstates.

Puremathematiciansworkinginpartialdifferentialequationsareoftendeterred bythetraditionalformalismofquantumfieldtheoryfoundinphysicstextbooks,and bythefactthattheconstructionofinteractingtheoriesis,atleastforthetimebeing, restrictedtoperturbativemethods.

Wehopethattheselecturenoteswillconvincethemthatquantumfieldtheory oncurvedspacetimesisfullofinterestingandphysicallyimportantproblems,witha niceinterplaybetweenalgebraicmethods,Lorentziangeometryandmicrolocalmethodsinpartialdifferentialequations.Ontheotherhand,mathematicalphysicistswith atraditionaleducation,whichmaylackfamiliaritywithmoreadvancedtoolsofmicrolocalanalysis,canusethistextasanintroductionandmotivationtotheuseof thesemethods.

Letusnowgiveamoredetaileddescriptionoftheselecturenotes.Thereader mayalsoconsulttheintroductionofeachchapterformoreinformation.

Forpedagogicalreasons,wehavechosentodevoteChapters 2 and 3 toabriefoutlineofthetraditionalapproachtoquantizationofKlein–GordonfieldsonMinkowski spacetime,buttheimpatientreadercanskipthemwithouttrouble.

Chapter 4 dealswithCCR -algebrasandquasi-freestates.AreaderwithaPDE backgroundmayfindthereadingofthischapterabittedious.Nevertheless,wethink itisworththeefforttogetfamiliarwiththenotionsintroducedthere.

InChapter 5 wedescribewell-knownconceptsandresultsconcerningLorentzian manifoldsandKlein–Gordonequationsonthem.Themostimportantarethenotion of globalhyperbolicity,apropertyofaLorentzianmanifoldimplyingglobalsolvabilityoftheCauchyproblem,andthe causalpropagator andthevarioussymplectic spacesassociatedtoit.

InChapter 6 wediscussquasi-freestatesforKlein–Gordonfieldsoncurvedspacetimes,whichisaconcreteapplicationoftheabstractformalisminChapter 4.Of interestarethetwopossibledescriptions ofaquasi-freestate,eitherbyitspacetime covariances,orbyitsCauchysurfacecovariances,whicharebothimportantinpractice.Anotherusefulpointisthediscussionofconformaltransformations.

Chapter 7 isdevotedtothemicrolocalanalysisofKlein–Gordonequations.We collectherevariouswell-knownresultsaboutwavefrontsets,H¨ormander’spropagationofsingularitiestheoremandits relatedstudywithDuistermaatof distinguished parametrices forKlein–Gordonoperators,whichplayafundamentalroleinquantizedKlein–Gordonfields.

InChapter 8 weintroducethemoderndefinitionofHadamardstatesduetoRadzikowskianddiscusssomeofitsconsequences.Weexplaintheequivalencewiththe olderdefinitionbasedonHadamardexpansionsandthewell-knownexistenceresult byFulling,NarcowichandWald.

InChapter 9 wediscussgroundstatesandthermalstates,firstinanabstractsetting,thenforKlein–Gordonoperatorsonstationaryspacetimes.Groundstatesshare thesymmetriesofthebackgroundstationaryspacetimeandarethenaturalanalogsof theMinkowskivacuum.Inparticular,theyarethesimplestexamplesofHadamard states.

Chapter 10 isdevotedtoanexpositionofaglobalpseudodifferentialcalculuson noncompactmanifolds,the Shubincalculus.Thiscalculusisbasedonthenotion ofmanifoldsofboundedgeometryandisanaturalgeneralizationofthestandard uniformcalculuson Rn .ItsmostimportantpropertiesaretheSeeleyandEgorov theorems.

InChapter 11 weexplaintheconstructionofHadamardstatesusingthepseudodifferentialcalculusinChapter 10.Theconstructionisdone,afterchoosingaCauchy surface,byamicrolocalsplittingofthespaceofCauchydataobtainedfromaglobal constructionofparametricesfortheCauchyproblem.Itcanbeappliedtomany spacetimesofphysicalinterest,liketheKerr–KruskalandKerr–deSitterspacetimes.

InChapter 12 weconstruct analytic Hadamardstatesby Wickrotation,awellknownprocedureinMinkowskispacetime.AnalyticHadamardstatesaredefined onanalyticspacetimes,byreplacingtheusual C 1 wavefrontsetbythe analytic

wavefrontset,whichdescribestheanalyticsingularitiesofdistributions.Likethe Minkowskivacuum,theyhavetheimportant Reeh–Schlieder property.AfterWick rotation,thehyperbolicKlein–GordonoperatorbecomesanellipticLaplaceoperator, andanalyticHadamardstatesareconstructedusingawell-knowntoolfromelliptic boundaryvalueproblems,namelythe Calder ´ onprojector.

InChapter 13 wedescribetheconstructionofHadamardstatesbythe characteristicCauchyproblem.Thisamountstoreplacingthespace-likeCauchysurfacein Chapter 11 byapastorfuture lightcone,choosingitsinteriorastheambientspacetime.Fromthetraceofsolutionsonthisconeonecanintroducea boundarysymplecticspace,anditturnsoutthatitisquiteeasytocharacterizestatesonthissymplectic spacewhichgenerateaHadamardstateintheinterior.Itsmainapplicationisthe conformalwaveequation onspacetimeswhichareasymptoticallyflatatpastorfuture nullinfinity.WealsodescribeinthischaptertheBMS group ofasymptoticsymmetriesofthesespacetimes,anditsrelationshipwithHadamardstates.

InChapter 14 wediscussKlein–GordonfieldsonspacetimeswithKillinghorizons.Ouraimistoexplainaphenomenonlooselyrelatedwiththe Hawkingradiation, namelytheexistenceofthe Hartle–Hawking–Israel vacuum,onspacetimeshavinga stationaryKillinghorizon.Theconstructionandpropertiesofthisstatefollowfrom theWickrotationmethodalreadyusedinChapter 12,theCalder´onprojectorsplaying alsoanimportantrole.

Chapter 15 isdevotedtotheconstructionofHadamardstatesbyscatteringtheory methods.Weconsiderspacetimeswhichareasymptoticallystaticatpastorfuture timeinfinity.Inthiscaseonecandefinethe in and outvacuumstates,whichare statesasymptotictothevacuumstateatpastorfuturetimeinfinity.Usingthetools fromChapters 10, 11 weprovethatthesestatesareHadamardstates.

InChapter 16 wediscussthenotionof Feynmaninverses.ItisknownthataKlein–Gordonoperatoronagloballyhyperbolicspacetimeadmits Feynmanparametrices, whichareuniquemodulosmoothingoperatorsandcharacterizedbythewavefrontset ofitsdistributionalkernels.Onecanaskifonecanalsodefineaunique,canonical trueinverse,havingthecorrectwavefrontset.Wegiveapositiveanswertothis questiononspacetimeswhichare asymptoticallyMinkowski.

Chapter 17 isdevotedtothequantizationofthe Diracequation andtothedefinitionofHadamardstatesforDiracquantumfields.TheDiracequationonacurved spacetimedescribesanelectron-positronfieldwhichisa fermionic field,andtheCCR -algebrafortheKlein–GordonfieldhastobereplacedbyaCAR -algebra.Apart fromthisdifference,thetheoryforfermionicfieldsisquiteparalleltothebosonic case.Wealsodescribethequantizationofthe Weylequation,whichoriginallywas thoughttodescribemasslessneutrinos.

1.2.1Acknowledgments. TheresultsdescribedinChapters 11, 12, 15,andpart ofthoseinChapters 10 and 13,originatefromcommonworkwithMichalWrochna, overaperiodofseveralyears.

IlearnedalotofwhatIknowaboutquantumfieldtheoryfrommylongcollaborationwithJanDerezinski,andseveralpartsoftheselecturenotes,likeChapters 4

1.3Notation 5

and 5 borrowalotfromourcommonbook[DG].Itaketheoccasionheretoexpress mygratitudetohim.

Finally,IalsogreatlyprofitedfromdiscussionswithmembersoftheAQFTcommunity.AmongthemIwouldliketoespeciallythankClaudioDappiagi,Valter Moretti,NicolaPinamonti,IgorKhavkine,KlausFredenhagen,DetlevBucholz,WojciechDybalski,KasiaRejzner,DorotheaBahns,RainerVerch,StefanHollandsand KoSanders.

1.3Notation

Wenowcollectsomenotationthatwewilluse.

Weset h iD .1 C 2 / 1 2 for 2 R.

Wewrite A b B if A isrelativelycompactin B

If X;Y aresetsand f W X ! Y wewrite f W X ! Y if f isbijective.If X;Y areequippedwithtopologies,wewrite f W X ! Y ifthemapiscontinuous,and f W X ! Y ifitisahomeomorphism.

1.3.1ScaleofabstractSobolevspaces. Let H arealorcomplexHilbert spaceand A aselfadjointoperatoron H .Wewrite A>0 if A 0 andKer A Df0g. If A>0 and s 2 R,weequipDom A s withthescalarproduct .ujv/ s D .A s ujA s v/ andthenorm kA s uk.Wedenoteby As H thecompletionofDom A s forthisnorm,whichisa(realorcomplex)Hilbertspace.

Chapter2

FreeKlein–GordonfieldsonMinkowski spacetime

Almostalltextbooksonquantumfieldtheorystartwiththequantizationofthefree (i.e.linear)Klein–GordonandDiracequationsonMinkowskispacetime.Thetraditionalexpositionrestsontheso-called frequencysplitting,whichamountstosplitting thespaceofsolutionsof,say,theKlein–Gordonequationintotwosubspaces,correspondingtosolutionshavingpositive/negativeenergy,orequivalentlywhoseFourier transformsaresupportedinthe upper/lowermasshyperboloid.

Onethenproceedswiththeintroductionof Fockspaces andthedefinitionofquantizedKlein–GordonorDiracfieldsusing creation/annihilationoperators.

SinceitreliesontheuseoftheFouriertransformation,thismethoddoesnotcarry overtoKlein–Gordonfieldsoncurvedspacetimes.Morefundamentally,ithasthe drawbackofmixingtwodifferentstepsinthequantizationoftheKlein–Gordonequation.

Thefirst,purelyalgebraicstepconsistsinusingthesymplecticnatureofthe Klein–GordonequationtointroduceanappropriateCCR -algebra.Thesecondstep consistsinchoosinga state onthisalgebra,whichontheMinkowskispacetimeisthe vacuumstate.

NeverthelessitisusefultokeepinmindtheMinkowskispacetimeasanimportant example.ThischapterisdevotedtotheclassicaltheoryoftheKlein–Gordonequation onMinkowskispacetime,i.e.toitssymplecticstructure.ItsFockquantizationwill bedescribedinChapter 3.

2.1Minkowskispacetime

InthesequelwewillusenotationintroducedlaterinSection 4.1

Theelementsof Rn D Rt Rd x willbedenotedby x D .t; x/,thoseofthedual .Rn /0 by D . ; k/.

2.1.1TheMinkowskispacetime.

Definition2.1. The Minkowskispacetime R1;d is R1Cd equippedwiththebilinear form 2 Ls .R1Cd ;.R1Cd /0 / givenby x x D t 2 C x 2 :

Definition2.2. (1) Avector 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˙ Dfx 2 R1;d W x x<0; ˙t>0g,resp. C ˙ Dfx 2 R1;d W x x 0; ˙t 0g arecalledthe open,resp. closedfuture/past(solid)lightcones.

(3) N Dfx 2 R1;d W x x D 0g,resp. N˙ D N \f˙t 0g arecalledthe null cone resp. future/pastnullcones.

Thereisasimilarclassificationofvectorsubspacesof R1;d

Definition2.3. Alinearsubspace V of R1;d is time-like ifitcontainsbothspacelikeandtime-likevectors, null ifitistangenttothenullcone N and space-like ifit containsonlyspace-likevectors.

Definition2.4. (1) If K R1;d , I˙ .K/ D K C C˙ ,resp. J˙ .K/ D K C C ˙ , iscalledthe time-like,resp. causalfuture/past of K ,and J.K/ D JC .K/ [ J .K/ the causalshadow of K .

(2) Twosets K1 , K2 arecalled causallydisjoint if K1 \ J.K2 / D; or,equivalently, if J.K1 / \ K2 D;.

(3) Afunction f on Rn iscalled space-compact,resp. future/pastspace-compact, if supp f J.K/,resp. supp f J˙ .K/ forsomecompactset K b Rn .The spacesofsmoothsuchfunctionswillbedenotedby C 1 sc .Rn /,resp. C 1 sc;˙ .Rn /.

2.1.2TheLorentzandPoincar ´ egroups.

Definition2.5. (1) Thepseudo-Euclideangroup O.R1Cd ; / isdenotedby O.1;d/ andiscalledthe Lorentzgroup

(2) SO.1;d/ isthesubgroupof L 2 O.1;d/ with det L D 1.

(3) If L 2 O.1;d/ onehas L.JC / D JC or L.JC / D J .Inthefirstcase L is called orthochronous andinthesecond anti-orthochronous.

(4) Thesubgroupoforthochronouselementsof SO.1;d/ isdenotedby SO " .1;d/ andcalledthe restrictedLorentzgroup.

Definition2.6. The .restricted/ Poincar´egroup istheset P.1;d/ D Rn SO " .1;d/ equippedwiththeproduct .a2 ;L2 / .a1 ;L1 / D .a2 C

ThePoincar ´ egroupactson Rn by ƒx D Lx C a for ƒ D .a;L/ 2 P.1;d/.

2.2TheKlein–Gordonequation 9

2.2TheKlein–Gordonequation

Werecallthatthedifferentialoperator

for m 0 iscalledthe Klein–Gordonoperator

Weset .k/ D .k 2 C m2 / 1 2 anddenoteby D .Dx / theFouriermultiplier definedby F . u/.k/ D .k/u.k/,where F u.k/ D .2 / d=2 ´ e ik x u.x/d xisthe (unitary)Fouriertransform.Notethat C m2 D @2 t C 2

The Klein–Gordonequation

isthesimplestrelativisticfieldequation.Itsquantizationdescribesa scalarbosonic field ofmass m.The waveequation (m D 0)isaparticularcaseoftheKlein–Gordon equation.Notethatsince C m2 preservesrealfunctions,theKlein–Gordonequationhasrealsolutions,whichareassociatedto neutralfields,correspondingtoneutral particles,whilethecomplexsolutionsareassociatedto chargedfields,corresponding tochargedparticles.

Itwillbemoreconvenientlatertoconsidercomplexsolutions,butinthischapter wewill,asisusualinthephysicsliterature,considermainlyrealsolutions.Thecase ofcomplexsolutionswillbebrieflydiscussedinSection 2.4

WereferthereadertoChapter 4 forageneraldiscussionoftherealvscomplex formalisminamoreabstractframework.

Weareinterestedinthespaceofits smooth,space-compact,real solutionsdenotedbySolsc;R .KG/.Solsc;R .KG/ isinvariantunderthePoincar´egroupifweset

2.2.1TheCauchyproblem. If 2 C 1 .Rn / and t 2 R weset .t/.x/ D .t; x/ 2 C 1 .Rd /.Anysolution 2 Solsc;R .KG/ isdeterminedbyitsCauchydata ontheCauchysurface †s Dft D s g Rd ,definedbythemap

TheuniquesolutioninSolsc;R .KG/ oftheCauchyproblem . C m2 / D

isdenotedby D Us f andgivenby

2FreeKlein–GordonfieldsonMinkowskispacetime

Themap Us iscalledthe Cauchyevolutionoperator.Thefollowingproposition expressestheimportant causalityproperty of Us .

Proposition2.7. Onehas supp Us f J.fs g supp f/:

2.2.2Advancedandretardedinverses. Letusnowconsiderthe inhomogeneousKlein–Gordonequation

whereforsimplicity v 2 C 1 0 .Rn /.Sincethereareplentyofhomogeneoussolutions, itisnecessarytosupplement(2.7)by supportconditions toobtainuniquesolutions, byrequiringthat vanishesforlargenegativeorpositivetimes.

Theorem2.8. (1) Thereexistuniquesolutions uret=adv D

of (2.7).Setting

where .t/ D Œ0;C1Œ .t/ istheHeavisidefunction,onehas

(2) onehas supp Gret=adv v J˙ .supp v/

Theoperators Gret=adv arecalledthe retarded/advancedinverses of P .Letus equip C 1 0 .Rn / withthescalarproduct

and C 1 0 .Rd I C

withthescalarproduct

Itfollowsfrom(2.8)that Gret=adv D Gadv=ret ; where A denotestheformaladjointof A withrespecttothescalarproduct . j /Rn . Theoperator G D Gret Gadv (2.12) iscalledinthephysicsliteraturethe Pauli–Jordan or commutatorfunction,oralso the causalpropagator.Notethat G D G ; supp Gv J.supp v/; (2.13)

2.2TheKlein–Gordonequation 11 and Gv.t; / D ˆR 1 sin. .t s//v.s; /ds: (2.14)

Thereisanimportantrelationshipbetween G and Us .Namely,ifwedenoteby %s W D 0 .Rd I R2 / !

.Rn / theformaladjointof

withrespecttothescalarproducts(2.10)and(2.11),then:

Thefollowinglemmafollowsfrom(2.6),(2.8)byadirectcomputation.

Lemma2.9. Onehas

for D 0 0 .

2.2.3Symplecticstructure. Itiswell-knownthattheKlein–Gordonequationis aHamiltonianequation.Indeedletusequip C 1 0 .Rd I R2 / withthesymplecticform:

Ifweidentifybilinearformson C 1 0 .Rd I R2 / withlinearoperatorsusingthescalar product . j /Rd ,wehave f g D .f j g/Rd ; wheretheoperator isdefinedinLemma 2.9.Ifweintroducethe classicalHamiltonian

weobtainthat

Setting f.t/ D %t U0 f for

wehave,byaneasycomputation f.t/ D etA f; (2.18)

whichshowsthat f ! f.t/ isthesymplecticflowgeneratedbytheclassicalHamiltonian E andthesymplecticform .Inparticular,if fi .t/ D etA fi , i D 1;2, f1 .t/ f2 .t/ isindependenton t

2FreeKlein–GordonfieldsonMinkowskispacetime

Equivalently,wecanequipSolsc;R .KG/ withthesymplecticform

wheretheright-handsideisindependenton t .FixingthereferenceCauchysurface †0 Rd ,weobtainthefollowingproposition:

Proposition2.10. TheCauchydatamapon †0

%0 W .Solsc;R .KG/; / ! .C 1 0 .Rd I R2 /; /; issymplectic,with % 1 0 D U0 ,wheretheCauchyevolutionoperator Us wasintroducedinSubsection 2.2.1.

Thisleadstoanotherinterpretationof(2.18):thespaceSolsc;R .KG/ isinvariant underthegroupoftimetranslations

/ D . s; x/; and s issymplecticon .Solsc;R .KG/; /.Then(2.18)canberewrittenas

2.3Pre-symplecticspaceoftestfunctions

ByProposition 2.10, .Solsc;R .KG/; / isasymplecticspace.Itiseasytoseethat ˛ƒ definedin(2.3)issymplecticif ƒ isorthochronous,forexampleusingTheorem 2.12 below.If ƒ isanti-orthochronous, ˛ƒ is anti-symplectic,i.e.transforms into . Identifying .Solsc;R .KG/; / with .C 1 0 .Rd I R2 /; / using %0 isconvenientfor concretecomputations,butdestroysPoincar´einvariance,sinceonefixestheCauchy surface †0 .Itwouldbeusefultohaveanotherisomorphicsymplecticspacewhich isPoincar´einvariantandatthesametimeeasiertounderstandthanSolsc;R .KG/.It turnsoutthatonecanusethespaceof testfunctions C 1 0 .Rn I R/,whichisafundamentalstepinformulatingthenotionof locality forquantumfields.

Proposition2.11. Considerthemap G W C 1 0 .Rn I R/ ! C 1 sc .Rn /.Then:

.1/ RanG D Solsc;R .KG/; .2/ Ker G D PC 1 0 .Rn I R/:

Moreover,wehave .3/.%0 G/ ı ı .%0 G/ D G:

2.3Pre-symplecticspaceoftestfunctions 13

Proof. (1)By P ı G D 0 andTheorem 2.8 (2),weseethatRanG Solsc;R .KG/ Converselylet 2 Solsc;R .KG/.If fs D %s ,thenbyLemma 2.9 weobtainthat

D G ı %s ı fs for s 2 R.Hence,if 2 C 1 0 .R/ with ´ .s/ds D 1 weobtain that

D ˆR .s/ dx D Gv; for v D ´R %s ı fs ds 2 C 1 0 .Rn /.

(2)Since G ı P D 0 wehave PC 1 0 .Rn I R/ Ker G .Converselylet v 2 C 1 0 .Rn I R/ with Gv D 0.Thenfor uret=adv D Gret=adv v wehave uret D uadv D u, u 2 C 1 0 .Rn / byTheorem 2.8 (2)and v D Pu since P ı Gret=adv D . (3)Wehave,using(2.14) %0 Gu D ´ 1 sin. s/u.s/ds ´ cos. s/u.s/ds ; hence ı .%0 G/u D ´ cos. s/u.s/ds ´ 1 sin. s/u.s/ds ; and .%0 G/ f D G%0 f

D ´ 1 sin. .t s//.ı0 .s/ ˝ f0 ı 0 0 .s/ ˝ f1 /ds

D 1 sin. t/f0 C cos. t/f1 ; whichyields .%0 G/ ı ı .%0 G/u

´ 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:

Thiscompletestheproofoftheproposition.

OnecansummarizePropositions 2.10 and 2.11 inthefollowingtheorem:

Theorem2.12. (1) Thefollowingspacesaresymplecticspaces:

C 1 0 .Rn I R/

PC 1 0 .Rn I R/ ;. jG /Rn ;.Solsc;R .KG/; /;.C 1 0 .Rd I R2 /; /:

(2) Thefollowingmapsaresymplectomorphisms:

C 1 0 .Rn I R/

PC 1 0 .Rn I R/ ;. jG /Rn G !.Solsc;R .KG/; / %0 !.C 1 0 .Rd I R2 /; /:

Thefirstandlastoftheseequivalentsymplecticspacesarethemostusefulforthe quantizationoftheKlein–Gordonequation.

2.4Thecomplexcase

LetusnowdiscussthespaceSolsc;C .KG/ of complex space-compactsolutions.We refertoSection 4.2 fornotationandterminology. Itismorenaturaltousethemap

asCauchydatamapandtoequipthespace C 1 0 .Rd I C2 / ofCauchydatawiththe Hermitianform

ThespaceSolsc;C .KG/ issimilarlyequippedwiththeform

; whichisindependenton t .TheCauchyevolutionoperatorbecomes

WehavethenthefollowinganalogofTheorem 2.12:

Theorem2.13. (1) ThefollowingspacesareHermitianspaces:

(2) Thefollowingmapsareunitary: C 1 0 .Rn I C/

Chapter3

FockquantizationonMinkowskispace

Wedescribeinthischapterthe Fockquantization oftheKlein–Gordonequationon Minkowskispacetime.Werecallthedefinitionofthe bosonicFockspace overa one-particlespaceandofthe creation/annihilationoperators,whichareubiquitous notionsinquantumfieldtheory.

Forexample,itiscommoninthephysicsorientedliteratureto specifyastatefor theKlein–Gordonfieldbydefiningfirstsomecreation/annihilationoperators.Wewill seeinChapter 4 thatthisisnothingelsethanchoosingaparticularK¨ahlerstructure onacertainsymplecticspace.

InthisapproachthequantumKlein–Gordonfieldsaredefinedaslinearoperators ontheFockspace,soonehastopayattentiontodomainquestions.Thesetechnical problemsdisappearifoneusesamoreabstractpointofviewandintroducestheappropriateCCR -algebra,aswillbedoneinChapter 4.Fockspaceswillreappear asthe(Gelfand–Naimark–Segal)GNS Hilbertspaces associatedtoapurequasi-free stateonthisalgebra.Apartfromthisfact,theycanbeforgotten.

3.1BosonicFockspace

3.1.1BosonicFockspace. Let h beacomplexHilbertspacewhoseunitvectors describethestatesofaquantumparticle.Ifthisparticleis bosonic,thenthestatesofa systemof n suchparticlesaredescribedbyunitvectorsinthe symmetrictensorpower ˝n s h,wherewetakethetensorproductsintheHilbertspacesense,i.e.completethe algebraictensorproductsforthenaturalHilbertnorm.

Asystemofanarbitrarynumberofparticlesisdescribedbythe bosonicFock space

wherethedirectsumisagaintakenintheHilbertspacesenseand ˝

s h D C by definition.Werecallthatthesymmetrizedtensorproductisdefinedby

Thevector vac D .1;0;:::/ iscalledthe vacuum anddescribeastatewithnoparticlesatall.Ausefulobservableon s .h/ isthe numberoperator N ,whichcountsthe

3FockquantizationonMinkowskispace

numberofparticles,definedby N j˝n s h D n :

Theoperator N isanexampleofa secondquantizedoperator,namely N D d . /, where d .a/j˝n s h D n X j D1 ˝j 1 ˝ a ˝ ˝n j ; for a alinearoperatoron h.

3.1.2Creation/annihilationoperators. Since s .h/ describesanarbitrary numberofparticles,itisusefultohaveoperatorsthatcreateorannihilateparticles. Onedefinesthe creation/annihilationoperators by a .h/‰n D pn C 1h ˝s ‰n ; a.h/‰n D pn.hj˝ ˝n 1 ‰n ;‰n 2˝n s .h/;h 2 h;

whereonesets .hju D .hju/ for u 2 h.Itiseasytoseethat a . / .h/ arewell definedonDom N 1 2 andthat .‰1 ja .h/‰2 / D .a.h/‰1 j‰2 /,i.e. a.h/ a .h/ on Dom N 1 2 .Moreover h 3 h ! a .h/; resp. a.h/ is C-linear,resp.anti-linear,(3.2) andasquadraticformsonDom N 1 2 onehas

Œa.h1 /;a.h2 / D Œa .h1 /;a .h2 / D 0; Œa.h1 /;a .h2 / D .h1 jh2 / ;h

where ŒA;B D AB BA,whichaversionofthe canonicalcommutationrelations, abbreviatedCCRinthesequel.

3.1.3FieldandWeyloperators. Onethenintroducesthe fieldoperators inthe Fockrepresentation F .h/ D 1 p2 .a.h/ C a .h//;h 2 h; (3.4)

whichcanbeeasilyshowntobeessentiallyselfadjointonDom N 1 2 .Onehas F .h1 C h2 / D F .h1 / C F .h2 /; 2 R;hi 2 h; onDom N 1 2 ; (3.5)

i.e. h ! F .h/ is R-linear,andthe Heisenbergform oftheCCRaresatisfiedas quadraticformsonDom N 1 2 ΠF .h1 /; F .h2 / D ih1 h2 : (3.6)

3.2FockquantizationoftheKlein–Gordonequation 17 for

Denotingagainby F .h/ theselfadjointclosureof F .h/,onecanthendefinethe Weyloperators

whichareunitaryandsatisfythe Weylform oftheCCR

If hR denotestherealformof h,i.e. h asarealvectorspace,then .hR ; / isa realsymplecticspace.Moreoveri,consideredasanelementof L.hR /,belongsto Sp.hR ; / andonehas

Re. j / 0:

3.1.4K ¨ ahlerstructures. Ingeneral,atriple .X ; ; j/,where .X ; / isareal symplecticspaceandj 2 L.X / satisfiesj2 D and ı j 2 Ls .X ; X 0 /,iscalled a pseudo-K ¨ ahlerstructure on X .If ı j 0,itiscalleda K ¨ ahlerstructure.The anti-involutionjiscalleda Kahleranti-involution.Wewillcomebacktothisnotion inSection 4.1.GivenaK¨ahlerstructureon X ,onecanturn X intoacomplexpreHilbertspacebyequippingitwiththecomplexstructurejandthescalarproduct:

Ifwechooseasone-particleHilbertspacethecompletionof X for . j /F ,wecan constructthe Fockrepresentation bythemap

X 3 x ! F .x/ whichsatisfies(3.5),(3.6).

3.2FockquantizationoftheKlein–Gordonequation

Fromtheabovediscussionweseethatthefirststepintheconstructionofquantum Klein–GordonfieldsistofixaK¨ahleranti-involutionononeoftheequivalentsymplecticspacesinTheorem 2.12,themostconvenientonebeing .C 1 0 .Rd I R2 /; /.

3.2.1TheK ¨ ahlerstructure. ThereareplentyofchoicesofK¨ahleranti-involutions.Themostnaturaloneisobtainedasfollows:letusdenoteby h thecompletion of C 1 0 .Rd I C/ withrespecttothescalarproduct .h1 jh2 /F D .h1 j 1 h2 /Rd :

3FockquantizationonMinkowskispace

If m>0,thisspaceisthe(complex)Sobolevspace H 1 2 .Rd / andif m D 0 the complexhomogeneousSobolevspace P H 1 2 .Rd /,exceptwhen d D 1,sincethe integral ´R jkj 1 d kdivergesatk D 0.Thisisanexampleoftheso-called infrared problem formasslessfieldsintwospacetimedimensions.

Toavoidasomewhatlengthydigression,wewillassumethat m>0 if d D 1. Letusintroducethemap

Aneasycomputationshowsthat:

Inotherwords,jisaK¨ahleranti-involutionon C 1 0 .Rd I R2 / andtheassociatedoneparticleHilbertspaceisunitarilyequivalentto h.Moreover,afteridentificationby V ,thesymplecticgroup fetA gt 2R becomestheunitarygroup feit gt 2R with positive generator .ThispositivityisthedistinctivefeatureoftheFockrepresentation.

3.3Quantumspacetimefields

Letusset

theintegralbeingforexamplenormconvergentin B.Dom N 1 2 ; s .h//.Weobtain from(2.14)and(3.7)that

and ˆF .Pu/ D 0.Settingformally

weobtainthe spacetimefields ˆF .x/,whichsatisfy

3.4Localalgebras 19

3.3.1Thevacuumstate. LetusdenotebyCCRpol .KG/ the -algebragenerated bythe ˆF .u/;u 2 C 1 0 .Rn I R/,seeSubsections 4.3.1 and 4.5.1 belowforaprecise definition.Thevacuumvector vac 2 s .h/ inducesa state !vac onCCRpol .KG/, calledthe Fockvacuumstate,by

Clearly, !vac induceslinearmaps

whicharecontinuousforthetopologyof C 1 0 .Rn I R/,andhenceonecanwrite

wherethedistributions !N 2 D 0 .RNn / arecalledinphysicsthe N -pointfunctions. Amongthemthemostimportantoneisthe 2-pointfunction !2 ,whichequals

Ifwewritesimilarlythedistributionalkernelof G ,weobtainby(2.14)

Thefactthat !2 .x;x 0 / and G.x x 0 / dependonlyon x x 0 reflectstheinvariance ofthevacuumstate !vac underspaceandtimetranslations.

3.4Localalgebras

Werecallthata doublecone isasubset

Wedenoteby A.O/ thenormclosureofVect.feiˆF .u/ W supp u O g/ in B. s .h// From(2.13)and(3.12)itfollowsthat

ŒA.O1 /; A.O2 / Df0g; if O1 ;O2 arecausallydisjoint:

3FockquantizationonMinkowskispace

WeobtainarepresentationofthePoincar´egroup P.1;d/ by -automorphismsof CCRpol .KG/ bysetting ˛ƒ ˆF .x/ D ˆF .ƒ 1 x/ for ƒ 2 P.1;d/.Fromtheinvarianceofthevacuumstateundertranslations,weobtainthat ˛.a; / .A/ D U.a/AU.a/ 1 for A 2 CCRpol .KG/,where Rn 3 a ! U.a/ isastronglycontinuousunitarygroup on s .h/.

Wehave ˛ƒ .A.O// D A.LO C a/,for ƒ D .a;L/ 2 P.1;d/

3.4.1TheReeh–Schliederproperty. Onemightexpectthattheclosedsubspacegeneratedbythevectors A vac for A 2 A.O/ dependson O ,sinceitdescribes excitationsofthevacuum vac localizedin O .Thisisnotthecase,andactuallythe following Reeh–Schlieder propertyholds:

Proposition3.1. Foranydoublecone O thespace fA vac W A 2 A.O/g isdensein s .h/.

Proof. Let u 2 s .h/ suchthat .ujA vac / D 0 forall A 2 A.O/.If O1 b O isa smallerdoubleconeand A 2 A.O1 /,thefunction f W Rn 3 x ! .ujU.x/A vac / has aholomorphicextension F to Rn C iCC ,i.e. f.x/ D F.x C iCC 0/,asdistributional boundaryvalues,seeSection 12.1.

Since U.x/AU .x/ 2 A.O/,wehave f.x/ D 0 for x closeto 0,hencebythe edgeofthewedgetheorem,seeSubsection 12.1.2, F D 0 and f D 0 on Rn .Vectors oftheform U.x/A vac for x 2 Rn ;A 2 A.O1 / aredensein s .h/,hence u D 0.