资源描述：
《Springer - Coherent States And Applications In Mathematical Physics， Bescure Rob.pdf》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。
CoherentStatesandApplicationsinMathematicalPhysics TheoreticalandMathematicalPhysicsTheseriesfoundedin1975andformerly(until2005)entitledTextsandMonographsinPhysics(TMP)publisheshigh-levelmonographsintheoreticalandmathematicalphysics.ThechangeoftitletoTheoreticalandMathematicalPhysics(TMP)signalsthattheseriesisasuitablepublicationplatformforboththemathematicalandthethe-oreticalphysicist.Thewiderscopeoftheseriesisreflectedbythecompositionoftheeditorialboard,comprisingbothphysicistsandmathematicians.Thebooks,writteninadidacticstyleandcontainingacertainamountofelementarybackgroundmaterial,bridgethegapbetweenadvancedtextbooksandresearchmono-graphs.Theycanthusserveasbasisforadvancedstudies,notonlyforlecturesandseminarsatgraduatelevel,butalsoforscientistsenteringafieldofresearch.EditorialBoardW.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg,GermanyP.Chrusciel,GravitationalPhysics,UniversityofVienna,Vienna,AustriaJ.-P.Eckmann,DépartementdePhysiqueThéorique,UniversitédeGenève,Geneva,SwitzerlandH.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,AustriaA.Kupiainen,DepartmentofMathematics,UniversityofHelsinki,Helsinki,FinlandH.Löwen,InstituteofTheoreticalPhysics,Heinrich-Heine-UniversityofDuesseldorf,Duesseldorf,GermanyM.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,USAN.A.Nekrasov,IHÉS,Bures-sur-Yvette,FranceM.Ohya,TokyoUniversityofScience,Noda,JapanM.Salmhofer,InstituteofTheoreticalPhysics,UniversityofHeidelberg,Heidelberg,GermanyS.Smirnov,MathematicsSection,UniversityofGeneva,Geneva,SwitzerlandL.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USAJ.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,AustriaForfurthervolumes:www.springer.com/series/720 MoniqueCombescureDidierRobertCoherentStatesandApplicationsinMathematicalPhysics MoniqueCombescureDidierRobertBatimentPaulDiracLaboratoireJean-LerayIPNLDepartementdeMathematiquesVilleurbanneNantesUniversityFranceNantesCedex03FranceISSN1864-5879e-ISSN1864-5887TheoreticalandMathematicalPhysicsISBN978-94-007-0195-3e-ISBN978-94-007-0196-0DOI10.1007/978-94-007-0196-0SpringerDordrechtHeidelbergLondonNewYorkLibraryofCongressControlNumber:2012931507©SpringerScience+BusinessMediaB.V.2012Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwrittenpermissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Printedonacid-freepaperSpringerispartofSpringerScience+BusinessMedia(www.springer.com) PrefaceThemaingoalofthisbookistogiveapresentationofvarioustypesofcoherentstatesintroducedandstudiedinthephysicsandmathematicsliteratureduringal-mostacentury.Wedescribetheirmathematicalpropertiestogetherwithapplicationtoquantumphysicsproblems.Itisintendedtoserveasacompendiumoncoherentstatesandtheirapplicationsforphysicistsandmathematicians,stretchingfromthebasicmathematicalstructuresofgeneralizedcoherentstatesinthesenseofGilmoreandPerelomov1viathesemiclassicalevolutionofcoherentstatestovariousspecificexamplesofcoherentstates(hydrogenatom,torusquantization,quantumoscilla-tor).Wehavetriedtoshowthatthefieldofapplicationsofcoherentstatesiswide,diversifiedandstillalive.Becauseofourownabilitylimitationswehavenotcoveredthewholefield.Besidesthiswouldbeimpossibleinonebook.Wehavechosensomepartsofthesubjectwhicharesignificantforus.Othercolleaguesmayhavedifferentopinions.Thereexistseveraldefinitionsofcoherentstateswhicharenotequivalent.Nowa-daysthemostwellknownistheGilmore–Perelomov[84,85,155]definition:aco-herentstatesystemisanorbitforanirreduciblegroupactioninanHilbertspace.Fromamathematicalpointofviewcoherentstatesappearlikeapartofgrouprep-resentationtheory.InparticularcanonicalcoherentstatesareobtainedwiththeWeyl–HeisenberggroupactioninL2(R)andthestandardGaussianϕ1/4−x2/20(x)=πe.Modulomul-tiplicationbyacomplexnumber,theorbitofϕ0isdescribedbytwoparameters(q,p)∈R2andtheL2-normalizedcanonicalcoherentstatesare−1/4−(x−q)2/2i((x−q)p+qp/2)ϕq,p(x)=πee.Waveletsareincludedinthegroupdefinitionofcoherentstates:theyareobtainedfromtheactionoftheaffinegroupofR(x→ax+b)ona“motherfunction”ψ∈L2(R).Thewaveletsystemhastwoparameters:ψa,b(x)=√1ψ(x−b).aa1Theyhavediscoveredindependentlytherelationshipwithgrouptheoryin1972.v viPrefaceOneofthemostusefulpropertyofcoherentsystemψzisthattheyarean“over-complete”systemintheHilbertspaceinthesensethatwecananalyzeanyη∈Hwithitscoefficientψz,ηandwehaveareconstructionformulaofηlikeη=dzη(z)ψ˜z,whereη˜isacomplexvaluedfunctiondependingonψz,η.Coherentstates(beinggivennoname)werediscoveredbySchrödinger(1926)whenhesearchedsolutionsofthequantumharmonicoscillatorbeingtheclosestpossibletotheclassicalstateorminimizingtheuncertaintyprinciple.Hefoundthatthesolutionsareexactlythecanonicalcoherentstatesϕz.Glauber(1963)hasextendedtheSchrödingerapproachtoquantumelectro-dynamicandhecalledthesestatescoherentstatesbecausehesucceededtoexplaincoherencephenomenainlightpropagationusingthem.AftertheworksofGlauber,coherentstatesbecameaverypopularsubjectofresearchinphysicsandinmathe-matics.Thereexistseveralbooksdiscussingcoherentstates.Perelomov’sbook[156]playedanimportantroleinthedevelopmentofthegroupaspectofthesubjectandinitsapplicationsinmathematicalphysics.Severalotherbooksbroughtcontributionstothetheoryofcoherentstatesandworkedouttheirapplicationsinseveralfieldsofphysics;amongthemwehave[3,80,126]butmanyotherscouldbequotedaswell.Thereisahugenumberoforiginalpapersandreviewpapersonthesubject;wehavequotedsomeoftheminthebibliography.Weapologizetheauthorsforforgottenreferences.Inthisbookweputemphasisonapplicationsofcoherentstatestosemi-classicalanalysisofSchrödingertypeequation(timedependentortimeindependent).Semi-classicalanalysismeansthatwetrytounderstandhowsolutionsoftheSchrödingerequationbehaveasthePlanckconstantisnegligibleandhowclassicalmechanicsisalimitofquantummechanics.Itisnotsurprisingthatsemi-classicalanalysisandcoherentstatesarecloselyrelatedbecausecoherentstates(whichareparticularquantumstates)willbechosenlocalizedclosetoclassicalstates.Neverthelesswethinkthatinthisbookwehavegivenmoremathematicaldetailsconcerningtheseconnectionsthanintheothermonographsonthatsubjects.Letusgivenowaquickoverviewofthecontentofthebook.Thefirsthalfofthebook(Chap.1toChap.5)isconcernedwiththecanonical(standard)GaussianCoherentStatesandtheirapplicationsinsemi-classicalanalysisofthetimedependentandthetimeindependentSchrödingerequation.ThebasicingredienthereistheWeyl–Heisenbergalgebraanditsirreduciblerep-resentations.TherelationshipbetweencoherentstatesandWeylquantizationisex-plainedinChaps.2and3.InChap.4wecomputethequantumtimeevolutionofcoherentstatesinthesemi-classicalrégime:theresultisasqueezedcoherentstateswhoseshapeisdeformed,dependingontheclassicalevolutionofthesystem.ThemainoutcomeisaproofoftheGutzwillertraceformulagiveninChap.5.Thesecondhalfofthebook(Chap.6toChap.12)isconcernedwithextensionsofcoherentstatessystemstoothergeometrysettings.InChap.6weconsiderquan-tizationofthe2-toruswithapplicationtothecatmapandanexampleof“quantumchaos”. PrefaceviiChapters7and8explainthefirstexamplesofnoncanonicalcoherentstateswheretheWeyl–HeisenberggroupisreplacedsuccessivelybythecompactgroupSU(2)andthenon-compactgroupSU(1,1).WeshallseethatsomerepresentationsofSU(1,1)arerelatedwithsqueezedcanonicalcoherentstates,withquantumdy-namicsforsingularpotentialsandwithwavelets.WeshowinChap.9howitispossibletostudythehydrogenatomwithcoherentstatesrelatedwiththegroupSO(4).InChap.10weconsiderinfinitesystemsofbosonsforwhichitispossibletoextendthedefinitionofcanonicalcoherentstates.Thisisusedtoprovemean-fieldlimitresultfortwo-bodyinteractions:thelinearfieldequationcanbeapproximatedbyanonlinearSchrödingerequationinR3inthesemi-classicallimit(largenumberofparticlesorsmallPlanckconstantaremathematicallyequivalentproblems).Chapters11and12areconcernedwithextensionofcoherentstatesforfermionswithapplicationstosupersymmetricsystems.FinallyintheappendiceswehaveatechnicalsectionAaroundthestationaryphasetheorem,andinsectionBwerecallsomebasicfactsconcerningLiealgebras,Liegroupsandtheirrepresentations.Weexplainhowthisisusedtobuildgeneral-izedcoherentsystemsinthesenseofGilmore–Perelomov.Thematerialcoveredinthesebookisdesignedforanadvancedgraduatestudent,orresearcher,whowishestoacquainthimselfwithapplicationsofcoherentstatesinmathematicsorintheoreticalphysics.Wehaveassumedthatthereaderhasagoodfoundinginlinearalgebraandclassicalanalysisandsomefamiliaritywithfunctionalanalysis,grouptheory,linearpartialdifferentialequationsandquantummechanics.WewouldliketothankourcolleaguesofLyon,Nantesandelsewhere,fordiscus-sionsconcerningcoherentstates.InparticularwethankourcollaboratorJimRalstonwithwhomwehavegivenanewproofofthetraceformula,StephanDebièvre,AlainJoyeandAndréMartinezforstimulatingmeetings.M.C.alsothanksSylvieFloresforofferingvaluablesupportinthebibliography.ToconcludewewishtoexpressourgratitudetoourspousesAlainandMarie-Francewhoseunderstandingandsupporthavepermittedtoustospendmanyhoursforthewritingofthisbook.LyonandNantes,FranceMoniqueCombescureDidierRobert Contents1IntroductiontoCoherentStates.....................11.1TheWeyl–HeisenbergGroupandtheCanonicalCoherentStates..21.1.1TheWeyl–HeisenbergTranslationOperator........21.1.2TheCoherentStatesofArbitraryProfile..........61.2TheCoherentStatesoftheHarmonicOscillator..........71.2.1DefinitionandProperties..................71.2.2TheTimeEvolutionoftheCoherentStatefortheHarmonicOscillatorHamiltonian..........111.2.3AnOver-completeSystem..................121.3FromSchrödingertoBargmann–FockRepresentation.......162WeylQuantizationandCoherentStates................232.1ClassicalandQuantumObservables.................232.1.1GroupInvarianceofWeylQuantization...........272.2WignerFunctions..........................282.3CoherentStatesandOperatorNormsEstimates...........352.4ProductRuleandApplications...................402.4.1TheMoyalProduct.....................402.4.2FunctionalCalculus.....................432.4.3PropagationofObservables.................452.4.4ReturntoSymplecticInvarianceofWeylQuantization...472.5HusimiFunctions,FrequencySetsandPropagation........492.5.1FrequencySets........................492.5.2AboutFrequencySetofEigenstates.............522.6WickQuantization..........................522.6.1GeneralProperties......................522.6.2ApplicationtoSemi-classicalMeasures...........553TheQuadraticHamiltonians......................593.1ThePropagatorofQuadraticQuantumHamiltonians........593.2ThePropagationofCoherentStates.................613.3TheMetaplecticTransformations..................69ix xContents3.4RepresentationoftheQuantumPropagatorinTermsoftheGeneratorofSqueezedStates.................713.5RepresentationoftheWeylSymboloftheMetaplecticOperators.783.6Traps.................................813.6.1TheClassicalMotion....................813.7TheQuantumEvolution.......................824TheSemiclassicalEvolutionofGaussianCoherentStates......874.1GeneralResultsandAssumptions..................874.1.1AssumptionsandNotations.................884.1.2TheSemiclassicalEvolutionofGeneralizedCoherentStates.............................914.1.3RelatedWorksandOtherResults..............1004.2ApplicationtotheSpreadingofQuantumWavePackets......1004.3EvolutionofCoherentStatesandBargmannTransform......1034.3.1FormalComputations....................1034.3.2WeightedEstimatesandFourier–BargmannTransform...1054.3.3LargeTimeEstimatesandFourier–BargmannAnalysis..1074.3.4ExponentiallySmallEstimates...............1104.4ApplicationtotheScatteringTheory................1145TraceFormulasandCoherentStates..................1235.1Introduction.............................1235.2TheSemi-classicalGutzwillerTraceFormula............1275.3PreparationsfortheProof......................1315.4TheStationaryPhaseComputation.................1355.5APointwiseTraceFormulaandQuasi-modes...........1435.5.1APointwiseTraceFormula.................1445.5.2Quasi-modesandBohr–SommerfeldQuantizationRules..1456QuantizationandCoherentStatesonthe2-Torus...........1516.1Introduction.............................1516.2TheAutomorphismsofthe2-Torus.................1516.3TheKinematicsFrameworkandQuantization...........1556.4TheCoherentStatesoftheTorus..................1626.5TheWeylandAnti-WickQuantizationsonthe2-Torus.......1666.5.1TheWeylQuantizationonthe2-Torus...........1666.5.2TheAnti-WickQuantizationonthe2-Torus........1686.6QuantumDynamicsandExactEgorov’sTheorem.........1706.6.1QuantizationofSL(2,Z)..................1706.6.2TheEgorovTheoremIsExact................1736.6.3PropagationofCoherentStates...............1746.7EquipartitionoftheEigenfunctionsofQuantizedErgodicMapsonthe2-Torus............................1756.8SpectralAnalysisofHamiltonianPerturbations...........177 Contentsxi7Spin-CoherentStates...........................1837.1Introduction.............................1837.2TheGroupsSO(3)andSU(2)....................1837.3TheIrreducibleRepresentationsofSU(2)..............1877.3.1TheIrreducibleRepresentationsofsu(2)..........1877.3.2TheIrreducibleRepresentationsofSU(2)..........1917.3.3IrreducibleRepresentationsofSO(3)andSphericalHarmonics..........................1967.4TheCoherentStatesofSU(2)....................1997.4.1DefinitionandFirstProperties................1997.4.2SomeExplicitFormulas...................2037.5CoherentStatesontheRiemannSphere...............2137.6ApplicationtoHighSpinInequalities................2167.6.1Berezin–LiebInequalities..................2167.6.2HighSpinEstimates.....................2177.7MoreonHighSpinLimit:FromSpin-CoherentStatestoHarmonic-OscillatorCoherentStates...............2208Pseudo-Spin-CoherentStates......................2258.1IntroductiontotheGeometryofthePseudo-Sphere,SO(2,1)andSU(1,1).............................2258.1.1MinkowskiModel......................2258.1.2LieAlgebra.........................2278.1.3TheDiscandtheHalf-PlanePoincaréRepresentationsofthePseudo-Sphere....................2298.2UnitaryRepresentationsofSU(1,1)................2318.2.1ClassificationofthePossibleRepresentationsofSU(1,1)..........................2338.2.2DiscreteSeriesRepresentationsofSU(1,1)........2348.2.3IrreducibilityofDiscreteSeries...............2388.2.4PrincipalSeries.......................2398.2.5ComplementarySeries....................2418.2.6BosonsSystemsRealizations................2438.3Pseudo-CoherentStatesforDiscreteSeries.............2458.3.1DefinitionofCoherentStatesforDiscreteSeries......2458.3.2SomeExplicitFormula...................2468.3.3BargmannTransformandLargekLimit..........2508.4CoherentStatesforthePrincipalSeries...............2528.5GeneratorofSqueezedStates.Application.............2528.5.1TheGeneratorofSqueezedStates..............2538.5.2ApplicationtoQuantumDynamics.............2548.6WaveletsandPseudo-Spin-CoherentStates.............2589TheCoherentStatesoftheHydrogenAtom..............2639.1TheS3SphereandtheGroupSO(4)................2639.1.1Introduction.........................263 xiiContents9.1.2IrreducibleRepresentationsofSO(4)............2649.1.3HypersphericalHarmonicsandSpectralDecompositionofΔS3............................2669.1.4TheCoherentStatesforS3.................2689.2TheHydrogenAtom.........................2719.2.1Generalities.........................2719.2.2TheFockTransformation:AMapfromL2(S3)tothePure-PointSubspaceofHˆ..............2739.3TheCoherentStatesoftheHydrogenAtom.............27710BosonicCoherentStates.........................28510.1Introduction.............................28510.2FockSpaces.............................28610.2.1BosonsandFermions....................28610.2.2Bosons............................28910.3TheBosonsCoherentStates.....................29210.4TheClassicalLimitforLargeSystemsofBosons..........29610.4.1Introduction.........................29610.4.2Hepp’sMethod........................29710.4.3RemainderEstimatesintheHeppMethod.........30310.4.4TimeEvolutionofCoherentStates.............30611FermionicCoherentStates........................31111.1Introduction.............................31111.2FromFermionicFockSpacestoGrassmannAlgebras.......31211.3IntegrationonGrassmannAlgebra.................31511.3.1MorePropertiesonGrassmannAlgebras..........31511.3.2CalculuswithGrassmannNumbers.............31711.3.3GaussianIntegrals......................31811.4Super-HilbertSpacesandOperators.................32011.4.1ASpaceforFermionicStates................32011.4.2IntegralKernels.......................32211.4.3AFourierTransform.....................32311.5CoherentStatesforFermions....................32411.5.1WeylTranslations......................32411.5.2FermionicCoherentStates..................32511.6RepresentationsofOperators....................32711.6.1Trace.............................32811.6.2RepresentationbyTranslationsandWeylQuantization...33111.6.3Wigner–WeylFunctions...................33411.6.4TheMoyalProductforFermions..............33911.7Examples...............................34111.7.1TheFermiOscillator.....................34111.7.2TheFermi–DiracStatistics.................34211.7.3QuadraticHamiltoniansandCoherentStates........34311.7.4MoreonQuadraticPropagators...............348 Contentsxiii12SupercoherentStates—AnIntroduction................35312.1Introduction.............................35312.2QuantumSupersymmetry......................35412.3ClassicalSuperspaces........................35612.3.1MorphismsandSpaces...................35612.3.2SuperalgebraNotions....................35712.3.3ExamplesofMorphisms...................35812.4Super-LieAlgebrasandGroups...................35912.4.1Super-LieAlgebras.....................35912.4.2Supermanifolds,aVeryBriefPresentation.........36112.4.3Super-LieGroups......................36212.5ClassicalSupersymmetry......................36612.5.1AShortOverviewofClassicalMechanics.........36612.5.2SupersymmetricMechanics.................36912.5.3SupersymmetricQuantization................37412.6SupercoherentStates.........................37612.7PhaseSpaceRepresentationsofSuperOperators..........37812.8ApplicationtotheDickeModel...................379AppendixAToolsforIntegralComputations...............383A.1FourierTransformofGaussianFunctions..............383A.2SketchofProofforTheorem29...................383A.3ADeterminantComputation.....................384A.4TheSaddlePointMethod......................387A.4.1TheOneRealVariableCase.................387A.4.2TheComplexVariablesCase................387A.5KählerGeometry...........................388AppendixBLieGroupsandCoherentStates...............391B.1LieGroupsandCoherentStates...................391B.2OnLieGroupsandLieAlgebras..................391B.2.1LieAlgebras.........................391B.2.2LieGroups..........................392B.3RepresentationsofLieGroups....................395B.3.1GeneralPropertiesofRepresentations............395B.3.2TheCompactCase......................397B.3.3TheNon-compactCase...................398B.4CoherentStatesAccordingGilmore–Perelomov..........398AppendixCBerezinQuantizationandCoherentStates.........401References...................................405Index......................................413 Chapter1IntroductiontoCoherentStatesAbstractInthisChapterwestudytheWeylHeisenberggroupintheSchrödingerrepresentationinarbitrarydimensionn.OneshowsthatitoperatesintheHilbertspaceofquantumstates(andonquantumoperators)asaphase-spacetranslation.ThenapplyingittoaSchwartzclassstateofarbitraryprofilewegetasetofgen-eralizedcoherentstates.WhenweapplytheWeylHeisenbergtranslationoperatortothegroundstateofthen-dimensionalHarmonicOscillator,onegetsthestan-dardcoherentstatesintroducedbySchrödinger(Naturwissenshaften14:664666,1926)intheearlydaysofquantummechanics(1926).LaterthecoherentstateshavebeenextensivelystudiedbyGlauber(Phys.Rev.131:27662788,1963;Phys.Rev.130:25292539,1963)forthepurposeofquantumopticsanditseemsthattheirnamecomesbacktothiswork.ThestandardcoherentstateshavebeengeneralizedbyPerelomov(GeneralisedCoherentStatesandTheirApplications,1986)tomoregeneralLiegroupsthantheWeylHeisenberggroup.Wealsointroducetheusualcreationandannihilationoperatorsindimensionnwhichareveryconvenientforthestudyofcoherentstates.Weshowthatcoherentstatesconstituteanon-orthogonalover-completesystemwhichyieldsaresolutionoftheidentityoperatorintheHilbertspaceandwhichallowsacomputationoftheHilbertSchmidtnormandofthetraceofrespectivelyHilbertSchmidtclassandtrace-classoperators.Westudytheirtime-evolutionforthequantumHarmonicOscillatorhamiltonianandshowthatatimeevolvedcoherentstatelocatedaroundphase-spacepointzisuptoaphaseacoherentstatelocatedaroundthephase-spacepointzt,whereztisthephase-spacepointoftheclassicalflowgovernedbytheHarmonicOscillator.ThispropertywasdescribedbySchrödingerasthenon-spreadingofthetimeevolutionofcoherentstatesunderthequantumHarmonicOscillatordynamics.WealsoshowhowtogofromtheSchrödingertotheFockBargmannrepresen-tationusingthestandardcoherentstates.M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,1TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_1,©SpringerScience+BusinessMediaB.V.2012 21IntroductiontoCoherentStates1.1TheWeylHeisenbergGroupandtheCanonicalCoherentStates1.1.1TheWeylHeisenbergTranslationOperatorConsiderquantummechanicsindimensionn.ThenthepositionoperatorQˆhasncomponentsQˆ1,...,QˆnwhereQˆjisthemultiplicationoperatorinL2(Rn)bythecoordinatexj.SimilarlythemomentumoperatorPˆhasncomponentsPˆjwhere∂Pˆj=−i(1.1)∂xjisthePlanckconstantdivideddy2π.QˆandPˆareselfadjointoperatorswithsuitabledomainsD(Q)ˆandD(P)ˆ.DQˆ=u∈L2Rnx2Rn,∀j=1,...,nju(x)∈L∂uDPˆ=u∈L2Rn∈L2Rn,∀j=1,...,n∂xjTheoperatorsQˆandPˆobeythefamousHeisenbergcommutationrelationPˆj,Qˆk=−δj,ki(1.2)onthedomainofQˆ·Pˆ−Pˆ·Qˆ.Thebracket[A,ˆBˆ]isthecommutator:A,ˆBˆ=AˆBˆ−BˆAˆOntheintersectionofthedomainsD(Q)ˆ∩D(P)ˆtheoperatorp·Qˆ−q·Pˆiswelldefinedforz=(q,p)∈R2n,wherethedotrepresentsthescalarproduct:np·Qˆ=pjQˆj1ItisselfadjointsoitisthegeneratorofaunitaryoperatorT(z)ˆcalledtheWeylHeisenbergtranslationoperator:iT(z)ˆ=expp·Qˆ−q·Pˆ(1.3)NowweusetheBakerCampbellHausdorffformulaLemma1Considertwoanti-selfadjointoperatorsAˆ,BˆintheHilbertspaceH,withdomainsD(A)ˆ,D(B)ˆ.Weassumethefollowingconditionsaresatisfied:(i)ThereexistsalinearsubspacespaceH0denseinH,whichisacoreforAˆandBˆ. 1.1TheWeylHeisenbergGroupandtheCanonicalCoherentStates3(ii)H0isinvariantforAˆ,Bˆ,etAˆ,etBˆ,∀t∈R.(iii)AˆandBˆcommutewith[A,ˆBˆ]inH0andi[A,ˆBˆ],welldefinedinH0,hasaselfadjointextensioninH.Thenwehave1expAˆ+Bˆ=exp−A,ˆBˆexpAˆexpBˆ(1.4)2ProofLetusintroduce−t2/2[A,ˆBˆ]tAˆtBˆF(t)u=eeeuwhereu∈H0isfixed.Letuscomputethetimederivative2F(t)u=−tA,ˆBˆe−t/2[A,ˆBˆ]etAˆetBˆu2+e−t/2[A,ˆBˆ]etAˆAˆ+BˆetBˆu(1.5)TheonlydifficultyistocommuteBˆwithetAˆ.Butwehave,usingthecommutationsassumptions,detAˆBˆe−tAˆ=etAˆA,ˆBˆe−tAˆ=A,ˆBˆdtSowegetthatF(t)=Aˆ+BˆF(t)(1.6)andtheformula(1.4)follows.Usingthisformula,onededucesthemultiplicationlawfortheoperatorsT(z)ˆ:T(z)ˆT(zˆiT(zˆ+z)=exp−σ(z,z))(1.7)2whereforz=(q,p),z=(q,p),σ(z,z)isthesymplecticproduct:σ(z,z)=q·p−p·q(1.8)andiT(z)ˆT(zˆ)=exp−σ(z,z)TˆzT(z)ˆwhichistheintegralformoftheHeisenbergcommutationrelation.Inparticularwehave:−1∗T(z)ˆ=T(z)ˆ=T(ˆ−z)sincethesymplecticproductofzbyitselfiszero.ThefactthattheWeylHeisenbergunitaryoperatorisatranslationoperatorcanbeseeninthefollowinglemma: 41IntroductiontoCoherentStatesLemma2Foranyz=(q,p)∈R2nonehasQˆQˆ−qT(z)ˆT(z)ˆ−1=(1.9)PˆPˆ−pProofLetusdenoteL(z)ˆ=p·Qˆ−q·Pˆifz=(q,p).WehaveeasilyditLˆ(z)−itL(z)ˆitLˆ(z)−itL(z)ˆieQˆe=eQ,ˆL(z)ˆedtButwehave[Q,ˆL(z)ˆ]=−iq.SowegettheformulaforQˆ.WiththesameproofwegettheformulaforPˆ.Corollary1−iq·p/2ip·Qˆ−iq·PˆT(z)ˆ=eee(1.10)ProofLet−it2q·p/2itp·Qˆ−itq·PˆU(t)ˆ=eeeUsingLemma1wegetddiT(tz)ˆ=U(t)ˆ=L(tz)ˆU(t)ˆdtdtHencethecorollaryfollows.Letusspecifythesituationindimension1.Weintroduce:ie1=√Pˆie2=√Qˆe3=i1Weeasilycheckthat[e1,e2]=e3,[e1,e3]=[e2,e3]=0ThismeansthattheoperatorsQ,ˆP,ˆ1generateaLiealgebradenotedbyh1whichistheWeylHeisenbergalgebra.Theelementsofthisalgebraaredefinedusingtripletsofcoordinates(s;x,y)∈R3by:W=xe1+ye2+se3(1.11) 1.1TheWeylHeisenbergGroupandtheCanonicalCoherentStates5Inquantummechanicsitismoreconvenienttousethefollowingcoordinates:itiW=−1+pQˆ−qPˆ2wheretherealnumberst,q,paredefinedas√√q=−x,p=y,t=−2sThenwecancalculatethecommutatoroftwoelementsW,WoftheLiealgebrah1:Lemma3e[W,W]=xy−yx3(1.12)σ((x,y),(x,y))=xy−xyissimplythesymplecticproductof(x,y)and(x,y).ProofWesimplyuseLemma1.ForanyWinh1wecandefinetheunitaryoperatoreWandwegetagroupusing(1.4).ThisgroupisdenotedH1.ItisaLiegroupanditsLiealgebraish1.TheLiegroupH1issimplyR3withthenoncommutativemultiplication(t,z)(t,z)=t+t+σ(z,z),z+z,wheret∈R,z∈R2(1.13)Wededuce(1.13)fromanelementarycomputation.IfW,W∈h1using(1.4)wehaveWWW1ee=e,whereW=[W,W]+W+W2Usingthe(t,q,p)and(t,q,p)coordinatesforWandWrespectively,wegetthecorrespondingcoordinates(t,q,p)forWsuchthatt=t+t+σ(z,z),z=z+zwhichistheWeylHeisenberggroupmultiplication(1.13).InthesamewaywedefinetheWeylHeisenbergalgebrahnanditsLieWeylHeisenberggroupHnforanyn≥1.TheWeylHeisenbergGroupHnandSchrödingerRepresentationinDimen-sionnTheWeylHeisenbergLiealgebrahnisareallinearspaceofdimension2n+1.AnyW∈hnhasthedecompositionitiW=−1+p·Qˆ−q·Pˆ,whereQˆ=Qˆ1,...,Qˆn,Pˆ=Pˆ1,...,Pˆn2(t;q,p)=(t;z)∈R×R2nisacoordinatessystemforW.TheLiebracketofWandW,inthesecoordinates,isi[W,W]=σ(z,z)1 61IntroductiontoCoherentStatesThisreflectstheHeisenbergcommutationrelations(1.2).Asforn=1agroupmultiplicationisintroducedinR×R2ntoreflectmultiplica-tionbetweenoperatorseW.So,HnisthesetR×R2nwiththegroupmultiplication(t,z)(t,z)=t+t+σ(z,z),z+z(1.14)whereσisthesymplecticbilinearforminR2n:σ(z,z)=q·p−q·p,ifz=(q,p),z=(q,p)HnisaLiegroupofdimension2n+1.TheSchrödingerrepresentationisdefinedasthefollowingunitaryrepresentationofHninL2(Rn):ρ(t,z)=e−it/2T(z),ˆ(t,z)∈HnInotherwordsthemap(t,z)→ρ(t,z)isagrouphomomorphismfromtheWeylHeisenberggroupHnintothegroupofunitaryoperatorsintheHilbertspaceL2(Rn).BytakingtheexponentialofWonerecoverstheWeylHeisenbergLiegroupdefinedabove:ieW=e−it/2exppQˆ−qPˆ=e−it/2T(z)ˆRecallthatz=(q,p).Remark1TheSchrödingerrepresentationisirreducible,thiswillbeaconsequenceoftheSchurLemma10.AccordingtothecelebratedStonevonNeumanntheorem(see[182])theSchrödingerrepresentationistheuniqueirreduciblerepresentationofHn,uptoconjugationwithaunitaryoperator,forevery>0.1.1.2TheCoherentStatesofArbitraryProfileTheactionoftheWeylHeisenbergtranslationoperatoronastateu∈L2(Rn)isthefollowing:iiT(z)uˆ(x)=exp−q·pexpx·pu(x−q)(1.15)2Physicallyittranslatesastatebyz=(q,p)inphasespace.Onehasasimilarfor-mulafortheFouriertransformthatwedenoteFdefinedasfollows:−n−ix·ξFu(ξ)=(2π)eu(x)dxRn 1.2TheCoherentStatesoftheHarmonicOscillator7iiFT(z)uˆ(ξ)=expq·pexp−q·ξF(u)(ξ−p)2whichsaysthatthestateistranslatedbothinpositionandmomentumbyrespec-tivelyqandp.Nowtakinganyfunctionu0intheSchwartzclassS(Rn)thecoherentstateassociatedtoitwillbesimplyuz(x)=T(z)uˆ0(x)(1.16)AusefulexampleforapplicationsisthefollowinggeneralizedGaussianfunction.LebeΓasymmetriccomplexn×nmatrixsuchthatitsimaginarypartΓispositive-definite.Thenwecantakeu0=ϕ(Γ),where(Γ)−n/41/4iΓx·xϕ(x)=(π)det(Γ)e2(1.17)1.2TheCoherentStatesoftheHarmonicOscillator1.2.1DefinitionandPropertiesTheyhavebeenintroducedbySchrödingerandhavebeenextensivelystudiedandused.Theyareobtainedbytakingasreferencestateu0thegroundstateofthehar-monicoscillatorx2u−n/4exp−(1.18)0(x)=ϕ0(x)=(π)2Thusϕz:=T(z)ϕˆ0issimplyaGaussianstateoftheform−n/4ii(x−q)2ϕz(x)=(π)exp−q·pexpx·pexp−(1.19)22−n/4iq·pq·ξ(ξ−p)2(Fϕz)(ξ)=(π)expexp−i−(1.20)22ϕzisastatelocalizedintheneighborhoodofaphase-spacepoint√z=(q,p)∈R2nofsizeinallthepositionandmomentumcoordinates.ThenitisaquantumstatewhichistheanalogofaclassicalstatezobtainedbytheactionoftheWeylHeisenberggroupHnonϕ0.Theyarealsocalledcanonicalcoherentstates.Theyhavemanyinterestingandusefulpropertiesthatweconsidernow.Itisusefultousethestandardcreationandannihilationoperators:1a=√Qˆ+iPˆ(1.21)21a=√Qˆ−iPˆ(1.22)2 81IntroductiontoCoherentStatesaissimplytheadjointofadefinedonD(Q)ˆ∩D(P)ˆ.Furthermoreasimplecon-sequenceoftheHeisenbergcommutationrelationisthat:aj,a=δj,k(1.23)kThentheHamiltonianofthen-dimensionalharmonicoscillatoroffrequency1isn122nHˆos=Pˆ+Qˆ=ajaj+=a·a+a·a(1.24)222j=1Itistrivialtocheckthatthegroundstateϕ0ofHˆosisaneigenstateofawitheigen-value0.Aquestionis:arethecoherentstatesϕzalsoeigenstatesofa?Theanswerisyesandiscontainedinthefollowingproposition:Proposition1Letz=(q,p)∈R2n.Wedefinethenumberα∈Cnas1α=√(q+ip)(1.25)2ThenthefollowingholdsT(z)ˆaT(z)ˆ−1=a−α(1.26)Moreoveraϕz=αϕz(1.27)ProofWesimplyuseLemma2toprove(1.26).Thenweremarkthat:T(z)ˆaT(z)ˆ−1ϕz=T(z)ˆaϕ0=0=(a−α)ϕzTheBakerCampbellHausdorffformula(1.4)isstilltrueforannihilation-creationoperatorsbutweneedtoadapttheproofwiththefollowingmodifications.LetH0bethelinearspacespannedbytheproductsφα(x)eη·xwhereα∈Nnandη∈Cn.WecanextendthedefinitionofT(z)uˆforeveryu∈H0andz∈C2n.Lemma4Foreveryu∈H0,z→T(z)uˆcanbeextendedanalyticallytoC2n.More-overT(z)uˆ∈H0andwehaveforeveryz,z∈C2nandeveryu∈H0,T(z)ˆT(zˆiT(zˆ+z)u=exp−σ(z,z))u(1.28)2whereσisextendedasabilinearformtoC2n×C2n.ProofUsingformula(1.15),wecanextendT(z)uˆanalyticallytoC2n.Sowecandefineiexpp·Qˆ−q·Pˆu:=T(z)u,ˆforz=(q,p)∈C2n 1.2TheCoherentStatesoftheHarmonicOscillator9Nowwiththesameproofasfor(1.4),wegetthat(1.28)isstilltrueforeveryz,z∈Cn.Using(1.28),inthecreationandannihilationoperatorsrepresentationwehavethat|α|2T(z)ˆ=expα·a−¯α·a=exp−expα·aexp(−¯α·a)(1.29)2Recallthatbyconventionofthescalarproduct·wehave:nnα¯·a=α¯jaj,α·a=αjajj=1j=1Using(1.29)wehave,sinceexp(−¯α·a)ϕ0=ϕ0:|α|2ϕexpα·az=exp−ϕ0(1.30)2Twodifferentcoherentstatesoverlap.Theiroverlappingisgivenbythescalarprod-uctinL2(Rn).Wehavethefollowingresult:Proposition2σ(z,z)|z−z|2ϕz,ϕz
=expiexp−(1.31)24ProofWefirstestablishausefullemma:Lemma5|z|2ϕ0,T(z)ϕˆ0=exp−(1.32)4ProofWeuse(1.29).Soweget|α|2α·a−¯α·a|z|2−¯α·a2ϕ0,T(z)ϕˆ0=exp−ϕ0,eeϕ0=exp−eϕ024Butsinceϕ0isaneigenstateofawitheigenvalue0,wesimplyhavee−¯α·aϕ0=1TheoperatorT(z)ˆtransformsanycoherentstateinanothercoherentstateuptoaphase: 101IntroductiontoCoherentStatesLemma6T(z)ϕˆiz=exp−σ(z,z)ϕz+z2ProofTheproofisimmediateusing(1.7).Theoverlapbetweenϕzandϕzisgivenby:iϕz,ϕz
=T(z)ϕˆ0,T(zˆ)ϕ0=expσ(z,z)ϕ0,Tˆz−zϕ02wherewehaveused(1.7).Nowusingthelemmaforthelastfactorwegetthere-sult.Intheparticularcaseofthedimensionnequalsone,thektheigenstateφkoftheharmonicoscillator(theHermitefunction,normalizedtounity)isgeneratedby(a)k:−1/2kφk=(k!)aϕ0sothatexpandingtheexponential,formula(1.30)givesrisetothefollowingwell-knownidentity:∞αk2φϕz=exp−|α|/2√kk!k=0Inarbitrarydimensionn,theoperator(a)kexcitesthegroundstateoftheharmonicjoscillatortothekthexcitedstateofthejthdegreeoffreedom.Morepreciselyletk=(k1,...,kn)∈Nnbeamultiindex.ThecorrespondingeigenstateofHˆosis:φk(x)=φk1(x1)...φkn(xn)(1.33)andithaseigenvalueEk=(k1+k2+···+kn+n/2).Notethatthiseigenvalueishighlydegenerate,exceptE0.WehaveLemma7n(a)kjjφk=ϕ0(1.34)kj!j=1Thephysicistsoftenusetheketnotationforthequantumstates.Letusdefineitforcompleteness:|0
=ϕ0|k
=φkandtheyalsodesignatethecoherentstatewiththeketnotation:|z
=ϕz 1.2TheCoherentStatesoftheHarmonicOscillator11Thenwehave:Lemma8∞|z|2αk|z
=exp−|k(1.35)4k!kj=1qj+ipjwhereαj=√and2αk=αk1αk2...αkn12nk!=k1!k2!...kn!1.2.2TheTimeEvolutionoftheCoherentStatefortheHarmonicOscillatorHamiltonianAremarkablepropertyofthecoherentstatesisthattheHarmonicOscillatordy-namicstransformsthemintoothercoherentstatesuptoaphase.ThispropertywasanticipatedbySchrödingerhimself[175]whodescribesitasthenon-spreadingofthecoherentstateswavepacketsundertheHarmonicOscillatordynamics.Further-morethetime-evolvedcoherentstateislocatedaroundtheclassicalphase-spacepointoftheharmonicoscillatorclassicaldynamics.Letz:=(q,p)∈R2nbetheclassicalphase-spacepointattime0.Thenitistrivialtoshowthatthephase-spacepointattimetisjustzt:=(qt,pt)givenbyzt=FtzwhereFtistherotationmatrixcostsintFt=−sintcostWehavethefollowingproperty:Lemma9DefineQ(t)ˆQˆ=e−itHˆos/eitHˆos/P(t)ˆPˆNotethatQ(ˆ−t),P(ˆ−t)aretheso-calledHeisenbergobservablesassociatedtoQ,ˆPˆ.Then: 121IntroductiontoCoherentStates(i)Q(t)ˆQˆ=F−t(1.36)P(t)ˆPˆ(ii)e−itHˆos/T(z)ˆeitHˆos/=T(zˆt)Proof(i)Onehas,usingtheSchrödingerequationandthecommutationpropertyofQ,ˆPˆthatdQ(t)ˆ−P(t)ˆ=dtP(t)ˆQ(t)ˆThenthesolutionis(1.36).(ii)Thene−itHˆos/p·Qˆ−q·PˆeitHˆos/=p·Q(t)ˆ−q·P(t)ˆ=pt·Qˆ−qt·PˆByexponentiationonegetstheresult.Proposition3Thequantumevolutionfortheharmonicoscillatordynamicsofacoherentstateϕzisgivenbye−itHˆos/ϕ−itn/2z=eϕztProofe−itHˆos/ϕz=e−itHˆos/T(z)ˆeitHˆos/×e−itHˆos/ϕ0=T(zˆ−itn/2−itn/2t)eϕ0=eϕztwherewehaveusedthatϕ0isaneigenstateofHˆoswitheigenvaluen.2InChap.3weshallseeasimilarpropertyforanyquadratichamiltonianwithpossibletime-dependentcoefficients.Thenthequantumtimeevolutionofacoherentstatewillbeasqueezedstateinsteadofacoherentstate,locatedaroundthephase-spacepointztfortheassociatedclassicalflowwhichislinear(sincetheHamiltonianisquadratic).1.2.3AnOver-completeSystemWehaveseenthatthecoherentstatesarenotorthogonal.Sotheycannotbecon-sideredasabasisoftheHilbertspaceL2(Rn)ofthequantumstates.Insteadthey 1.2TheCoherentStatesoftheHarmonicOscillator13willconstituteanover-completesetofcontinuousstatesoverwhichthestatesandoperatorsofquantummechanicscanbeexpanded.WecannowintroducetheFourierBargmanntransformthatwillbestudiedinmoredetailsinSect.1.3.Westartfromu0∈L2(Rn),u02:=n|u0(x)|2dx=1.RLetusdefinetheFourierBargmanntransformbythefollowingformulaBv(z)=:v(z)=(2π)−n/2u2nFuz,v,z=(q,p)∈R(1.37)Ifu0isthestandardGaussianϕ0,theassociatedFourierBargmanntransformwillbedenotedFB.Proposition4FuBisanisometryfromL2(Rn)intoL2(R2n)ProofWehaveip·q−ix·p/uz,v
=e2v(x)u0(x−q)edx(1.38)RnFromPlanchereltheoremweget−nu2v(x)u2(2π)(q,p),vdp=0(x−q)dx(1.39)RnRnThenweintegrateinqvariableandchangethevariables:q=x−q,x=x,sowegettheresult.Thenbypolarizationwegetthatthescalarproductoftwostatesψ,ψ∈L2(Rn)canbeexpressedintermsofψ,(ψ):ψ,ψ
=dz(ψ)(z)ψ(z)(1.40)Wededuce,usingFubinitheoremthatthefunctionψ(z)determinesthestateψcompletely:(z)ϕψ=dzψz(1.41)ThisimpliesthattheSchrödingerrepresentationisirreducible.ThenweuseSchurslemma:Lemma10IfAˆisaboundedoperatorinL2(Rn)suchthatAˆT(z)ˆ=T(z)ˆA,ˆ∀z∈R2nthenAˆ=C1ˆforsomeC∈C. 141IntroductiontoCoherentStatesWededucethatthecoherentstatesprovidearesolutionofunity.Definethefol-lowingmeasure:dμ(z)=Cdz=Cdq1dq2...dqndp1dp2...dpn(1.42)whereandC∈Cisaconstanttobedeterminedlater.Let|zz|betheprojectionoperatoronthestate|z.WeconsidertheoperatorAˆ=dμ(z)|zz|Wehavethefollowingresult:Proposition5AˆcommuteswithalltheoperatorsT(z)ˆ.ProofUsing(1.7)weget:iA,ˆT(z)ˆ=dμ(z)exp−σ(z,z)|zz−z|2i−exp−σ(z,z)|z+zz|2Nowusingthechangeofvariablez=z+zinthelasttermwegetzero.ThereforeinviewoftheSchurslemmaAˆmustbeamultipleoftheidentityoperator:Aˆ=d−11WedeterminetheconstantdbycalculatingtheaverageoftheoperatorAˆinthecoherentstate|z:2|z|2d−1= z|Aˆ|z
=dμ(z)z|z=dμ(z)exp−2TheconstantCcanbechosensothatd=1.Thereforetheresolutionoftheidentitytakestheform:dμ(z)|zz|=1(1.43)wheredμ(z)isgivenby(1.42)andtheconstantCissuchthat|z|2Cdzexp−=1R2n2ThisgivesC=(2π)−n(1.44) 1.2TheCoherentStatesoftheHarmonicOscillator15Theresolutionofidentity(1.43)allowstocomputethetraceofanoperatorintermsofitsexpectationvalueinthecoherentstates.Letusrecallwhatthetraceofanoperatoriswhenitexists.Definition1AnoperatorBˆissaidtobeoftraceclasswhenforsome(andthenany)eigenbasisekoftheHilbertspaceonehasthattheseriesek,(Bˆ∗B)ˆ1/2ekisconvergent.ThenthetraceofBˆisdefinedasTrBˆ=ek,Beˆk(1.45)k∈NAnoperatorBˆissaidtobeofHilbertSchmidtclassifBˆ∗Bˆisoftraceclass.Proposition6LetBˆbeanHilbert–SchmidtoperatorinL2(Rn)thenwehaveBˆ2−nBuˆ2=(2π)zdz(1.46)HSR2nIfBˆisatrace-classoperatorinL2(Rn)thenwehave−nuTrBˆ=(2π)z,Buˆzdz(1.47)R2nProofLet{ej}beanorthonormalbasisforL2(Rn)(forexampletheHermitebasisφj).Wehave22Bˆ=BeˆjHSj2=Beˆj(1.48)jButwehaveBeˆBeˆ∗j(z)=j,uz=ej,Bˆuz(1.49)UsingParsevalformulaforthebasis{ej}weget2−n∗2Beˆj=(2π)Bˆuzdz(1.50)R2nj≥0UsingthatBˆ2=Bˆ∗2wegetthefirstpartofthecorollary.HSHSForthesecondpartweusethateveryclasstraceoperatorcanbewrittenasBˆ=Bˆ∗BˆBˆBˆ21where1,2areHilbertSchmidt.MoreovertheHilbertSchmidtnormisassociatedwiththescalarproductBˆ2,Bˆ1
=Tr(Bˆ∗Bˆ1).Soweget2 161IntroductiontoCoherentStatesTrBˆ=TrBˆ∗Bˆ=(2π)−nBˆBˆdz212uz,1uzR2n=(2π)−nuz,Buˆzdz(1.51)R2nTheseformulaswillappeartobeveryusefulinthesequel.1.3FromSchrödingertoBargmannFockRepresentationThisrepresentationiswelladaptedtothecreation-annihilationoperatorsandtotheHarmonicoscillator.ItwasintroducedbyBargmann[17].InthisrepresentationthephasespaceR2nisidentifiedtoCn:q−ip(q,p)→ζ=√2andastateψisrepresentedbythefollowingentirefunctiononCn:p2+q2ψ(ζ)=ψ(q,p)e4HolRecallthatψ(z)=(2π)−n/2ϕz,ψ,z=(q,p).2nProposition7Themapψ→ψisanisometryfromL(R)intotheFockspaceHolF(Cn)ofentirefunctionsfonCnsuchthatf(ζ)2−ζ·ζ¯dζ∧dζ¯<+∞eCnF(Cn)isanHilbertspaceforthescalarproductζ·ζ¯f2,f1
=f1(ζ)f2(ζ)e−dζ∧dζ¯(1.52)CnProofAdirectcomputationshowsthatψHolisholomorphic:∂ζ¯ψHol=0.Recallthattheholomorphicandantiholomorphicderivativesaredefinedasfollows.11∂ζ=√(∂q+i∂p),∂ζ¯=√(∂q−i∂p)22Wecaneasilygetthefollowingexplicitformulaforψ:Hol
1x2√ζ2ψ(ζ)=(π)−3n/42−n/2ψ(x)exp−−2x·ζ+dx(1.53)HolRn22 1.3FromSchrödingertoBargmannFockRepresentation17Thetransformationψ→ψiscalledtheBargmanntransformandisdenotedHolbyB.ItskernelistheBargmannkernel:
1x2√ζ2B(x,ζ)=(π)−3n/42−n/2exp−−2x·ζ+,22wherex∈Rn,ζ∈Cn(1.54)Recallthenotationsx2=x·x,ζ2=ζ·ζ,ζ·ζ¯=|ζ|2.Usingthatψ→ψisanisometryfromL2(Rn)intoL2(R2n),weeasilygetthatψ2−ζ·ζ¯dζ∧dζ¯=ψ2(ζ)e2(1.55)HolCnHenceBisanisometryformL2(Rn)intoF(Cn).−ζ·ζ¯nForconvenienceletusintroducetheGaussianmeasureonC,dμB=e|dζ∧dζ¯|.ItisnotdifficulttoseethatF(Cn)isacompletespace.If{fk}isaCauchysequenceinF(Cn)then{fk}convergestofinL2(Cn,dμB).Sowegetinaweaksensethat∂ζ¯f=0sofisholomorphichencef∈F(Cn).LetusnowcomputethestandardharmonicoscillatorintheBargmannrepresen-tation.Wefirstgetthefollowingformula√x2∂xψ(x)B(x,ζ)dx=ψ(x)−ζB(x,ζ)dx(1.56)RnRn√2ζ∂ζψ(x)B(x,ζ)dx=ψ(x)x−B(x,ζ)dx(1.57)RnRnHence1B(xψ)(ζ)=√(∂ζ+ζ)Bψ(ζ)(1.58)21B(∂xψ)(ζ)=√(∂ζ−ζ)Bψ(ζ)(1.59)2ThenwegettheBargmannrepresentationforthecreationandannihilationoperatorsBaψ(ζ)=ζB[ψ](ζ)(1.60)B[aψ](ζ)=∂ζB[ψ](ζ)(1.61)SothestandardharmonicoscillatorHˆnos=(aa+),hasthefollowingBargmann2representationHˆos=ζ·∂ζ+n(1.62)2 181IntroductiontoCoherentStatesRemark2ItisveryeasytosolvethetimedependentSchrödingerequationforHˆos.nn−it−itIfF∈F(C)suchthatζ∂ζF∈F(C),thenF(t,ζ)=e2F(e2ζ)satisfiesi∂F(t),F(0,ζ)=F(ζ)(1.63)tF(t)=HˆosMoreoverif=1andifwepute−t/2inplaceofe−it/2wesolvetheheatequation∂tF(t)=HˆosF(t).WeshallseenowthattheHermitefunctionsφαhaveaverysimpleshapeintheBargmannrepresentation.Letusdenoteφα#(ζ)=Bφα.ThenwehaveProposition8Foreveryα∈Nn,ζ∈Cn,φ#(ζ)=(2π)−n/2(α!)−1/2ζα(1.64)α#nMoreover{φα(ζ)}α∈NnisanorthonormalbasisinF(C).ProofLetusfirstrecallthenotationsindimensionn.Forα=(α1···αn)∈N,α!=αnαα1αn1!···αn!andforζ=(ζ1,...,ζn)∈C,ζ=ζ1···ζn.Wegeteasilythatζα,ζβ
=0ifα=β.Itisenoughtocompute,forn=1,ζk2andthisisaneasycomputationF(C)withtheGammafunction.αnLetusprovenowthatthesystem{ζ}α∈NnistotalinF(C).Letf∈F(Cn)besuchthatζα,f
=0forallα∈Nn.fisentiresowehavefαf(ζ)=αζαwherefαaretheTaylorcoefficientoffat0.ThesumisuniformlyconvergentoneveryballofCn.OntheothersidefromBesselinequality,weknowthattheαnαTaylorseriesαfαζconvergesinF(C).Butwecanseethat{ζ}α∈Nnisalsoanorthogonalsystemineachballwithcenterat0.Thenwegetthatfα=0foreveryαhencef=0.#Letusremarkherethatwecouldalsoprovethatthesystem{φα(ζ)}α∈Nnisor-thogonalusingthatHermitefunctionsisanorthonormalsystemandBisanisome-try.Finally,letusproveformula(1.64).Itisenoughtoassumen=1.Wegeteasilythatφ#=√1.Soforeveryk≥1,wehave,using(1.60),02π(a)kζk#φ(ζ)=φk(ζ)=B√0√k!2πk!Thenwegetthefollowinginterestingresult. 1.3FromSchrödingertoBargmannFockRepresentation19Corollary2TheBargmanntransformBisanisometryfromL2(Rn)ontoF(Cn).TheintegralkernelofB−1is
1x2√ζ¯2B−1(ζ,x)=(π)−3n/42−n/2exp−−2x·ζ¯+(1.65)22wherex∈Rn,ζ∈Cn.WealsogetthattheBargmannkernelisageneratingfunctionfortheHermitefunctions.Corollary3Foreveryx∈Rnandζ∈CnwehaveζαB(x,ζ)=φα(x)(1.66)((2π)nα!)1/2α∈NnProofComputetheFouriercoefficientintheHermitebasisofx→B(x,ζ).ThestandardcoherentstatesalsohaveasimpleexpressionintheBargmannFockspace.LetϕXbethenormalizedcoherentstateatX=(x,ξ).Proposition9WehavethefollowingBargmannrepresentationforthecoherentstateϕX−n/2η¯(ζ−η)B[ϕX](ζ)=(2π)e2(1.67)x−iξwhereη=√.2ProofAdirectcomputationgives
√−n1√ζ2B[ϕ2πdyexp−y2−yx+iξ+2ζ−(1.68)X](ζ)=Rn2ThenwegettheresultbyFouriertransformoftheGaussiane−y2.OneofthenicepropertiesofthespaceF(Cn)isexistenceofareproducingkernel.Proposition10Foreveryf∈F(Cn)wehave−nη¯·ζnf(ζ)=(2π)ef(η)dμB(η),∀ζ∈C(1.69)Cn 201IntroductiontoCoherentStatesProofItisenoughtoassumethatfisapolynomialinζandthat=1.Sowehaveζαη¯αf(ζ)=cαn/21/2,withcα=n/21/2f(η)dμB(η)(2π)(α!)Cn(2π)(α!)α(1.70)Hencewegetζαη¯αf(ζ)=(2π)−nf(η)dμ−neη¯·ζf(η)dμB(η)=(2π)B(η)Cnαα!Cn(1.71)η¯·ζ−nRemark3Thefunctioneζ(η)=(2π)eisarepresentationoftheDiracdeltafunctioninthepointζ.NotethateζisnotinF(Cn).Moreoverwehavef(ζ)=−neζ,fand|f(ζ)|≤(2π)fF(Cn).UsingtheBargmannrepresentationwecangiveaproofofthewell-knownMehlerformulaconcerningtheHermiteorthonormalbasis{φk}inL2(R).Itissuf-ficienttoassumethat=1.Theorem1Foreveryw∈Csuchthat|w|<1wehaveφ(x)φ(y)wkkkk∈Nn−n/22−n/21+w2222w=π(1−w)exp−x+y+x·y(1.72)2(1−w2)1−w2wherek=|k|=k1+···+kk.ProofThecasen≥2canbeeasilydeducedfromthecasen=1.Soletn=1.Theleftandrightsideof(1.72)areholomorphicinwintheunitdisc{w∈C,|w|<1}.Sobyanalyticcontinuationprincipleitisenoughtoproveitforw=e−t/2foreveryt>0.Hencetherightsideof(1.72)istheheatkerneldenotedKos(t;x,y)oftheharmonicoscillatorHˆos.UsingRemark2andinverseBargmanntransformwegeteasilythefollowingintegralexpressionforK(t;x,y):Kos(t;x,y)x2+y2−1−3/2−=2πe2√1×exp2x·ζ¯+wy·ζ−w2ζ2+ζ¯2−ζζ¯dζ∧dζ¯(1.73)C2 1.3FromSchrödingertoBargmannFockRepresentation21ThelastintegralisaFouriertransformofaGaussianfunctionasitisseenusingrealq−ipcoordinatesζ=√,z=(q,p).Wehave2−1−3/2−1Az·z−iz·YKos(t;x,y)=2πe2dY(1.74)R2wherei(x+wy)13+w2i(1−w2)−t/2Y=,A=22,w=e(1.75)wy−x2i(1−w)1−wAisasymmetricmatrix,itsrealpartispositivedefiniteanddet(A)=1−w2.Sowehave(see[117]orAppendicesA,BandC)x2+y21−1−1/22−1/2−−AY·YKos(t;x,y)=π(1−w)e2e2(1.76)TheMehlerformulafollows. Chapter2WeylQuantizationandCoherentStatesAbstractItiswellknownfromtheworkofBerezin(Commun.Math.Phys.40:153174,1975)in1975thatthequantizationproblemofaclassicalmechani-calsystemiscloselyrelatedwithcoherentstates.InparticularcoherentstateshelptounderstandthelimitingbehaviorofaquantumsystemwhenthePlanckconstantbecomesnegligibleinmacroscopicunits.Thisproblemiscalledthesemi-classicallimitproblem.InthischapterwediscusspropertiesofquantumsystemswhentheconfigurationspaceistheEuclideanspaceRn,sothatintheHamiltonianformalism,thephasespaceisRn×Rnwithitscanonicalsymplecticformσ.Thequantizationproblemhasmanysolutions,sowechooseaconvenientone,introducedbyWeyl(TheClas-sicalGroups,1997)andWigner(GroupTheoryandItsApplicationstoQuantumMechanicsofAtomicSpectra,1959).WestudythesymmetriesofWeylquantization,theoperationalcalculusandap-plicationstopropagationofobservables.WeshowthatWickquantizationisanaturalbridgebetweenWeylquantizationandcoherentstates.Applicationsaregivenofthesemi-classicallimitafterintroduc-inganefficientmoderntool:semi-classicalmeasures.Weillustratethegeneralresultsprovedinthischapterbyexplicitcomputationsfortheharmonicoscillator.Moreapplicationswillbegiveninthefollowingchap-ters,inparticularconcerningpropagatorsandtraceformulasforalargeclassofquantumsystems.2.1ClassicalandQuantumObservablesThequantizationproblemcomesfromquantummechanicsandisamathematicalsettingfortheBohrcorrespondenceprinciplebetweentheclassicalworldandthequantumworld.Letusconsiderasystemwithndegreesoffreedom.AccordingtheBohrcorre-spondenceprinciple,itisnaturaltocheckawaytoassociatetoeveryrealfunctionAonthephasespaceR2n(classicalobservable)aself-adjointoperatorAˆintheHilbertspaceL2(Rn)(quantumobservable).Accordingthequantummechanicalprinciples,themapA→Aˆhastosatisfysomeproperties.M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,23TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_2,©SpringerScience+BusinessMediaB.V.2012 242WeylQuantizationandCoherentStates(1)A→Aˆislinear,Aˆisself-adjointifAisrealand1ˆ=1L2(Rn).(2)positionobservables:xj→ˆxj:=QˆjwhereQˆjisthemultiplicationoperatorbyxj.(3)momentumobservables:ξj→ξˆj:=PˆjwherePˆjisthedifferentialoperator∂.i∂xj(4)commutationruleandclassicallimit:foreveryclassicalobservablesA,BwehaveilimA,ˆBˆ−{A,B}=0.→0Letusrecallthat[A,ˆBˆ]=AˆBˆ−BˆAˆisthecommutatorofAˆandBˆ,{A,B}isthePoissonbracketdefinedasfollows:{A,B}(x,ξ)=(∂n.xA·∂ξB−∂xB·∂ξA)(x,ξ),x,ξ∈RLetusremarkthatifweintroduce∇A=(∂xA,∂ξA)thenwehave{A,B}(x,ξ)=σ(∇A(x,ξ),∇B(x,ξ))(σisthesymplecticbilinearform).IftheobservablesA,Bdependonlyonthepositionvariable(oronthemomen-tumvariables)thenAˆ·Bˆ=A.Bbut,thisisnolongertrueforamixedobservable.Thisisrelatedtothenon-commutativityforproductofquantumobservablesandtheidentity:[ˆxj,ξˆj]=iso,thequantumobservablecorrespondingtox1ξ1isnotdeterminedbytherules(1)to(4).Wedonotwanttodiscussherethequantizationprobleminitsfullgenerality(seeforexample[77]).Onewaytochooseareasonableandconvenientquantizationprocedureisthefollowing,whichiscalledWeylquantization(see[117]formoredetails).LetLzbeareallinearformonthephasespaceR2n,wherez=(p,q),Lz(x,ξ)=σ(z,(x,ξ))(everylinearformonR2nislikethis).ItisnotdifficulttoseethatLˆzisawelldefinedquantumHamiltonian(i.e.anessentiallyself-adjoint2n−itLoperatorinL(R)).ItspropagatorezhasbeenstudiedinChap.1.RemarkthatwehaveLˆz=−L(z)ˆ,withthenotationofChap.1.Forψ∈S(Rn),wehaveexplicitly−itL−it2q·pitx·pezψ(x)=e2eψ(x−tq).(2.1)So,theWeylprescriptionisdefinedbytheconditions(1)to(4)andthefollowing:(5)e−iLz(x,ξ)→e−iLz=T(z)ˆWeshallusefreelytheSchwartzspaceS(Rn)1anditsdualS(Rn)(temperatedis-tributionsspace).1Recallthatf∈S(Rn)meansthatfisasmoothfunctioninRnandforeverymultiindicesα,β,xα∂βuisboundedinRn.Ithasanaturaltopology.S(Rn)isthelinearspaceofcontinuouslinearxformonS(Rn). 2.1ClassicalandQuantumObservables25Proposition11ThereexistsauniquecontinuousmapA→AˆfromS(R2n)intoL(S(Rn),S(Rn))satisfyingconditions(1)to(5).MoreoverifA∈S(R2n)andψ∈S(Rn)wehavethefamiliarformulaAψ(x)ˆ=(2π)−nx+yi−1(x−y)·ξA,ξeψ(y)dydξ,(2.2)R2n2andAˆisacontinuousmapfromS(Rn)toS(Rn).ThehermitianconjugateofAˆisthequantizationofthecomplexconjugateofA∗ˆ¯i.e.(A)ˆ=A.InparticularAˆisHermitianifandonlyifAisreal.ProofHereitisenoughtoassumethat=1.LetusconsiderthesymplecticFouriertransforminS(R2n).AssumefirstthatA∈S(R2n).A(z)˜=A(ζ)e−iσ(z,ζ)dζ.(2.3)R2nWehavetheinverseformulaA(X)=(2π)−nA(z)˜eiσ(z,X)dz.(2.4)R2nForψ,η∈S(Rn)wehaveψ,Aηˆ=(2π)−nA(z)˜eiLˆzψ,ηdz.(2.5)R2nInotherwordswegetAψˆ=(2π)−nA(z)˜T(z)ψdz.ˆ(2.6)R2nDefinition2ForagivenoperatorAˆ,thefunctionAiscalledthecontravariantsym-bolofAˆandthefunctionA˜isthecovariantsymbolofAˆ.LetusremarkthatwehavetheinverseformulaProposition12IfAˆisacontinuousmapfromS(Rn)toS(Rn)thenwehaveforeveryX∈R2n,A(X)˜=TrAˆT(ˆ−X).(2.7)ProofForX=0theformulaisaconsequenceoftheFourierinversionformula.ForanyXweusethattheWeylsymbolofT(ˆ−X)isz→e−iσ(z,X). 262WeylQuantizationandCoherentStatesAsaconsequencewehaveafirstnormoperatorestimate.IfA˜∈L1(R2n)wehaveAˆ≤(2π)−nA(z)˜dz.(2.8)R2nTher.h.s.informula(2.2)canbeextendedbycontinuityinAtothedistributionspaceS(R2n).LetuscomputenowtheSchwartzkernelKAoftheoperatorAˆdefinedinformula(2.6).WehaveKA(x,y)=A(x˜−y,p)eip·(x+y)/2dp.(2.9)RnUsinginverseFouriertransforminpvariables,weget−nx+yi(x−y)·ξKA(x,y)=(2π)A,ξedξ(2.10)Rn2thisgives(2.2).Theotherpropertiesareeasytoproveandlefttothereader.Letusfirstremarkthatfrom(2.10)wegetaformulatocomputethe-WeylsymbolofAˆifweknowitsSchwartzkernelK−iu·ξuuA(x,ξ)=eKx+,x−du.(2.11)Rn22Sometimes,weshallusealsothenotationAˆ=OpwA(-WeylquantizationofA).HenceweshallsaythatAˆisan-pseudodifferentialoperatorsandthatAisitsWeylsymbol.ForapplicationsitisusefultobeabletoreadpropertiesoftheoperatorAˆonitsWeylsymbolA.AfirstexampleistheHilbertSchmidtproperty.Proposition13LetAˆ∈L(S(Rn),S(Rn)).ThenAˆisHilbertSchmidtinL2(Rn)ifandonlyifA∈L2(R2n)andwehaveAˆ2−nA(x,ξ)2=(2π)dxdξ.(2.12)HSR2nInparticularifAˆandBˆaretwoHilbertSchmidtoperatorsthenA.ˆBˆisatraceoperatorandwehaveTrA.ˆBˆ=(2π)−nA(x,ξ)B(x,ξ)dxdξ.(2.13)R2nProofWeknowthat22AˆHS=KA(x,y)dxdy.R2nThenwegetthepropositionusingformula(2.10)andPlanchereltheorem. 2.1ClassicalandQuantumObservables27WeshallseelatermanyotherpropertiesconcerningWeylquantizationbutmostoftimeweonlyhavesufficientconditionsonAtohavesomepropertyofAˆ,likeforexampleL2continuityortrace-classproperty.LetusgiveafirstexampleofcomputationofaWeylsymbolstartingfromanintegralkernel.Weconsidertheheatsemi-groupe−tHˆos,oftheharmonicoscillatorHˆos.LetusdenoteKw(t;x,ξ)theWeylsymbolofe−tHˆosandK(t;x,y)itsintegralkernel.Fromformula(2.11)weget−iu·ξuuKw(t;x,ξ)=eKt;x+,x−du.(2.14)Rn22UsingMehlerformula(1.72)wehavetocomputetheFouriertransformofagen-eralizedGaussianfunction,soaftersomecomputations,wegetthefollowingniceformula:−n/2−tanh(t/2)(x2+ξ2)Kw(t;x,ξ)=cos(t/2)e.(2.15)Recallthatx2=x·x=|x|2.2.1.1GroupInvarianceofWeylQuantizationLetusfirstremarkthataneasyconsequenceofthedefinitionofWeylquantizationistheinvariancebytranslationsinthephasespace.Moreprecisely,wehave,foranyclassicalobservableAandanyz∈R2n,T(z)ˆ−1AˆT(z)ˆ=A·T(z),whereA·T(z)(z)=A(z−z).(2.16)HamiltonianclassicalmechanicsisinvariantbytheactionofthegroupSp(n)ofsymplectictransformationsofthephasespaceR2n.Anaturalquestiontoaskistoquantizelinearsymplectictransformations.Weshallseelaterhowitispossible.Inthissectionwestatethemainresults.RecallthatthesymplecticgroupSp(n)isthegroupoflineartransforma-tionsofR2nwhichpreservesthesymplecticformσ.SoF∈Sp(n)meansthatσ(FX,FY)=σ(X,Y)forallX,Y∈R2n.Ifweintroducethematrix01J=−10thenF∈Sp(n)⇐⇒FtJF=J,(2.17)whereFtisthetransposedmatrixofF.Ifn=1thenFissymplecticifandonlyifdet(F)=1. 282WeylQuantizationandCoherentStatesLinearsymplectictransformationscanbequantizedasunitaryoperatorsinL2(Rn)Theorem2ForeverylinearsymplectictransformationF∈Sp(n)andeverysym-bolA∈Σ(1)wehaveR(F)ˆ−1AˆR(F)ˆ=A·F.(2.18)MoreoverR(F)ˆisuniqueuptomultiplicationbyacomplexnumberofmodulus1Definition3ThemetaplecticgroupisthegroupMet(n)generatedbyR(F)ˆandλ1,λ∈C,|λ|=1.Remark4AconsequenceofTheorem2isthatRˆisaprojectiverepresentationofthesymplecticgroupSp(n)intheHilbertspaceL2(Rn).Itisaparticularcaseofamoregeneralsetting[193].Morepropertiesofthemetaplecticgroupwillbestudiedinthenextchapter.Letusgiveheresomeexamplesofthemetaplectictransform.•TheFouriertransformFisassociatedwiththesymplectictransformation(x,ξ)→(ξ,−x).•ThepartialFouriertransformFj,invariablexj,isassociatedwiththesymplectictransform:(xj,ξj)→(ξj,−xj),(xk,ξk)→(xk,ξk),ifk
=j.•LetAbealineartransformationonRn,thetransformationψ→|det(A)|1/2×ψ(Ax)isassociatedwiththesymplectictransformxAxFA=.ξ(At)−1ξ•LetAbearealsymmetricmatrix,thetransformationψ→eiAx·x/2ψisassociatedwiththesymplectictransform10F=.A12.2WignerFunctionsLetϕ,ψ∈L2(Rn).TheydefinearankoneoperatorΠψ,ϕη=ψ,ηϕ.ItsWeylsymbolcanbecomputedusing(2.11). 2.2WignerFunctions29Definition4TheWignerfunctionofthepair(ψ,ϕ)istheWeylsymboloftherankoneoperatorΠψ,ϕ.ItwillbedenotedWϕ,ψ.Moreexplicitlywehave−iu·ξuuWϕ,ψ(x,ξ)=eϕx+ψx−du.(2.19)Rn22AnequivalentdefinitionoftheWignerfunctionisthefollowing:W−n−iσ(z,z)/ϕ,ψ(z)=(2π)ϕ,T(zˆ)ψedz,(2.20)R2nwhereT(z)ˆ=e−iLˆz.Wecaneasilyseethat(2.19)and(2.20)areequivalentusingformula(2.6)andPlancherelformulaforsymplecticFouriertransform.TheWignerfunctionsareveryconvenienttouse.Inparticularwehavethefol-lowingniceproperty:Proposition14LetusassumethatAˆisHilbertSchmidtandψ,ϕ∈L2(Rn).Thenwehaveψ,Aϕˆ=(2π)−nA(X)Wψ,ϕ(X)dX.(2.21)R2nIfA∈S(R2n)andifψ,ϕ∈S(Rn),theformula(2.21)isstilltrueintheweaksenseoftemperatedistributions.ProofLetusfirstremarkthatψ,Aϕˆ=Tr(AΠˆψ,ϕ).Hencethefirstpartofthepropositioncomesfrom(2.13).Nowifψ,ϕ∈S(Rn)thenweeasilygetWψ,ϕ∈S(R2n).OntheothersidethereexistsAj∈S(R2n)suchthatAj→AinS(R2n).Soweapply(2.21)toAjandwegotothelimitinj.WhatWignerwaslookingforwasanequivalentoftheclassicalprobabilitydis-tributioninthephasespaceR2n.Thatis,associatedtoanyquantumstateadistri-butionfunctioninphasespacethatimitatesaclassicaldistributionprobabilityinphasespace.Recallthataclassicalprobabilitydistributionisanon-negativeBorelfunctionρ;Z→R+,Z:=R2n,normalizedtounity:ρ(z)dz=1,ZandsuchthattheaverageofanyobservableA∈C∞issimplygivenbyρ(A)=A(z)ρ(z)dz.Z 302WeylQuantizationandCoherentStatesFromProposition14weseethatapossiblecandidateisρ(z)=(2π)−nWϕ,ϕ.Actuallyinthephysicalliteraturetheexpressionabove(withthefactor(2π)−n)istakenasthedefinitionoftheWignerfunctionbutwedonottakethisconvention.InthefollowingwedenotebyWϕtheWignertransformforϕ,ϕ.Whatabouttheexpectedpropertiesof(2π)−nWϕasapossibleprobabilitydis-tributioninphasespace?Namely:•positivity•normalizationto1•correctmarginaldistributionsProposition15Letz=(x,ξ)∈R2nandϕ∈L2(Rn)withϕ=1.Wehave(i)−nϕ(x)2(2π)Wϕ(x,ξ)dξ=,Rnwhichistheprobabilityamplitudetofindthequantumparticleatpositionx.(ii)−nϕ(ξ)˜2(2π)Wϕ(x,ξ)dx=,Rnwhichistheprobabilityamplitudetofindthequantumparticleatmomentumξ.(iii)−nW(2π)ϕ(x,ξ)dxdξ=1.R2n(iv)Wϕ(x,ξ)∈R.Proof(i)Letf∈Sbeanarbitrarytestfunction.WehaveWϕ(x,ξ)f(ξ)dξRnyy−iξ·y/=dyϕ¯x+ϕx−dξef(ξ)22nyy=(2π)dyϕ¯x+ϕx−(Ff)(y).(2.22)Rn22BytakingfortheusualFouriertransformFfanapproximationoftheDiracdistributionaty=0wegettheresult. 2.2WignerFunctions31(ii)Isprovensimilarly.(iii)Followsfromthenormalizationtounityofthestateϕ.(iv)WehaveW∗=(2π)−ndzϕ,T(ˆ−z)ϕeiσ(z,z)/ϕ(z)andtheresultfollowsbychangeoftheintegrationvariablez→−zandbynotingthatσ(z,−z)=−σ(z,z).LetusnowcomputetheWignerfunctionWz,zforapair(ϕz,ϕz)ofcoherentstates.Proposition16ForeveryX,z,z∈R2nwehave2n1z+zi1Wz,z(X)=2exp−X−−σX−z,z−z.(2.23)22ProofItisenoughtoconsiderthecase=1.Letusapplyformula(2.20):−n−iσ(X,z)Wz,z(X)=(2π)ϕz,Tˆzϕzedz.(2.24)R2nUsingformula(1.7)fromChap.1,wehaveiσ(z,z)ϕz,Tˆzϕz=ϕz,ϕz+ze2−1|z−z−z|2iσ(z,z+z)+σ(z,z)=e4e2.(2.25)Usingthechangeofvariablesz=z−z+u,wehavetocomputetheFouriertransformofthestandardGaussiane−|u|2/4and(2.23)follows.WehavethefollowingpropertiesoftheWignertransform:Proposition17Letϕ,ψ∈L2(Rn)betwoquantumstates.ThenWϕ,ψ∈L2(R2n)∩L∞(R2n)andwehave(i)Wnϕ,ψL∞≤2ϕ2ψ2.(ii)Wn/2ϕ,ψL2≤(2π)ϕ2ψ2.(iii)Letϕ,ψ∈L2(Rn).Thenwehaveϕ,ψ2−n=(2π)Wϕ,WψL2(R2n). 322WeylQuantizationandCoherentStatesProof(i)isasimpleconsequenceofthedefinitionoftheWignertransformandoftheCauchySchwartzinequality.Fortheproofof(ii)wenotethat2W2iξ·y/yydzϕ,ψ(z)=dxdξdyeϕ¯x+ψx−.22Usinganapproximationargument,wecanassumethatϕ,ψ∈L1(Rn)∩L∞(Rn).Sowehaveϕ(x¯+y)ψ(x−y)∈L2(Rn,dy).AccordingtothePlanchereltheorem22wehave2yy(2π)−ndξdyeiξ·y/ϕ¯x+ψx−222yy=dyϕ¯x+ψx−22sothat2W2nyydzϕ,ψ(z)=(2π)dxdyϕ¯x+ψx−22=(2π)nϕ2ψ2.(2.26)TheWignertransformoperateasonewishesinphasespace,namelyaccordingtotheschemeofclassicalmechanics:Proposition18Letϕ,ψ∈L2(Rn)andT(z),ˆR(F)ˆbe,respectively,operatorsoftheWeylHeisenbergandmetaplecticgroups,corresponding,respectively,toaphase-spacetranslationbyvectorz∈R2nasymplectictransformationinphasespaceWehaveWTˆ(z)ϕ,T(zˆ)ψ(z)=Wϕ,ψ(z−z),(2.27)WRˆ(F)ϕ,R(F)ψˆ(z)=Wϕ,ψF−1z.(2.28)ProofWehavethenicegrouppropertyoftheWeylHeisenbergtranslationopera-tor:T(ˆ−ziT(X)ˆ)T(X)ˆT(zˆ)=exp−σ(X,z)sothatiW(z)=(2π)−ndXexp−σ(z−z,X)ϕ,T(X)ψˆT(zˆ)ϕ,T(zˆ)ψ=Wϕ,ψ(z−z). 2.2WignerFunctions33AsaresultofthepropertyofthemetaplectictransformationwehaveR(F)ˆ−1T(zˆ)R(F)ˆ=TˆF−1z.ThereforeW(z)=(2π)−ndzϕ,T(Fzˆ)ψe−iσ(z,z)/Rˆ(F)ϕ,R(F)ψˆ=(2π)−ndzϕ,T(zˆ)ψe−iσ(z,Fz)/−n−iσ(F−1z,z)/=(2π)dzϕ,T(zˆ)ψe,wherewehaveusedthechangeofvariableFz=zandthefactthatasymplecticmatrixhasdeterminantone.NowwegetaformulatorecovertheWeylsymbolofanyoperatorAˆ∈L(S(Rn),S(Rn)).Proposition19EveryoperatorAˆ∈L(S(Rn),S(Rn))hasacontravariantWeylsymbolAandacovariantWeylsymbolA˜inS(R2n).Wehave,inthedistributionsenseingeneral,intheusualsenseifAˆisboundedinL2(Rn),A(X)=(2π)−2nϕz,AϕˆzWz,z(X)dzdz,(2.29)R4niA(X)˜=(2π)−nϕz+X,Aϕˆze−σ(X,z)dz.(2.30)R2nProofWecomputeformally.Itisnotverydifficulttogiveallthedetailsforarigor-ousproof.WeapplyinverseformulafortheFourierBargmanntransform(seeChap.1).Soforanyψ∈S(Rn),wehaveAψ(x)ˆ=(2π)−2nϕz,Aϕˆzϕz,ψϕz(x)dzdz.(2.31)R4nSowegetaformulafortheSchwartzkernelKAforAˆ,KA(x,x)=(2π)−2nϕz,Aϕˆzϕz(x)ϕz(x)dzdz.(2.32)R4nThenweapplyformula(2.11)togetthecontravariantsymbolA.Theformulaforthecovariantsymbolfollowsfrom(2.7)andtracecomputationwithcoherentstates. 342WeylQuantizationandCoherentStatesTheonlybutimportantmissingpropertytohaveaniceprobabilisticsettingwiththeWignerfunctionsispositivitywhichisunfortunatelynotsatisfiedbecausewehavethefollowingresult,provedbyHudson[120]forn=1,thenextendedton≥2bySotoClaverie[181].Theorem3W2n(Γ)ψ(X)≥0onRifandonlyifψ=CϕzwhereCisacomplexnumber,Γacomplex,symmetricn×nmatrixwithapositivenondegenerateimag-inarypartΓ,z∈R2n,wherewedefinetheGaussian(Γ)−n/41/4iϕ(x)=(π)detΓexpΓx·x.(2.33)2ProofWemoreorlessfollowthepaperofSotoClaverie[181].WecancheckbydirectcomputationthattheWignerdensityofϕz(Γ)ispositive(accordingthedefinitionwehavetocomputetheFouriertransformoftheexpo-nentofaquadraticform).Wecanalsogivethefollowingmoreelegantproof.First,itisenoughtoconsiderthecasez=(0,0).Second,itispossibletofindameta-plectictransformationFsuchthatϕz(Γ)=R(F)ϕˆ0(seethesectiononsymplecticinvarianceandChap.3formorepropertiesonthemetaplecticgroup).HencewegetWR(F)ϕˆ(X)=Wϕ(F−1(X)).ButwehavecomputedaboveWϕ,whichisa000standardGaussian,soitispositive.Conversely,assumenowthatWψ(X)≥0onR2n.WeshallprovethattheFourierBargmanntransformψ#(z)isaGaussianfunctiononthephasespace.HenceusingtheinverseBargmanntransformformula,weshallseethatψisaGaus-sian.Letusfirstprovethetwofollowingproperties:ψ#(z)
=0,∀z∈R2n,(2.34)2ψ#(z)≤Ceδ|z|,∀z∈R2n,forsomeC,δ>0.(2.35)Wehaveseenthatψ,ϕ2−nz=(2π)Wψ(X)Wϕz(X)dXR2nn−1|X−z|2=2Wψ(X)edX.(2.36)R2nThelastintegralispositivebecausebyassumptionWψ(X)≥0andWψ(X)dX=1.Usingagain(2.36)weeasilyget(2.34).ThesecondstepistouseapropertyofentirefunctionsinCn.LetusrecallthatinChap.1,wehaveseenthatthefunction#p2+ip·q#ψa(ζ):=expψ(q,p)(2.37)2 2.3CoherentStatesandOperatorNormsEstimates35isanentirefunctioninthevariableζ=q−ip∈Cn.Moreoverwegeteasilythatψa#(ζ)satisfiesproperties(2.34).ToachievetheproofofTheorem3weapplythefollowinglemma,whichisaparticularcaseofHadamardfactorizationtheoremforn=1,extendedforn≥2in[181].Lemma11LetfbeanentirefunctioninCnsuchthatf(ζ)
=0forallζ∈CnandforsomeC>0,δ>0,mf(ζ)≤Ceδ|ζ|,∀ζ∈Cn.(2.38)Thenf(ζ)=eP(ζ),wherePisapolynomialofdegree≤m.2.3CoherentStatesandOperatorNormsEstimatesLetusgivenowafirstapplicationofcoherentstatestoWeylquantization.Weas-sumefirstthat=1.Theorem4(CalderonVaillancourt)ThereexistsauniversalconstantCnsuchthatforeverysymbolA∈C∞(R2n)wehaveAˆ≤Cnsup∂γA(X).(2.39)L(L2,L2)X|γ|≤2n+1,X∈R2nBeginningoftheProofFrom(2.32)wegettheformulaψ,Aηˆ=(2π)−nϕz,Aϕˆzψ#(z)η#(z)dzdz.(2.40)R4nWeshallget(2.39)byprovingthattheBargmannkernelKB(z,z):=ϕz,AϕˆzisAthekernelofaboundedoperatorinL2(R2n).LetusfirstrecallaclassicallemmaLemma12Let(Ω,μ)beameasured(σ-finite)space,KameasurablefunctiononΩ×ΩsuchthatmK:=maxsupK(z,z)dz,supK(z,z)dz.z∈ΩΩz∈ΩΩThenKistheintegralkernelofaboundedoperatorTKonL2(Ω)andwehaveTK≤mK.SotheCalderonVaillancourttheoremwillbeaconsequenceofthefollowing. 362WeylQuantizationandCoherentStatesLemma13ThereexistsauniversalconstantCnsuchthatforeverysymbolA∈C∞(R2n)wehaveKB≤C−2n−1∂γ.(2.41)A(z,z)n1+|z−z|supXA(X)|γ|≤2n+1,X∈R2nProofWehavealreadyseenthatBKA(z,z)=A(X)Wz,z(X)dXRn2=2nA(X)exp−X−z+z−iσX−1z,z−zdX.Rn22(2.42)Firstremarkthatwehaveϕz,Aϕˆz≤supA(X).(2.43)X∈R2nSoweonlyhavetoconsiderthecase|z−z|≥1.Theestimateisprovedbyinte-grationbyparts(asisusualforanoscillatingintegral).Letusintroducethephasefunction2Φ=−X−z+z−iσX−1z,z−z.(2.44)22Wehave|∂XΦ|≥|z−z|hence∂XΦ·∂XΦΦe=e.(2.45)|∂XΦ|2Sowegetthewantedestimatesperforming2n+1integrationsbypartsintheinte-gral(2.42)usingformula(2.45).ThisachievestheproofoftheCalderonVaillancourttheorem.Corollary4AˆisacompactoperatorinL2(Rn)ifAisC∞onR2nandsatisfiesthefollowingcondition:lim∂γA(z)=0,∀γ∈N2d,|γ|≤2n+1.(2.46)z|z|→+∞ProofLetusintroduceχ∈C∞(R2n)suchthatχ(X)=1if|X|≤1andχ(X)=02if|X|≥1.LetusdefineAR(X)=χ(X/R)A(X).ForeveryR>0,AˆRisHilbert 2.3CoherentStatesandOperatorNormsEstimates37Schmidthencecompact.UsingtheCalderonVaillancourtestimate,wegetlimAˆ−AR=0.|R|→+∞SoAˆiscompact.UsingthesameideaasforprovingCalderonVaillancourttheorem,wegetnowasufficienttrace-classcondition.Theorem5ThereexistsauniversalconstantτnsuchthatforeveryA∈C∞(R2n)wehaveAˆ≤τn∂γA(X)dX.(2.47)TrXR2n|γ|≤2n+1Inparticularifther.h.s.isfinitethenAˆisinthetraceclassandwehaveTrAˆ=(2π)−nA(X)dX.(2.48)R2nProofRecallthat=1.From(2.29)weknowthatAˆhasthefollowingdecompo-sitionintorankoneoperators:Aˆ=(2π)−nϕz,AϕˆzΠz,zdzdz.(2.49)R4nButweknowthatΠz,zTR=1.SowehaveAˆ≤(2π)−nϕz,Aϕˆzdzdz.(2.50)TRR4nUsingintegrationbypartsasintheproofofCalderonVaillancourt,wehaveAϕˆ≤C−N−|X−(z+z)/2|2∂γdX(2.51)ϕz,zN1+|z−z|eXA(X)R2n|γ|≤NwithN=2n+1.Nowperformthechangeofvariablesu=(z+z)/2,v=z−zandusingYounginequalitywegetγϕz,Aϕˆzdzdz≤τn∂XA(X)dX(2.52)R4nR2n|γ|≤Nhence(2.47)follows.Wecanget(2.48)byusingapproximationswithcompactsupportARlikeintheproofofCorollary4. 382WeylQuantizationandCoherentStatesRemark5UsinginterpolationresultsitispossibletogetsimilarestimatesfortheSchattennormAˆpfor10andforallN≥1thereexistsCNsuchthatfor|z−X0|≥2r0wehaveAϕˆ≤CNz−N,for|z|≥R.(2.55)zNProofItisconvenientheretoworkonFourierBargmannnside.Soweestimate−nϕz,Aϕˆz=(2π)A(Y)Wz,z(Y)dY.(2.56)R2nAswehavealreadyseen,wehave 2.3CoherentStatesandOperatorNormsEstimates39A(Y)Wz,z(Y)dYR2n2=2nexp−1Y−z+z−iσY−1z,z−zA(Y)dY.(2.57)R2n22Usingintegrationsbypartsasabove,consideringthephasefunctionΨ(Y)=−|Y−z+X|2−iσ(Y−1X,z−X)andthedifferentialoperator∂YΨ∂22|∂Ψ|2Y,wegetforeveryYM,Mlargeenough,−MM−M|Y−z||z−z|Aϕˆz,ϕz≤CM,M1+√1+√dY.(2.58)[|Y|≤r0]ThereforeweeasilygettheestimatechoosingM,MconvenientlyandusingthattheFourierBargmannntransformisanisometry.WeneedtointroducesomepropertiesfortheWeylsymbolsA.Definition5ApositivefunctionmonRdisatemperateweightifitsatisfiesthefollowingproperty.ThereexistN,CsuchthatNdm(X+Y)≤m(X)1+|X−Y|,∀X,Y∈R.(2.59)AsymbolAisaclassicalobservableofweightmifforeverymultiindexαthereexistsCαsuchthat∂αA(X)≤C2n.Xαm(X),∀X∈RThespaceofsymbolsofweightmisdenotedΣ(m).Abasicexampleoftemperateweightismμ(X)=(1+|X|)μ,μ∈R.WeshalldenoteΣμ=Σ(mμ).ForexampleΣ0=Σ(1).Remark6Theproductoftwotemperateweightsisatemperateweightandifmisatemperateweightthenm−1isalsoatemperateweight.AsprovedbyUnterberger[186]andrediscoveredbyTataru[183],itispossibletocharacterizetheoperatorclassΣ(1)onthematrixelementϕz,Aϕˆz.Westatenowasemi-classicalversionofUnterbergerresult.Theorem6LetAˆbea-dependentfamilyofoperatorsfromS(Rn)toS(Rn).ThenAˆ=Opw(A)withA∈Σ(1)withuniformestimate2ifandonlyifforevery2Thismeansthatforeveryγ,sup∈]0,1]∂γA∞<+∞. 402WeylQuantizationandCoherentStatesNthereexistsCNsuchthatwehave−Nϕ,Aϕˆ≤C1+|z√−z|,∀∈]0,1),z,z∈R2n.(2.60)zzNProofSupposethatAˆ=Opw(A),withA∈Σ(1)isaboundedfamily.Wegetestimate(2.60)byintegrationsbypartsasabove.Converselyifwehaveestimates(2.60),using(2.23)and(2.29)wehave2−nAˆ1z+zA(X)=(π)ϕz,ϕzexp−X−R4n2z+iJX−·(z−z)dzdz.(2.61)2√Usingthechangeofvariablesz+z=uandz−z=vwegeteasilythatthere2existsC>0suchthatA(X)≤C,∀X∈R2n,∈]0,1].(2.62)InthesamewaywecanestimateeveryderivativesofA,afterderivationinXintheintegral(2.61).TheothermainfactinWeylquantizationisexistenceofanoperationalcalculus.Weshallrecallitspropertiesinthenextsection.2.4ProductRuleandApplications2.4.1TheMoyalProductOneofthemostusefulpropertiesofWeylquantizationisthatwehaveanoperationalcalculusdefinedby:TheProductRuleforQuantumObservablesLetusstartwithA,B∈S(R2n).WelookforaclassicalobservableCsuchthatAˆ·Bˆ=Cˆ.LetusfirstremarkthattheintegralkernelofCˆisKC(x,y)=KA(x,s)KB(s,y)ds.(2.63)RnUsingrelationshipbetweenintegralkernelsandWeylsymbols,wegetC(X)=(π)−2ne2iσ(Y,Z)A(X+Z)B(X+Y)dYdZ,(2.64)R4n 2.4ProductRuleandApplications41whereσisthesymplecticbilinearformintroducedabove.NowletusapplyPlancherelformulainR4nandthefollowingFouriertransformformula:iB.T,TmLemma16Letf(T)=e2,forT∈RwhereBisanondegeneratesym-metricm×mmatrix.ThentheFouriertransformf˜ism/2−1/2iπsgnB−iB−1ζ,ζf(ζ)˜=(2π)|detB|ee2,(2.65)wheresgnBisthesignatureofthematrixB.ProofSee[117,163].HencewegetiC(x,ξ)=expσ(Dx,Dξ;Dy,Dη)A(x,ξ)B(y,η).(2.66)2(x,ξ)=(y,η)Wecanseeeasilyonformula(2.66)thatC∈S(R2n).Sothat(2.64)definesanon-commutativeproductonclassicalobservables.WeshalldenotethisproductC=AB(Moyalproduct).Insemi-classicalanalysis,itisusefultoexpandtheexponentin(2.66),sowegettheformalseriesin:CjC(x,ξ)=j(x,ξ),wherej≥0j1iCj(x,ξ)=σ(Dx,Dξ;Dy,Dη)A(x,ξ)B(y,η).(2.67)j!2(x,ξ)=(y,η)WecaneasilyseethatingeneralCisnotaclassicalobservablebecauseofthedependence.Itcanbeprovedthatitisasemi-classicalobservableinthefollowingsense.Definition6WesaythatAisasemi-classicalobservableofweightm,wheremistemperateweightonR2n,ifthereexist0>0andasequenceAj∈Σ(m),j∈N,sothatAisamapfrom]0,0]intoΣ(m)satisfyingthefollowingasymptoticcon-dition:foreveryN∈Nandeveryγ∈N2nthereexistsCN>0suchthatforall∈]0,1[wehave∂γsupm−1(z)A(,z)−jAj(z)≤CNN+1,(2.68)∂zγR2n0≤j≤NA0iscalledtheprincipalsymbol,A1thesub-principalsymbolofAˆ.Thesetofsemi-classicalobservablesofweightmisdenotedbyΣsc(m).ItsrangeinL(S(Rn),S(Rn))isdenotedΣsc(m). 422WeylQuantizationandCoherentStatesμWemayusethenotationΣsc=Σsc(mμ).NowwestatetheproductruleforWeylquantization.Theorem7Letm,mbetwotemperateweightsinR2n.ForeveryA∈Σ(m)andB∈Σ(m),thereexistsauniqueC∈Σsc(mp)suchthatAˆ·Bˆ=CˆwithCj≥0jCj.TheCjaregivenby1(−1)|β|βααβCj(x,ξ)=2jα!β!Dx∂ξA·Dx∂ξB(x,ξ).|α+β|=jProofThemaintechnicalpointistocontroltheremaindertermsuniformlyinthesemi-classicalparameter.Thisisdetailedintheappendixofthepaper[31].Corollary5Undertheassumptionofthetheorem,wehavethewellknowncor-respondencebetweenthecommutatorforquantumobservablesandthePoissoni[A,ˆBˆ]∈Σbracketforclassicalobservables,sc(mm)anditsprincipalsymbolisthePoissonbracket{A,B}.AveryusefulapplicationoftheMoyalproductisthepossibilitytogetsemi-classicalapproximationsforinverseofellipticsymbol.Definition7LetA()beasemi-classicalobservableinΣsc(m)andX0∈R2n.WeshallsaythatAisellipticatX0ifA0(X0)
=0.WeshallsaythatAisuniformlyellipticifthereexistsc>0suchthatA(X)≥cm(X),∀X∈R2n.(2.69)Theorem8LetA∈Σsc(m)beanuniformlyellipticsemi-classicalsymbol.ThenthereexistsB∈Σsc(m−1)suchthatBA=1(inthesenseofasymptoticexpansioninΣsc(1)).Moreover,wehaveBˆ·Aˆ=1+O∞,(2.70)wheretheremainderisestimatedintheL2normofoperators.Moreoverthesemi-classicalsymbolBofBˆisB=j≥0jBjwithB−1−20=A,B1=−A1A.(2.71)00ProofLetusdenotebyCj(E,F)thejthtermintheMoyalproductEF.ThemethodconsiststocomputebyinductionB0,...,BNsuchthatjBN+1.(2.72)jA(h)=O0≤j≤N 2.4ProductRuleandApplications43WestartwithB10=.ThenextstepistocomputeB1suchthatB1A0+A1B0=0.A0ThentocomputeB2suchthatC2(A0,B0)+C1(A1,B1)+B2A0=0.SowegetalltheBjbyinductionusingtheasymptoticexpansionfortheMoyalproduct.Theremaindertermin(2.70)isestimatedusingtheCalderonVaillancourttheo-rem.Wegivenowalocalversionoftheabovetheorem,whichcanbeprovedbythesamemethod.Theorem9LetA∈Σsc(m)beanellipticsymbolinanopenboundedsetΩofR2n.Thenforeveryχ∈C∞(Ω)thereexistsBχ∈Σsc−∞suchthat0BˆAˆ=ˆχ+O∞.(2.73)χRemark7ForapplicationitisusefultonotethatifAdependsinauniformwayofsomeparameterε∈[0,1]thenBalsodependsuniformlyinε.Inparticularεmaydependon.2.4.2FunctionalCalculusAnusefulconsequenceofthealgebraicpropertiesofsymbolicquantizationisafunctionalcalculus:undersuitableassumptionsifHˆisanHermitiansemi-classicalobservablethenforeverysmoothfunctionf,f(H)ˆisalsoasemi-classicalobserv-able.ThetechnicalstatementisTheorem10LetHˆbeauniformlyellipticsemi-classicalHamiltonian.Letfbeasmoothrealvaluedfunctionsuchthat,forsomer∈R,wehave∀k∈N,∃Ck,f(k)(s)≤Cr−k,∀s∈R.ksThenf(H)ˆisasemi-classicalobservablewithasemi-classicalsymbolHf(,z)givenbyjHHf(,z)f,j(z).(2.74)j≥0InparticularwehaveHf,0(z)=fH0(z),(2.75)Hf,1(z)=H1(z)fH0(z),(2.76) 442WeylQuantizationandCoherentStatesandfor,j≥2,H(k)f,j=dj,k(H)f(H0),(2.77)1≤1≤2j−1γwheredj,k(H)areuniversalpolynomialsin∂zH(z)with|γ|+≤j.Aproofofthistheoremcanbefoundin[68,Chap.8],[107].Inparticularwecantakef(s)=(λ+s)−1forλ
=0(theproofbeginswiththiscase)orfwithacompactsupport.FromthistheoremwecangetthefollowingconsequencesonthespectrumofHˆ(see[107]).Theorem11LetHˆbelikeinTheorem10.AssumethatH−1[E−,E+]isacom-0pactsetinRn×Rn.ConsideraclosedintervalI⊂[E−,E+].Thenwehavethefollowingproperties.(i)∀∈]0,0],0>0,thespectrumofHˆisdiscreteandisafinitesequenceofeigenvaluesE1()≤E2()≤···≤EN()whereeacheigenvalueisrepeatedIaccordingitsmultiplicity.MoreoverNI=O(−n)as0.(ii)Forallf∈C∞(I),f(H)ˆisatrace-classoperatorandwehave0 TrfHˆj−dτj(f),(2.78)j≥0whereτ−1(I).Inparticular,wehavejaredistributionssupportedinH0τ−d0(f)=(2π)fH0(z)dz,(2.79)R2nτ−d1(f)=(2π)fH0(z)H1(z)dz.(2.80)R2nAneasyconsequenceofthisisthefollowingWeylasymptoticformula:Corollary6IfI=[λ−,λ+]suchthatλ±arenoncriticalvaluesforH03thenwehavelim(2π)nNI=dqdp.(2.81)→0[H0(q,p)∈I]Remark8Formula(2.81)isverywellknownandcanbeprovedinmanyways,undermuchweakerassumptions.Foraproofusingthefunctionalcalculussee[163,pp.283287].3Thatλisanon-criticalvalueforHmeansthat∇H(z)
=0ifH(z)=λ. 2.4ProductRuleandApplications45UnderourassumptionsweshallseeinChap.4thatwehaveaWeylasymptoticwithanaccurateremainderestimate:N−n1−nI=(2π)dqdp+O,[H(q,p)∈I]usingatimedependentmethodduetoHörmanderandLevitan([116]anditsbiblio-graphy).Formoreaccurateresultsaboutspectralasymptoticssee[122].2.4.3PropagationofObservablesNowwecometothemainapplicationoftheresultsofthissection.Weshallgiveaproofofthecorrespondence(inthesenseofBohr)betweenquantumandclassicaldynamics.Asweshallseethistheoremisausefultoolforsemi-classicalanalysisalthoughitsproofisaneasyapplicationofWeylcalculusrulesstatedabove.ThemicrolocalversionofthefollowingresultisoriginallyduetoEgorov[73].R.Beals[18]foundanicesimpleproof.Theorem12(TheSemi-classicalPropagationTheorem)Letusconsideratimede-pendentHamiltonianH(t)∈Σsc2satisfying:∂γHj(t,z)≤Czγ,for|γ|+j≥2;(2.82)−2H(t)−H00(t)−H1(t)∈Σsc.(2.83)WeassumethatH(t,z)iscontinuousfort∈Randthatalltheestimatesareuniformintfort∈[−T,T].1γ0LetusintroduceanobservableA∈Σ,suchthat∂A∈Σif|γ|≥1.ThenweXhavethefollowing.(a)Forsmallenoughandforeveryψ∈S(Rn),theSchrödingerequationi∂tψt=H(t)ψˆt,ψt=s=ψ(2.84)hasauniquesolutionwhichwedenoteψt=U(t,s)ψˆ.MoreoverU(t,s)ˆcanbeextendedasaunitaryoperatorinL2(Rn).(b)ThetimeevolutionA(t,s)ˆofAˆ,fromtheinitialtimesisA(t,s)ˆ=U(s,t)ˆAˆ×U(t,s)ˆandhasasemi-classicalWeylsymbolA(t,s)suchthatA(t,s)∈Σsc1.MorepreciselywehaveA(t,s)j≥0jAj(t,s),inΣsc0,whichisuniformint,s,fort,s∈[−T,T].MoreoverAj(t,s)canbecomputedbythefollowingformulas:AΦt,s(z),(2.85)0(t,s;z)=At AΦτ,t,Ht,τA1(t,s;,z)=1(τ)Φ(z)dτ(2.86)sandforj≥2,Aj(t,s;z)canbecomputedbyinductiononj. 462WeylQuantizationandCoherentStatesProofProperty(a)willbeprovedlater.ItiseasiertoproveitifHistimeindepen-dentbecausewecanproveinthiscasethatHˆisessentiallyself-adjoint(foraproofsee[163]).ThenwehaveitU(t)ˆ:=U(t,ˆ0)=exp−Hˆ.Letusremarkthat,undertheassumptionofthetheorem,theclassicalflowforH0existsglobally.Indeed,theHamiltonianvectorfield(∂ξH0,−∂xH0)hasasublineargrowingatinfinityso,noclassicaltrajectorycanblowupinafinitetime.Moreover,usingusualmethodsinnonlinearO.D.E.(variationequation)wecanprovethatA(Φt,s)∈Σ(1)withsemi-normuniformlyboundedfort,sbounded.Now,fromtheHeisenbergequationandtheclassicalequationsofmotionweget∂U(s,τ)ˆAU(τ,s)ˆ0(t,τ)∂τi =U(sˆ;τ)H(τ),ˆA−H(τ),A0Φt,τU(τ,s),ˆ(2.87)0(t,τ)whereA0(t,s)=A(Φt,s).But,fromthecorollaryoftheproductrule,theprincipalsymbolofiH(τ),ˆA−H(τ),A0Φt,τ0(t,τ)vanishes.So,inthefirststep,usingtheproductruleformula,wegettheapproxima-tionU(s,t)ˆAˆU(t,s)ˆ−A0(t,s)tiˆ=U(s,τ)ˆH(τ),ˆA0(t,τ)−H(τ),A0Φt,τU(τ,s)dτ.ˆ(2.88)sNow,itisnotdifficulttoobtain,byinduction,thefullasymptoticsin.Forj≥2,t Γ(α,β)∂α∂βHαβΦt,τAj(t,s;z)=ξxk(τ)·∂ξ∂xA(z)dτ,s|(α,β)|+k=j+10≤≤j−1(2.89)with(−1)|β|−(−1)|α|−1−|(α,β)|Γ(α,β)=i.α!β!2|α|+|β|Themaintechnicalpointistoestimatetheremainderterms.Foraproofwithmoredetailssee[31]wheretheauthorsgetauniformestimateuptoEhrenfesttime(oforderlog−1).WegiveinAppendixBthenecessarydetailsforuniformestimatesonfinitetimesintervals. 2.4ProductRuleandApplications47Remark9IfH(t)=H0(t)isapolynomialfunctionofdegree≤2inzonthephasespaceR2nthenthepropagationtheoremassumesasimplerform:A(t,s)=A(Φt,s)andtheremaindertermisnull.Thisisaconsequenceofthefollowingexactformula:iH,ˆBˆ={H,B},(2.90)whereB∈Σ+∞.Nowwegiveanapplicationofthepropagationtheoremandcoherentstatesinsemi-classicalanalysis:werecovertheclassicalevolutionfromthequantumevolu-tion,intheclassicallimit0.Corollary7ForeveryobservableA∈Σ0andeveryz∈R2n,wehavelimU(t,s)ϕˆAˆU(t,s)ϕˆ=AΦt,s(z)(2.91)z,z0andthelimitisuniformin(t,s;z)oneveryboundedsetofRt×Rs×R2zn.ProofU(t,s)ϕˆz,AˆU(t,s)ϕˆz=ϕz,U(s,t)ˆAˆU(t,s)ϕˆz=A(t,s;X)Wz,z(X)dXR2n|X−z|2−n−=(π)A(t,s;X)edX.(2.92)R2nSobythepropagationtheoremweknowthatA(t,s;X)=A(Φt,s(X))+O().Hencethecorollaryfollows.Remark10ThelastresulthasalonghistorybeginningwithEhrenfest[74]andcon-tinuingwithHepp[113],BouzouinaRobert[31].InthislastpaperitisprovedthatthecorollaryisstillvalidfortimessmallerthantheEhrenfesttimeTE:=γE|log|,forsomeconstantγE>0.2.4.4ReturntoSymplecticInvarianceofWeylQuantizationLetusgivenowafirstconstructionofmetaplectictransformations.Otherequivalentconstructionsandmorepropertieswillbegivenlater(chapteronquadratichamilto-nians).Lemma17ForeveryF∈Sp(n)wecanfindaC1-smoothcurveFt,t∈[0,1],inSp(n),suchthatF0=1andF1=F. 482WeylQuantizationandCoherentStatesProofAnexplicitwaytodothatistousethepolardecompositionof√F,F=V|F|whereVisasymplecticorthogonalmatrixand|F|=FtFispositivesymplecticmatrix.Eachofthesematriceshavealogarithm,soF=eKeLwithK,LHamilto-nianmatrices,andwecanchooseFt=etKetL.Ftisclearlythelinearflowdefined1S−1bythequadraticHamiltonianHt(z)=2tz·zwhereSt=−JF˙tFt.Nowweusethe(exact)propagationtheorem.U(t,s)ˆdenotesthepropagatordefinedbythequadraticHamiltonianbuiltintheproofofLemma17andTheo-rem12.ThenwedefineR(F)ˆ=U(ˆ1,0).RecallthatU(t,ˆ0)isthesolutionoftheSchrödingerequationdiU(t,ˆ0)=H(t)ˆU(t,ˆ0),U(ˆ0,0)=1.(2.93)dtThefollowingtheoremtranslatesthesymplecticinvarianceoftheWeylquantiza-tion.Theorem13ForeverylinearsymplectictransformationF∈Sp(n)andeverysym-bolA∈Σ(1)wehaveR(F)ˆ−1AˆR(F)ˆ=A·F.(2.94)ProofThisisadirectconsequenceoftheexactpropagationformulaforquadraticHamiltoniansU(ˆ0,t)AˆU(t,ˆ0)=AΦt,0.(2.95)Wecangetanotherproofofthefollowingresult(seeformulas(2.27)).Corollary8Letψ,η∈L2(Rn).ForeverylinearsymplectictransformationF∈Sp(n),wehavethefollowingtransformationformulafortheWignerfunction:W(z)=WF−12nR(F)ψ,ˆR(F)ηˆψ,η(z),∀z∈R.(2.96)ProofForeveryA∈S(R2n),wehaveR(F)η,ˆAˆR(F)ψˆ=A(z)WR(F)ψ,ˆR(F)ηˆ(z)dzR2n=η,R(F)ˆ−1AˆR(F)ψˆ=A(F·z)Wψ,η(z)dz.(2.97)R2nThecorollaryfollows.Wehavethefollowinguniquenessresult. 2.5HusimiFunctions,FrequencySetsandPropagation49Proposition20GiventhelinearsymplectictransformationF∈Sp(n),thereex-istsauniquetransformationR(F)ˆ,uptoacomplexnumberofmodulus1,satisfy-ing(2.18).ProofIfVˆsatisfiesVˆ−1AˆVˆ=A·FthenifBˆ=Vˆ−1·R(F)ˆ,weseethatBˆcom-muteswitheveryAˆ,A∈Σ(1).InparticularBˆcommuteswiththeHeisenbergWeyltranslationsT(z)ˆ,henceT(z)ˆ−1BˆT(z)ˆ=Bˆ.ButweknowsthatT(z)ˆ−1BˆT(z)ˆ=B(·+z).SotheWeylsymbolofBˆ(itisatemperatedistribution)isaconstantcom-plexnumberλ.ButhereBˆisunitary,so|λ|=1.2.5HusimiFunctions,FrequencySetsandPropagation2.5.1FrequencySetsTheHusimitransformofsometemperatedistributionsu∈S(Rn)isdefinedasfollows:Definition8TheHusimitransformofu∈S(Rn)isthefunctionHu(z)definedonthephasespaceR2nby−nu,ϕ22nHu(z)=(2π)z,z∈R.(2.98)TheHusimitransformincontrastwiththeWignertransformisalwaysnon-negative.WeshallseebelowthattheHusimidistributionisaregularizationoftheWignerdistribution.Proposition21Foreveryϕ∈L2(Rn)wehaveHϕ=Wϕ∗G0,whereG0isagaussianfunctioninphasespacenamely−n−|z|2/G0(z)=(π)e.OnehasR2nG0(z)dz=1.ThismeansthattheHusimidistributionisaregular-izationoftheWignerdistribution.ProofAccordingtotheProposition17(iii)wehave−nWHϕ(z)=(2π)ϕz,WϕL2(R2n).ButweknowthatWϕ(X)=Wϕ(X−z).z0 502WeylQuantizationandCoherentStatesWeuseProposition16:n|X−z|2Wϕ,Wϕ=2exp−Wϕ(X)dX.zR2nThisyieldstheresult.Insemi-classicalanalysis(orinhighfrequencyanalysis)itisimportanttoun-derstandwhatistheregionofthephasespaceR2nwheresomestatesψ∈L2(Rn)dependingon,essentiallyliveswhenissmall.Forthatpurposeletusintroducethefrequencysetofψ.Definition9Letψ∈L2(Rn),dependingon,suchthatψ≤1.WesaythatψisnegligiblenearapointX0∈R2n,ifthereexistsaneighborhoodVXsuchthat0H∞ψ(z)=O,∀z∈VX.(2.99)0LetusdenoteN[ψ]theset{X∈R2n,ψisnegligiblenearX}.ThefrequencysetFS[ψ]isdefinedasthecomplementofN[ψ]inR2n.Example1•Ifψ=ϕzthenFS[ϕz]={z}.iS(x)n•Letψ=a(x)ewhereaandSaresmoothfunctions,a∈S(R),Sreal.ThenwehavetheinclusionFS[ψ]⊆(x,ξ)|ξ=∇S(x).(2.100)Thereareseveralequivalentdefinitionsofthefrequencysetthatwenowgive.Proposition22Letψbesuchthatψ≤1andX0=(x0,ξ0)∈R2n.Thefol-lowingpropertiesareequivalent:(i)H+∞ψ(X)=O,∀X∈VX.0(ii)ThereexistsA∈S(R2n),suchthatA(X0)=1andAψˆ=O+∞.(2.101)(iii)ThereexistsaneighborhoodVXofX0suchthatforallA∈C∞(VX),000Aψˆ=O+∞.(2.102)(iv)Thereexistχ∈C∞(Rn)suchthatχ(x0)=1andaneighborhoodVξofξ000suchthatix·ξ+∞χ(x)e,ψ=O(2.103)forallξ∈Vξ.0 2.5HusimiFunctions,FrequencySetsandPropagation51ProofLetusassume(i).Thenwehave∞,|z−XHψ(z)=O0|r0/2.(2.105)zNWehave,usinglinearityofintegration,Aψˆ=(2π)−ndzϕz,ψAϕˆz.Fromthetriangleinequality,wehaveAψˆ≤(2π)−ndzψ,ϕzAϕˆz≤(2π)−ndz+dz.(2.106)[|z−X0|0,r0>0suchthat|H(X)−E|≥δ,foreveryX∈B(X0,r0).Letuschoosesomeχ∈C∞(B(X0,r0)),0χ(X0)=1.Usingtheorem9andtheremarkfollowingthistheorem(hereattheendε=),wecanfindBsuchthatBˆHˆ−E=ˆχ+O+∞,(2.110)sowegetχψˆ=O(+∞)henceX0∈/FS[ψ].AssumenowthatHˆsatisfiestheassumptionsofthePropagationtheoremandψsatisfiestheSchrödingerequation(2.109).Proposition24ThefrequencysetFS[ψ]isinvariantundertheclassicalflowΦt,foreveryt∈R.ProofLetX0∈/FS[ψ].ThereexistsacompactsupportsymbolAellipticatX0suchthatAψˆ=O(+∞).ForeverytwehaveitEU(ˆ−t)Aψˆ=O+∞=eA(t)ψˆ.RecallthattheprincipalsymbolofA(t)ˆisA·Φt.SowefindthatifzisnearΦ−t(X0),thenA(t)ψˆ=O(+∞),henceΦ−tX0∈/FS[ψ].SoweseethatFS[ψ]isinvariant.2.6WickQuantization2.6.1GeneralPropertiesFollowingBerezinShubin[23]westartwiththefollowinggeneralsetting.LetMbealocallycompactmetricspace,withapositiveRadonmeasureμandHanHilbertspace.Foreachm∈Mweassociateaunitvectorem∈Hsuchthat 2.6WickQuantization53themapm→emisstronglycontinuousfromMintoH.MoreoverweassumethatthefollowingPlancherelformulaissatisfied,forallψ∈H,2e2ψ=m,ψdμ(m).(2.111)MLetusdenoteψ#(m)=em,ψ.Themapψ→ψ#(m):=Iψ(m)isanisometryfromHintoL2(M).ThecanonicalcoherentstatesintroducedinChap.1areex-amplesofthissettingwhereM=R2n,H=L2(Rn),z→ϕz,withthemeasuredμ(z)=(2π)−ndqdp,z=(q,p)∈R2n.Definition10LetAˆ∈L(H).(i)ThecovariantsymbolofAˆisthefunctiononMdefinedbyAc(m)=em,Aeˆm.(ii)ThecontravariantsymbolofAˆisthefunctiononM,ifitexists,suchthatAψˆ=Ac(m)Πmψdm,ψ∈H.(2.112)MForthestandardcoherentstatesexample,thecovariantsymboliscalledWicksymbolandthecontravariantsymboltheanti-Wicksymbol.ThecovariantsymbolsatisfiestheequalityAc(m)=Tr(AΠˆm).Letuscomputetheanti-WicksymbolofsomeoperatorAˆwithWeylsymbolA.Weknowthatthe-WeylsymboloftheprojectorΠzistheGaussian|X−z|2−n−(π)e.SowefindthattheWeylsymbolofAˆistheconvolutionofitsanti-WicksymbolandastandardGaussianfunction:|X−z|2−nc−A(X)=(π)A(X)edz.(2.113)R2nThisformulashowsthatifAˆhasaboundedanti-Wicksymbol(Ac∈L∞(R2n))thenitsWeylsymbolisanentirefunctioninC2n,whichisarestrictionforagivenoperatortohaveananti-Wicksymbol.LetusremarkthattheWicksymbolisaninverseformulaassociatedwith(2.113):|X−z|2n−Ac(z)=2A(X)edX.(2.114)R2nNowwegiveanotherinterpretationofthecontravariantsymbol.LetusfirstremarkthatwehaveI∗·I=1H,(2.115)I·I∗=ΠH,(2.116)whereΠHistheorthogonalprojectorinL2(M)onHidentifiedwithI(H). 542WeylQuantizationandCoherentStatesProposition25LetusassumethatAˆhasacontravariantsymbolAcsuchthatAc∈L∞(M).ThenwehaveAˆ=I∗·Ac·I,(2.117)whereAcisherethemultiplicationoperatorinL2(M).ProofForeveryψ,η∈Hwehaveη,Aψˆ=η,emem,Aψˆdμ(m)(2.118)Mandcem,Aψˆ=A(m)em,Πmψdμ(m)Mc=A(m)Πmem,Πmψdμ(m).(2.119)MSowegetcη,Aψˆ=A(m)Πmem,Πmψη,emdμ(m)dμ(m).(2.120)M×MWegettheconclusionusingtheequalityη,em=em,emη,emdμ(m).(2.121)MEstimatesonoperatorswithcovariantandcontravariantsymbolsareeasiertoprovethanforWeylsymbols.Moreovertheycanbeusedasafirststeptogetesti-matesinthesettingofWeylquantizationasweshallseeforpositivity.Thefollowingpropositioniseasytoprove.Proposition26LetAˆbeanoperatorinHwithacontravariantsymbolAc.SupposethatAc∈L∞(M).ThenAˆisboundedinHandwehaveAc∞≤Aˆ≤Ac.(2.122)∞MoreoverAˆisself-adjointifandonlyifAcisrealandAˆisnon-negativeifAcisμ-almosteverywherenon-negativeonM.ForourbasicexampleH=L2(Rn),itisconvenienttousethefollowingnotation.IfAisaclassicalobservable,A∈Σ(1),Opw(A)denotestheWeylquantizationofAandOpaw(A)denotestheanti-WickquantizationofA.InotherwordsOpaw(A)admitsAasananti-Wicksymbol.Thefollowingpropositionisaneasyconsequenceoftheaboveresults. 2.6WickQuantization55Proposition27LetA∈Σ(1)(moregeneralsymbolscouldbeconsidered).Thenwehave|X|2aww−n−Op(A)=Op(A∗G),whereG(X)=(π)e,(2.123)aw(A)ψ=(2π)−nA(z)Hψ,Opψ(z)dz,(2.124)R2nwhereHψ(z)istheHusimifunctionofψ.WegetnowthefollowingusefulconsequenceforWeylquantization.Proposition28(Semi-classicalGardinginequality)LetA∈Σ(1),A≥0onR2n.ThenthereexistsC∈Rsuchthatforevery∈]0,1]wehaveψ,Aψˆ≥C,∀ψ∈L2Rn.(2.125)ProofWeknowthatOpw(A∗G)isanon-negativeboundedoperator.Sothepropo-sitionwillbeprovedifOpw(A∗G−A)=O().(2.126)Usingastandardargumentforsmoothingwithconvolution,weget−1(A∗G−A)∈Σ(1),withuniformestimatesin∈]0,1].Henceweget(2.126)asaconse-quenceoftheCalderonVaillancourttheorem.Theseresultsareusefultostudythematrixelementsψ,Aψˆ,forafamily{ψ}inthesemi-classicalregime[106].ThissubjectisrelatedwithanefficienttoolintroducedbyLionsPaul[137]andP.Gérard[82](seealso[35]):thesemi-classicalmeasures.Thisisanapplicationofanti-Wickquantizationasweshallseenow.2.6.2ApplicationtoSemi-classicalMeasuresSemi-classicalmeasureswereintroducedtodescribelocalizationandoscillationsoffamiliesofstates{ψ},ψ=1(oratleastboundedinL2(Rn)).LetusfirstremarkthatA→ψ,OpawAψisaprobabilitymeasureμinR2n.MoreoverthisprobabilitymeasurehasadensitygivenbytheHusimifunctionofψ,dμ=(2π)−nHψ(z)dz. 562WeylQuantizationandCoherentStatesInparticularwehaveψ,OpawAψ≤A∞foreveryA∈Cb(R2n)(spaceofcontinuous,boundedfunctionsonR2n).Definition11Asemi-classicalmeasureforthefamilyofnormalizedstates{ψ}isaprobabilitymeasureμonthephasespaceR2nforwhichthereexistsatleastonesequence{k},limk=0suchthatforeveryA∈Σ(1),wehavek→+∞limψOpawAψ=Adμ.(2.127)kkkk→+∞R2nInotherwords,themeasuresequenceμkweaklyconvergestowardthemeasureμ.Remark11Semi-classicalmeasurescanalsobedefinedforstatesψ∈L2(Rn,K)whereKisanHilbertspace.BythewayinthissettingWeylsymbolsandanti-WicksymbolsareoperatorsinK.Wecanalsodefinesemi-classicalmeasuresforstatisticalmixedstatesρˆ,whereρˆisanon-negativeoperatorsuchthatTrρˆ=1.Formoreapplicationsandpropertiesoftheseextensionsseethehugeliteratureonthissubject;forexamplesee[135].ThefollowingpropositionisastraightforwardapplicationofthepropertiesoftheHusimifunction.Proposition29Letμbeasemi-classicalmeasurefor{ψ}.Thenthesupportsupp(μ)ofthemeasureμisincludedinthefrequencysetFS[ψ],supp(μ)⊆FS[ψ].Example2(i)Letψ=ϕz,astandardcoherentstate.Thenthisfamilyhasonesemi-classicalmeasure,μ=δz(Diracprobability).(ii)Letusassumethatthestatesfamily{ψ}istightinthefollowingsense.Thereexistsasmoothsymbolχ,withcompactsupport,suchthatχψˆ=ψ+O().ThenusingLemma15,wecanseethatthefamilyofprobabilities{μ}istight,soapplyingtheProkhorovcompacitytheorem,thereexistsatleastonesemi-classicalmeasure.Oneofachallengingprobleminquantummechanicsistocomputethesesemi-classicalmeasuresforfamilyofboundstatessatisfying(2.109).Ifforsomeε>0,H−1[E−ε,E+ε]isaboundedset,thisfamilyistight.Forclassicallyergodicsystemsitisconjecturedthatthereexistsonlyonesemi-classicalmeasure,whichistheLiouvillemeasure[106].Oneimportantpropertyofsemi-classicalmeasuresisthefollowingpropagationresult. 2.6WickQuantization57LetusconsiderthetimedependentSchrödingerequationi∂tψ(t)=Hψˆ(t),ψ(0)=ψ,(2.128)whereHisatimeindependentHamiltonian.WeassumethatHisreal,subquadraticandindependent(forsimplicity).γ∞2n∂H∈LR,forallγsuchthat|γ|≥2.(2.129)XLetμbeasemi-classicalmeasurefor{ψ}.Theorem14Foreveryt∈R,{ψ(t)}hasasemi-classicalmeasuredμtforthesamesubsequencekgivenbythetransportofdμbytheclassicalflow:Φt,μ(t)=(Φt)∗μ.ProofForeveryA∈C0∞(R2n),thesemi-classicalEgorovtheoremandcomparisonbetweenanti-WickandWeylquantizationgiveψ(t),Opaw(A)ψ(t)=A·Φtdμψ+O().(2.130)R2nHencewegettheresultgoingtothelimitforthesequencek.WehavethefollowingconsequenceforthestationarySchrödingerequation.Corollary10Letμbesemi-classicalmeasureforafamilyofboundstates{ψ},satisfyingHψˆ=Eψ.ThenμisinvariantbytheclassicalflowΦtforeveryt∈R.−itEProofψ(t)=eψsatisfiesthetimedependentSchrödingerequationsous-ingtheTheoremweget(Φt)∗μ=μ.NowweillustrateCorollary10onHermiteboundstatesoftheharmonicoscilla-tor.Weassumen=1.WecaneasilycomputeHusimifunctionHjoftheHermitefunctionφj.2(q2+p2)j−1(q2+p2)Hj(q,p)=ϕX,φj=je2.(2.131)2j!Wewanttostudythequantummeasuresdμj=(2π)−1Hj(q,p)dqdpwhentheenergiesE1j=(j+)havealimitE>0.Sowehave→0andj→+∞.For2simplicitywefixE>0andchoose=Ej=.jLetfbeintheSchwartzclassS(R2).Wehavetocomputethelimitoff(X)dμj(X)forj→+∞.Usingpolarcoordinatesandachangeofvariables 582WeylQuantizationandCoherentStateswehavetostudythelargeklimitfortheLaplaceintegral∞√1j√j−uI(j):=ueEf2ucosθ,2usinθdu,θ∈[0,2π[.(j+1)!0Wecanassumethatfhasaboundedsupportand(0,0)isnotinthesupportoff.UsingtheLaplacemethodweget√limI(j)=f2E(cosθ,sinθ).(2.132)j→+∞So,wehave2π√1limf(X)dμj(X)=√f2E(cosθ,sinθ)dθ.(2.133)j→+∞2π2E0Onther.h.s.of(√2.133)werecognizetheuniformprobabilitymeasureonthecircleofradius2E.Thismeasureisasemi-classicalmeasureforthequantumharmonic√oscillator.Letusremarkthattheclassicaloscillatorofenergy√2Emovesonthecircleofradius2Einthephasespace. Chapter3TheQuadraticHamiltoniansAbstractTheaimofthischapteristoconstructthequantumunitarypropagatorforHamiltonianswhicharequadraticinpositionandmomentumwithtime-dependentcoefficients.Weshowthatthequantumevolutionisexactlysolvableintermsoftheclassicalflowwhichislinear.Thisallowstoconstructthemetaplectictransfor-mationswhichareunitaryoperatorsinL2(Rn)correspondingtosymplectictrans-formations.SimpleexamplesofsuchmetaplectictransformationsaretheFouriertransform,whichcorrespondstothesymplecticmatrixJdefinedin(3.4)andthepropagatoroftheharmonicoscillator,correspondingtorotationsinthephasespace.Themainresultsofthischapterarecomputationsofthequantumevolutionop-eratorsforquadraticHamiltoniansactingoncoherentstates.WeshowthatthetimeevolvedcoherentstatesarestillGaussianstateswhicharerecognizedtobesqueezedstatescenteredattheclassicalphasespacepoint(seeChap.8).Fromthesecompu-tationswecandeducemostofpropertiesconcerningquantumquadraticHamiltoni-ans.InparticularwegettheexplicitformoftheWeylsymbolsofthemetaplectictransformations.TheseformulasaregeneralizationsoftheMehlerformulafortheharmonicoscillator.QuadraticHamiltoniansareveryimportantinquantummechanicsbecausemoregeneralHamiltonianscanbeconsideredasnon-trivialperturbationsoftime-dependentquadraticonesasweshallseeinChap.4.3.1ThePropagatorofQuadraticQuantumHamiltoniansAclassicalquadraticHamiltonianHisaquadraticformdefinedinthephasespaceR2n.Weassumethatthisquadraticformistimedependent,sowehaveH(t,z)=cj,k(t)zjzk1≤j,k≤nwherez=(q,p)∈R2nisthephasespacevariableandtherealcoefficientscj,k(t)arecontinuousfunctionsoftimet∈R.Itcanberewrittenas1qH(t,q,p)=(q,p)S(t)(3.1)2pM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,59TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_3,©SpringerScience+BusinessMediaB.V.2012 603TheQuadraticHamiltonianswhereS(t)isa2n×2nrealsymmetricmatrixoftheblockformGtLTS(t)=t(3.2)LtKtGt,Lt,Ktaren×nrealmatriceswithGt,Ktbeingsymmetric,andLTtdenotesthetransposeofLt.TheclassicalequationsofmotionforthisHamiltonianarelinearandcanbewrittenasq˙q=JS(t)(3.3)p˙pwhereJisthesymplecticmatrix01nJ=(3.4)−1n0LetF(t)betheclassicalflowfortheHamiltonianH(t).Itmeansthatitisasymplectic2n×2nmatrixobeyingF(t)˙=JS(t)F(t)(3.5)withF(0)=1.Thenthesolutionof(3.3)withq(0)=q,p(0)=pissimplyq(t)q=F(t)p(t)pWenowconsiderthequantumHamiltonianQH(t)=Q,P·S(t)(3.6)PThequantumevolutionoperatorU(t)1issolutionoftheSchrödingerequationdiU(t)=H(t)U(t)(3.7)dtwithU(0)=1.ThefollowingresultwasalreadyprovedinChap.2asaparticularcaseofamoregeneralresult.Weshallgivehereasimpledirectproof.Theorem15OnehasforalltimestQQU(t)∗U(t)=F(t)(3.8)PP1WeshallexplainlaterwhythisquantumpropagatorisawelldefinedunitaryoperatorinL2(Rn). 3.2ThePropagationofCoherentStates61ProofDefineQt=U(t)∗QU(t),andsimilarlyforP(Heisenbergobservables).UsingtheSchrödingerequationonehasdQt∗Q−i=U(t)H(t),U(t)dtPtPButQQH(t),=−iJS(t)PPThismeansthatQt,PtmustsatisfythelinearequationdQtQt=JS(t)dtPtPtwhichistriviallysolvedby(3.8).3.2ThePropagationofCoherentStatesInthissectionwegivetheexplicitformofthetimeevolvedcoherentstatesintermsoftheclassicalflowF(t)givenbythe2n×2nblockmatrixform:AtBtF(t)=(3.9)CtDtItwillbeshownthatthecomplexn×nmatrixAt+iBtisalwaysnon-singular.Thenweestablishthefollowingresult:U−n/4T(z−1/2i−1tϕz=(π)t)det(At+iBt)exp(Ct+iDt)(At+iBt)x·x2(3.10)wherezt=F(t)zisthephasespacepointoftheclassicaltrajectoryandT(z)istheWeyl–Heisenbergtranslationoperatorbythevectorz=(x,ξ)∈R2n:iT(z)=expξ·Q−x·P(3.11)ThismeansthatUtϕzisasqueezedstatecenteredatthephasespacepointzt,sothesqueezedstatemovesontheclassicaltrajectory.Wetake=1forsimplicity.Asimpleexampleistheharmonicoscillator1d21H+x2(3.12)os=−22dx2 623TheQuadraticHamiltoniansItiswellknownthatfort=kπ,k∈Zthequantumpropagatore−itHoshasanexplicitSchwartzkernelK(t;x,y)(Mehlerformula,Chap.1).Itiseasiertocomputedirectlywiththecoherentstatesϕz.ϕ0isthegroundstateofHos,sowehavee−itHϕ−it/20=eϕ0(3.13)Letuscomputee−itHϕz,∀z∈R2,withthefollowingansatz:e−itHϕz=eiδt(z)T(z−it/2ϕ0(3.14)t)ewherezt=(qt,pt)isthegenericpointontheclassicaltrajectory(acirclehere),comingfromzattimet=0.Letψt,zbethestateequaltother.h.s.in(3.14),anddϕ=Hϕ,ϕ|letuscomputeδt(z)suchthatψt,zsatisfiestheequationit=0=ψ0,z.dtWehaveT(zi(ptx−qtpt/2)u(x−qt)t)u(x)=eandψi(δt(z)−t/2+ptx−qtpt/2)t,z(x)=eϕ0(x−qt)(3.15)So,aftersomecomputationslefttothereader,usingpropertiesoftheclassicaltra-jectoriesq˙2222t=pt,p˙t=−qt,pt+qt=p+qtheequationd1i22ψψt,z(x)=Dx+xt,z(x)(3.16)dt2issatisfiedifandonlyif1δt(z)=(ptqt−pq)(3.17)2Letusnowintroducethefollowinggeneralnotationsforlateruse.Ftistheclassicalflowwithinitialtimet0=0andfinaltimet.Itisrepresentedasa2n×2nmatrixwhichcanbewrittenasfourn×nblocks:AtBtFt=(3.18)CtDtLetusintroducethefollowingsqueezedstates:ϕΓdefinedasfollows:Γiϕ(x)=aΓexpΓx·x(3.19)2whereΓ∈Σn,ΣnistheSiegelspaceofcomplex,symmetricmatricesΓsuchthat(Γ)ispositiveandnon-degenerateandaΓ∈CissuchthattheL2-normofϕΓisone.WealsodenoteϕzΓ=T(z)ϕΓ.ForΓ=i1,wedenoteϕ=ϕi1. 3.2ThePropagationofCoherentStates63Theorem16Wehavethefollowingformulas,foreveryx∈Rnandz∈R2n,UΓ(x)=ϕΓt(x)(3.20)tϕUΓ(x)=T(Ftz)ϕΓt(x)(3.21)tϕzwhereΓt=(Ct+DtΓ)(At+BtΓ)−1andaΓ=aΓ(det(At+BtΓ))−1/2.tBeginningoftheProofThefirstformulacanbeprovenbytheansatziUtϕ0(x)=a(t)expΓtx·x2whereΓt∈Σnanda(t)isacomplexvaluestime-dependentfunction.WegetfirstaRiccatiequationtocomputeΓtandalinearequationtocomputea(t).Thesecondformulaiseasytoprovefromthefirst,usingtheWeyltranslationoperatorsandthefollowingknownpropertyUT(z)U∗=T(Ftz)ttLetusnowgivethedetailsoftheproofforz=0.WebeginbycomputingtheactionofaquadraticHamiltonianonaGaussian(=1).Lemma18iΓx·xTiiΓx·xLx·Dxe2=Lx·Γx−TrLe22ProofThisisastraightforwardcomputation,using1xjDk+DkxjLx·Dx=Ljki21≤j,k≤nand,forω∈Rn,iΓx·xiΓx·x(ω·Dx)e2=(Γx·ω)e2Lemma19iiΓx·xΓx·x(GDx·Dx)e2=GΓx·Γx−iTr(GΓ)e2ProofAsabove,wegetiΓx·x11iiΓx·xHe2=Kx·x+x·LΓx+GΓx·Γx−Tr(L+GΓ)e2(3.22)222 643TheQuadraticHamiltoniansWearenowreadytosolvetheequation∂iψ=Hψ(3.23)∂twith−n/4−x2/2ψ|t=0(x)=g(x):=πeForsimplicityweassumeherethatΓ=i1,theproofcanbeeasilygeneralizedtoΓ∈Σn.WetrytheansatziΓtx·xψ(t,x)=a(t)e2(3.24)whichgivestheequationsΓ˙TL−ΓtGΓt(3.25)t=−K−2Γt1a(t)˙=−Tr(L+GΓt)a(t)(3.26)2withtheinitialconditionsΓ−n/40=i1,a(0)=(π)WenotethatΓTLetLΓdeterminethesamequadraticforms.SothefirstequationisaRicattiequationandcanbewrittenasΓ˙T−LΓt−ΓtGΓt(3.27)t=−K−ΓtLwhereLTdenotesthetransposedmatrixforL.Weshallnowseethat(3.27)canbesolvedusingHamiltonequationKLF˙t=JTFt(3.28)LGF0=1(3.29)WeknowthatAtBtFt=CtDtisasymplecticmatrix∀t.Sousingthenextlemma,wehavedet(At+iBt)=0∀t.LetusdenoteMt=At+iBt,Nt=Ct+iDt(3.30)WeshallprovethatΓ−1.Byaneasycomputation,wegett=NtMt 3.2ThePropagationofCoherentStates65M˙TMt+GNtt=L(3.31)N˙t=−KMt−LNtNow,computedNtM−1=NM˙−1−NM−1MM˙−1dtt=−K−LNM−1−NM−1LTM+GNM−1=−K−LNM−1−NM−1LT−NM−1GNM−1(3.32)whichisexactly(3.27).Nowwecomputea(t),usingthefollowingequality:T+G(C+iD)(A+iB)−1=Tr(M)M˙−1=Tr(L+GΓTrLt)usingTrL=TrLT.LetusrecalltheLiouvilleformuladlog(detMt)=TrM˙−1(3.33)tMtdtwhichgivesdirectly−n/4−1/2a(t)=(π)det(At+iBt)(3.34)Tocompletetheproof,weneedtoprovethefollowing.Lemma20LetFbeasymplecticmatrix.ABF=CDThendet(A+iB)=0and(C+iD)(A+iB)−1ispositivedefinite.WeshallproveamoregeneralresultconcerningtheSiegelspaceΣn.Lemma21IfABF=CDisasymplecticmatrixandZ∈ΣnthenA+BZandC+DZarenon-singularand(C+DZ)(A+BZ)−1∈ΣnProofLetusdenoteM:=A+BZ,N:=C+DZ.Fissymplectic,sowehaveFTJF=J.UsingMI=FNZ 663TheQuadraticHamiltonianswegetTTMIM,NJ=(I,Z)J=0(3.35)NZwhichgivesMTN=NTMInthesameway,wehave1TTM¯1TIM,NJ=(I,Z)FJF2iN¯2iZ¯1I1=(I,Z)J=Z¯−Z=−Z(3.36)2iZ¯2iWegetthefollowingequation:NTM¯−MTN¯=2iZ(3.37)BecauseZisnon-degenerate,from(3.37),weseethatMandNareinjective.Ifx∈Cn,Ex=0,wehaveM¯x¯=xTMT=0hencexTZx¯=0thenx=0.So,wecandefineα(F)(Z)=(C+DZ)(A+BZ)−1(3.38)Letusprovethatα(F)Z∈Σn.Wehaveα(F)Z=NM−1T−1TT−1TT−1−1⇒α(F)Z=MN=MMNM=NM=α(F)ZWehavealso:TNM−1−N¯M¯−1NTM¯−MTN¯MM¯==Z2i2iandthisprovesthat(α(F)(Z))ispositiveandnon-degenerate.ThisfinishestheproofoftheTheoremforz=0.ThemapF→α(F)definesarepresentationofthesymplecticgroupSp(n)intheSiegelspaceΣn.Forlateruseitisusefultointroducethedeterminant:δ(F,Z)=det(A+BZ),F∈Sp(n),Z∈Σn.Thefollowingresultsareeasyal-gebraiccomputations. 3.2ThePropagationofCoherentStates67Proposition30Wehave,foreveryF1,F2∈Sp(n),(i)α(F1F2)=α(F1)α(F2).(ii)δ(F1F2,Z)=δ(F1,α(F2)(Z))δ(F2,Z).(iii)ForeveryZ1,Z2∈ΣnthereexistsF∈Sp(n)suchthatα(F)(Z1)=Z2.InotherwordstherepresentationαistransitiveinΣn.Manyotherpropertiesoftherepresentationαarestudiedin[139]and[77].Forcompleteness,westatethefollowing.Corollary11ThepropagatorUtiswelldefinedanditisaunitaryoperatorinL2(Rn).ProofForeverycoherentstateϕz,UtϕzissolutionoftheSchrödingerequation.AswehaveseeninChap.1,thefamily{ϕz}z∈R2nisovercompleteinL2(Rn).Sofor-mula(3.20)whollydeterminestheunitarygroupUt.InapreliminarystepwecanseethatUtψiswelldefinedforψ∈S(Rn)usinginverseFourier–Bargmanntrans-form,thatUtψ∈S(Rn),andthatUtψ = ψ.SowecanextendUtinL2(Rn).InparticularitresultsthatUtisaunitaryoperatorandthatHthasauniqueself-adjointextensioninL2(Rn).ItwillbeusefultocomputetheFourier–BargmanntransformofUtϕz.RecallthatU(t)=R(Ft)whereR(F)isthemetaplecticoperatorcorrespondingtothesymplectic2n×2nmatrixFandthatFthasafourblocksdecompositionAtBtFt=CtDtNowwedefinen×ncomplexmatricesYt,Ztasfollows:Yt=At+iBt−i(Ct+iDt),Zt=At+iBt+i(Ct+iDt)Onehasthefollowingproperty,usingthesymplecticityofFt:Lemma22Z∗Z=Y∗Y−41Ytisinvertible.OnecandefinethematrixWtasfollows:W−1t=ZtYtwhichsatisfiesthefollowingproperty. 683TheQuadraticHamiltoniansLemma23(i)W0=0(ii)∗WWtt<1(iii)−11(Γ+i1)=(1+W)2iInparticularWisasymmetricmatrix.Proof(i)IsaneasyconsequenceofthefactthatF0=1,henceY0=21,Z0=0.(ii)Wehave∗∗−1∗−1∗−1∗−1∗−1WW=YZZY=YYY−41Y=1−4YY.(iii)Isasimplealgebraiccomputation.Theorem17Thematrixelementsϕz,Utϕzaregivenbythefollowingformula:n/2−1/2ϕz,Utϕz=2detAt+Dt+i(Bt−Ct)−i/2σ(Ftz,z)−(x2+ξ2)/4−W(ξ+ix)·(ξ+ix)/4×eee(3.39)wherez−Ftz=(x,ξ)and−1Wt=At−Dt+i(Bt+Ct)At+Dt+i(Bt−Ct)ProofForsimplicityweforgetthetimeindexteverywhere.Itisenoughtoassumethatz=0.Fromthemetaplecticinvariance,weget(Γ)ϕz,Uϕz=ϕz,TFzϕ−iσ(Fz,z)/2(Γ)=eϕz−Fz,ϕ(3.40)SowehavetocomputeϕX,ϕ(Γ).Wehave12(Γ)−n/2−1/2(ip·q−q)ϕX,ϕ=πdet(A+iB)e2i(Γ+i)x·x−ix·(p+iq)×e2edx(3.41)RnSotheresultfollowsfromcomputationoftheFouriertransformofageneralizedGaussian(orsqueezedstate). 3.3TheMetaplecticTransformations693.3TheMetaplecticTransformationsRecallthatametaplectictransformationassociatedwithalinearsymplectictrans-formationF∈Sp(n)inR2n,isaunitaryoperatorR(F)inL2(Rn)satisfyingoneofthefollowingequivalentconditionsR(F)∗AR(F)=A◦F,∀A∈SR2n(3.42)R(F)∗T(X)R(F)=TF−1(X),∀X∈R2n(3.43)R(F)∗AR(F)=A◦F,forA(q,p)=qj,1≤j≤nandA(q,p)=pk,1≤k≤n(3.44)AistheWeylquantizationoftheclassicalsymbolA(q,p)andwerecallthattheoperatorT(X)isdefinedbyiT(X)=expξ·Q−x·P(3.45)whenX=(x,ξ)∈R2n.WeshallprovebelowthatforeveryF∈Sp(2n)thereexistsametaplectictrans-formationR(F).Thistransformationisuniqueuptoamultiplicationbyacomplexnumberofmodulus1.Lemma24IfR1(F)andR2(F)aretwometaplecticoperatorsassociatedtothesamesymplecticmapFthenthereexistsλ∈C,|λ|=1,suchthatR1(F)=λR2(F).ProofDenoteR=R1(F)R2(F)−1.ThenwehaveR∗T(X)R=T(X)forallX∈R2n.ApplyingtheSchurlemma10wegetR=λ1,λ∈C.ButRisunitarysothat|λ|=1.WeshallproveherethatF→R(F)definesaprojectiverepresentationoftherealsymplecticgroupSp(n)withsignindeterminationonly.Moreprecisely,letusdenotebyMp(n)thegroupofmetaplectictransformationsandπpthenaturalprojection:Mp→Sp(2n)thenthemetaplecticrepresentationisagrouphomo-morphismF→R(F),fromSp(n)ontoMp(n)/{1,−1},suchthatπp[R(F)]=F,∀F∈Sp(2n)Formoredetailsaboutthemetaplectictransformationssee[133].Proposition31ForeveryF∈Sp(n)wecanfindaC1-smoothcurveFt,t∈[0,1],inSp(n),suchthatF0=1andF1=F.ProofAnexplicitwaytodothatistousethepolardecompositionofF,F=V|F|√whereVisasymplecticorthogonalmatrixand|F|=FTFispositivesymplecticmatrix.Eachofthesematriceshavealogarithm,soF=eKeLwithK,LHamilto-nianmatrices,andwecanchooseFt=etKetL. 703TheQuadraticHamiltoniansLetFtbeasinProposition31.FtisthelinearflowdefinedbythequadraticHamiltonianH1−1t(z)=2Stz·zwhereSt=−JF˙tFt.Sousingaboveresults,wedefineR(F)=U1.HereUtisthesolutionoftheSchrödingerequationdiUt=H(t)Ut(3.46)dtthatobeysU0=1.NamelyitisthequantumpropagatorofthequadraticHamilto-nianH(t).ThatthemetaplecticoperatorsodefinedsatisfiestherequiredpropertiesfollowsfromTheorem15.Proposition32LetusconsidertwosymplecticpathsFtandFtjoining1(t=0)toF(t=1).ThenwehaveU1=±U(withobviousnotation).1Moreover,ifF1,F2∈Sp(2n)thenwehaveRF1RF2=±RF1F2(3.47)ProofWefirstremarkthatthepropagatorofaquadraticHamiltonianisdeterminedbyitsactiononsqueezedstatesϕΓanditsclassicalflow.Sousing(3.20)weseethatthephaseshiftbetweenthetwopathscomesfromvariationofargumentbetween0and1ofthecomplexnumbersb(t)=det(At+iBt)andb(t)=det(At+iBt).tb(s)˙Wehavearg[b(t)]=(ds)andbyacomplexanalysisargument,wehave0b(s)1b(s)˙1b˙(s)ds=ds+2πN0b(s)0b(s)withN∈Z.Sowegetb(1)−1/2=eiNπb(1)−1/2ThesecondpartofthepropositionisaneasyconsequenceofTheorem16concern-ingpropagationofsqueezedcoherentstatesandProposition30.Moreprecisely,thesignindeterminationin(3.47)isaconsequenceofvariationsforthephaseofdet(A+iB)concerningF=F1andF=F2.TocomparewithF1F2weapplyProposition30.Remark12AgeometricalconsequenceofProposition30isthefollowing.ThemapF→R(F)inducesagroupisomorphismbetweenthesymplecticgroupSp(n)andthequotientofthemetaplecticgroupMp(n)/{−1,1}.InotherwordsthegroupMp(n)isatwo-coverofSp(n).Aninterestingpropertyofthemetaplecticrepresentationisthefollowing.Proposition33ThemetaplecticrepresentationRhastwoirreductiblenon-equivalentcomponentsinL2(Rn).ThesecomponentsarethesubspacesL2(Rn)odofoddstatesandL2ev(Rn)ofevenstates. 3.4RepresentationoftheQuantumPropagatorinTermsoftheGenerator71ProofLetusfirstremarkthatthesubspacesL2(Rn)areinvariantforR(F)be-od,evcausequadraticHamiltonianscommutewiththeparityoperatorΠψ(x)=ψ(−x).NowwehavetoprovethatL2(Rn)areirreducibleforR.od,evLetusbeginbyconsideringthesubspaceL2ev(Rn).LetBbeaboundedoperatorinL2ev(Rn)suchthatR(F)B=BR(F)foreveryF∈Sp(n).AccordingtotheSchurlemma,wehavetoprovethatB=λ1,λ∈C.InparticularBcommuteswiththepropagatoroftheharmonicoscillatorUt=eitHos.WecansupposethatBisHermitian.SoBisdiagonalintheHermitebasisφα(seeChap.1).WehaveBφα=λαφαforeveryα∈Nn.Toconcludewehavetoprovethatλα=λβif|α|and|β|areeven.Assumeforsimplicitythatn=1(theproofisalsovalidforn≥2).itx2LetusconsiderthemetaplectictransformationRt=eassociatedtothesym-plectictransformF0t.t=exp00WehaveRtφk=c(t,k,j)φjj≥0UsingthatRtB=BRtwegetc(t,k,j)λj=c(t,k,j)λk(3.48)Nowweshallprovethatifk−jiseventhenc(t,k,j)=0forsomethenceλj=λk.Thisaconsequenceofthefollowingitx2c(t,k,j)=eφk(x)φj(x)dxIfc(t,k,j)=0foreverytthenx2mφk(x)φj(x)dx=0foreverym∈N.ButthisRisnotpossibleifk−jiseven(seepropertiesofHermitefunctions).SowehaveprovedthatL2ev(R)isirreducible.WiththesameproofwealsofindthatL2(R)isalsoirreducible.odAssumenowthatBR(F)=R(F)BandthatBisalineartransformationfromL2ev(R)inL2(R).WestillhaveBφk=λkφk.ButanHermitefunctionisoddoroddevensoλk=0forallkandB=0.Sotheserepresentationsarenon-equivalent.3.4RepresentationoftheQuantumPropagatorinTermsoftheGeneratorofSqueezedStatesInthissectionouraimistorevisitsomeresultsobtainedin[49]and[51].LetusstartwithclassicalHamiltonianmechanicsinthecomplexmodelCn,ζ=q−ip√.Asabove,Fisaclassicalflowforaquadratic,time-dependentHamiltonian.2LetusdenotebyFcthesameflowinCn.Weeasilyget1Fcζ=Yζ+Z¯ζ¯(3.49)2 723TheQuadraticHamiltonianswhereY=A+D+i(B−C),Z=A−D+i(B+C)(3.50)RecallthatallthesematricesaretimedependentandYisinvertible.SowehaveY=|Y|V(polardecomposition),where|Y|2=YY∗andVisaunitarytransformationofCn(V∈SU(n)).WealreadyintroducedW=ZY−1andweknowthat0≤W∗W<1.SowecanfactorizeFcinthefollowingway.Fc=Dc·Sc(3.51)Scζ=Vζ(3.52)c∗−1/2∗−1/2∗ζ¯(3.53)Dζ=1−WWζ+1−WWWComingbacktotherealrepresentationinR2n,Sisanorthogonalsymplectictrans-formation(arotationinthephasespace).LetuscomputeR(S).Todothat,wewriteV=eiL(thisispossiblelocallyintime).LisanHermitiancomplexmatrix.Vistheflowattime12ofthequadraticHamiltonian1Hcζ,ζ¯=ζ·LTζ¯+ζ¯·LζS2LetHbetherealrepresentationofHc,thenwehaveR(S)=e−iHS.SoR(S)isaquantumrotationbecausewehave,foreveryobservableO,eiHSOe−iHS=O·SItismoredifficulttocomputeR(D)(thedilationorsqueezingpart).WewritedownthepolardecompositionofW,W=U|W|,|W|2=W∗W,UunitarytransformationinCn.Wearelookingforageneratorat(new)time1forthetransformationD.LetusintroduceacomplextransformationBinCnwiththepolardecompositionB=U|B|(|B|2=B∗B).Afterstandardcomputations,weget0B∗cosh|B|−sinh|B|U∗exp=∗(3.54)B0Usinh|B|−Ucosh|B|UComparingwithpreviouscomputationofDc,weget∗−1/2cosh|B|=1−WW(3.55)sinh|B|U∗=cosh|B|W∗(3.56)sinh|B|=cosh|B||W|(3.57)2Thistimeisanewtime,whichhasnothingtodowitht. 3.4RepresentationoftheQuantumPropagatorinTermsoftheGenerator73Wecansolvethelastequation:|B|=argtanh|W|(3.58)Moreexplicitlywehave(W∗W)nB=W(3.59)2n+1n≥0InparticularBisasymmetricmatrix(BT=B)becauseWisasymmetricmatrix.Asfortherotationpart,wecangetnowadilationgenerator,iHc=ζ·Bζ−ζ¯·B∗ζ¯D2suchthatthe(complex)equationofmotionisζ˙=B∗ζ¯Finallywerestorethetimet.WehaveadecompositionFt=D(Bt)S(Lt)suchthatUD(BS(Lt=λtt)t),whereλtisacomplexnumber,|λt|=1and1(a†·B†∗ta−a·Bta)D(Bt)=e2(3.60)isthegeneratorofsqueezedstates.MorepropertiesofD(B)willbegivenattheendofthissection.WecangetnowProposition34ForeverytimetwehaveU1/2VtφB(3.61)tϕ0=dettwhereφBisthesqueezedstatedefinedbytφB=D(Bt)ϕ0(3.62)t1/2anddetVtisdefinedbycontinuity,startingfromt=0(V0=1).ProofItisenoughtoshowthatϕ0isaneigenstateofS(L)witheigenvalueγ=1Tr(L).Clearly2a†t†ϕ††·La+a·La0=Lj,iaai+Li,jaiaϕ0=Li,iϕ0jji,jisinceaiϕ0=0,∀i=1,...,n.Wegettheresultbyexponentiating.LetusremarkherethatevenifLisdefinedinasmalltimeintervalwecanconcludebecausetheprefactorofφBmustbecontinuousintimet.tWecanalsodemonstratethatthequantumevolutionofacoherentstateϕzissimplyadisplacedalongtheclassicalmotionofasqueezedstate: 743TheQuadraticHamiltoniansProposition35Letzt=(qt,pt)bethephasespacepointoftheclassicalflowattimetstartingwithinitialconditionsz=(q,p).Thusqtq=FtptpOnehasU(t)ϕ1/2T(zz=detVtt)ΦBtProofOneusesthefactthat,duetothatU(t)=R(Ft),U(t)ϕU(t)T(z)ϕT(zU(t)ϕz=0=t)0Moreonn-DimensionalSqueezedStatesConsidernowanycomplexsymmetricn×nmatrixWsuchthatW∗W<1.TakeasbeforethepolardecompositionofWtobe∗1/2W=U|W|,|W|=WWUbeingunitary.Wedefinethen×ncomplexsymmetricmatrixBtobeB=Uargtanh|W|MoreexplicitlywritingtheTaylorexpansionofargtanhuat0wefind∞(W∗W)nB=Wandwehave|W|=tanh|B|(3.63)2n+1n=0NowweconstructtheunitaryoperatorD(B)inL2(Rn)as1D(B)=expa†·Ba†−a·B∗a2WehaveLemma25(i)D(B)isunitarywithinverseD(−B).(ii)a(1−WW∗)−1/2−W(1−W∗W))−1/2aD(B)a†D(−B)=−(1−W∗W)−1/2W∗(1−W∗W)−1/2a†ProofLetusdenote††aB(t)=D(tB)aD(−tB),a(t)=D(tB)aD(−tB)B 3.4RepresentationoftheQuantumPropagatorinTermsoftheGenerator75Computingd1a††∗B(t)=D(tB)a·Ba−a·Ba,aD(−tB)dt2weget,usingthecommutingrelations,daB(t)=−BaB(t)dtandthesamefora†:d†∗∗aB(t)=BaB(t)dt∗Wegettheresultbycomputingexp0Basin(3.54).B0Wedefinethen-dimensionalsqueezedstateasψ(B)=D(B)ϕ0whereϕ0isthestandardGaussian(groundstateoftheHarmonicoscillator).Weshallfirstcomputeψ(B)intheFock–Bargmannrepresentation.WerecallthatintheFock–BargmannrepresentationonehasB[ϕ−n/20](ζ)=(2π)(independentofζ∈Cn).Thenwetrythefollowingansatz:1Bψ(B)(ζ)=aexpζ·Mζ(3.64)2whereMisacomplexsymmetricmatrixthatwewanttocompute.Fromnowonweassume=1.LetusconsidertheantihermitianHamiltonianHBtobe1Ha†·Ba†−a·B∗aB=2Takeafictitioustimettovarybetween0and1andconsidertheevolutionoperatorUB(t)=etHB.Thenψ(B)(t)=UB(t)ϕ0satisfiesthedifferentialequationd(B)(B)ψ(t)=HBψ(t)dtwiththefollowinglimitingvalues:ψ(B)(0)=ϕ(B)(B)0,ψ(1)=ψ 763TheQuadraticHamiltoniansIntheFock–BargmannrepresentationHBhasthefollowingform:1∗HB(ζ,∂ζ)=(ζ·Bζ−∂ζ·B∂ζ)2Usingtheansatz(3.64)forψ(B)(t)weseethatonehas111a˙+aζ·Mζ˙expζ·Mζ=a(t)HB(ζ,∂ζ)expζ·Mζ222Nowwecomputetherighthandsideusingtheconventionofsummationoverre-peatedindices;weget111∗1∗1a(t)ζ·Bζ−∂ζjBijMikζk−Bij∂ζjζkMkiexpζ·Mζ2222211=aζ·Bζ−Tr(B∗M)−ζ·MB∗Mζexpζ·Mζ22IdentifyingwegetM˙=B−MB∗1∗M,a˙=−aTr(BM)2Letussolvethedifferentialequation∂∗tM=−MBM+B(3.65)withMt=0=0.WeconsiderNsuchasM=UNwithUindependentoftgivenbythepolardecompositionofB.Thenequation(3.65)becomesU∂tN=−UN|B|N+U|B|Thusitreducesto∂tN=−N|B|N+|B|Attimet=0,N=0thusthesolutionattimet=1isafunctionof|B|(thuscommutingwith|B|)givenbyN=tanh|B|ThisimpliesthatM=Utanh|B|=U|W|=W 3.4RepresentationoftheQuantumPropagatorinTermsoftheGenerator77Nowconsiderthedifferentialequationsatisfiedbya(t).Weget2a˙=−aTr(B∗M)=−aTr|B|U∗UN=−aTr|B|tanh|B|Sincea(0)=(2π)−n/2weget1ta(t)=(2π)−n/2exp−dsTr|B|tanhs|B|20−n/21=(2π)exp−Trlogcosht|B|2−n/2−1/2−n/221/4a(1)=(2π)detcosh|B|=(2π)det1−|W|wherewehaveusedthateTrA=deteA.Thusthesqueezedstateψ(B)=D(B)ϕ0hasthefollowingFock–Bargmannrep-resentation(B)−n/2−1/21ψ(ζ)=(2π)detcosh|B|expζ·Wζ(3.66)2NowusingtheinverseFock–Bargmanntransformwegobacktothecoordinaterepresentationofψ(B).Onehastocomputethefollowingintegral:x2√ζ¯21dqdpexp−−x2ζ¯+−ζ·ζ¯+ζ·Wζ(3.67)222Theargumentoftheexponentialcanberewrittenasx212ip·qp2122−−q−++x·(q+ip)−q+p242421+(q·Wq−ip·Wq−iq·Wp−p·Wp)4x2311=−+x(q+ip)−q2−p2+(q·Wq−ip·Wq−iq·Wp−p·Wp)2444Thusitappearsaquadraticforminq,pwhichcanbewritteninmatrixformas1q−(q,p)M2pwith131−W−i(W+1)M=2−i(W+1)W+1Onehas−111iM=−12i(31−W)(1+W) 783TheQuadraticHamiltoniansThustheintegraloverq,pin(3.67)willbetheexponentialofthefollowingquadraticform:1xx2−(x,ix)M−1−(3.68)2ix2withaphasewhichisexactly−1/2n/2−1/2det(M/2π)=(2π)det(1+W)Itiseasytocomputetheexpressionin(3.68).Itgives1−1−x·(1−W)(1+W)x2Thusrestoringthedependenceandthefactorsπwegetthefollowingresult.Proposition36Thesqueezedstateψ(B)isactuallyaGaussianinthepositionrep-resentationgivenby(B)iψ(x)=aΓexpx·Γx(3.69)2withΓ=i(1−W)(1+W)−1and−n/22−1/2−1/2aΓ=(π)det1−|W|(1+W)3.5RepresentationoftheWeylSymboloftheMetaplecticOperatorsSeeChap.2forthedefinitionsofcovariantandcontravariantWeylsymbols.WehaveshownthatR(F)=U1whereUtisthequantumpropagatorofthequadraticHamiltonian,1QH(t)=Q,PMt(3.70)2P−1,FwithMt=−JF˙tFttbeingacontinuouspathinthespaceSp(n)joining1att=0toFatt=1.In[52]theauthorsshowthefollowingresult. 3.5RepresentationoftheWeylSymboloftheMetaplecticOperators79Theorem18(i)Ifdet(1+F)=0thecontravariantWeylsymbolofR(F)hasthefollowingform:iπνdet(1+F)−1/2−1R(F,X)=eexp−iJ(1−F)(1+F)X·X(3.71)whereν∈Zifdet(1+F)>0andν∈Z+1/2ifdet(1+F)<0.(ii)Ifdet(1−F)=0thecovariantWeylsymbolofR(F)hasthefollowingform:#iπμdet(1−F)−1/2i−1R(F,X)=eexp−J(1+F)(1−F)X·X(3.72)4whereμ=¯ν+nandν¯∈Z.2ThisformulahasbeenheuristicallyproposedbyMehligandWilkinson[143]withoutthecomputationofthephase.Seealso[61].WecanrestorethedependenceofR(F,X)andR#(F,X)byputtingafactor−1intheargumentoftheexponentials.ProofLetusstatethefollowingpropositionwhichisadirectconsequenceof(3.39)afteralgebraiccomputations.Proposition37ThematrixelementsofR(F)oncoherentstatesϕz,aregivenbythefollowingformula:R(F)ϕn−1/2ϕz+Xz=2det1+F+iJ(1−F)2X1X−iJX×exp−z++iσ(X,z)+KFz+222X−iJX×z+(3.73)2where−1KF=(1+F)1+F+iJ(1−F)(3.74)NowwecancomputethedistributioncovariantsymbolofR(F)bypluggingformula(3.73)intoformula(2.29).Letusbeginwiththeregularcasedet(1−F)=0andcomputethecovariantsymbol.UsingProposition37andformula(3.39),wehavetocomputeaGaussianintegralwithacomplex,quadratic,non-degeneratecovariancematrix(see[117]).ThiscovariancematrixisKF−1andwehaveclearly−1−1KF−1=−iJ(1−F)1+F+iJ(1−F)=−(1−iΛ) 803TheQuadraticHamiltonianswhereΛ=(1+F)(1−F)−1Jisarealsymmetricmatrix.Sowehave2−12−1(KF−1)=−1+Λ,(KF−1)=−Λ1+Λ(3.75)Sothat1−KFisintheSiegelspaceΣ2nandTheorem7.6.1of[117]canbeapplied.Theonlyseriousproblemistocomputetheindexμ.Letusdefineapathof2n×2nsymplecticmatricesasfollows:Gt=etπJ2nifdet(1−F)>0,andGt=G2t⊗etπJ2n−2ifdet(1−F)<0,whereη(t)0G2=01η(t)whereηisasmoothfunctionon[0,1]suchthatη(0)=1,η(t)>1on]0,1]andwhereJ2nisthe2n×2nmatrixdefiningthesymplecticmatrixontheEuclideanspaceR2n.G1andFareinthesameconnectedcomponentofSp(2n)whereSp(2n)={F∈Sp(2n),det(1−F)=0}.Sowecanconsiderapaths→FsinSp(2n)suchthatF=G1andF=F.01Letusconsiderthefollowing“argumentofdeterminant”functionsforfamiliesofcomplexmatrices:θ[Ft]=argcdet1+Ft+iJ(1−Ft)(3.76)β[F]=argdet(1−K−1+F)(3.77)whereargcmeansthatt→θ[Ft]iscontinuousintandθ[1]=0(F0=1),andS→arg+[det(S)]istheanalyticdeterminationdefinedontheSiegelspaceΣ2nsuchthatarg+[det(S)]=0ifSisreal(see[117],vol.1,Sect.3.4).Withthesenotationswehaveβ[F]−θ[F]μ=(3.78)2πLetusconsiderfirstthecasedet(1−F)>0.UsingthatJhasthespectrum±i,wegetdet(1+Gt+iJ(1−Gt))=4nentπiand1−KG=1.1Letusremarkthatdet(1−KF)−1=det(1−F)−1det(1−F+iJ(1+F)).Letusintroduce(E,M)=det(1−E+M(1+E))forE∈Sp(2n)andM∈sp+(2n,C).LetconsidertheclosedpathCinSp(2n)definedbyadding{Gt}0≤t≤1and{Fs}0≤s≤1.Wedenoteby2πν¯thevariationoftheargumentfor(•,M)alongC.Thenwegeteasilyβ(F)=θ[F]+2πν¯+nπ,n∈Z(3.79)Whendet(1−F)<0,byanexplicitcomputation,wefindarg+[det(1−KG1)]=0.Sowecanconcludeasabove.Theformulaforthecontravariantsymbolcanbeeasilydeducedfromthecovari-antformulausingasymplecticFouriertransform. 3.6Traps81Remark13WhenthequadraticHamiltonianHistimeindependentthenFt=etJS,Sisasymmetricmatrix.Soifdet(etJS+1)=0,thenwegetapplying(3.71),−1/2ttRetJS,X=eiπνdetcoshJSexpiσtanhJSX·X(3.80)22Thisformulawasobtainedin[118].InparticulartheMehlerformulafortheHar-monicoscillatorisobtainedwithS=1.In[61]theauthordiscusstheMaslovindexrelatedwiththemetaplecticrepre-sentation.Remark14Inthepaper[52]theauthorsgiveadifferentmethodtocomputethecon-travariantWeylsymbolR(F,X)inspiredby[76].TheyconsiderasmoothfamilyFtoflinearsymplectictransformationsassociatedwithafamilyoftime-dependentquadraticHamiltoniansHt.AfterquantizationwehaveaquantumpropagatorUtwithitscontravariantWeylsymbolUtw(X).WemaketheansatzUtw(X)=αteX·MtXwhereαtisacomplexnumber,Mtisasymmetricmatrix.UsingtheSchrödingerequation∂tUtw=HtUtw,andtheMoyalproduct,wefindforMtaRiccatiequa-tionwhichissolvedwiththeclassicalmotion.Afterwards,αtisfoundbysolvingaLiouvilleequationhencewerecoverthepreviousresults(see[52]fordetails).Thisapproachwillbeadaptedlaterinthisbookinthefermionicsetting.3.6TrapsWenowgiveanapplicationinphysicsofourcomputationsconcerningquadraticHamiltonians.Thequantummotionofanioninaquadrupolarradio-frequencytrapissolvedexactlyintermsoftheclassicaltrajectories.ItisproventhatthequantumstabilityregionscoincidewiththestabilityregionsoftheassociatedMathieuequation.Byquantumstabilitywemeanthatthequantumevolutionoveroneperiod(theso-calledFloquetoperator)hasonlypure-pointspectrum.Thusthequantummotionis“trapped”inasuitablesense.WeexhibitthesetofeigenstatesoftheFloquetoperator.3.6.1TheClassicalMotionLetusconsiderathree-dimensionalHamiltonianofthefollowingform:p2e2122H(t)=+z−x+yV1−V0cos(ωt)(3.81)2mr220Herep=(px,py,pz)isthethree-dimensionalmomentum,mthemassoftheion,andr=(x,y,z)isthethree-dimensionalposition.r0isthesizeofthetrapande 823TheQuadraticHamiltoniansthechargeoftheelectron.V1(resp.V0)istheconstant(resp.alternating)voltage.Suchtime-periodicHamiltoniansarerealizedbyusingtrapsthatarehyperboloidsofrevolutionalongthez-axisthataresubmittedtoadirectcurrentplusanalternatingcurrentvoltage.AlsoknownasPaultrapstheyallowtoconfineisolatedionslikecesiumforratherlongtimes,andalsoafewionstogetherinthesametrap.ItisobvioustoseethatthisHamiltonianispurelyquadraticandthatitdecouplesintothreeone-dimensionalHamiltonians:H(t)=hx(t)⊗1⊗1+1⊗hy(t)⊗1+1⊗1⊗hz(t)p2x2p2wherehx(t)=hy(t)=x−(α−βcos(ωt)),hz(t)=z+z2(α−βcos(ωt))2m22meVewithα=21,β=2V0.rr00Thusx(t),y(t),z(t)evolveaccordingtotheMathieuequations:x(t)¨−x(t)α/m−cos(ωt)β/m=0,z(t)¨+z(t)2α/m−cos(ωt)2β/m=0(3.82)Itisknownthateachoftheseequationshavestabilityregionsparametrizedby(α,β,ω),inwhichthemotionremainsbounded.FurthermoreitcanbeshownthatitisquasiperiodicwithFloquetexponentρ.See[142].Theorem19Thereexistρ,ρ∈Randrapidlyconvergingsequencescn,cn∈Rdependingona=4α/mω2,b=2β/mω2suchthatthesolutionof(3.82)withx(0)=u,x(t)˙=v/m,z(0)=u,z(˙0)=v/maregivenbyu+∞ωt2v+∞ωtx(t,u,v)=cncos(2n+ρ)+cnsin(2n+ρ)c2mωd2−∞−∞u+∞ωt2v+∞ωtz(t,u,v)=ccos(2n+ρ)+csin(2n+ρ)cn2mωdn2−∞−∞wherec=Zcn,d=Z(2n+ρ)cn,andsimilarlyforc,d.NotethattheFloquetexponentρisthesameforx,ybutρ=ρ.TheproofdependsheavilyonthelinearityofMathieu’sequations.NotethatthestabilityregionsaredelimitedbythecurvesCjforwhich(3.82)hasperiodicsolu-tions,i.e.forwhichρ=j∈N.Forgivenα,β,thereexistsω1,ω2∈R+suchthatforanyω∈]ω1,ω2[theclassicalequationsofmotion(3.82)havestablesolutions.ω1hastobelargeenoughhencethename“radio-frequencytraps”.3.7TheQuantumEvolutionSinceionsareactuallyquantumobjectsitisrelevanttoconsidernowthequantumproblem.AsknownfromthegeneralconsiderationsofthisChapteronquadratic 3.7TheQuantumEvolution83HamiltoniansthequantumevolutionforHamiltoniansoftheform(3.81)iscom-pletelydeterminedbythequantummotion.TheHilbertspaceofquantumstatesisH=L2(R3).Onethusfindsthatthetime-periodicHamiltonian22x2+y2H(t):=−Δ+α−βcos(ωt)z−2m2whereΔisthe3-dimensionalLaplacianandgeneratesanunitaryoperatorU(t,s)thatevolvesaquantumstatefromtimestotimet.TheFloquetoperatoristheoperatorontimeevolutionoveroneperiodT=2π/ω:U(T,0)=UUUx(T,0)y(T,0)z(T,0)WeshalldenotebyUF(resp.U)theoperatorUx(T,0)(resp.Uz(T,0)).ThenoneFhasthefollowingresults:Theorem20(i)Givenα,β∈Rthereexistsω1,ω2∈R+asintheprevioussectionsuchthatforanyψ∈Handany>0thereexistsR∈R+suchthatsupF|r|>RU(t,s)ψ<tF(|r|>R)beingthecharacteristicfunctionoftheexterioroftheball|r|≤R.(ii)Forα,β,ωasabove,UFhaspure-pointspectrumoftheform{exp(−iρπ(k+1/2))}k∈NwhereρisasintheprecedingsectiontheclassicalFloquetexponent.SimilarlyforU.FRemark15Acompleteproofcanbeseenin[47].(i)saysthatthetimeevolutionofanyquantumstateremainsessentiallylocalizedalongthequantumevolution.ProofToprove(i)itisenoughtostatetheresultinonedimension,andtoas-sumem==1.Weestablishtheresultforψ∈C∞whichisadenseset.0LetW(−2x,ξ,t)betheWignerfunctionofthestateψ(t):=U(t,0)ψ.ThenW(x,.,t)∈L1(R),∀t∈R,∀x∈R.Furthermorewehave2xψt,−=dξW(x,ξ,t)2Itfollowsthat2F|x|>Rψ(t)=dxdξW(x,ξ,t)=dxdξWx(t),x(t),˙0|x|>2R|x|>2Rwherex(t)isasolutionof(3.82)withx(0)=x,x(˙0)=ξ.Thenweperformachangeofvariableswithuniform(int)Jacobian,thelinearityofMathieu’sequation 843TheQuadraticHamiltoniansandtheboundednessintimeofitssolutioninthestableregion,togetherwiththefactthatW(x,ξ,t)∈L1(R2)toconclude.AnextensionofTheorem20toHermite-likewavefunctionsisthefollowing:define∗k/2√2−1/2LtiNtxΦk(t,x)=(Lt)Hkx/|Lt|expLt2LtwhereHkarethenormalizedHermitepolynomialsandthedeterminationofthesquare-rootisfollowedbycontinuityfromt=0.WehaveU(t,0)Φk(0,.)=Φk(t,.),∀t∈RThiscanbeprovenbyusingtheTrotterproductformula,asteplikeapproximationfN(t)ofthefunctionf(t)=α−βcos(ωt)andthecontinuityofsolutionsof(3.82)whenfN→f.See[47]and[99,100]fordetails.ThenormalizedeigenstatesofUFaregeneralizedsqueezedstates.Letusassumem=1forsimplicity.LetF(t)bethesymplectic2×2matrixsolutionofF˙=JMFwhere10M(t)=0f(t)f(t)=α−βcos(ωt)AtBtF(t)=CtDtg0WechoosesuitableinitialdataF(0)=−1with0g−1/2ωdg=2cWedefineLt=At+iBt,Nt=Ct+iDt.Fortheseinitialdata,wehaveg+∞1+∞Liρωt/2inωtiρωt/2inωtt=ecne,Nt=ecnecgd−∞−∞Itisclearthat|LNtt|,areT-periodicwhereT=2π/ω,andfurthermoreLt∗k/2i(k+1/2)ρωt−1/2Lte2(Lt)Lt 3.7TheQuantumEvolution85isalsoT-periodic.Thenweintroducetheself-adjointquasi-energyoperator∂K=−i+hx(t)∂tactingintheHilbertspace2(R)⊗L2(TK=Lω)offunctionsdependingonbothxandt,T-periodicint.ThisformalismhasbeenintroducedbyHowland[119]andYajima[204].Kiscloselyrelatedtothequantumevolutionoperator.Wehavethefollowingresult:Lemma26(i)AssumeΨ∈KisaneigenstateofKwitheigenvalueλ.Thenforanyt∈Tω,Ψ(t,.)∈L2(R)andsatisfiesU(t+T,t)Ψ(t)=e−iλT/Ψ(t)(ii)Converselyletψ∈L2(R)satisfyU(T,0)ψ=e−iTλ/ψThenΨ=eiλt/U(t,0)ψ∈KandsatisfiesKΨ=λΨProofDefine(k+1/2)ρωt/2ΦΨk(t,x)=ek(x)ThenusingtheperiodicitypropertieswehaveΨk∈Kand1U(T,0)Φk(0,.)=exp−iπρk+Φk(0,.)2andthus1ρωKΨk=k+Ψk22 Chapter4TheSemiclassicalEvolutionofGaussianCoherentStatesAbstractInthisChapterweconsidersemiclassicalasymptoticsofthequantumevolutionofcoherentstatesatanyorderinthePlanckconstant.Weconsideracon-trolintimeoftheremaindertermdependingexplicitlyonandonthestabilitymatrix.WefindthatthequantumevolvedcoherentstateisinL2-normwellapprox-imatedbyasqueezedstatelocatedaroundthephase-spacepointztoftheclassicalflowreachedattimet,withadispersioncontrolledbythestabilitymatrixatpointzt.TheideagoesbacktoHepp(Commun.Math.Phys.35:265–277,1974)andwasfurtherdevelopedbyG.Hagedorn(Ann.Phys.135:58–70,1981;Ann.Inst.HenriPoincaré42:363–374,1985).Themethodthatwedevelopherefollowsthepaper(CombescureandRobertinAsymptot.Anal.14:377–404,1997)whereweconsidergeneraltime-dependentHamiltoniansandusethesqueezedstatesformalismandthemetaplectictransformation(seeChap.3).ThedifferencebetweentheexactandthesemiclassicalevolutionisestimatedintimetandinthesemiclassicalparametergivinginparticularthewellknownEhrenfesttimeoforderlog(−1).Wethenprovidetwoapplicationsofthesemiclassicalestimates:thefirstonecon-cernsthesemiclassicalestimateofthespreadingofquantumwavepacketswhicharecoherentstatesintermsoftheLyapunovexponentsoftheclassicalflow.Thesec-ondapplicationistothescatteringtheoryforgeneralshortrangeinteractions:thenthelargetimeasymptoticscanbecontrolledandthequantumscatteringoperatoractsoncoherentstatesfollowingtheclassicalscatteringtheorywithgoodestimatesin.MoreaccurateestimatescanbeobtainedusingtheFourier–Bargmanntrans-form(RobertinPartialDifferentialEquationsandApplications,2007).WeconsiderGevreytypeestimatesforthesemiclassicalcoefficientsand-exponentiallysmallremainderestimatesinSobolevnormsforsolutionsoftime-dependentSchrödingerequations.4.1GeneralResultsandAssumptionsWeshallconsiderthequantumHamiltonianH(t)ˆofapossiblytime-dependentproblem.Weassumethatthecorrespondingtime-dependentSchrödingerequationdefinesauniquequantumunitarypropagatorU(t,s).Thenweconsiderthecanoni-calGaussiancoherentstatesϕzwherez=(q,p)∈R2nisaphase-spacepoint.ThenweletitevolvewiththequantumpropagatorthatisweconsiderthequantumstateM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,87TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_4,©SpringerScience+BusinessMediaB.V.2012 884TheSemiclassicalEvolutionofGaussianCoherentStatesattimetdefinedbyΨz(t):=U(t,0)ϕzSemiclassically(whenissmall)Ψ(z,t,)iswellapproximatedinL2-normbyasuperpositionof“squeezedstates”centeredaroundthephase-spacepointzt=Φ(t)zoftheclassicalflow.ThisstudygoesbackinthepioneeringpaperbyHepp[113]andwaslaterdevelopedbyG.Hagedorninaseriesofpapers[99,100].In[49]anapproachisdevelopedconnectingthesemiclassicalpropagationofcoherentstatestotheso-calledsqueezedstates.Wedevelophereageneralizationoftheseresults,followingthepaper[51],allowinggeneraltime-dependentHamiltoniansandestimatingtheerrortermwithrespecttotimet,tozandto.TheknowledgeofthetimeevolutionforanyGaussianϕzisawaytogetmanypropertiesforthefullpropagatorU(t,0)(wecanalwaysassumethattheinitialtimeis0).Thisiseasytounderstandusingthatthefamily{ϕz,z∈R2n}isovercomplete(Chap.1).LetusdenotebyKt(x,y)theSchwartz-distributionkernelofU(t,0).Fromovercompletenesswegetthefollowingformula:K−nt(x,y)=(2π)dzU(t,0)ϕz(x)ϕz(y)(4.1)R2nThisequalityholdsasSchwartzdistributionsonR2nandexplainswhyitisveryusefultosolvetheSchrödingerequationwithcoherentstateϕzasinitialstate:i∂tΨz(t)=H(t)Ψˆz(t),Ψz(0)=ϕz(4.2)Severalapplicationswillbegivenlateraswellfortime-dependentandtime-independentSchrödingerequationstype.4.1.1AssumptionsandNotationsLetH(t)ˆbeaself-adjointSchrödingerHamiltonianinL2(Rn)obtainedbyquan-tizingageneraltime-dependentsymbolH(x,ξ,t)calledclassicalHamiltonian.Weusethe-Weylquantization(seeChap.2).HisassumedtobeaC∞-smoothfunc-tionforx∈Rn,ξ∈Rn,t∈]−T,T[0≤T≤+∞satisfyingaglobalestimate:(A.0)Thereexistsomenonnegativeconstantsm,M,KH,Tsuchthat22−M/2∂γγ≤K1+|x|+|ξ|x∂ξH(x,ξ,t)H,Tuniformlyin(x,ξ)∈R2n,t∈]−T,T[for|γ|+|γ|≥m.SoHmaybeaverygeneralHamiltonianincludingtime-dependentmagneticfieldsornonEuclideanmetrics.WefurthermoreassumeH(x,ξ,t)tobesuchthattheclas-sicalandquantumevolutionsexistfromtime0totimetfortinsomeinterval]−T,T[whereT<+∞orT=+∞.Moreprecisely: 4.1GeneralResultsandAssumptions89(A.1)Givensomez=(q0,p0)∈R2nthereexistsapositiveTsuchthattheHamil-tonequations∂H∂Hq˙t=(qt,pt,t),p˙t=−(qt,pt,t)∂p∂qhaveauniquesolutionforanyt∈]−T,T[startingfrominitialdataz:=(q0,p0).Wedenotezt=(qt,pt):=Φ(t)zthephase-spacepointreachedattimetstartingbyzattime0.(A.2)ThereexistsauniquequantumpropagatorU(t,s),(t,s)∈R2withthefol-lowingproperties:(i)U(t,s)isunitaryinL2(Rn)withU(t,s)=U(t,r)U(r,s),∀(r,s,t)∈R3(ii)U(.,.)isastronglycontinuousoperator-valuedfunctionfortheoperatornormtopologyinL2(Rn).TheusualnorminL2(Rn)isdenotedby..(iii)Let2nβα22B(k)=u∈LR:x∂xu:=uB(k)<∞|α|+|β|≤kandletB(−k)bethestandarddualspaceofB(k).Thenweassumethatthereexistssomek∈Nsuchthatforanyψ∈L2(Rn)andanys∈[−T,T]U(t,s)ψisB(−k)-valuedabsolutelycontinuousintandsatisfiesthetime-dependentSchrödingerequation∂iU(t,s)ψ=H(t)U(t,s)ψˆ∂tinB(−k)atalmosteveryt∈]−T,T[.IfHisindependentontimetsatisfying(A.0),(A.1)issatisfiedifthetrajectoryisonacompactenergylevel:H(qt,pt)=EwithH−1(E)boundedinRnq×Rnpand(A.2)issatisfiedifHˆisself-adjoint.ButifHdependsontimet,nogeneralconditionsareknownthatensureproperties(A.1),(A.2)tobetrue.However,weshallindicateintheusualSchrödingercasewithtime-dependentpotentialsorintheSchrödingercasewithtime-dependentelectricandmagneticfieldssomegeneraltechnicalconditionsprovidedbyYajima[202,205]suchthat(A.2)holdstruefork=2:1ξ2+V(t,x),I(1)LetH(x,ξ,t)=T:=[−T,T]and2∂VV∈CIT,Lp2Rn+CIT,L∞Rn,∈Lp1,α1(IT)+L∞,β(IT)∂twhereβ>1,p2=Max(p,2),p1=2np/(n+4p)ifn≥5,p1>2p(p+1)ifp=4andp1=2p/(p+1)ifn≤3,α1>4p/(4p−n)andp/mm,pu(t,x)mL(I)=u:dtdx<∞IRn 904TheSemiclassicalEvolutionofGaussianCoherentStates(NotethatV(t,x)canbelessregularinthetimevariableifitismoreregularinthespacevariables.)Thenproperty(A.2)issatisfiedfors,t∈IT×ITandforanyquantizationofH(t).(2)LetH(x,ξ,t)=1(ξ−A(t,x))2+V(t,x)whereV(t,x)andA(t,x)=2{Aj(t,x)}j=1,...,naretheelectricandmagneticvectorpotentials.IfB(t,x)isthestrengthtensorofthemagneticfieldi.e.theskew-symmetricmatrix∂Ak∂AjBj,k=−∂xj∂xkweassumethefollowing:(i)Aj:Rn+1→Rissuchthatforanymultiindexα,∂xαAjisC1in(t,x)∈Rn+1.(ii)Thereexistsε>0suchthat∂α≤C−1−εxB(t,x)α1+|x|,|α|≥1∂αA(t,x)+∂α∂tA(t,x)≤Cn+1xxα,|α|≥1,(t,x)∈R(iii)V:Rn+1→RbelongstoLp,α(R)+L∞(R)forsomep>n/2withp≥1andα=2p/(2p−n).InChap.1wehavedescribedtheconstructionofstandardcoherentstatesbyapplyingtheWeyl–HeisenbergoperatortothegroundstateΨ0ofthen-dimensionalharmonicoscillatorwithHamiltonian1K0:=Pˆ2+Qˆ22(4.3)ϕz=T(z)Ψˆ0InChap.3wehavecomputedanexplicitformulaforthetimeevolutionofcoherentstatesdrivenbyanyquadraticHamiltonianin(q,p).WeshallnowdefinegeneralizedcoherentstatesbyapplyingT(z)ˆtotheex-citedstatesΨνofthen-dimensionalharmonicoscillator;givenamultiindexν=(ν1,...,νn),Ψνisthenormalizedeigenstateof(4.3)witheigenvalue|ν|+n/2.Werecallthenotationn|ν|=νjj=1Wethusdefine(ν)=T(z)ΨˆΨzν(0)andϕzissimplyΨz.(ν)SimilarlywedefinegeneralizedsqueezedstatesΦcenteredaroundthephase-z,Bspacepointz.LetWbeasymmetricn×nmatrixwithpolardecompositionW=U|W|where|W|=(W∗W)1/2andUaunitaryn×nmatrix.WeassumethatW∗W≤1 4.1GeneralResultsandAssumptions91anddefineB:=UArgtanh|W|(4.4)NowasinChap.3weconstructtheunitaryoperatorD(B)ˆinL2(Rn)as1D(B)ˆ=expa∗·Ba∗−a·B∗a2a∗andabeingthecreationandannihilationoperators.Wedefine(ν)Φ:=T(z)ˆD(B)Ψˆνz,BOfcoursewehave(ν)(B)Φ=ψ0,Bwhereψ(B)isthestandardsqueezedstate,usingthenotationsofChap.2and(0)Φ=ϕzz,0usingthenotationofChap.1.4.1.2TheSemiclassicalEvolutionofGeneralizedCoherentStatesInthissectionweconsiderthequantumevolutionofsuperpositionsofgeneralized(ν)coherentstatesoftheformΨzandproveundertheaboveassumptionsthat,uptoanerrortermwhichcanbecontrolledint,z,,itiscloseinL2-normtoasuper-(ν)positionofsqueezedstatesoftheformΦwherezt:=Φ(t)zisthephase-spacezt,BtpointreachedattimetbytheclassicalflowΦ(t)ofH(t),andBtiswell-definedthroughthelinearstabilityproblematpointzt.Wefollowtheapproachdevelopedin[51]whereweuse:–thealgebraofthegeneratorsofcoherentandsqueezedstates–theso-calledDuhamelprinciple,whichisnothingbutthefollowingidentity:t1U1(t,s)−U2(t,s)=dτU1(t,τ)Hˆ1(τ)−Hˆ2(τ)U2(τ,s)(4.5)iswhereUi(t,s)isthequantumpropagatorgeneratedbythetime-dependentHamiltonianHˆi(t),i=1,2.WetakeHˆ1(t)=H(t)ˆandHˆ2(t)tobethe“Tay-lorexpansionuptoorder2”ofH(t)ˆaroundtheclassicalpathzt.Moreprecisely:∂H∂HHˆ2(t)=H(qt,pt,t)+Qˆ−qt·(qt,pt,t)+Pˆ−pt·(qt,pt,t)∂q∂p1Qˆ−qt+Qˆ−qt,Pˆ−ptMt(4.6)2Pˆ−pt 924TheSemiclassicalEvolutionofGaussianCoherentStatesMtbeingtheHessianofH(t)computedatpointzt=(qt,pt):∂2HMt=(4.7)∂z2z=ztTheinterestingpointisthatsinceHˆ2(t)isatmostquadratic,itsquantumpropa-gatoriswrittenuniquelythroughthegeneratorsofcoherentandsqueezedstates.InChap.3wehaveshownthelinkbetweenthequantumpropagatorofpurelyquadraticHamiltoniansandthemetaplectictransformations.ItisshownthatthequantumpropagatorofpurelyquadraticHamiltonianscanbedecomposedintoaquantumrotationtimesasqueezinggenerator.Letusbemoreexplicit:LetHˆQ(t)beapurelyquadraticquantumHamiltonianoftheformQˆHˆQ(t)=Q,ˆPˆS(t)PˆwithS(t)a2n×2nsymmetricmatrixoftheformGtL˜tS(t)=LtKtwhereGt,KtaresymmetricandL˜denotesthetransposeofL.InwhatfollowsS(t)willbesimplyMt.LetF(t)bethesymplecticmatrixsolutionofF(t)˙=JMtF(t)(4.8)withinitialdataF0=1,where01J=−10Ithasthefour-blockdecompositionA(t)B(t)F(t)=C(t)D(t)InChap.3ithasbeenestablishedthatthequantumpropagatorUq(t,0)ofHˆQ(t)isnothingbutthemetaplecticoperatorR(F(t))ˆassociatedtothesymplecticma-trixF(t).ItimpliesthattheHeisenbergobservablesQ(t)ˆ,P(t)ˆobeytheclassicalNewtonequationsfortheHamiltonianHˆQ(t)asexpected.ThereforewehaveLemma27QˆQˆUq(0,t)Uq(t,0)=F(t)PˆPˆPassingfromthe(Q,ˆP)ˆrepresentationtothe(a∗,a)representationweeasilyget 4.1GeneralResultsandAssumptions93Lemma28−1a∗1YtZ¯ta∗Uq(t)Uq(t)=a2ZtY¯taFurthermoreonecanshowthatR(F(t))ˆdecomposesintoaproductofarotationpartanda“squeezing”part.Definingthecomplexmatrices:Yt=A(t)+iB(t)−iC(t)+iD(t),Zt=A(t)+iB(t)+iC(t)+iD(t)(4.9)wehavethefollowingidentity:Z∗Z=Y∗Y−41andY−1tisinvertible.ThematrixWt=ZtYtissuchthatW∗Wtt<1,W0=0FurthermoreithasbeenshowninChap.3(Lemmas21and23)thatitisasymmetricmatrix.ThusonecandefinethematrixBtaccordingto(4.4);notethatBtisnottobeconfusedwiththematrixB(t)ofthefour-blockdecompositionofF(t).IntroducingthepolardecompositionofYt:Y∗t=|Yt|Vtwhere|Y|2=YY∗weseethatVtisasmoothfunctionoftandwecandefine(atleastlocallyintime)asmoothself-adjointmatrixΓtbyVt=exp(iΓt)LetR(t)ˆbethefollowingunitaryoperatorinL2(Rn)(metaplectictransformation): i∗0Γ˜ta∗R(t)ˆ=expa,a2Γt0aIthasthefollowingproperty:Lemma29a∗−1Vta∗R(t)ˆR(t)ˆ=a(V˜t)∗aRemark16R(t)ˆwillbetherotationpartofthemetaplectictransformationUq(t,0)whileD(Bˆt)willbethesqueezingpart. 944TheSemiclassicalEvolutionofGaussianCoherentStatesOnehasthefollowingproperty:Proposition38Thequantumpropagatorsolvingthetime-dependentSchrödingerequationdiUq(t,s)=HˆQ(t)Uq(t,s),U(s,s)=1dtisgivenbyUq(t,s)=D(Bˆt)R(t)ˆR(s)ˆ−1D(ˆ−Bs)DuetothechainruleitisenoughtoshowthatUq(t,0)=D(Bˆt)R(t)ˆThedetailedproofcanbefoundinthepaper[51].NowwecanderiveanexplicitformulaforthequantumpropagatorofHˆ2(t).LetSt(z)betheclassicalactionalongthetrajectoryfortheclassicalHamilto-nianH(x,ξ,t)startingatphase-spacepointz=(q,p)attime0andreachingzt=(qt,pt)attimet:tSt(z)=dsx˙s·ξs−H(xs,ξs,s)0anddefineqt·pt−q·pδt=St(z)−2Thefollowingresultholdstrue:Proposition39LetU2(t,s)bethequantumpropagatorfortheHamiltonianHˆ2(t)givenby(4.6).ThenwehaveU2(t,s)=expi(δt−δs)/T(zˆD(BˆR(t)ˆR(s)ˆ−1D(ˆ−BT(ˆ−zt)t)s)s)(4.10)q+ipProofHereweusethenotationz:=√.2UsingtheBaker–Campbell–Hausdorffformula(andomittingtheindextinthefollowingformulas)wegetd1T(z)ˆ=T(z)ˆz˙¯·a∗−˙z·a+z·z˙¯−˙z·¯zdt2But1i1z·z˙¯−˙z·¯z=(p˙·q−˙q·p),andiδ˙=iS˙−(p·q+˙p·q)222 4.1GeneralResultsandAssumptions95sothattakingthederivativeof(4.10)withrespecttotimetweget∂U211i=−S˙+(p·˙q+˙p·q)−(p˙·q−˙q·p)∂t22+˙q·Pˆ−p−˙p·Qˆ−qU2+T(zˆt)Hq(t)T(ˆ−zt)U2=q˙·p−S˙+˙q·Pˆ−p−˙p·Qˆ−q 1Qˆ−q+Qˆ−q,Pˆ−pMtU22Pˆ−p=Hˆ2(t)U2(4.11)wherewehaveusedthatq˙·p−S˙=H(q,p,t).TheimportantfactherewillbethatU2propagatescoherentstatesintoWeyltranslatedsqueezedstatessothatusing(4.5)wegetacomparisonbetweenthequantumevolutionofcoherentstatesandtheWeyl-displacedsqueezedstatecen-teredaroundthephase-spacepointzt.Considerasaninitialstateacoherentstate(0)Φ=T(z)Ψˆ0.Wegetz,0U2(t,0)Φ(0)=eiδt/T(zˆD(BˆR(t)Ψˆi(δt/+γt)Φ(0)z,0t)t)0=ezt,Btusingthefactthat1R(t)Ψˆ0=exp(iγt)Ψ0,whereγt=tr(Γt)2(0)ThereforeapplyingDuhamel’sformula(4.5)toΦwegetz,0t(0)i(δt/+γt)(0)1H(s)ˆ−Hˆi(δs/+γs)(0)U(t,0)Φz,0−eΦzt,Bt=dsU(t,s)2(s)eΦzs,Bsi0(4.12)Thiswillbethestartingpointofoursemiclassicalestimate.Takinganarbitrarymultiindexμ=(μ1,...,μn)wedenoteΦμ(t):=T(zˆt)D(Bˆt)R(t)ΨˆμStartingfrom(4.12)wededucethatforanyintegerl≥1thereexistindexedfunc-tionscν(t,)suchthat3(l−1)iδt/l/2U(t,0)Φ0(0)−cν(t,)Φν(t)e≤Ct(4.13)|ν|=0 964TheSemiclassicalEvolutionofGaussianCoherentStatesFurthermoretheconstantCtcanbecontrolledinthetimetandinthecenterzoftheinitialstate.LetusnowexplicittheTaylorexpansionoftheHamiltonianaroundtheclassicalphase-spacepointattimetzt:wedenotef(ν)(ζ)2n1∂νjf(ζ)ννj·ζ=νjζj,ν=(ν1,...,ν2n)ν!νj!∂ζ1jforfbeingarealfunctionofζ∈R2n.Takingforζthephase-spacepointzwecanwritetheTaylorexpansionofH(q,p,t)aroundthephase-spacepointzt=(qt,pt)attimetasl+1H(ν)(zt,t)νH(ζ,t)=·(ζ−zt)ν!|ν|=01H(ν)(zt+θ(ζ−zt),t)νl+1+·(ζ−zt)(1−θ)dθ0(ν−1)!|ν|=l+2l+1H(ν)(zt,t)ν+rν=·(ζ−zt)ν,t(ζ−zt,t)·(ζ−zt)(4.14)ν!|ν|=0|ν|=l+2Nowweperformthe-quantizationof(4.6)denotingΩ:=(Q,ˆP)ˆ;wegetl+1Hν(zt,t)H(t)ˆ−Hˆ·(Ω−zt)ν+Rν(t)(4.15)2(t)=ν!|ν|=3|ν|=l+2whereRν(t)=T(ˆ−zt)Opwζν·rν,t(ζ)T(zˆt)√Wenowinsert(4.15)into(4.12)andobtainthesemiclassicalestimateoforderforthepropagationofgeneralizedcoherentstates,usinginparticulartheCalderon–Vaillancourtestimate(Chap.2).Letusintroducesomenotation:weconsideronlynonnegativetimeanddefineσ(z,t):=sup1+|zt|0≤s≤t∗1/2θ(z,t)=suptrF(s)F(s)0≤s≤twhereF(t)isthesymplecticmatrixsolutionof(4.8).LetM1beanyfixedintegernotsmallerthan(M+(m−2)+)/2.Letusdefinej√lM1|t|2j+lρl(z,t,)=σ(z,t)θ(z,t)1≤j≤l 4.1GeneralResultsandAssumptions97(werecallthattheconstantsM,m,KH,TweredefinedinAssumption(A.0)).Notethatif(A.0)issatisfiedwithm=2andM=0(H(t)issaidtobesubquadratic)thenwehaveM1=0.Theresultwillbeageneralizationof(4.13)usingasinitialstateasuperpositionofgeneralizedcoherentstates(definedwithhigherorderHermitefunctions).Theorem21AssumeH(x,ξ,t)andzbesuchthat(A.0)–(A.2)aresatisfied.Thenforanyintegersl≥1,J≥1andanyrealnumberκ>0thereexistsauniversalconstantΓ>0suchthatforeveryfamilyofcomplexnumbers{cμ,μ∈Nn,|μ|≤J}√thereexistcν(t,)forν∈Nn,|ν|≤3(l−1)+J,suchthatfor0<+θ(t)<κthefollowingL2-estimateholds:JJ+3(l−1)(j)iδt/U(t,0)cjΦz,0−ecμ(t,)Φμ(t)|j|=0|μ|=01/2≤ΓK2H,Tρl(z,,t)|cμ|(4.16)0≤|μ|≤JMoreoverthecoefficientscμ(t,)canbecomputedbythefollowingformula:cμ(t,)−cμ=(k1+···+kp)/2−pap,μ,ν(t)cν(4.17)|ν|≤J1≤p≤l−1k1+···+kp≤2p+l−1|μ−ν|≤3l−3ki≥3wheretheentriesap,μ,ν(t)aregivenbytheevolutionoftheclassicalsystemandareuniversalpolynomialsinH(γ)(zt,t)for|γ|≤l+2satisfyingap,μ,ν(0)=0.Remark17(CommentsontheerrorestimateandtheEhrenfesttime)Theerrortermseemsaccuratebutnotveryexplicitinourgeneralsetting.Letusassumeforsim-plicitythatT=+∞andthattheclassicaltrajectoryztisboundedandunstablewithaLyapunovexponentλ>0.SothereexistssomeconstantC>0suchthatθ(z,t)≤Ceλt,∀t≥0.Thenforeveryε>0thereexistCεandhε>0suchthat1−3ε1ε00andanyintegerl≥1andanyf∈S(Rn)thereexistsΓ>0suchthatthefollowingL2-normestimateholds:U(t,0)T(z)Λˆf−U2(t,0)ΛPl(f,t,)≤ΓKH,tρl(z,t,)(4.22)wherePl(.,t,)isthe(,t)-dependentdifferentialoperatordefinedbyPk/2−jl(f,t,)=f+pjk(x,Dx,t)f(k,j)∈IlwithIl={(k,j)∈N×N:1≤j≤l−1,k≥3j,1≤k−2j≤l}. 1004TheSemiclassicalEvolutionofGaussianCoherentStatesMoreoverthepolynomialspkj(x,ξ,t)canbecomputedexplicitlyintermsoftheWeylsymbolofthefollowingdifferentialoperatorsdefinedabovettlt2dtldtl−1···dt1Π(t1,...,tl;k1,...,kl)000Remark19Ifweconsideran-metaplectictransformationVinL2(Rn)wecon-siderthefollowingobjectV˜=Λ−1VΛ(withnoconfusionwiththenotationforthetransposeofamatrix).BydefinitionsinceVhasaquadraticgeneratorV˜is-independent.SowefindthatU2(t,0)ΛPl(f,t,)isactuallyacoherentstatecenteredatztwithprofileD(B˜t)R(t)f˜.MoreexplicitlyU2(t,0)ΛPl(f,t,)=eiδt/T(zˆj/2pj(x,Dx,t)D(B˜t)R(t)f˜t)Λ0≤j≤l−1wherepj(x,Dx,t)aredifferentialoperatorswithpolynomialcoefficientsdepend-ingsmoothlyontaslongastheclassicalflowz→ztexists.4.1.3RelatedWorksandOtherResultsInthephysicsliteraturethequantumpropagationofcoherentstateshasbeenconsid-eredbymanyauthors,inparticularbyHeller[110,111]andLittlejohn[138].InthemathematicalliteratureGaussianwavepacketshavebeenintroducedandstudiedinmanyrespects,particularlyunderthename“Gaussianbeams”(see[8,159,160]).SomewhatrelatedtothesubjectofthisChapteristhestudybyPaulandUribe[152,153]ofthe-asymptoticsoftheinnerproductsoftheeigenfunctionsofaSchrödingertypeHamiltonianwithacoherentstateandof“semiclassicaltracefor-mulas”(seeChap.5).However,theirapproachdiffersfromtheonepresentedherebytheuseofFourier-integraloperators,whichwereintroducedinconnectionwithwavepacketspropagationintheclassicalpaperbyCordobaandFefferman[54].4.2ApplicationtotheSpreadingofQuantumWavePacketsInthissectionwegiveanapplicationoftheestimateoftheprecedingsectiontothespreading(inphasespace)ofaquantumwavepacketwhichis,attime0,local-izedintheneighborhoodofafixedpointofthecorrespondingclassicalmotion.Letz=(q,p)besuchafixedpointandtakeasaninitialquantumstatethecoherentstateϕz.ThequantumstateattimetisΨ(t)=U(t,0)ϕzWehaveseenthatwecanapproximateΨ(t)byaGaussianwavepacketagainlocal-izedaroundzt=zbutwithaspreadinggovernedbythestabilitymatrixM0ofthe 4.2ApplicationtotheSpreadingofQuantumWavePackets101correspondingclassicalmotion(givenby(4.7)).AwayofmeasuringthespreadingofwavepacketsaroundthepointzinphasespaceistocomputenS(t)=T(ˆ−z)Ψ(t),a∗aj+aja∗T(ˆ−z)Ψ(t)jjj=1aT(ˆ−z)Ψ(t)2a∗T(ˆ−z)Ψ(t)2=+(4.23)Theintuitionbehindthisdefinitionisthefollowing.LetWz,tbetheWignerfunctionofthestatesΨ(t).AccordingtothepropertiesseeninChap.2,(2π)−nWz,t(X)isaquasi-probabilityonthephasespaceR2nXandwehave−ndXW(2π)z,t(X)A(X)=Ψ(t),AΨ(t)ˆ.ApplyingthisrelationtoAˆ=T(z)(ˆa∗·a+a·a∗)Tˆ(−z)whichhastheWeyl-symbolA(q,p)=|q−x|2+|p−ξ|2,weseethatS(t)isthevarianceofthequasi-probability(2π)−nWz,t(X).LetusnoticethatS(t)iswelldefinediftheestimate(4.13)holdsintheSobolevspaceΣ(2)andwithsomemoreassumptiononthequantumevolutionU(t,0)onecangettheestimate(4.13)inΣ2-normasweshallseenow.MorerefinedestimatesinotherSobolevnormswillbegiveninthelastsectionofthischapter.Letusconsidersomesymbolgsatisfyingassumption(A.0)withm=0,suchthatOpw(g)isinvertibleintheSchwartzspaceS(Rn).LetusassumethatthefollowingL2-operatornormestimateholds:Opww−1≤CgU(t,0)Opgt,gThenanestimateanalogousto(4.13)holdstrue:OpwgU(t,0)T(z)Λˆf−U2(t,0)ΛPl(f,t,)≤Ct,gΓKH,t,ρl(z,t,)ObviouslyS(0)=nandweareinterestedinthedifferenceΔS(t):=S(t)−S(0)LetusfirstcalculatenT(t):=T(ˆ−z)Φ(t),a∗aj+aja∗T(ˆ−z)Φ(t)jjj=1whereΦ(t)istheapproximantofΨ(t)givenbyΦ(t)=eiδt/T(z)Uˆ0(t)Ψ0ThenaU21∗T(t)=n+20(t)Ψ0=n+trZtZt2 1024TheSemiclassicalEvolutionofGaussianCoherentStateswhereZtisdefinedby(4.9)andweuseLemma28.Therefore1T(t)−T(0)=∗ZtrZtt2WeshownowthatthisisthedominantbehaviorofΔS(t)uptosmallcorrectiontermsthatwecanestimate.Letusassumethattheclassicalflowatphase-spacepointzhasfiniteLyapunovexponents,withagreatestLyapunovexponentλ∈R(fornotionsconcerningthesta-bilityandLyapunovexponentsforordinarydifferentialequationswereferto[39]).ThenbydefinitionthereexistssomeconstantC>0suchthatF(t)≤Ceλt,∀t≥0whereCisindependentoft.InwhatfollowswedenotebyCagenericconstantindependentoft,.Thenundertheaboveassumptionsweget√Ψ(t)−Φ(t)≤Cte3λt,∀t≥0Σ(2)Wededucethefollowingresult:Theorem23UndertheaboveassumptionswehavethelongtimeasymptoticsΔS(t)=ΔT(t)+O(ε)ifoneofthetwofollowingconditionsisfulfilled:(i)λ≤0(stablecase)and0≤t≤ε−1/2(ii)λ>0(unstablecase)and∃ε>εsuchthat0≤t≤1−2ε/6λlog(1/)InparticularwehaveCorollary12LetusassumethattheHamiltonianHistimeindependentandthatthegreatestLyapunovexponentisλ>0.ThenS(t)−S(0)behaveslikee2λtast→+∞and→0aslongast[log(1/)]−1stayssmallenough.(i)MorepreciselythereexistsC>0suchthate2λt≤ΔT(t)≤Ce2λt,∀t≥0Cε1−2ε1ΔS(t)=ΔT(t)+O,for0≤t≤logforsomeε>ε6λ(ii)Inparticularforn=1wehaveamoreexplicitresult:4b2+(a−c)22εΔS(t)=sinh(λt)+O(4.24)2(b2−ac)abundertheaboveconditionfortwhere=M0istheHessianmatrixofHbcatz. 4.3EvolutionofCoherentStatesandBargmannTransform103ProofWeget(i)usingthatthematrixJM0hasatleastoneeigenvaluewithrealpartλ.Toprove(4.24)wecomputeexplicitlytheexponentialofthematrix√tJM0whichgivesF(t).Itseigenvaluesareλ=b2−acand1/λ.Sowegettheformulacosh(λt)+bsinh(λt)csinh(λt)exp(tJMλλ0)=−asinh(λt)cosh(λt)−bsinh(λt)λλhencewehave∗4b2+(a−c)22ΔT(t)=trZtZt=2sinh(λt).2(b−ac)4.3EvolutionofCoherentStatesandBargmannTransformInSect.4.1wehavestudiedtheevolutionofcoherentstatesusingthegeneratorsofcoherentstatesandtheDuhamelformula.Herewepresentadifferentapproachfol-lowing[164],workingessentiallyontheFourier–Bargmannside(seeChap.1).ThisapproachisusefultogetestimatesinseveralnormsofBanachspacesoffunctionsandalsotogetanalytictypeestimates.WekeepthenotationsofSect.4.1.Werevisitnowthealgebraiccomputationsofthissectioninadifferentpresentation.RecallthatwewanttosolvetheCauchyproblem∂ψ(t)i=H(t)ψ(t),ˆψ(0)=ϕz,(4.25)∂twhereϕzisacoherentstatelocalizedatapointz∈R2n.Ourfirststepistotransformthisproblemwithsuitableunitarytransformationssuchthatthesingularperturba-tionprobleminbecomesaregularperturbationproblem.4.3.1FormalComputationsWerescaletheevolvedstateψz(t)bydefiningftsuchthatψz(t)=T(zˆt)Λft.Thenftsatisfiesthefollowingequation:i∂tft=Λ−1T(zˆ−1H(t)ˆT(zˆT(zˆΛft(4.26)t)t)−i∂tt)−n/4−1|x|2withtheinitialconditionft=0=gwhereg(x)=πe2.Weeasilygettheformula√√Λ−1T(zˆ−1H(t)ˆT(zˆwHt,x+qt,ξ+pt(4.27)t)t)Λ=Op1UsingtheTaylorformulawegettheformalexpansion√√√√Ht,x+qt,ξ+pt=H(t,zt)+∂qH(t,zt)x+∂pH(t,zt)ξj/2−1K+K2(t;x,ξ)+j(t;x,ξ)(4.28)j≥3 1044TheSemiclassicalEvolutionofGaussianCoherentStateswhereKj(t)isthehomogeneousTaylorpolynomialofdegreejinX=(x,ξ)∈R2n.1γγKj(t;X)=∂H(t;zt)Xγ!X|γ|=jWeshallusethefollowingnotationfortheremaindertermoforderk≥1:√Rj/2k(t;X)=Ht,zt+X−Kj(t;X)(4.29)ja2thereexistsC>0suchthatforallu∈S(Rn)wehaveea|x|u(x)≤Ceb|X|FBu(X)(4.40)Lp(Rnx)L2(R2n)X√Moregenerally,foreverya≥0andeveryb>a2thereexistsC>0suchthatfor|S|allu∈S(Rn)andallS∈Sp(2n)wehaveea|x|R(S)uˆ(x)≤Ceb|X|FBu(X)(4.41)Lp(Rnx)L2(R2n)XWeneedtocontrolthenormsofHermitefunctions(seeChap.1)insomeweightedLebesguespaces.LetμbeaC∞-smoothandpositivefunctiononRmsuchthatlimμ(x)=+∞(4.42)|x|→+∞∂γμ(x)≤θ|x|2,∀x∈Rm,|x|≥Rγ(4.43)forsomeRγ>0andθ<1.Lemma30Foreveryrealp∈[1,+∞],forevery∈N,thereexistsC>0suchthatforeveryα,β∈Nmwehaveeμ(x)αβ−|x|2|α+β|+1|α+β|x∂xe,p≤C (4.44)2where•,pisthenormontheSobolevspace1W,p, istheEulerGammafunction.2Moregenerally,foreveryrealp∈[1,+∞],forevery∈N,thereexistsC>0suchthateμ((Γ)−1/2x)αβ−|x|2x∂e,p|α+β|+1(Γ)1/2+(Γ)−1/2 |α+β|≤C(4.45)21Recallthatu∈W,pmeansthat∂xαu∈Lpforevery|α|≤.2TheEulerclassicalGammafunction mustbenotconfusedwiththecovariancematrixΓt. 4.3EvolutionofCoherentStatesandBargmannTransform1074.3.3LargeTimeEstimatesandFourier–BargmannAnalysis(N)Inthissectionwetrytocontrolthesemi-classicalerrortermRz(t,x),forlargetime,intheFourier–Bargmannrepresentation.ThisisalsoapreparationtocontroltheremainderoforderNin,NforanalyticHamiltoniansconsideredinthefol-lowingsubsection.LetusintroducetheFourier–Bargmanntransformofbj(t)g,BB2nj(t,X)=Fbj(t)g(X)=bj(t)g,gX,forX∈R.Theinductionequation(4.32)becomes,forj≥1,w#∂tBj(t,X)=Op1K(t)gX,gXBkt,XdX(4.46)R2nk+=j+2≥3|X|2withinitialconditionBj(0,X)=0forj≥1andwithB0(t,X)=exp(−).4WehaveseeninSect.4.1thatwehavew −n Op1K(t)gX,gX=(2π)K(t,Y)WX,X(Y)dY,(4.47)R2nwhereWX,XistheWignerfunctionofthepair(gX,gX).LetusnowcomputetheremaindertermintheFourier–Bargmannrepresentation.UsingthatFBisanisometrywegetBOpwR(X)F(t)◦Ft,tbj(t)g10w=Bjt,XOp1R(t)◦Ft,t0gX,gXdX(4.48)R2nwhereR(t)isgivenbytheTaylorintegralformula(4.49):/2−11√γγ−1R(t,X)=∂Ht,zt+θXX(1−θ)dθ(4.49)k!X0|γ|=(N)Weshalluse(4.48)toestimatetheremaindertermRz,usingestimates(4.40)and(4.41).NowweshallconsiderlongtimeestimatesforBj(t,X).Lemma31Foreveryj≥0,everys∈N,r≥1,thereexistsC(j,α,β)suchthatfor|t|≤T,wehaveeμ(X/4)Xα∂βBj(t,X)≤C(j,α,β)σ(t,z)NM1|F|3j(1+T)j(4.50)Xs,rT 1084TheSemiclassicalEvolutionofGaussianCoherentStateswhere|F|T=sup|t|≤T|Ft|.M1andσ(t,z)weredefinedinSect.4.1,M1dependsonassumption(A.0)onH(X,t).•s,risthenormintheSobolevspaceWs,r(R2n)3ProofThemainideaoftheproofisasfollows(see[164]fordetails).Weproceedbyinductiononj.Forj=0(4.50)resultsfrom(4.44).Letusassumeinequalityproveduptoj−1.Wehavetheinductionformula(j≥1)∂tBj(t,X)=Kt,X,XBkt,XdX(4.51)R2nk+=j+2≥3where1γwγKt,X,X=∂XH(t,zt)Op1(FtY)gX,gX,and(4.52)γ!|γ|=wγ2nγOp1(FtY)gX,gX=2(FtY)WX,X(Y)dY(4.53)R2nByaFouriertransformcomputationonGaussianfunctions,wegetthefollowingmoreexplicitexpression:X+Xγ−βwγγ−|β|Op1(FtY)gX,gX=Cβ2Ft2β≤γJ(X−X)−|X−X|2/4−(i/2)σ(X,X)×HβFtee2(4.54)Estimate(4.50)followseasily.Nowwehavetoestimatetheremainderterm.LetuscomputetheFourier–Bargmanntransformoftheerrorterm:R˜(N+1)(t,X)=FBj/2OpwRk(t)◦Ftbj(t)g(X)z1j+k=N+3k≥3w=Bjt,XOp1Rk(t)◦FtgX,gXdXR2nj+k=N+3k≥3UsingestimatesontheBjwegetthefollowingestimatefortheerrorterm: 3TheSobolevnormisdefinedhereasfs,r=(|α|≤sdx|f(x)|r)1/rfors∈N,r≥1. 4.3EvolutionofCoherentStatesandBargmannTransform109Lemma32Foreveryκ>0,forevery∈N,s≥0,r≥1,thereexistsCN,suchthatforallTandt,|t|≤T,wehaveXαβR˜(N+1)3N+3N+1N+3∂z(t,X)≤CN,MN,(T,z)|F|(1+T)2(4.55)Xs,rT√for|F|T≤κ,|α|+|β|≤,whereMN,(T,z)isacontinuousfunctionofγsup|t−t0|≤T|∂XH(t,zt)|.3≤|γ|≤NProofAsaboveforestimationoftheBj(t,X),letusconsidertheintegralkernelswNkt,X,X=Op1Rk(t)◦FtgX,gX(4.56)Wehave1(k+1)/21kNkt,X,X=(1−θ)k!0|γ|=k+1√γγ×∂YHt,zt+θFtY(FtY)·WX,X(Y)dYdθR2n(4.57)LetusdenotebyNk,ttheoperatorwiththekernelNk(t,X,X).UsingthechangeX+XofvariableZ=Y−andintegrationsbypartsinXasabove,wecanes-2timateNk,t[Bj(t,•)](X).ThenusingtheestimatesontheBj(t,X)wegetesti-mate(4.55).Now,itisnotdifficulttoconverttheseresultsintheconfigurationspace,using|x−qt|2+11/2(4.40).Letusdefineλ,t(x)=(|Ft|2).Theorem24Letusassumethat(A.0)issatisfied.Thenwehavefortheremainderterm,(N)∂(N)(N)Rz(t,x)=iψz(t,x)−H(t)ψˆz(t,x)∂tthefollowingestimate.Foreveryκ>0,forevery,M∈N,r≥1thereexistCN,M,andNsuchthatforallTandt,|t|≤T,wehaveλMR(N)(t)≤CN,(N+3−)/2σ(z,t)NM1|F|3N+3(1+T)N+1(4.58),tz,rT√forevery∈]0,1],|Ft|≤κ.Moreover,ifH(t)ˆadmitsaunitarypropagator(seecondition(A.2)),thenunderthesameconditionsasabove,wehaveU(N)(t)≤CN,σ(z,t)NM1|F|3N+3(1+T)N+2(N+1)/2(4.59)tϕz−ψz2T 1104TheSemiclassicalEvolutionofGaussianCoherentStatesProofUsingtheinverseFourier–Bargmanntransform,wehaveR(N)(t,x)=T(zˆt)ΛRˆ[F(x)R˜(N)(t,X)dXzt]ϕXzR2nLetusremarkthatusingestimatesonthebj(t,x),wecanassumethatNisarbi-trarylarge.WecanapplypreviousresultsontheFourier–Bargmannestimatestoget(4.58).ThesecondpartisaconsequenceofthefirstpartandoftheDuhamelprinciple.Weseethattheestimate(4.58)ismuchmoreaccurateinnormthanestimate(4.59),wehavelostmuchinformationapplyingthepropagatorUt.ThereasonisthatingeneralweonlyknowthatthepropagatorisboundedonL2andnomore.Sometimesitispossibletoimprove(4.59)ifweknowthatUtisboundedonsomeweightedSobolevspaces.Letusgiveherethefollowingexample.LetHˆ=−2+V(x).AssumethatVsatisfies:V∈C∞Rn,∂αV(x)≤CαV(x),MV≥1,V(x)−V(y)≤C1+|x−y|forsomeM∈R.Sothetime-dependentSchrödingerequationforHˆhasaunitarypropagatorUit−1Hˆmt=e.ThedomainofHˆcanbedeterminedforeverym∈N(seeforexample[163]). DHˆm=u∈W2m,2Rn,Vmu∈L2RnItisanHilbertspacewiththenormdefinedbyu2=|α|∂αu2+Vu22m,VL2(Rn)L2(Rn)|α|≤2mUsingtheSobolevtheoremwegetthesupremumnormestimatefortheerror:sup(U(N)(t,x)≤CNM1|F|3N+3(1+T)N+2(N−n/2)/2tϕz)(x)−ψzN,σ(z,t)Tx∈Rn(4.60)4.3.4ExponentiallySmallEstimatesUptonowtheorderNofthesemi-classicalapproximationswasfixed,evenarbi-trarylarge,buttheerrortermwasnotcontrolledforNlarge.HereweshallgiveestimateswithacontrolforlargeN.Themethodisthesameasontheprevioussec-tion,usingsystematicallytheFourier–Bargmanntransform.Theproofarenotgivenhere,wereferto[164]formoredetails.Foradifferentapproachsee[101,102].To 4.3EvolutionofCoherentStatesandBargmannTransform111getexponentiallysmallestimatesforasymptoticexpansionsinsmallitisquitenaturaltoassumethattheclassicalHamiltonianH(t,X)isanalyticinX,whereX=(x,ξ)∈R2n.Thisproblemwasstudiedinadifferentcontextin[87]concern-ingBorelsummabilityforsemi-classicalexpansionsforbosonssystems.So,inwhatfollowsweintroducesuitableassumptionsonH(t,X).AsbeforeweassumethatH(t,X)iscontinuousintimetandC∞inXandthatthequantumandclassicaldynamicsarewelldefined.LetusdefineacomplexneighborhoodofR2ninC2n, Ω2nρ=X∈C,|X|<ρ(4.61)whereX=(X1,...,X2n)and|·|istheEuclideannorminR2northeHermi-tiannorminC2n.Ourmainassumptionsarethefollowing.(Aω)(Analyticassumption)Thereexistsρ>0,T∈]0,+∞],C>0,ν≥0,suchthatH(t)isholomorphicinΩρandfort∈IT,X∈Ωρ,wehaveH(t,X)≤Ceν|X|,and(4.62)∂γH(t,zt+Y)≤Rγγ!eν|Y|,∀t∈R,Y∈R2nXforsomeR>0andallγ,|γ|≥3.WebeginbygivingtheresultsontheFourier–Bargmannside.ItisthemainstepandgivesaccurateestimatesforthepropagationofGaussiancoherentstatesinthephasespace.WehaveseenthatitisnotdifficulttotransfertheseestimatesintheconfigurationspacetogetapproximationsofthesolutionoftheSchrödingerequation,byapplyingtheinverseFourier–BargmanntransformaswedidintheC∞case.Themainresultsarestatedinthefollowingtheorem.Theorem25Letusassumethatconditions(A0)and(Aω)aresatisfied.Thenthefollowinguniformestimateshold.Xα∂βBj(t,X)XL2(R2d,eλ|X|dX)3j+1+|α|+|β|3jj−j3j+|α2|+|β|≤C|F|1+|t−t0|j3j+|α|+|β|(4.63)λ,TTwhereCλ>0dependsonlyonλ≥0andisindependentonj∈N,α,β∈N2nand|t|≤T.ConcerningtheremaindertermestimatewehaveXα∂βR˜(N)(t,X)XzL2(R2n,eλ|X|dX)(N+3)/2N+13N+33N+3+|α|+|β|−N−1≤(1+|t|)|F|TCλ(N+1)3N+3+|α|+|β|×3N+3+|α|+|β|2(4.64)√whereλ<ρ,α,β∈N2n,N≥1,ν|F|T≤2(ρ−λ),Cdependsonλandisλindependentontheotherparameters(,T,N,α,β). 1124TheSemiclassicalEvolutionofGaussianCoherentStatesFromTheorem25weeasilygetweightedestimatesforapproximatesolutionsandremaindertermforthetime-dependentSchrödingerequation.LetusrecalltheSobolevnormsintheSobolevspaceWm,r(Rn).1/ru=|α|/2∂αu(x)rdxr,m,xRn|α|≤mandafunctionμ∈C∞(Rd)suchthatμ(x)=|x|for|x|≥1.Proposition42Foreverym∈N,r∈[1,+∞],λ>0andε≤min{1,λ},there|F|TexistsCr,m,λ,ε>0suchthatforeveryj≥0andeveryt∈ITwehaveRˆ[Fεμj+13j+2dj/2jt]bj(t)ger,m,1≤(Cr,m,λ,ε)1+|F|Tj1+|t|(4.65)Theorem26WiththeabovenotationsandundertheassumptionsofTheorem25,(N)ψz(t,x)satisfiestheSchrödingerequationi∂(N)(t,x)=H(t)ψˆ(N)(t,x)+R(N)(t,x),where(4.66)tψzzzψ(N)(t,x)=eiδt/T(zˆRˆ[Fj/2bj(t)g(4.67)zt)Λt]0≤j≤NisestimatedinProposition42andtheremaindertermiscontrolledwiththefollow-ingweightedestimates:R(N)(t)eεμ,tzr,m,N+1(N+1)/2√3N+3−mN+1≤C(N+1)|F|T1+|t|(4.68)whereCdependsonlyon√m,r,εandnotonN≥0,|t|≤Tand>0,withthex−qtcondition|F|T≤κ.Theexponentialweightisdefinedbyμ,t(x)=μ(√).≥0andhρmε0,C>0.Sowehavefoundthattherenormalizedevolvedstateb(t,x)g(x)obtainedfromψz(t,x)hasaGevrey-2asymptoticexpansionin1/2.Recallthataformalcomplexseriesj≥0cjκjisaGevreyseriesofindexμ>0ifthereexistconstantsC0>0,C>0suchthat|cj (j)1/μ,∀j≥1.j|≤C0CAnyholomorphicfunctionf(κ)inacomplexneighborhoodof0hasaconvergentGevrey-1Taylorseries.Butinmanyphysicalexampleswehaveanon-convergentGevreyasymptoticseriesf[κ]forafunctionfholomorphicinsomesectorwithapex0.UndersometechnicalconditionsonfitispossibletodefinetheBorelsum 4.3EvolutionofCoherentStatesandBargmannTransform113Bf(τ)fortheformalpowerseriesf[κ]andtorecoverf(κ)fromitsBorelsumperformingaLaplacetransformonBf(τ)(see[180]fordetailsandbibliography).WhenitisnotpossibletoapplyBorelsummability,thereexistsawellknownmethodtominimizetheerrorbetween1≤j≤Ncjκjandf(κ).Itiscalledtheas-tronomersmethodandconsistsofstoppingtheexpansionafterthesmallesttermoftheseries(itisalsocalled“theleasttermtruncationmethod”foraseries).Concern-ingthesemiclassicalexpansionfoundforb(t,x)g(x)itisnotclearthatitisBorelsummableorsummableinsomeweakersense.AsufficientconditionforthatwouldbethatthepropagatorUtcanbeextendedholomorphicallyinκ:=1/2ina(small)sector{reiθ,00,a0>0,a>0,ε>0,suchthatifwechooseN=[]−1wehavecR(N)(t)eεμ,t≤exp−(4.69)zL2forevery|t|≤T,∈]0,0].Moreover,wehavecψ(N)(t)−U(t,t0)ϕz≤exp−(4.70)zL2Alsowehavethefollowing.Corollary14(LargeTime,LargeN)LetusassumethatT=+∞andthereex-istγ≥0,δ≥0,C1≥0,suchthat|Ft,t|≤exp(γ|t|),|zt|≤exp(δ|t|)forevery0θ∈]0,1[thereexistsa=[aθ]−1thereexistθ>0suchthatifwechooseN,θθcθ>0,ηθ>0suchthat(N,θ+2)/2(N,θ+1)εμ,tcθRz(t)eL2≤exp−θ(4.71)forevery|t|≤1−θlog(−1),∀∈∈]0,ηθ].Moreoverwehave6γψ(N,θ)cθz(t)−U(t,t0)ϕzL2≤exp−θ(4.72)undertheconditionsof(4.71).Remark21WehaveconsideredherestandardGaussian.AlltheresultsaretrueandprovedinthesamewayforGaussiancoherentstatesdefinedbygΓ,foranyΓ∈Σn+.Theseresultshavebeenprovedin[164]andin[101,102]usingdifferentmethods. 1144TheSemiclassicalEvolutionofGaussianCoherentStatesAlltheresultsinthissubsectioncaneasilybededucedfromTheorem25.Propo-sition42andTheorem26areeasilyprovedusingtheestimatesofSect.2.2.TheproofofthecorollariesareconsequencesofTheorem26andtheStirlingformulafortheEulerGammafunction.4.4ApplicationtotheScatteringTheoryInthissectionweassumethattheinteractionsatisfiesashortrangeassumptionandweshallproveresultsfortheactionofthescatteringoperatoractingonthesqueezedstates.Onegetsasemiclassicalasymptoticsfortheactionofthescatteringoperatoronasqueezedstatelocatedatpointz−intermsofasqueezedstatelocatedatpointz+wherez+=Scl(z−),Sclbeingtheclassicalscatteringmatrix.Forthebasicclassicalandquantumscatteringtheorieswereferthereaderto[66,162].Letusfirstrecallsomebasicfactsonclassicalandquantumscatteringtheory.WeconsideraclassicalHamiltonianHforaparticlemovinginacurvedspaceandinanelectromagneticfield:1nnH(q,p)=g(q)p·p+a(q)·p+V(q),q∈R,p∈R(4.73)2g(q)isasmoothpositivedefinitematrixandthereexistc>0,C>0suchthatc|p|2≤g(q)p·p≤C|p|2,∀(q,p)∈R2na(q)isasmoothlinearformonRnandV(q)asmoothscalarpotential.Inwhat(0)p2followsitwillbeassumedthatH(q,p)isashortrangeperturbationofH=2inthefollowingsense:thereexistρ>1,andCα,forα∈Nnsuchthat∂α1−g(q)+∂αa(q)+∂αV(q)≤C−ρ−|α|,∀q∈Rn(4.74)qqqαqHandH(0)definetwoHamiltonianflowsΦt,ΦtonthephasespaceR2nforall0t∈R.TheclassicalscatteringtheoryestablishesacomparisonofthetwodynamicsΦt,Φtinthelargetimelimit.Notethatthefreedynamicsisexplicit:0Φt(q,p)=(q+tp,p)0Themethodsof[66,162]canbeusedtoprovetheexistenceoftheclassicalwaveoperatorsdefinedbyΩclX=limΦ−tΦtX(4.75)±0t→±∞ThislimitexistsforeveryX∈Z0whereZ0={(q,p)∈R2n,p=0}andisuniformoneverycompactofZ0.WealsohaveforallX∈Z0limΦtΩcl(X)−Φt(X)=0±0t→±∞ 4.4ApplicationtotheScatteringTheory115Moreover,Ω±clareC∞-smoothsymplectictransformations.Theyintertwinethefreeandtheinteractingdynamics:H◦ΩclX=Ωcl◦H(0)(X),∀X∈Ztclclt±±0,andΦ◦Ω±=Ω±◦Φ0ThentheclassicalscatteringmatrixSclisdefinedbyclcl−1clS=Ω+Ω−Thisdefinitionmakessensesinceonecanprove(see[162])thatmoduloaclosedsetN0ofLebesguemeasurezeroinZ(ZZ0⊆N0)onehascl(ZclΩ+0)=Ω−(Z0)MoreoverSclissmoothinZN0andcommuteswiththefreeevolution:SclΦt=ΦtScl00Thescatteringoperatorhasthefollowingkinematicinterpretation:letusconsiderX−∈Z0anditsfreeevolutionΦtX−.Thereexistsauniqueinteractingevolution0Φt(X)whichisclosetoΦt(X−)fort−∞.Moreoverthereexistsauniquepoint0X+∈Z0suchthatΦt(X)isclosetoΦt(X+)fort+∞.X,X+aregivenby0clXclX=Ω−−andX+=SX−Using[66]wecangetamorepreciseresult.LetIbeanopenintervalofRandassumethatIis“nontrapping”forHwhichmeansthatforeveryXsuchthatH(X)∈Iwehavelimt→±∞|Φt(X)|=+∞.ThenwehaveProposition43IfIisanontrappingintervalforHthenSclisdefinedeverywhereinH−1(I)andisaC∞smoothsymplecticmap.Onthequantumsideonecandefinethewaveoperatorsandthescatteringop-eratorinasimilarway.LetusnotethatthequantizationHˆofHisessentiallyself-adjointsothattheunitarygroupU(t)=exp(−itH)ˆiswelldefinedinL2(Rn).ThefreeevolutionU0(t):=exp(−itHˆ(0))isexplicit:iξ2U(x)=(2π)−nexp0(t)ψ−t+(x−y)·ξψ(y)dydξ(4.76)R2n2Theassumption(4.74)impliesthatwecandefinethewaveoperatorsΩ±andthescatteringoperatorS()=(Ω+)∗Ω−(see[66,162]).RecallthatΩ±=limU(−t)U0(t)t→±∞TherangesofΩ±areequaltotheabsolutelycontinuoussubspaceofHˆandwehave()U()Ω±U0(t)=U(t)Ω±,S0(t)=U0(t)S(4.77) 1164TheSemiclassicalEvolutionofGaussianCoherentStatesOnewantstoobtainacorrespondencebetweenlim→0S()andScl.Therearemanyworksonthesubject(see[99,102,165,203]).Herewewanttocheckthisclassicallimitusingthecoherentstatesapproachlikein[99,102].Wepresenthereadiffer-enttechnicalapproachextendingtheseresultstomoregeneralperturbationsoftheLaplaceoperator.WerecallsomenotationsofChap.3:ΣnistheSiegelspacenamelythespaceofcomplexsymmetricn×nmatricesΓsuchthatΓispositiveandnondegenerate.GivenFany2n×2nsymplecticmatrixtheunitaryoperatorR(F)ˆisthemetaplectictransformationassociatedtoF.gΓistheGaussianfunctionofL2norm1definedbyΓig(x)=aΓexpΓx·x(4.78)2andwedenoteϕΓ=T(z)gˆΓzFinallyΛistheunitaryoperatordefinedin(4.21).Themainresultofthissectionstatesarelationshipbetweenthequantumscatter-ingandtheclassicalscattering.Theorem27ForeveryN≥1,everyz−∈ZN0andeveryΓ−∈ΣnwehavethefollowingsemiclassicalapproximationforthescatteringoperatorS()actingontheΓ−Gaussiancoherentstateϕz−:S()ϕΓ−=eiδ+/T(zˆR(Gˆj/2bjgΓ−+O(N+1)/2(4.79)z−+)Λ+)0≤j≤Nwherewedefinezcl+=Sz−,z±=(q±,p±)zt=(qt,pt)istheinteractingscatteringtrajectoryzt=Φt(Ω−clz−),δ+= +∞q+p+−q−p−∂z+(ptqt−H(zt))dt−,G+=,bjisapolynomialofdegree−∞2∂z−≤3j,b0=1.TheerrortermO((N+1)/2)isestimatedintheL2-norm.LetusdenoteΓ−()ψ−=ϕz−,andψ+:=Sψ−UsingthedefinitionofS()wehave ψ+=limlimU0(t)U(t−s)U0(s)ψ−(4.80)t→+∞s→−∞Thestrategyoftheproofconsistsofapplyingtheestimate(4.13)atfixedtimettoU(t−s)in(4.80)andthentoseewhathappensinthelimitss→−∞,t→+∞. 4.4ApplicationtotheScatteringTheory117LetusdenotebyFt0theJacobistabilitymatrixforthefreeevolutionandbyFt(z)theJacobistabilitymatrixalongthetrajectoryΦt(z).Wehave01nt1nFt=01nWeneedlargetimeestimatesconcerningclassicalscatteringtrajectoriesandtheirJacobistabilitymatrices.Proposition44UndertheassumptionsofTheorem27thereexistsauniquescat-teringsolutionoftheHamiltonequationz˙t=J∇H(zt)suchthatz˙t−ρt−∂tΦ0z+=Ot,fort→+∞z˙t−ρt−∂tΦ0z−=Ot,fort→−∞Proposition45LetusdenoteΦszF0Gt,s:=Ft−s0−sThenwehave(i)lims→−∞Gt,s=Gtexists,∀t≥0−tG(ii)limt→+∞Ft=G+exists0∂zt∂z+(iii)Gt=,andG+=∂z−∂z−Thesetwopropositionswillbeprovenlatertogetherwiththefollowingone.ThemainstepintheproofofTheorem27willbetosolvethefollowingasymptoticCauchyproblemfortheSchrödingerequationwithdatagivenattimet=+∞:N)(N)(N+3)/2i∂sψz−=Hψˆz−(s)+OfN(s)(4.81)(N)Γ−lims→−∞U0(−s)ψz−(s)=ϕz−wherefN∈L1(R)∩L∞(R)isindependentof.Thefollowingresultisanexten-sionforinfinitetimesofresultsproveninSect.4.1forfinitetimes.Proposition46Theproblem(4.81)hasasolutionwhichcanbecomputedinthefollowingway:ψ(N)(t,x)=eiδ(zt)/T(zˆR(Gˆj/2bj(t,z−)gΓ−z−−)Λt)0≤j≤NThebj(t,z−,x)areuniquelydefinedbythefollowinginductionformulaforj≥1startingwithb0(t,x)≡1:w ∂tbj(t,z−,x)g(x)=Op1Kl(t)bk(t,.)g(x)(4.82)k+l=j+2,l≥3limbj(t,z−,x)=0(4.83)t→−∞ 1184TheSemiclassicalEvolutionofGaussianCoherentStateswith 1γγ2nK(t,X)=Kjt,Gt(X)=∂H(zt)(GtX),X∈Rjγ!X|γ|=jbj(t,z−,x)isapolynomialofdegree≤3jinvariablex∈Rnwithcomplextime-dependentcoefficientsdependingonthescatteringtrajectoryztstartingfromz−attimet=−∞.Moreoverwehavetheremainderuniformestimatei∂(N)(N)(N+3)/2−ρ(4.84)tψz−(t)=Hψˆz−(t)+Otuniformlyin∈]0,1],andt≥0.ProofofTheorem27Withoutgoingintothedetailswhicharesimilartothefinitetimecase,weremarkthatintheinductionformula(4.82)wecanusethefollowingestimatestogetuniformdecreaseintimeestimatesforbj(t,z−,x).Firstthereexistc>0andT0>0suchthatfort≥T0wehave|qt|≥ct.Usingtheshortrangeassumptionandconservationoftheclassicalenergyweseethatfor|γ|≥3thereexistsCγ>0suchthat∂γH(zt)≤C−ρ−1(4.85)XγtThereforewededuce(4.84)from(4.82)and(4.85).UsingProposition46andDuhamel’sformulawegetU(t)ψ(N)(s)=ψ(N)(t+s)+O(N+1)/2z−z−uniformlyint,s∈R.Butwehaveψ(N)(t)−U(t−s)U0(s)ψ−z−≤ψ(N)(t)−U(t−s)ψ(N)(s)+U(N)(s)z−z−0(s)ψ−−ψz−WeknowthatlimU(N)(s)=00(s)ψ−−ψz−s→−∞Thengoingtothelimits→−∞wegetuniformlyint≥0ψ(N)(t)−U(t)Ω−ψ−=O(N+1)/2z−(N)ThenwecancomputeU0(−t)ψz−(t)inthelimitt→+∞andwefindoutthatS()ψ(N)(N+1)/2−=ψ++O()where(N)(N)ψ+=limU0(−t)ψz−(t)t→+∞SowehaveprovedTheorem27. 4.4ApplicationtotheScatteringTheory119LetusnowproveProposition44followingthebook[162].ProofLetusdenoteu(t):=zt−Φt0z−.Wehavetosolvetheintegralequationt0(zJ∇Hu(s)+Φ0u(t)=Φt−)+s(z−)ds−∞WecanchooseT1<0suchthatthemapKdefinedbytKu(t)=J∇Hu(s)+Φ0(zs−)ds−∞isacontractioninthecompletemetricspaceCTofcontinuousfunctionsufrom1]−∞,T1]intoR2nsuchthatsupt≤T|u(t)|≤1,withthenaturaldistance.Sowecan1applythefixedpointtheoremtoproveProposition44usingstandardtechnics.Proposition45canbeprovedbythesamemethod.LetusnowproveProposition46.ProofLetusdenotezs0:=Φs0(z−).FurthermoreifSisasymplecticmatrixABS=CDandΓ∈Σn(ΣnistheSiegelspace)wedefineΣ−1S(Γ)=(C+DΓ)(A+BΓ)∈ΣnThenlet0=Σ(ΓΓsF0−)sOnehasforeveryN≥0:i∂(N)(t,s,x)=H(t)ψˆ(N)(t,s,x)+R(N)(t,s,x)tψzzzwhereψ(N)(t,s,x)=eiδss,t/T(zˆRˆFt,sF0j/2bj(t,s)gΓ−(4.86)zt)Λs0≤j≤NandR(N)(t,s,x)=eiδs,t/(N+3)/2z−×T(zˆRˆFt,sF0−)Λsj+k=N+2,k≥3wR0Γ−×Op1k(t,s)◦Ft,sFsbj(t,s)g(4.87) 1204TheSemiclassicalEvolutionofGaussianCoherentStatesOnedenotesFt,s=Ft−s(Φsz−)thestabilitymatrixatΦt−s(Φs0(z−)).Moreover0thepolynomialsbj(t,s,x)areuniquelydefinedbythefollowinginductionformulaforj≥1startingwithb0(s,s,x)≡1:w Γ−∂tbj(t,s,x)=Op1Kl(t,s)bk(t,.)g(x)k+l=j+2,l≥3bj(s,s,x)=0where 1γt−s00γ2nKl(t,s,X)=∂XHΦΦsz−Ft−sFsX,X∈Rγ!|γ|=lSousingPropositions44and45wecancontrolthelimits→−∞in(4.86)and(4.87)andwegettheproofoftheProposition46.ThefollowingcorollaryisanimmediateconsequenceofTheorem27andofthepropertiesofthemetaplectictransformation:Corollary15ForeveryN∈Nwehave()Γ−iδ+/j/2x−q+Γ+∞Sϕz−=eπj√ϕz+(x)+O0≤j≤Nwherez+=Scl(z−),Γ+=ΣG(Γ−),πj(y)arepolynomialsofdegree≤3jin+y∈Rn.Inparticularπ0=1.RecallthatΣmisthespaceofsmoothclassicalobservablesLsuchthatforeveryγ∈R2nthereexistsCγ≥0suchthat∂γL(X)≤Cm,∀X∈R2nXγXTheWeylquantizationLˆofLiswelldefined(seeChap.2).Onehasthefollowingresult:Corollary16ForanysymbolL∈Σ(m),m∈R,wehave√()ϕ()=LSclSz,LSˆϕz(z−)+O−−Inparticularonerecoverstheclassicalscatteringoperatorfromthequantumscat-teringoperatorinthesemiclassicallimit.ProofUsingCorollary15onegets()()Γ+Γ+√Sϕz−,LSˆϕz−=ϕz−,Lϕˆz−+OandtheresultfollowsfromatrivialextensionofLemma14ofChap.2. 4.4ApplicationtotheScatteringTheory121Remark22Asimilarresultwasprovenforthetime-delayoperatorin[192].Theproofgivenhereisdifferentanddoesn’tuseaglobalnon-trappingassumption.FurtherstudyconcernsthescatteringevolutionofLagrangianstates(alsocalledWKBstates).ItwasconsideredbyYajima[203]inthemomentumrepresentationandbyS.Robinson[167]forthepositionrepresentation.Theapproachdevelopedhereprovidesamoredirectandgeneralproofthatisdetailedin[164].NotealsothatundertheanalyticandGevreyassumptiononecanrecoverthere-sultof[102]forthesemiclassicalpropagationofcoherentstateswithexponentiallysmallestimate. Chapter5TraceFormulasandCoherentStatesAbstractThemostknowntraceformulainmathematicalphysicsiscertainlytheGutzwillertraceformulalinkingtheeigenvaluesoftheSchrödingeroperatorHˆasPlancksconstantgoestozero(thesemi-classicalrégime)withtheclosedorbitsofthecorrespondingclassicalmechanicalsystem.Gutzwillergaveaheuristicproofofthistraceformula,usingtheFeynmanintegralrepresentationforthepropagatorofHˆ.InmathematicsthiskindoftraceformulawasfirstknownasPoissonformula.ItwasprovedfirstfortheLaplaceoperatoronacompactmanifold,thenformoregeneralellipticoperatorsusingthetheoryofFourier-integraloperators.Ourgoalhereistoshowhowtheuseofcoherentstatesallowsustogivearathersimpleanddirectrigorousproof.5.1IntroductionAquantumsystemisdescribedbyitsHamiltonianHˆanditsadmissibleenergiesaretheeigenvaluesEj()(wesupposethatthespectrumoftheself-adjointoperatorHˆintheHilbertspaceH=L2(Rn)isdiscrete).ThefrequencytransitionbetweenEk()−Ej()energiesEj()andEk()isωj,k=.Ifn=1,orifthesystemisintegrable,itispossibletoprovesemi-classicalex-pansionforindividualeigenvaluesEj()when0.Formoregeneralsystemsitisverydifficultandalmostimpossibletoanalyzeindividualeigenvalues.Butitispossibletogiveastatisticaldescriptionoftheenergyspectruminthesemi-classicalregimebyconsideringmeanvaluesTrfHˆ=fEj()(5.1)Afirstresultcanbeobtainedifwesupposethatthe-WeylsymbolHofHˆissmoothandsatisfiestheassumptionofthefunctionalcalculusinChap.2(Theo-rem10).ConsideranintervalIε=[λ1−ε,λ2+ε]suchthatH−1(I)iscompactforεsmallenough.Thefollowingresultisprovedin[107]:Proposition47(i)ForeverysmoothfunctionfsupportedinIεwehavetheasymptoticexpansionatanyorderinM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,123TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_5,©SpringerScience+BusinessMediaB.V.2012 1245TraceFormulasandCoherentStatesTrfHˆ(2π)−ndXfH(X)+j−nCj(f)(5.2)R2nj≥1whereCj(f)arecomputabledistributionsinthetestfunctionf.(ii)Ifλ1andλ2arenoncriticalvaluesforH1andifNIdenotesthenumberofeigenvaluesofHˆinI:=[λ1,λ2]thenwehavetheWeylasymptoticformulaN−nVolH−1(I)+O1−n(5.3)I=(2π)Remark23(i)ThefirstpartofthePropositionisaneasyapplicationofthefunc-tionalcalculus.Using(i)itispossibletoproveaWeylformulawithanerrortermO(θ−n)withθ<2.Theerrortermwithθ=1isoptimal(ingeneral)andcanbeobtainedusing3amethodinitiatedbyHörmander[42,107,116].FurthermoreusingatrickinitiatedbyDuistermaatGuillemin[71]theremaindertermcanbeimprovedino(1−n)ifthemeasureofclosedclassicalpathonH−1(λi)iszero,fori=1,2(see[158]).ThedensityofstatesofaquantumsystemHˆisthesumofdeltadistributionD(E)=δ(E−Ej()).Theintegrateddensityofstatesisthespectralreparti-tionfunctionN(E)={j,Ej()0andsatisfiestheproperty(H.0)inthevariableq.(ii)Thetechnicalcondition(H.0)impliesinparticularthatHˆisessentiallyself-adjointonL2(Rn)forsmallenoughandthatχ(H)ˆisa-pseudodifferentialoperatorifχ∈C∞(R)(seeChap.2and[107]).0LetusdenotebyφttheclassicalflowinducedbyHamiltonsequationswithHamiltonianH,andbyS(q,p;t)theclassicalactionalongthetrajectorystartingat(q,p)attimet=0,andevolvingduringtimettS(q,p;t)=ps·˙qs−H(q,p)ds(5.10)0where(qt,pt)=φt(q,p),anddotdenotesthederivativewithrespecttotime.Weshallalsousethenotation:αt=φt(α)whereα=(q,p)∈R2n,isaphase-spacepoint.RecallthattheHamiltonianHisconstantalongtheflowφt. 1285TraceFormulasandCoherentStatesIfγisaperiodictrajectoryparametrizedast →αt,αT∗=α0whereTγ∗istheγprimitiveperiod(thesmallestpositiveperiod),theclassicalactionalongγisT∗γSγ=dtptq˙t:=pdq0γAnimportantroleinwhatfollowsisplayedbythelinearizedflowaroundtheclassicaltrajectory,whichisdefinedasfollows.Let∂2HH(αt)=2(5.11)∂αα=αtbetheHessianofHatpointαt=φt(α)oftheclassicaltrajectory.LetJbethesymplecticmatrix01J=(5.12)−10where0and1are,respectively,thenullandidentityn×nmatrices.LetFtbethe2n×2nrealsymplecticmatrixsolutionofthelineardifferentialequationF˙t=JH(αt)Ft10(5.13)F0==101Ftdependsonα=(q,p),theinitialpointfortheclassicaltrajectory,αt.LetγbeaclosedorbitonΣEwithperiodTγ,andletusdenotesimplybyFγthematrixFγ=F(Tγ).Fγisusuallycalledthemonodromymatrixoftheclosedorbitγ.Ofcourse,Fγdoesdependonα,butitseigenvaluesdonot,sincethemonodromymatrixwithadifferentinitialpointonγisconjugatetoFγ.Fγhas1aseigenvalueofalgebraicmultiplicityatleastequalto2.Inallthatfollows,weshallusethefollowingdefinition:Definition12Wesaythatγisanon-degenerateorbitiftheeigenvalue1ofFγhasalgebraicmultiplicity2.LetσdenotetheusualsymplecticformonR2nσα,α=p·q−p·q,α=(q,p);α=q,p(5.14)(·istheusualscalarproductinRn).Wedenoteby{α1,α}abasisfortheeigenspace1ofFγbelongingtotheeigenvalue1,andbyVitsorthogonalcomplementinthesenseofthesymplecticformσV=α∈R2n:σ(α,α=0(5.15)1)=σα,α1Then,therestrictionPγofFγtoViscalledthe(linearized)Poincarémapforγ. 5.2TheSemi-classicalGutzwillerTraceFormula129InmoregeneralcasestheHamiltonianflowwillcontainmanifoldsofperiodicorbitswiththesameenergy.Whenthishappens,theperiodicorbitswillnecessarilybedegenerate,butthetechniquesweuseherecanstillapply.Theprecisehypothesisforthis(HypothesisC)willbegivenlater.FollowingDuistermaatandGuilleminwecallthisacleanintersectionhypothesis,itismoreexplicitthanotherversionsofthisassumption.Sincethestatementofthetraceformulaissimplerandmoreinformativewhenonedoesassumethattheperiodicorbitsarenon-degenerate,wewillgiveonlythatformulainthiscase.Weshallnowassumethefollowing.Let(ΓE)TbethesetofallperiodicorbitsonΣEwithperiodsTγ,0<|Tγ|≤T(includingrepetitionsofprimitiveorbitsandassigningnegativeperiodstoprimitiveorbitstracedintheoppositesense).•(H.1)ThereexistsδE>0suchthatH−1([E−δE,E+δE])isacompactsetofR2nandEisanoncriticalvalueofH(i.e.H(z)=E⇒∇H(z)=0).•(H.2)Allγin(ΓE)Tarenon-degenerate,i.e.1isnotaneigenvalueforthecor-respondingPoincarémap,Pγ.Inparticular,thisimpliesthatforanyT>0,(ΓE)Tisadiscreteset,withperiods−T≤Tγ<···0(γ∈N2n),suchthat∂γ≤Cδ2nzA(z)γH(z),∀z∈R•(H.4)gisaC∞functionwhoseFouriertransformg˜isofcompactsupportwithSuppg˜⊂[−T,T]andletχbeasmoothfunctionwithacompactsupportcon-tainedin]E−δE,E+δE[,equalto1inaneighborhoodofE.ThenthefollowingregularizeddensityofstatesρA(E)iswelldefined:E−HˆρA(E)=TrχHˆAχˆHˆg(5.16)Notethat(H.1)impliesthatthespectrumofHˆispurelydiscreteinaneighborhoodofEsothatρA(E)iswelldefined.Wehavealso,moreexplicitly,E−EjρA(E)=gχ2(Ej)Aϕˆ(5.17)j,ϕj1≤j≤NwhereE1≤···≤ENaretheeigenvaluesofHˆin]E−δE,E+δE[(withmulti-plicities)andϕjisthecorrespondingeigenfunction(Hϕˆj=Ejϕj).LetusremarkE−Ejherethatthescaling:istherightonetohaveanicesemi-classicallimit.Thefirstargumentisthatifn=1(andforintegrablesystems),inregularcase,eigen-valuesaregivenbyBohrSommerfeldformula[108]andtheirmutualdistanceisoforder.Thesecondargumentisincludedinthefollowingresult[106,158]. 1305TraceFormulasandCoherentStatesUnderassumption(H.1)theLiouvillemeasuredLEiswelldefinedontheenergysurfaceΣE:dΣEdLE=(dΣEistheEuclideanmeasureonΣE)|∇H|NowwecanstatetheGutzwillertraceformula.Theorem28(Gutzwillertraceformula)Assume(H.0)(H.2)aresatisfiedforH,(H.3)forAand(H.4)forg.Thenthefollowingasymptoticexpansionholdstrue,moduloO(∞),ρ−n/2−(n−1)kA(E)≡(π)gˆ(0)A(α)dσE(α)+ck(gˆ)ΣEk≥−n+2+(2π)n/2−1ei(Sγ/+σγπ/2)gˆ(Tγ)det(1−P−1/2γ)γ∈(ΓE)TTγ∗γj×A(αs)ds+d(g)ˆ(5.18)j0j≥1whereTγ∗istheprimitiveperiodofγ,σγistheMaslovindexofγ(σγ∈Zandisγcomputedintheproof),ck(g˜)aredistributionsing˜withsupportin{0},d(g˜)arejdistributionsing˜withsupport{Tγ}.Remark25WecanincludemoregeneralHamiltoniansdependingexplicitlyin,Kj(j)(0)H=j=1HsuchthatHsatisfies(H.0)andforj≥1,∂γH(j)(z)≤CH(0)(z)(5.19)γ,jItisusefulforapplicationstoconsiderHamiltonianslikeH(0)+H(1)whereH(1)maybe,forexample,aspinterm.Inthatcasetheformula(5.18)istruewithdifferentcoefficients.InparticularthefirstterminthecontributionofTγismultipliedbyT∗γH(1)(αexp(−is)ds).0Remark26ForSchrödingeroperatorsweonlyneedsmoothnessofthepotentialV.Inthiscasethetraceformula(5.18)isstillvalidwithoutanyassumptionsatin-finityforVwhenwerestrictourselvestoacompactenergysurface,assumingE0thereexistsCN,Tsuchthatiδ(α,t)U(t)ϕα−expT(αˆRˆ(Ft)ΛPN(x,Dx,t,)ψ0≤CN,TN(5.28)t)whereψ0(x)=π−n/4exp(−|x|2/2),andPN(t,)isthe(,t)-dependentdifferen-tialoperatordefinedbyPk/2−jwN(x,Dx,t,)=1+pkj(x,D,t)(k,j)∈INwithIN=(k,j)∈N×N,1≤j≤2N−1,k≥3j,1≤k−2j<2N(5.29)wherethedifferentialoperatorspkj(x,Dx,t)areproductsofjWeylquantizationofhomogeneouspolynomialsofdegreekswith1≤s≤jks=k(see[52],Theorem3.5anditsproof).Sothatwegetpw(x,Dkjx,t)ψ0=Qkj(x)ψ0(x)(5.30)whereQkj(x)isapolynomial(withcoefficientsdependingon(α,t))ofdegreekhavingthesameparityask.Thisisclearfromthefollowingfacts:homogeneouspolynomialshaveadefiniteparity,andWeylquantizationbehaveswellwithrespecttosymmetries:Opw(A)commutestotheparityoperatorΣf(x)=f(−x)ifandonlyifAisanevensymbolandanticommuteswithΣifandonlyifAisanoddsymbol)andψ0(x)isanevenfunction.Sowegetk+|γ|iδ(t,α)−jm(α,t)=ck,j,γ2exp(j,k)∈IN;|γ|≤2N×T(α)ΛˆQγψ0,T(αˆt)ΛQk,jR(Fˆt)ψ0+ON(5.31)whereQk,j,respectively,Qγarepolynomialsinthexvariablewiththesameparityask,respectively,|γ|.Thisremarkwillbeusefulinprovingthatwehaveonlyentirepowersinin(5.18),eventhoughhalfintegerpowersappearnaturallyintheasymptoticpropagationofcoherentstates.ByaneasycomputationwehaveT(α)ΛˆQγψ0,T(αˆt)ΛQk,jR(Fˆt)ψ01α−αt=exp−iσ(α,αt)Tˆ1√Qγψ0,Qk,jR(Fˆt)ψ0(5.32)2whereTˆ1(·)istheWeyltranslationoperatorwith=1. 5.3PreparationsfortheProof133Wesetα−αtmk,j,γ(α,t)=Tˆ1√Qγψ0,Qk,jR(Fˆt)ψ0(5.33)α−αtm0(α,t)=Tˆ1√ψ0,R(Fˆt)ψ0(5.34)Wecomputem0(α,t)first.Weshallusethefactthatthemetaplecticgrouptrans-formsGaussianwavepacketsintoGaussianwavepacketsinaveryexplicitway.IfwedenotebyAt,Bt,Ct,Dtthefourn×nmatricesoftheblockformofFtAtBtFt=(5.35)CtDtWehavealreadyseeninChaps.3and4,sinceFtissymplectic,thatUt=At+iBtisinvertible.Sowehavedefined−1,whereVΓt=VtUtt=(Ct+iDt)(5.36)WehavefromourChap.3(seealso[77],Chap.4)−1/2−n/2m0(α,t)=[detUt]cπiiq−qt×exp(Γt+i1)x·x−√x−Rn22×p−pt+i(q−qt)dx(5.37)ButtheintegrationinxisaFouriertransformofaGaussianandcanbeperformed(seeinAppendixA,Sect.A.1).ThecomplexmatrixΓt+i1isinvertibleandwehave−11+Wt(Γt+i1)=(5.38)2iwhereweusethefollowingnotation:W−1t=ZtYt,Zt=Ut+iVt,Yt=Ut−iVt(5.39)ItisclearthatYisinvertible(seeChap.3).Sowegetnn/2−1−1/2−1/2iΨE(t,α)m0(α,t)=2πdetYtUt[detUt]ce(5.40)∗wherethephaseΨE(t,α)isgivenbyt1ΨE(t,α)=tE−H(α)+σ(αs−α,α˙s)ds20i+(1−Wt)(α˘−˘αt)·(α˘−˘αt)(5.41)4withα˘=q+ipifα=(q,p). 1345TraceFormulasandCoherentStates−1/2In(5.40)wehaveaproductofsquarerootofdeterminant.[detUt]cisabranchfor[detUt]−1/2withthephase(orargument)obtainedbycontinuityintimefromUt=0=1.ForacomplexsymmetricmatrixMwithdefinite-positiverealpart,[detM]−1/2isabranchfor(detM)−1/2withthephaseobtainedbycontinuityalong∗apathjoiningMtoM,theeigenvaluesofM1/2havingpositiverealpart.Follow-ingcarefullythesephaseswillgivetheMaslovcorrectionindex.Remark27Thereishereadifferencewiththepaper[53]wherethephasewasobtainedbeforeintegrationiny∈Rn,socomputationsherewillbealittlebitmorenaturalandeasier.ThesamephaseΨE(t,α)appearswhencomputingmk,j,γ(α,t)withnon-trivialamplitudes.Thentheformulafortheregularizeddensityofstatesin(5.22)takestheformiΨ(t,α)ρA(E)=dtdαa(t,α,)eE(5.42)RR2nwhereΨEisgivenin(5.41)anda(t,α,)j∈Naj(t,α)j.OurplanistoproveTheorem28byexpanding(5.42)bythemethodofsta-tionaryphase.ThenecessarystationaryphaselemmaforcomplexphasefunctionscaneasilybederivedfromTheorem7.7.5in[117].Thereisalsoanextendeddis-cussionofcomplexphasefunctionsdependingonparametersin[117]leadingtoTheorem7.7.12,buttheformofthestationarymanifoldherepermitsustousethefollowingresultprovedinSect.A.2.Theorem29(Stationaryphaseexpansion)LetO⊂Rdbeanopenset,andleta,f∈C∞(O)withf≥0inOandsuppa⊂O.WedefineM=x∈O,f(x)=0,f(x)=0andassumethatMisasmooth,compactandconnectedsubmanifoldofRdofdi-mensionksuchthatforallx∈MtheHessian,f(x),offisnon-degenerateonthenormalspaceNxtoMatx.Undertheconditionsabove,theintegralJ(ω)=deiωf(x)a(x)dxhasthefol-Rlowingasymptoticexpansionasω→+∞,moduloO(ω−∞):d−k2π2J(ω)≡c−jjωωj≥0Thecoefficientc0isgivenby−1/2iωf(m0)f(m)|Nmc0=edeta(m)dVM(m)(5.43)Mi∗wheredVM(m)isthecanonicalEuclideanvolumeinM,m0∈Misarbitrary,and−1/2[detP]∗denotestheproductofthereciprocalsofsquarerootsoftheeigenvalues 5.4TheStationaryPhaseComputation135ofPchosenwithpositiverealparts.Notethat,sincef≥0,theeigenvaluesoff(m)|Nmlieintheclosedrighthalfplane.i5.4TheStationaryPhaseComputationInthissectionwecomputethestationaryphaseexpansionof(5.22)withphaseΨEgivenby(5.41).Notethata(t,α,)isactually,accordingto(5.31),apolynomialin1/2and−1/2.Hencethestationaryphasetheorem(with-independentsym-bola)appliestoeachcoefficientofthispolynomial.WeneedtocomputethefirstandsecondorderderivativesofΨE(t,α).Letusintroducethe2n×2ncomplexsymmetricmatrixWt−iWtWt=−iWt−WtItisenoughtocomputefirstderivativesforΨEuptoO(|αt−α|2):11∂tΨE(t,α)≡E−H(α)−(αt−α)·Jα˙t+W−1α˙t·(αt−α)(5.44)22i11∂TTαΨE(t,α)≡1+FJ(αt−α)+F−1W−1(αt−α)(5.45)22iThecriticalsetforthestationaryphasetheoremisdefinedas(α,t)∈R2n×R,ΨCE=E(α,t)=0,∂tΨE(t,α)=0,∂αΨE(t,α)=0WehaveseeninChap.2thatsinceFissymplectic,onehasW∗W<1,soif(ΨE(α,t))=0thenαt=α.Using(5.44)weget(α,t)∈R2n×R,H(α)=E;αCE=t=αHence(t,α)isacriticalpointmeansthatαisonaperiodicpathofenergyE,fortheHamiltonianH,andperiodt.ThesecondderivativesofΨErestrictedonCEcanbecomputedasfollows:21∂t,tΨE(t,α)=Wt−1α˙·˙α(5.46)2i21T∂t,αΨE(t,α)=−∂αH(α)+Ft−1Wt−1(α)˙(5.47)2i21T1T∂α,αΨE(t,α)=JFt−FtJ+Ft−1Wt−1(Ft−1)(5.48)22iLetΨ(t0,α0)betheHessianmatrixofΨEatpoint(t0,α0)ofCE.WehavetoEcomputethekernelofΨ(α0,t0).E 1365TraceFormulasandCoherentStatesLemma33Forevery(t0,α0)∈CEwehavekerΨ(t2n0,α0)=(τ,v)∈R×R,v·∂αH=0,(Ft−1)v+τα˙=0(5.49)E0ProofUsingtheTaylorformulawehave1=(tψE(t,α)ΨE0,α0)(t−t0,α−α0)·(t−t0,α−α0)23+|α−α3(5.50)+O|t−t0|0|FromW∗W<1weget,forsomec>0,ψ2E(t,α)≥c|α−αt|(5.51)Usingthatα−αt=(α−α0)+(α0−α0,t)+(α0,t−αt)+(αt−αt)weget0000)(α−α22α−αt=(1−Ft0)+(t0−t)α˙t+O|t−t0|+|α−α0|0Thenfrom(5.50)and(5.51)weget,forsomec>0,Ψ≥c(F2E(t0,α0)(t−t0,α−α0)·(t−t0,α−α0)t0−1)(α−α0)+(t−t0)α˙(5.52)Sowehaveprovedthepart⊆in(5.49).Thepart⊇isobvious.Thefirstthingtocheck,inordertoapplythestationaryphasetheoremisthatthesupportofαin(5.42)canbetakenascompact,uptoanerrorO(∞).Wedothisinthefollowingway:letusrecallsomepropertiesof-pseudodifferentialcalculusprovedin[68,107].Thefunctionm(z)=H(z)isaweightfunction.In[68]itisprovedthatχ(H)ˆ=HˆχwhereHχ∈S(m−k),foreveryk(χislikein(H.4)).Moreprecisely,wehaveintheasymptoticsenseinS(m−k),HjHχ=χjj≥0andsupport[Hχ,j]isinafixedcompactsetforeveryj(see(H.4)and[107]forthecomputationsofHχ,j).LetusrecallthatthesymbolspaceS(m)isequippedwiththefamilyofsemi-norms∂γsupm−1(z)u(z)∂zγz∈R2nNowwecanprovethefollowinglemma.Lemma34ThereisacompactsetKinR2nsuchthatform(α,t)=ϕα,AˆχU(t)ϕα 5.4TheStationaryPhaseComputation137wehavem(α,t)dα=O+∞R2n/Kuniformlyineveryboundedintervalint.ProofLetχ˜∈C∞(]E−δE,E+δE[)suchthatχχ˜=χ.Using(H.3)andthecom-0positionrulefor-pseudodifferentialoperatorswecanseethatAˆχ(H)ˆisboundedonL2(Rn).SothereexistsaC>0suchthat2m(α,t)≤Cχ˜HˆϕαButwecanwrite22χ˜Hˆϕα=χ˜Hˆϕα,ϕαLetusintroducetheWignerfunction,wα,forϕα(i.e.theWeylsymboloftheor-thogonalprojectiononϕα).WehaveHˆ2−nχ˜ϕα,ϕα=(π)Hχ2(z)wα(z)dzwhere|z−α|2−n−wα(z)=(π)eUsingremainderestimatesfrom[107]wehave,foreveryNlargeenough,HˆHj+N+1RN()χ2=χ2,j0≤j≤NwherethefollowingestimateinHilbertSchmidtnormholds:supRN()HS<+∞0<≤1NowthereisanR>0suchthatforeveryj,wehaveSupp[Hχ2,j]⊆{z,|z|0,|α−r|2c−−edzdα≤Ce{|z|≤R,|α|≥R+1}From(5.49)weseethatasufficientconditiontoapplythestationaryphasetheo-remisthefollowingcleanintersectionconditionfortheHamiltonianflowφt. 1385TraceFormulasandCoherentStatesCleanIntersectionCondition(CI)WeassumethatCEisaunionofsmoothcom-pactconnectedcomponentsandoneachcomponent,thetangentspaceT(t0,α0)CEtoCEat(t0,α0)coincideswiththelinearspace{(τ,v)∈R×R2n,v·∂αH=0,(Ft1)v+τα˙=0}.0Soundercondition(CI)thekernelofΨ(t0,α0)coincideswiththetangentspaceET(t,α)(CE).HenceΨ(t0,α0)isnon-degenerateonthenormalspaceatCEon00E(t0,α0),asisrequiredtoapplythestationaryphasetheorem.AtthispointwehavealreadyprovedthatthereexistsanasymptoticexpansionfortheregularizeddensityofstatesρA(E).Amoredifficultproblemistocomputethisasymptoticsingeneral.Thesimplercaseistheperiod0oftheflow:CE={0,α),H(α)=E}.Thentheproperty(CI)issatisfiedifEisnoncriticalforH.Remarkthat0isnotanaccumulationpointintheperiodsofclassicalpaths.TheHessianmatrixonCEis−1∇Hi∇HΨ(0,α)=2E−∇H0where∇H=∂αH.ThenormalspaceNαtoCEhasthebasis{(1,0),(0,∇H)}.SothedeterminantofΨ(0,α)restrictedtoN(0,α)is ∇H 4.ThestationaryphaseEtheoremgivesusProposition48Letgbesuchthatg˜issupportedin]−T0,T0[whereT0>0andφthasnoperiodictrajectoryonΣEwithaperiodin]−T0,T0[{0}.Thenwehaveχ(H(α))−ndΣ1−n+O2−n(5.53)ρA(E)=˜g(0)(2π)EΣE ∇H Moreovertheasymptoticscanbeextendedasafullasymptoticsin.Asanapplicationof(5.53)wehavethefollowing.Theorem30(Weylasymptoticformula)AssumethatHsatisfiescondition(H.0).Considerλ1<λ2suchthatH−1[λ1−ε,λ2+ε]iscompactandarenoncriticalvaluesforH.LetNIbethenumberofeigenvaluesEj()ofHˆinI=[λ1,λ2].ThenwehaveN−nVolα∈R2n,H(α)∈I+O1−n(5.54)I=(2π)ProofUsingapartitionofunityitisenoughtoconsiderσ(λ)=χEj()Ej()≤λwhereχissupportedinasmallneighborhoodofλ1orλ2(betweenλ1andλ2wecanapplythefunctionalcalculustogetanasymptoticexpansion;see[107]). 5.4TheStationaryPhaseComputation139Toproveanasymptoticexpansionforσ(λ)weuse(5.53)choosingA=χ(H),g˜even,g˜(0)=1,g≥0andg(λ)≥δ0for|λ|≤ε0forsomeδ0>0,ε0>0.Wehave1μ−λσ(λ)−gσ(μ)dμ=σ(λ)−σ(λ+τ)g(τ)dτ(5.55)From(5.53)wehave,afterintegration,−1μ−λ−n−ngσ(μ)dμ=(2π)dαχH(α)+OH(α)≤λUsingthefollowingestimate,forsomeC>0:σ(λ+τ)−σ(λ)≤C1+|τ|1−n(5.56)wegetσ(λ)=(2π)−ndαχH(α)+O−nH(α)≤λthen(5.54)follows.Nowweprove(5.56).Itisenoughtoconsiderthecaseτ≥0.Supposeτ≤ε0.Thenλ+τμ−λ1−nδ0σ(μ)≤dμg=OλForτ=ε0with∈N,usingthetriangleinequality,wegetσ(λ+ε≤C1−n0)−σ(λ)Finallyforε0<τ<(+1)ε0usingagainthetriangleinequalityweget(5.56).Remark28AssumingthatthesetofallperiodicaltrajectoriesofHinΣEisofLiouville-measure0,itispossibletoprovebythesamemethodthefollowingresult.ForeveryC>0wehavelimn−1j,E−C≤Ej()≤E−C=dLE=:LE(ΣE)(5.57)0ΣEThisresultwasalreadyprovedin[106,158]usingFourier-integraloperators.NowwecometotheproofoftheGutzwillertraceformula(5.18).NotethatforisolatedperiodicorbitsonΣEthenon-degenerateassumptionisequivalenttothecondition(CI).SoitresultsfromourdiscussionthatinthiscasetheHessianmatrixΨat(t0,α0),whereγisaperiodicpathwithperiodt0=kTγ∗Eandα0∈γisnon-degenerateonthenormalspaceNt0,α0atCE.HereNt0,α0isthelinearspace{R(1,0)+{(0,v),v∈R2n,σ(v,∇H)=0}.OurmainproblemistocomputethedeterminantoftherestrictionΨ(t0,α0)ofΨ(t0,α0)toNt,α.WeE,⊥E00 1405TraceFormulasandCoherentStatesshalldenoteΠα˙theorthogonalprojectioninR2nonJ∇H(α0):=˙α(tangentvectortoγ).Itisconvenienttointroducethenotations1TiG=Wt0−1,K=Ft0−1G+J+iJ22UsingthatFtissymplecticwehave02Ψ∂α,αE(t0,α0)=K(Ft0−1)Sowehave−1Gα˙·˙αKα˙ΨE(t0,α0)=i(5.58)KαK˙(Ft−1)0Thisformulaisgeneral.Furthermorewehavetheveryusefulresult:Lemma35Kisa2n×2ninvertiblematrixandwehave−11U−1−i(1+U)K=−(5.59)2i1+V−(1+iV)Inparticularwehave−1nYdetK=(−1)det(5.60)2whereU=Ut,V=Vt,Y=Yt.000ProofWehave,usingdefinitionofW,W−1−i(W−1)W−1+iJ=−i(W+1)−(W+1)2iV2V(U−iV)−10=−1−2iU−2U0(U−iV)Aftersomealgebraiccomputations,usinginparticularthesymplecticrelations,wefind−1−iVi(1+U)(U−iV)−10K=−1−i1−V−1+U0(U−iV)Sowegetthelemma.Nowwebegintousethenon-degeneracyconditiontocomputethedeterminantofΨ(t0,α0).WehaveE,⊥−1−1dKα˙detiΨE,⊥(t0,α0)=idet(5.61)KαK(˙Ft−1)+iΠα˙0whered=1(W−1).2 5.4TheStationaryPhaseComputation141Letusintroducenowconvenientcoordinates.WedefineaPoincarésectionSbytheequationT(α)=0whereTisaclassicalobservablesuchthat{T,H}(α)=1,T(α0)=0,TisdefinedinanopenneighborhoodV0ofα0.ThefirstreturnPoincarémapP(α)=φT(α)(α)isdefinedinV0∩SsuchthatT(φT(α)(α))=0withT(α0)=t0,T(α)isthefirstreturntime.InV0wecandefinenewsymplecticcoordinates:(e,τ,),wheree=H(α),τ=T(α),(α)∈R2(n−1).ThedifferentialP(α0)ofPatα0isrelatedwiththestabilitymatrixF=(∂αφt0)(α0):P(αT0)v=Fv−F∇T·vα˙(5.62)ForenearE,thePoincarémapPeisdefinedinV0∩S∩ΣeintoV1∩S∩Σe,whereV1isaneighborhoodofα0,byP(α)=φT(α)(α),T(α0)=t0.Itisasymplecticmapandfore=EitsdifferentialPγistherestrictionofP(α0)toNγ:=Tα(S∩ΣE)0(formoredetailswereferto[103]).NotethatNγ={v∈R2n,v·∇H=v·∇T=0}.WhentheenergyeisvaryingaroundEwehaveasmoothfamilyofclosedtra-jectoriesofperiodT(e)parametrizedbyα(e)∈V0∩Ssuchthatα(E)=α0andT(E)=t0.Tandαaresmoothine.ThisresultisknownasthecylinderTheorem[103].ItisaconsequenceoftheimplicitfunctiontheoremappliedtotheequationφT(e)(α(e))=α(e).Inparticularwehave(F−1)α(E)=T(E)α˙(5.63)Notethatα(E)·∇H=1soα(E)=0andthenon-degeneracyassumptionimpliesthat{˙α,α(E)}isabasisforthegeneralizedeigenspaceE1fortheeigenvalue1ofF(wehaveanon-trivialJordanblockifT(E)=0).LetVbethesymplecticorthogonalofE1:V={v|σ(v,α)˙=σ(v,α(E))=0}.TherestrictionofFtoVisthealgebraiclinearPoincarémapPγ(al).Using(5.62)wecaneasilyprovethatPγandPγ(al)areconjugate:MPγ=Pγ(al)whereMisaninvertiblelinearmapfromNγontoV.SowehaveP(al)det(Pγ−1)=detγ−1Inparticularifγisnon-degeneratethenPγ−1isinvertible.Thestrategyistosimplifyasfaraspossiblether.h.s.in(5.61).TosimplifyourdiscussionweshallassumethatT(E)=0.ItisnotarestrictionbecauseifT=0wecanperturbalittleFbyFε,ε>0,suchthatFεα˙=FonV,Fεα˙=α,Fεα(E)=α(E)+εα˙ThedeterminantwehavetocomputedependsonlyonthesymplecticmapF,sowecancomputewithFεα˙andtakethelimitasε→0.Thefirststepistofindv∈C2nsuchthat(F−1)v+v·˙αK−1α˙=˙α(5.64) 1425TraceFormulasandCoherentStatesWiththisv:=v0weget−1−1d−v0·KαK˙α˙detiΨE,⊥(t0,α0)=idet(5.65)0K(F−1)+iΠα˙whereΠ(v·˙α)α˙.α˙=2|˙α|Adirectcomputationgives−1iKα˙=−(F+1)∇H2so(5.64)istransformedintoi(F−1)v=v·˙α(F+1)∇H2Using(F−1)T∇H=0wehavev·˙α=0,sowehavetosolve(F−1)v0=˙α(5.66)Wearelookingforv0=λα˙+μα(E)andwefind1α(E)·˙αv0=α˙−α(E)(5.67)T(E)|˙α|2Soourfirstsimplificationgivestheexpressiondeti−1−1d−v−1ΨE,⊥(t0,α0)=i0K˙α˙detKdetF−1+iKΠα˙Forthefirsttermwegetσ(α(E),α)˙d−v0K˙α˙=−iv0·Jα˙=iT(E)detKisalreadycomputed.Weshallcomputedet(F−1+iK−1Πα˙)inasymplecticbasis{v1α(E)where1,v2,v3,...,v2n}wherev1=˙α,v2=∇H·α(E)σ(v2j−1,v2j)=1for1≤j≤nandσ(vj,vk)=0if|j−k|=1.Inthisbasiswehave−1vj·˙α−1KΠα˙vj=KΠα˙α˙|˙α|2SocombiningwiththefirstcolumnwecaneliminatethetermsK−1Πα˙vjforj≥2andusingthatF−1isinvertibleonVwecanassumethatinthefirstcolumnonlythetwofirsttermsarenotzero.Finallywehaveobtained⎛⎞x1δ−1⎝0⎠=−δxdetF−1+iKΠα˙=detx202det(Pγ−1)0[Pγ−1](5.68) 5.5APointwiseTraceFormulaandQuasi-modes143Itislefttocomputex2andδ.WehaveT(E)δ=σ(F−1)v=−σα,α˙(E)(5.69)2,v2(∇H·α(E))21σ(F+1)∇H,v=|∇H|2(5.70)x2=−12Soweget−1Ψ(tn2−1detiE,⊥0,α0)=2 ∇H det(Y)det(Pγ−1)(5.71)Usingtheexpression(5.40)wefindtheleadingtermρ1,γ(E)forthecontributionoftheperiodicpathγinformula(5.18),assumingforsimplicitythatA=1,n/2−1−1−1/2−1/21/2ρ1,γ(E)=(2π)detYU[detU]c[detY]∗−1/2iψE−1׈g(Tγ)det(Pγ−1)e ∇H (5.72)where[u]1/2denotesasuitablebranchforthesquareroot.Sowegetn/2−1i(Sγ/+σγπ/2)∗det(1−P−1/2ρ1,γ(E)=(2π)egˆ(Tγ)Tγγ)(5.73) withσγ∈ZandSγ=pdq.γLetusremarkthat,becausePγissymplecticand1isnoteigenvalueofPγ,weσ|det(Phavedet(Pγ−1)=(−1)γ−1)|whereσisthenumberofeigenvaluesofPγsmallerthan1.Soweseethatiσγπ/2iσπ/2e=±e(5.74)ThuswegetthatthecontributionoftheMaslovindexinTheorem28istodeterminethesignin(5.74).Wehavegivenhereananalyticalmethodforitscomputation.Wedonotconsideritsgeometricalinterpretation(Maslovcycle)forwhichwerefertotheliteratureonthissubject[60,134,166]andreferencesintheseworks.γTheothercoefficients,darespectralinvariantswhichhavebeenstudiedbyjGuilleminandZelditch.Inprinciplewecancomputethemusingthisexplicitap-proach.ThiscompletestheproofofTheorem28.5.5APointwiseTraceFormulaandQuasi-modesFromthewellknownBohrSommerfeldquantizationrulesitisbelievedthatthereexiststrongconnectionsbetweenperiodictrajectoriesofaclassicalsystemHandboundstatesofitsquantizationHˆ.Inthissectionwediscusssomepropertiesoflo-calizationforboundstatesorapproximateboundstates(quasi-modes)nearperiodictrajectoriesinthesimplestcases.Moregeneralresultsareprovedin[153]. 1445TraceFormulasandCoherentStates5.5.1APointwiseTraceFormulaTheideaofthisformulahasappearedin[153].Wegivehereaproofofthemainre-sultof[153]forGaussiancoherentstates.Weassumethatproperties(H.0)and(H.1)aresatisfied.Considerthelocaldensityofstatesdefinedforeveryα∈R2nbyE−Ej()2ρE(α)≡gχEj()ψα,ψjjwhereψarethenormalizedeigenfunctionsforHˆ,Hψˆ=Ej()ψ.jjjTheorem31ThelocaldensityofstatesρE(α)hasthefollowingasymptoticbehav-ioras→0:1+kρE(α)≡k(g,α)2(5.75)k∈NThecoefficientsk(g,α)aresmoothinαandaredistributionsing˜.Theirexpres-sionsdependonthebehaviorofthepatht →φtα.(i)Ifthepatht →φtαhasnoperiodicpointwithperiodinsuppg˜thenk(g,α)aredistributionsing˜supportedin{0}.Inparticulartheleadingtermis1−n+110(g,α)=√π2g˜(0)(5.76)2 ∇H(α) (ii)Ift →φtαhasaprimitiveperiodT∗,k(g,α)aredistributionsing˜supportedin{mT∗,m∈Z}.Inparticulartheleadingtermis1n+1−∗0(g,α)=√π2g˜mTC(m)(5.77)2m∈Zwhere−1C(m)=1−W∗α˙·˙α2.mTRecallthatWtdependsonthemonodromymatrixFt.ProofAsforthetraceformula(5.18),wefirstgiveatime-dependentformulafor−itHˆρE(α)withthepropagatorU(t)=e.IfΠzistheorthogonalprojectiononthecoherentstateϕzwehaveE−HˆρE(α)=TrgχHˆΠαbycomputingthetraceonthebasisψ.Sowegetj1tρE(α)=dtg˜(t)eiEϕα,U(t)χHˆϕα(5.78)2π 5.5APointwiseTraceFormulaandQuasi-modes145In(5.78)theintegrandisthesameasintheproofofthetraceformula.Thedifferenceisthatherewehaveonlyatimeintegration.Sothestationaryphasetheoremismuchsimplertoapply:αisfixedsuchthatH(α)=E,thecriticalsetofthephaseψE,isdefinedbytheequationφtα=αsowehavet=mT∗whereT∗istheprimitiveperiod(T∗=0ifαisnotaperiodicpointoftheflow).ψ¨E(t,α)isherethesecondderivativeintimeofψE.Sowehaveiψ¨E(t,α)=1−Wtα˙·˙α2Fort=0wehaveψ¨i ∇H 2.Usingthat∇H=0,thestationaryphaseE(α)=2theoremgivesthepart(i)oftheTheorem.FortheperiodiccasewehavetorecallthatWtisacomplexsymmetricn×nmatrixandthatW∗W<1.WiththispropertieswehaveeasilythatforeveryT>0thereexistscT>0suchthat21−Wtα˙·˙α≥ct ˙α fort∈[−T,T]Sothecriticalpointst=mT∗,m∈Z,arenondegenerateandthestationaryphasetheoremgivesthepart(ii)oftheTheorem.5.5.2Quasi-modesandBohr–SommerfeldQuantizationRulesQuasi-modes(orapproximatedeigenfunctions)canbeconsideredinmoregeneralandmoreinterestingcases(see[125,153,159,160])butforsimplicityweshallconsiderheremainlythefullyperiodiccase.Wealwaysassumethat(H.0)and(H.1)aresatisfied.Weintroduce:(H.P)ForeveryE∈[E−,E+],ΣEisconnectedandtheHamiltonianflowΦtisHperiodiconΣEwithaperiodTE.Remark29Forn=1theperiodicityconditionisalwayssatisfied.Ford>1thisconditionisratherstrong.Neverthelessitissatisfiedforintegrablesystemsandforsystemswithalargegroupofsymmetries.Letusfirstrecallaresultinclassicalmechanics(GuilleminSternberg,[95]):Proposition49Letusassumethataboveconditionsaresatisfied.LetγbeaclosedpathofenergyEandperiodTE.ThentheactionintegralJ(E)=pdqdefinesγafunctionofE,C∞in]E−,E+[andsuchthatJ(E)=TE.InparticularforonedegreeoffreedomsystemswehaveJ(E)=dzH(z)≤E 1465TraceFormulasandCoherentStatesNowwecanextendJtoanincreasingfunctiononR,linearoutsideaneigh-borhoodofI.LetusintroducetherescaledHamiltonianKˆ=(2π)−1J(H)ˆ.Usingpropertiesconcerningthefunctionalcalculus[107],wecanseethatKˆhasallthepropertiesofHˆandfurthermoreitsHamiltonianflowhasaconstantperiod2πinK0=K−1(λ)forλ∈[λ1Σ−,λ+]whereλ±=J(E±).Soinwhatfollowswere-λ02πplaceHˆbyKˆ,itsenergyrenormalization.Indeed,themapping1Jisabijective2πcorrespondencebetweenthespectrumofHˆin[E−,E+]andthespectrumofKˆin1J(E[λ−,λ+],includingmultiplicities,suchthatλj=j).2πK0LetusdenotebymtheaverageoftheactionofaperiodicpathonΣandbyλμ∈ZitsMaslovindex(m=1pdx−2πF).Undertheaboveassumptionsthe2πγfollowingresultswereprovedin[107],usingsemi-classicalFourier-integralopera-torsandideasintroducedbeforebyColindeVerdière[44]andWeinstein[195].5.5.2.1StatementsofResultsConcerningSpectralAsymptoticsTheorem32[44,107,195]ThereexistC0>0and1>0suchthat spectKˆ∩[λ−,λ+]⊆Ik()(5.79)k∈Zwithμ2μ2Ik()=−m+k−−C0,−m+k−+C044for∈]0,1].Letusremarkforsmallenough,theintervalsIk()donotintersectandthistheoremgivestheusualBohrSommerfeldquantizationconditionsfortheenergyspectrum,moreexplicitly,1μJ(Ek−−m+O2λk=k)=2π4Underastrongerassumptionontheflow,itispossibletoestimatethenumberofstatesineachclusterIk().tK0(H.F)ΦhasnofixedpointinΣ,∀λ∈[λ−−ε,λ++ε]and∀t∈]0,2π[.K0FLetusdenotebydk()thenumberofeigenvaluesofKˆintheintervalIk().Theorem33[42,44,108]Undertheaboveassumptions,forsmallenoughandμ−m+(k−)∈[λ−,λ+],wehave4μdj−dk()≡Γj−m+k−(5.80)4j≥1 5.5APointwiseTraceFormulaandQuasi-modes147withΓj∈C∞([λ−,λ+]).Inparticular−ddνΓ1(λ)=(2π)λΣλIntheparticularcasen=1wehaveμ=2andm=−min(H0)hencedk()=1.FurthermoretheBohrSommerfeldconditionstakethefollowingmoreaccurateform:Theorem34[107]Letusassumen=1andm=0.Thenthereexistsasequencefk∈C∞([F−,F+]),fork≥2,suchthat1λk∞+hfk(λ)=++O(5.81)2k≥21)∈[λfor∈Zsuchthat(+−,λ+].2Inparticularthereexistsgk∈C∞([λ−,λ+])suchthat11+hkg∞(5.82)λ=+k++O22k≥21)∈[Fwhere∈Zsuchthat(+−,F+].2WecandeducefromtheabovetheoremandTaylorformulatheBohrSommer-feldquantizationrulesfortheeigenvaluesEnatallorderin.Corollary17Thereexistλ →b(λ,)andC∞functionsbjdefinedon[λ−,λ+]suchthatb(λ,)=j∈Nbj(λ)j+O(∞)andthespectrumEnofHˆisgivenby1En+,+O∞(5.83)n=b2fornsuchthat(n+1)∈[λ−,λ+].Inparticularwehaveb−10(λ)=J(2πλ)and2b1=0.WhenH−1(I)isnotconnectedbutsuchthattheMconnectedcomponentsaremutuallysymmetric,underlinearsymplecticmaps,thentheaboveresultsstillhold[107].Remark30Forn=1,themethodsusuallyusedtoproveexistenceofacompleteasymptoticexpansionfortheeigenvaluesofHˆarenotsuitabletocomputetheco-efficientsbj(λ)forj≥2.Thiswasdonerecentlyin[46]usingthecoefficientsdjkappearinginthefunctionalcalculus. 1485TraceFormulasandCoherentStates5.5.2.2AProofoftheQuantizationRulesandQuasi-modesWeshallgivehereadirectprooffortheBohrSommerfeldquantizationrulesbyusingcoherentstates,following[26].Asimilarapproach,withmorerestrictiveas-sumptions,wasconsideredbeforein[151].Thestartingpointisthefollowingremark.Letr>0andsupposethatthereexistsCrsuchthatforevery∈]0,1],thereexistE∈Randψ∈L2(Rd),suchthatHˆ−Eψ≤Cr,andliminf ψ :=c>0(5.84)r→0Iftheseconditionsaresatisfied,weshallsaythatHˆhasaquasi-modeofenergyEwithanerrorO(r).Withquasi-modeswecanfindsomepointsinthespectrumofHˆclosetotheenergyE.Moreprecisely,ifδ>Cr,theinterval[E−δr,E+δr]cmeetsthepointspectrumofHˆ.Thisiseasilyprovedbycontradiction,usingthatHˆisself-adjoint.SoifthespectrumofHˆisdiscreteinaneighborhoodofE,thenweknowthatHˆhasatleastoneeigenvaluein[E−δr,E+δr].LetusassumethattheHamiltonianHˆsatisfiesconditions(H.0),(H.1),(H.P).UsingProposition49,wecanassumethattheHamiltonianflowΦtHhasacon-stantperiod2πinH−1]E−−ε,E++ε[,forsomeε>0.Followinganoldideainquantummechanics(A.Einstein),letustrytoconstructaquasi-modeforHˆwithenergiesE()closetoE∈[E−,E+],relatedwitha2πH0periodictrajectoryγE⊂Σ,bytheAnsatzE2π()itEψγ=eU(t)ϕzdt(5.85)E0wherez∈γE(ψγisastatelivingonγE).LetusintroducetherealnumbersE2π1μσ()=q(t)p(t)˙−H0q(t),p(t)dt+2π04wheret →(q(t),p(t))isa2π-periodictrajectoryγEinH0−1(E),E∈[E−,E+],μistheMaslovindexofγ.InorderthattheAnsatz(5.85)providesagoodquasi-mode,wemustfirstcheckthatitsmassisnottoosmall.Proposition50Assumethat2πistheprimitiveperiodofγE.ThenthereexistsarealnumbermE>0suchthat ψ1/41/2γ =mE+O(5.86)EProofUsingthepropagationofcoherentstatesandtheformulagivingtheactionof√metaplectictransformationsonGaussians,uptoanerrortermO(),wehave 5.5APointwiseTraceFormulaandQuasi-modes1492π2π2−d/2iΦ(t,s,x)−1/2 ψγ =(π)edet(At+iBt)E00Rd−1/2×det(As+iBs)dtdsdxwherethephaseΦis1Φ(t,s,x)=(t−s)E+(δt−δs)+(qs·ps−qt·pt)+x·(pt−ps)21+Γt(x−qt)·(x−qt)−Γs(x−qs)·(x−qs)(5.87)2Γtisthecomplexmatrixdefinedin(5.36).Letusshowthatwecancomputeanasymptoticsfor ψγ 2withthestationaryEphaseTheorem.Usingthat(Γt)ispositivenon-degenerate,wefindthatΦ(t,s,x)≥0,andΦ(t,s,x)=0⇔{x=qt=qs}(5.88)Ontheset{x=qt=qs}wehave∂xΦ(t,s,x)=pt−ps.Soif{x=qt=qs}thenwehavet=s(2πistheprimitiveperiodofγE)andwegeteasilythat∂sΦ(t,s,x)=0.Inthevariables(s,x)wehavefoundthatΦ(t,s,x)hasonecriticalpoint:(s,x)=(2)(t,qt).LetuscomputetheHessianmatrix∂s,xΦat(t,t,qt):(2)−(Γtq˙t−˙pt)·˙qt[Γt(q˙t−˙pt)]T∂s,xΦ(t,t,qt)=(5.89)Γt(q˙t−˙pt)2iΓtTocomputethedeterminant,weusetheidentity,forr∈C,u∈Cd,R∈GL(Cd)ruT10r−uT·R−1uuT−1=(5.90)uR−Ru10RThenweget2det−i∂2Φ(t,t,qs,xt)−1=Γtq˙t·˙qt+(Γt)(Γtq˙t−˙pt)·(Γtq˙t−˙pt)det[2Γt](5.91)ButEisnotcritical,so(q˙t,p˙t)=(0,0)andwefindthatdet[−i∂s,x2Φ(t,t,qt)]=0.ThestationaryphaseTheorem(seeAppendixA)gives√ 2=m2 ψγE+O()(5.92)Ewith2π2(d+1)/2√det(A−1/2det2−1/2mE=2πt+iBt)−i∂s,xΦ(t,t,qt)dt0(5.93)WenowgiveoneformulationoftheBohrSommerfeldquantizationrule. 1505TraceFormulasandCoherentStatesTheorem35LetusassumethattheHamiltonianHˆsatisfiesconditions(H.0),(H.1),(H.P)withperiodTE=2πandthat2πisaprimitiveperiodforaperiodictrajectoryγE⊆ΣE.Then−1/4ψγisaquasi-modeforHˆ,withanerrortermO(7/4),ifEsatisfiesEthequantizationcondition:μ1σ():=+pdq∈Z(5.94)42πγEMoreover,thenumberλ:=1pdq−Eisconstanton[E−,E+].Havingcho-2πγEsenC>0largeenough,theintervalsμ7/4μ7/4I(k,)=+b+k+λ−C,+b+k+λ+C44satisfy:ifI(k,)∩[E−,E+]=∅thenHˆhasaneigenvalueinI(k,).ProofWeuse,oncemore,thepropagationofcoherentstates.Usingperiodicityoftheflow,wehave,ifH(z)=E,2iπσ()ϕU(2π)ϕz=ez+O()(5.95)√Herewehavetoremarkthattheterminhasdisappeared.Thisneedsacalcula-tion.Byintegrationbyparts,weget2πitEHψˆγE=ie∂tU(t)ϕzdt02iπE=ieU(2π)ϕz−ϕz+EψγE=Eψ2(5.96)γ+OESo,wefinallygetaquasi-modewithanerrorO(7/4),using(5.86).Moreaccurateresultsonquasi-modes,usingcoherentstates,areprovedinpar-ticularin[125,164]. Chapter6QuantizationandCoherentStatesonthe2-TorusAbstractThetwodimensionaltorusT2isaverysimplesymplecticspace.Nev-erthelessitgivesnontrivialexamplesofchaoticdynamicalsystems.Thesesystemscanbequantizedinanaturalway.Weshallstudysomedynamicalandspectralpropertiesofthem.6.1IntroductionThe2-torusT2,withitscanonicalsymplecticform,isseenhereasaphasespace.ItisusefultoconsiderclassicalsystemsandquantumsystemsbuiltonT2foratleasttwopurposes.Dynamicalpropertiesofclassicalnon-integrableHamiltoniansystemsinthephasespaceRd×Rd(d≥2)arequitedifficulttostudy.Inparticulartherearenotsomanyexplicitmodelsofchaoticsystems.Butonthe2-torusitisveryeasytogetadiscretechaoticsystembyconsideringa2×2matrixFwithentriesinZandsuchthat|TrF|>2.Sowegetadiscreteflowt→Ftzfort∈Z,z∈T21,whereT2=R2/Z2isthe2-dimensionaltorus.In1980Hannay–Berrysucceededtoconstructa“goodquantization”R(F)ˆcor-respondingtothe“classicalsystem”(T2,F).FromthisstartingpointmanyresultswereobtainedconcerningconsequencesofclassicalchaosonthebehavioroftheeigenstatesoftheunitaryfamilyofoperatorsR(F)ˆaswellbyphysicistsandmathe-maticians.Inthissectionweshallexplainsomeoftheseresultsandtheirrelationshipwithperiodiccoherentstates.6.2TheAutomorphismsofthe2-TorusWehavealreadyseeninChap.1thatR2isasymplecticlinearspacewiththecanon-icalsymplecticbilinearformσ=dq∧dp.T2isalsoasymplectic(compact)man-ifoldwiththesymplectictwo-formσ=dq∧dpidentifiedwiththeplaneLebesguemeasure.1Fort∈N,FtzmeansthatweapplyFt-timesstartingfromz,andift>0thenFt=(F−t)−1.M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,151TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_6,©SpringerScience+BusinessMediaB.V.2012 1526QuantizationandCoherentStatesonthe2-TorusHerewecallautomorphismofthe2-torusT2anymapFinducedbyasymplecticmatrixF∈SL(2,Z).LetFbeoftheformabF=(6.1)cdwithentriesinZsatisfyingdet(F)=ad−bc=1;thecorrespondingmapofthe2-torusisgivenby(q,p)∈T2−→q,p∈T2,withq=aq+bp(mod1),p=cq+dp(mod1)SoFisasymplecticdiffeomorphismofT2.InparticularitpreservestheLebesguemeasuremLonT2.WeshallconsidernowthediscretedynamicalsysteminT2generatedbyF.Letusfirstrecallthedefinitionsandpropertiesconcerningclassicalchaos(er-godicity,mixing).Formoredetailswerefertothebooks[55,123,140].A(discrete)dynamicalsystemisatriplet(X,Φ,m)whereXisameasurablespace,maprobabilitymeasureonXandΦameasurablemaponXpreservingthemeasurem:ForanymeasurablesetE⊂Monehasm(Φ−1E)=m(E).Theorbit(ortrajectory)ofapointx∈XisO(x):={Φk(x),k∈Z}.TheorbitisperiodicifΦT(x)=xforsomeT∈Z,T=0.Definition13ForadynamicalsystemD=(X,Φ,m)letusconsiderthetimeav-1t=TterageorBirkhoffaverageET(f,x)=Tt=0f(Φ(x)),wherefismeasurable.Φisergodicifforanyfunctionf∈L1(X,m)onehaslimET(f,x)=m(f),m—everywhere(6.2)T→∞wherem(f):=fdmisthespatialaverage.XRemark31Ifadynamicalsystemisergodicitstimeaverage(inthesenseofthelefthandsideof(6.2))equalsthe“spaceaverage”,anddoesnotdependontheinitialpointx∈Xalmostsurely.Proposition51AdynamicalsystemD=(X,Φ,m)isergodicifandonlyifoneofthefollowingstatementsissatisfied:(i)AnymeasurablesetE⊂XwhichisΦ-invariantissuchthatm(E)=0orm(XE)=0.(ii)Iff∈L∞(X,m)isΦ-invariant(f◦Φ=f)thenitisconstantm-everywhere.See[123,139]forproofs.Thismeansinparticularthattheperiodicorbitsofanergodicdynamicalsystemarerather“rare”fromameasurablepointofview: 6.2TheAutomorphismsofthe2-Torus153Proposition52LetΦbeacontinuousmaponacompacttopologicalspaceXen-dowedwithaprobabilitymeasuremwhichisΦ-invariantandsuchthatm(U)>0foranyopensetU.If(X,Φ,m)isergodic,thenthesetofperiodicorbitsinXisofmeasure0.Althoughrelativelyrare,theperiodicorbithaveastrongimportanceintheframe-workofergodictheorysincetheyallowtheconstructionofinvariantmeasuresinthefollowingway.Letx∈XandletO(x)beaperiodicorbitofperiodp(x)∈R.Thefollowingprobabilitymeasureisclearlyinvariant:p(x)1mx=δΦk(x)p(x)k=1whereδaistheDiracdistributionatpointa∈R.GivenamapΦinXandmaninvariantmeasure,itisnotalwaystruethatmistheuniqueinvariantmeasure.IfitisthecasethemapΦissaid“uniquelyergodic”andtheuniqueinvariantmeasureisergodic(see[123]).Wellknownexamplesareirrationalrotationsonthecircle(ortranslationsonthetorusT1).Ifαisanirrationalnumber,Φ(x)=x+α,mod.1definesauniqueergodictransformationinT1,theuniqueinvariantmeasureistheLebesguemeasure(see[123]).Inthetopologicalframeworkonehasacharacterizationofsuchmaps[55]:Proposition53LetD=(X,Φ,m)beadynamicalsystemwithXacompactmetricspace,andΦacontinuousmap.Disuniquelyergodicifandonlyif∀f∈C(X):1T−1llimf◦Φ−m(f)=0k→∞Tl=0∞where
·
∞isthenormoftheuniformconvergence.Thereisastrongerpropertyofdynamicalsystemswhichisthe“mixing”prop-erty:Definition14AdynamicalsystemD=(X,Φ,m)issaidtobemixingif∀f,g∈L2(X,m)onehaslimfΦk(x)g(x)dm(x)=m(f)m(g)k→∞XThefollowingresultisusefulandeasytoprove.Proposition54D=(X,Φ,m)ismixingifandonlyifthereexistsatotalsetTinL2(X,m)suchthatforeveryf,g∈TwehavelimfΦk(x)g(x)dm(x)=m(f)m(g)k→∞X 1546QuantizationandCoherentStatesonthe2-TorusIntheframeworkwhereXisadifferentiablemanifold,onecandefinethenotionofAnosovsystem(see[123,139]):Definition15AdiffeomorphismΦofadifferentiablemanifoldMisAnosovif∀x∈M,thereexistsadecompositionofthetangentspaceatxindirectsumoftwosubspacesExuandExsandconstantsK>0,0<λ<1satisfying(DssuuxΦ)Ex=EΦ(x),(DxΦ)Ex=EΦ(x)andDxΦnEs≤Kλn,DxΦ−nEu≤Kλnxx∀x∈M,n∈N.Wehavethefollowingusefulstabilityresult(see[139]).Theorem36LetMbeacompactmanifoldandΦanAnosovdiffeomorphismonM.Thereexistsε>0smallenoughsuchthatifΦ−ΨC1(M)<εthenΨisanAnosovdiffeomorphismonM,whereΨC1(M)=supΦ(x)+DxΦ(x)x∈MTheorem37IfΦisadiffeomorphismAnosovonT2thenthedynamicalsystemD=(T2,Φ,μ)ismixing,μbeingthenormalizedLebesguemeasureonT2.LetF∈SL(2,Z).Thehyperbolicautomorphismofthe2-torusdefinedbyFrep-resentsthesimplestexamplesofhyperbolicdynamicalsystemswhen|TrF|>2.NamelyifthisissatisfiedthenFhastwoeigenvaluesλ+=λ>λ−=λ−1withλ>1.DenoteTx(T2)thetangentspaceatpointx∈T2,Ex+(resp.Ex−)theeigenspaceassociatedtotheeigenvalueλ(resp.λ−1)andDxF:Tx(T2)−→TFx(T2)thedifferentialofF.OnehasD=|λ|
vifv∈E+xF(v)xD=λ−1vifv∈E−xF(v)xwhere
·
isthenormassociatedtotheRiemannianmetricds2=dq2+dp2onT2.ThisprovesthatFisanAnosovdiffeomorphism,andisthereforeergodicandmix-ing.WecanalsogiveamoredirectproofthatFismixingusingProposition54.LetusconsiderthetotalfamilyinL2(T2),ek(x)=e2iπk·x,wherek∈Z2.WehaveeΦnk(x)e(x)dm(x)=ek(x)e(ΦT)n(x)dm(x)T2T2 6.3TheKinematicsFrameworkandQuantization155If=0,usingthatAhaseigenvaluesλandλ−1withλ>1,weseethat(ΦT)nislargefor±nlargehenceweget2ek(x)e(Φn(x))dm(x)=0.WecanconcludeTusingProposition54.OnecaneasilyidentifytheperiodicpointsofF:Proposition55Theperiodicpoints(q,p)∈T2ofanhyperbolicautomorphismofT2areexactlypoints(q,p)suchthat(q,p)∈Q2/Z2.ProofLetAbeanhyperbolicautomorphismofT2,andn∈N∗.ThenthefinitesetLrsn={(,),r,s=1,...,n}isinvariantunderAandsoallelementsofLnarennperiodicforA.Letm=n∈N∗.Sinceonehas rsLm,n=,,r=1,...,m,s=1,...,n⊂LmnmnallpointsofLm,narealsoperiodic.ThusallpointsinQ2/Z2areperiodicforA.Nootherpointcanbeperiodic.Namelyapoint(q,p)∈T2isperiodicofperiodk∈N∗ifandonlyifthereexists(m,n)∈Z2suchthatkqmA−1=pnButthematrixAk−1isinvertibleandhasonlyrationalentries.Thusqk−1mrk=A−1=pnskwith(rk,sk)∈Q2.Thiscompletestheproof.Remark32AnhyperbolicautomorphismofT2isalwaysmixing(soergodic)butneveruniquelyergodicsinceeveryperiodicpointxgivesaninvariantprobabilitymeasuremx.6.3TheKinematicsFrameworkandQuantizationWecloselyfollowtheapproachesof[10,27,29,30,59,104].Letusrecallthatweconsiderasphasespacethe2-torusT2=R2/Z2withitscanonicalsymplectictwoform.Usingthecorrespondenceprinciplebetweenclassicalandquantummechanics,itseemsnaturaltolookforthequantumstatesψhavingthesameperiodicityinpositionandmomentum(q,p)astheunderlyingclassicalsystem.TheWeyl–HeisenbergtranslationoperatorsT(q,p)ˆ“translate”thequantumstatebyavectorz=(q,p)∈R2.SowearelookingforsomeHilbertspaceH, 1566QuantizationandCoherentStatesonthe2-TorusincludedintheSchwartztemperatedistributionspaceS(R),suchthatforeveryψ∈HwehaveT(ˆ1,0)ψ=e−iθ1ψ(6.3)T(ˆ0,1)ψ=eiθ2ψ(6.4)whereweallowaphaseθ=(θ1,θ2)sincetwowavefunctionsψ1,ψ2satisfyingψ2=eiαψ1definethesamequantumstateand,moreimportantly,weshallrecovertheplanemodelasθrunsoverthesquare[0,2π[×[0,2π[.(6.4)meansthatthe-FouriertransformFψsatisfiesFψ(p+1)=e−iθ2Fψ(p)(6.5)RecallthatFψ(p)=(2π)−1/2e−iqp/ψ(q)dq.RFrom(6.3),(6.4)weseethatψmustbeajointeigenvectorfortheWeyl–HeisenbergoperatorsT(q,p)ˆandwegetT(ˆ0,1)T(ˆ1,0)ψ=T(ˆ1,0)T(ˆ0,1)ψSincewehaveT(ˆ0,1)T(ˆ1,0)=ei/T(ˆ1,0)T(ˆ0,1)conditions(6.3),(6.4)entailthefollowingquantificationcondition1=Nwhere2πN∈NandisthePlanckconstant.Moreover,thequantumstatesψliveinaN-dimensionalcomplexvectorspace.Thisresultcanbeobtainedusingthepowerfulmethodsofthegeometricquanti-zation[59].Herewefollowamoreelementaryapproachasin[29,30].LetusdenotebyHN(θ)thelinearspaceoftemperatedistributionsψsatisfyingperiodicityconditions(6.3),(6.4)with=1(remarkthatif=1andifψ2πN2πNsatisfies(6.3),(6.4)thenψ=0).Soinallthischapteritisassumedthat=12πNforsomeN∈N.Proposition56HN(θ)isaNdimensionalcomplexlinearsubspaceofthetemper-atedistributionspaceS(R).ProofLetψ∈HN(θ).Fromcondition(6.4)wefindthatthesupportofψisinthediscreteset{q2πj+θ2j=,j∈Z}.SoψisasumofderivativesofDiracdistributions2πNα(α)(α)ψ=cjδqj.Usinguniquenessofthisdecompositionwecanprovethatcj=0(0)forα=0,sowehaveψ=cjδqwherecj=c.Now,using(6.3)wegetajjperiodicityconditiononthecoefficientcj.Sowehaveciθ1ikθ1j+N=ecj,∀j∈Zandψ=cjeδqj+k(6.6)0≤j≤N−1k∈ZConverselyitiseasytoseethatifψsatisfies(6.6)thenψ∈HN(θ).Sothepropo-sitionisproven. 6.3TheKinematicsFrameworkandQuantization157Fromtheproofoftheproposition,wegetabasisofHN(θ):e(θ)=N−1/2eikθ1δjqj+k,0≤j≤N−1k∈Z(θ)eobviouslysatisfies(6.3).Letuscheckthatitsatisfies(6.5)bycomputingitsjFouriertransform.AsaconsequenceoftheusualPoissonformula:e2iπkx=δ(x)k∈Z∈ZwegetaftersomeeasycomputationsFe(θ)(p)=N−1e−2iπp(j+θ2/2π)δ(6.7)j+θ1N2π∈ZLetusintroducepθ1(θ)−1/2−ikθ2=N+2πandε=Nk∈Zeδ+θ1+k,for=N2π0,...,N−1.Wehavenow(θ)(θ)Fej=Fj,ε(6.8)0≤≤N−1wherethematrixelementFj,isgivenby−1/2iθ1θ2Fj,=Nexp−2πj+θ2+θ1j+N2π(θ)WeputonHN(θ)theuniqueHilbertspacestructuresuchthat{e}0≤j≤N−1isanjorthonormalbasis.SoweseethatFisaunitarytransformationfromHN(θ1,θ2)ontoHN(−θ2,θ1).Inparticularifθ=(0,0),thematrix{Fj,}isthematrixofthediscreteFouriertransform.Forallψ∈H(θ1,θ2)wehaveN−1(θ)ψ=cj(ψ)ejj=0ThenthevectorN−1cj(ψ)j=0isinterpretedphysicallyasthequantumstateoftheparticleinthepositionrepresen-tation.Similarlyinthemomentumrepresentationoneseesthatψ˜θ∈HN(−θ2,θ1)isdecomposedasN−1ψ˜θ=dj(ψ)ε(θ)jj=0 1586QuantizationandCoherentStatesonthe2-TorusOnegoesfromthepositiontothemomentumrepresentationviaageneralizedDis-creteFourierTransform:N−1θ2θ112πkθ1dkψˆ=exp−i+k√cj(ψ)exp−ij+N2πNNNj=0Forθ=(0,0)werecognizethediscreteFourieroperatorthatwehaveintroducedabove.AconvenientrepresentationformulaforelementsofHN(θ)canbeobtainedus-ingthefollowingsymmetrizationoperator:Σ(θ)=(−1)Nz1z2ei(θ1z1−θ2z2)T(z)ˆ(6.9)Nz∈Z2Letusremarkthatψ∈HN(θ)ifandonlyifψ∈S(R)satisfiesT(z)ψˆ=(−1)Nz1z2eiσ((θ2,θ1),(z1,z2))ψ(6.10)(θ)Proposition57ΣdefinesalinearcontinuousmapfromS(R)inS(R).ItsrangeNisHN(θ).Moreoverforeveryψ∈S(R)wehave(θ),Σ(θ)ψ=N−1/2eiθ1ψ(q(θ)ej−)=e(ψ)(6.11)jNj∈Zandψ(x)21e(θ)2dx=(ψ)dθ(6.12)2jR4π[0,2π[2(θ)2Themapψ→{e(ψ)}0≤j≤N−1canbeextendedasanisometryfromL(R)ontojtheHilbertspaceL2([0,2π[2,CN,dθ).4π2ProofRecallthatwehaveT(z)ψ(x)ˆ=e−iz1z2/2eixz2/ψ(x−z1)SowehaveΣ(θ)ψ=ei(θ1z1−θ2z2)eixz2/ψ(x−z1)Nz1,z2∈ZWefirstcomputethez2-sumusingthePoissonformula:1ei(xz2/−θ2z2)=δkθ2NN+2πNz2∈Zk∈Z 6.3TheKinematicsFrameworkandQuantization159weget1Σ(θ)ψ=eikθ1eiθ1ψ(qNj−)δqj+k(6.13)Nk∈Z0≤j≤N−1∈ZTheequalities(6.11)and(6.12)followeasilyfrom(6.13).(θ)(θ)Inparticularweseethatforeveryj=0,...,N−1wehavee=ΣψjjNwhereψ∞11j(x)=ψ0(qj−x),ψ0isC,withsupportin[−,]andψ0(0)=1.4N4N(θ)ThisprovedthatΣ(S(R))=HN(θ).N(θ)LetusdefinethemapI(ψ)={e(ψ)}0≤j≤N−1.WeknowthatIdefinesanjisometryfromL2(R)intoL2([0,2π[2,CN,dθ).WehavetoprovenowthatIis4π2onto.ItisenoughtoprovethattheconjugateoperatorI∗isinjectiveonL2([0,2π[2,CN,dθ).TodothatwehavetocomputeI∗f,ψwheref=(f20,...,fN−1),fj4πareperiodicalfunctionsonthelattice2πZ×2πZandψ∈S(R).Thisisanexerciselefttothereader.ThisleadstoadirectintegraldecompositionofL2(R):22π2π21L(R)∼=dθHN(θ)2π0022π2π1(θ)ψ∼=dθψ(θ),whereψ(θ)=Σψ2πN00ThisisaBlochdecompositionofL2(R)analogoustothedescriptionofelectronsinaperiodicstructure.ItappearsthatHN(θ)isequippedwiththenaturalinnerproductandthespacesHN(θ)arethenaturalquantumHilbertspacesofstateshavingthetorusasphasespace.Letusexplainnowinmoredetailtheidentification22π2π21L(R)∼=dθHN(θ)2π00(θ)2Foreveryψ∈S(R)wedefineψ(θ,j)˜=e(ψ)whereθ∈[0,2π[andj∈Z.jWehaveseenthatψ→ψ˜isanisometryfromL2(R)ontoL2([0,2π[2×(Z/NZ),dθ⊗dμ2N)whereμNistheuniformprobabilityonZ/NZ.4πLetAˆbesomeboundedoperatorinL2(R).AssumethatAˆisalinearcontinuousoperatorfromS(R)toS(R)andfromS(R)toS(R)andthatAˆcommuteswithΣ(θ)(AΣˆ(θ)=Σ(θ)Aˆ),foreveryθ∈[0,2π[2.ThenAˆisadecomposableoperatorNNN(seeReed–Simon[162],t.1,p.281).Morepreciselywehavethefollowingusefulresult. 1606QuantizationandCoherentStatesonthe2-TorusProposition58LetusdenotebyAˆN,θtherestrictionofAˆtoHN(θ).Thenwehave,foreveryψ1,ψ2∈L2(R),dθψ2,Aψˆ12=ψ2(θ),AˆN,θψ1(θ)(6.14)L(R)[0,2π[2HN(θ)4π2(θ)whereψ1(θ)=Σψ1.N(θ)ProofThisiseasilyprovedusingthatψ(θ)=0≤j≤N−1ψ(θ,j)e˜and(6.12).jWeshallapplythefollowingresultsprovedusingReed–Simon[162].Corollary18LetAˆbeadecomposableoperatorlikeabove.ThenwehaveAˆL2(R)=supAˆN,θH(θ)(6.15)θ∈[0,2π[2NandAˆisanisometryinL2(R)ifandonlyifAˆN,θisanisometryinHN(θ)foreveryθ∈[0,2π[2.FirstexamplesaretheWeyl–Heisenbergtranslations.Lemma36Letz=(z1,z2)∈R2.ThenT(z)Σˆ(θ)=Σ(θ)T(z)ˆifandonlyifNN2.Moreoverifzn1n2Nz∈Z1=andz2=wehaveNNTˆn1n2(θ)n1n2n2(θ)N,θN,Nej=expiπNexpi(θ2+2πj)Nej+n1(6.16)ProofExercise.Corollary19Theunitary(projective)representation(nn1n21,n2)→TˆN,θ(,)ofNN2(θ)thegroupZinHisirreducible.N(θ)(θ)ProofLetVbeaninvariantsubspaceofHNandv=0≤j≤N−1ajej,v=0.Ifan1,0)wegete(θ)∈V∀j.Butplayingj=0forj=j0,thenusingtranslationTˆN,θ(Njn2),ifmcoefficientsawithTˆN,θ(0,jarenot0thereexistsanonzerovectorofVNwithm−1nonnullcoefficients.SowecanconcludethatV=HN(θ).Remark33Ithasbeenprovedthatallirreducibleunitaryrepresentationsofthedis-creteHeisenberggroupareequivalentto(Tˆn1n2N,θ(,),HN(θ)),forsome(N,θ)∈NNN∗×[0,2π[2[63].Fortheparticularcaseθ=(0,0),thestatese0canbeidentifiedwiththenaturaljbasisinCN.ThenthetranslationoperatorsT(ˆ1/N,0),T(ˆ0,1/N)aresimplyN×N 6.3TheKinematicsFrameworkandQuantization161matricesofthefollowingform:T(ˆ0,1/N):=Z=diag1,ω,ω2,...,ωN−1(6.17)whereω=e2iπ/NistheprimitiveNthrootofunity.⎛⎞00...01⎜10...00⎟T(ˆ1/N,0):=X=⎜⎜....⎟⎟(6.18)⎝.......⎠....00...10Theseoperators(matrices)havebeenintroducedbySchwinger[175]as“general-izedPaulimatrices”andareintensivelyusedinquantuminformationtheoryforthemutuallyunbiasedbasesprobleminCN.See[50,175].Theyhavethefollowingproperties:Proposition59(i)XandZareunitary.(ii)Theyareidempotent,namelyXN=ZN=1(theidentitymatrixinCN).(iii)Theyω-commute:XZ=ωZX.(iv)XisdiagonalizedbythediscreteFouriertransformF:F∗XF=ZwhereFj,k=√1ωjk,∀j,k=1,...,N.NRemark34AcomplexN×NmatrixisanHadamardmatrixifallitsentrieshaveequalmodulus.NotethatFisanunitaryHadamardmatrixoftheVandermondeform,andthatXanditspowersgeneratethecommutativealgebraofthe“circulant”matrices.AN×NmatrixCissaidtobecirculantifallitsrowsandcolumnsaresuccessivecircularpermutationsofthefirst:⎛⎞c1c2...cN⎜cNc1...cN−1⎟⎜⎟C=circ(c1,c2,...,cN)=⎜........⎟⎝....⎠(6.19)c2c3...c1C=c11+cNX+···+c2XN−1(see[57]). 1626QuantizationandCoherentStatesonthe2-TorusTheDiscreteFouriertransforminCNisverynaturalinthiscontextsinceittransformsanybasisvectorinthepositionrepresentationintoanybasisvectorinthemomentumrepresentation.Lemma37ForanycirculantmatrixCthereexistsadiagonalmatrixDsuchthatF∗CF=DFurthermore√N−1D−jkj,j=Ncˆj=ck+1ω0ProofUseProposition59(iv)and(6.19).ThesepropertiesareveryusefultoconstructtheN+1mutuallyunbiasedbasesinQuantumInformationTheoryforNaprimenumber.See[50].6.4TheCoherentStatesoftheTorusAlreadyusedinthephysicalliteraturein[131]weintroducenowthecoherentstatesadaptedtothetorusstructureofthephasespace.Theywillbetheimagebythe(θ)periodisationoperatorΣoftheusualGaussiancoherentstatesstudiedinChap.1.NIndimension1onehas,forz=(q,p)∈R2andγ∈C,γ>0,1/42γiqpixp(x−q)ϕγ,z(x)=exp−++iγ(6.20)π22(θ)(θ)ϕγ,z=ΣNϕγ,z(6.21)(θ)ItiseasilyseenfromthedefinitionpropertiesofΣandtheproductrulesforT(z)ˆNthatiϕ(θ)=(−1)Nn1n2ei(θ1n1−θ2n2)+2σ(n,z)T(nˆ+z)ϕγ,zγ,0(6.22)n1,n2∈ZForeveryz=(z,z)∈Z2weget12(θ)Nzzi(zθ2−zθ1)iπNσ(z,z)(θ)ϕγ,z+z=(−1)12e21eϕγ,z(6.23)(θ)(θ)Thusthestatesϕγ,z+z,ϕγ,zareequalmodulophasefactor,sotheydescribethesamephysicalsystemandwecanidentifythem.Recallthatσisthesymplecticform:σ(a,b),(c,d)=ad−bc 6.4TheCoherentStatesoftheTorus163(θ)Theset{ϕγ,z}z∈T2thereforeconstitutesacoherentstatessystemadaptedtothetorus.(θ)Inthebasiseofthepositionrepresentationwehavej(θ)(θ)cj(q,p):=ej,ϕq,p1/4γ1−iqpiθmi(xmp)1m2=√e2e1ejexpiγxj−q(6.24)πN2m∈Zwherexm=j+θ2−m.jN2πNSimilarlywehaveinthemomentumrepresentation(forγ=i):1/411−iqp−imθ+iqξk−1k2dk(q,p)=√e2e2mexpξm−pπN2m∈Zwithξk=k+θ1−m.mN2πNAnimportantpropertywhichisinheritedfromtheovercompletenesscharacterofthesetofcoherentstatesinL2(Rn)isthatthe{ϕ(θ)}γ,zz∈T2formanovercompletesystemofHN(θ)witharesolutionoftheidentityoperator1HN(θ):Proposition60∀θ∈[0,2π)2and∀=1/2πNwehavedqdp1=ϕ(θ)ϕ(θ)HN(θ)γ,q,pγ,q,pT22π(θ)whereweusethebra–ketnotationfortheprojectoronthecoherentstateϕq,p.ProofForsimplicityweassumeγ=i.SinceHN(θ)isfinitedimensionalitisenoughtoprovethat∀(j,k)∈[0,N−1]2wehavedqdpϕθθeθθ=δq,p,ekj,ϕq,pj,kT22πNowusing(6.24)togetherwithFubini’sTheoremweget1dqdpc¯iθ1(m−n)k(q,p)cj(q,p)=edpexp2πij−k−N(m−n)pT22πm,n01×dqϕ¯θxm−qϕθxm−q(6.25)0,0j0,0j0Ifj=kthenthefirstintegralintherighthandsideof(6.25)iszeroexceptform=ninwhichcaseweget1.Thusweget1dqdpc2=dqϕθxj−q2=ϕθ2=1j(q,p)0,0m0,0(q,p)∈T22π0m∈ZIfj=kthesameintegraliszerosincej−k∈N∗. 1646QuantizationandCoherentStatesonthe2-TorusWealsogettheFourier–Bargmanntransformψofanystateψ∈HN(θ)as√ψ=Nϕθ,ψq,p√θ:ψ∈Hθ22dqdpThemapWN(θ)−→Wψ=Nψ∈L(T,)isobviouslyiso-2πmetric.Thequantityθθ2H(q,p)=ϕq,p,ψiscalledtheHusimifunctionofψ∈HN(θ).WehaveasacorollaryananalogousresultasProposition6:Corollary20LetAˆθ∈L(Hθ).ThendqdpTrAˆ=ϕθ,Aˆθϕθθq,pq,pT22πWehavethefollowingveryusefulsemi-classicalresult.Proposition61Foreverycomplexnumbersγ,γwithpositiveimaginarypartwehave:(i)ThereexistconstantsC>0,c>0suchthatforanyz,z∈T2,N≥1,√2ϕθθ≤CNe−d(z,z)cN(6.26)γ,z,ϕγ,zwhered(z,z)isthedistancebetweenzandzonthetorusT2.Inparticularforγ=γ=iwecanchoosec=π.(ii)Thereexistsc>0suchthat∀θ∈[0,2π)2wehave∀z=(q,p)∈T2ϕθ2−cNγ,z=1+OeProofForsimplicity,letassumethatγ=γ=i.Theproofisthesameforarbitraryγ,γ.Werecallthatinthecontinuouscaseonehas2|z−z|2ϕz,ϕz=exp−2sothatϕz
=1Thusweshallprovethattheanalogousproperties(i)and(ii)holdforthecoherentstatesofthe2-torusbutonlyinthesemi-classicallimitN→∞.Aweakerresultisgivenin[29],hereweshallgiveadifferentproof.Werewrite(6.23):foreveryz∈T2,m=(m1,m2)∈Z2,(θ)iπN(σ(z,m)+m1m2)i(m2θ2−m1θ1)iπNσ(z,z)(θ)ϕz+m=eeeϕz(6.27) 6.4TheCoherentStatesoftheTorus165θθLetusdenotefz,z(θ)=ϕz,ϕzandconsiderfz,zasaperiodicfunctioninθforthelattice(2πZ)2.ItsFouriercoefficientcm(z,z)canbecomputedusing(6.27),dθc−im·θθθmz,z=eϕz,ϕz2(6.28)[0,2π[24πdθ=eiπN(σ(z,m)+m1m2)ϕθθ(6.29)z,ϕz+ˇm2[0,2π[24πiπN(σ(z,m)+m1m2)=eϕz,ϕz+ˇm(6.30)wheremˇ=(m1,−m2).fz,zbeingasmoothfunctioninθ,wegetfz,z(θ)≤cmz,z(6.31)m∈Z2−1|z−z−ˇm|2But|cm(z,z)|=(π)exp(−).So2−1|z−z−ˇm|2cmz,z=(π)exp−2m∈Z2m∈Z2Nowwehave|z−z−ˇm|2≥|z−z|2+|m|2−2|m||z−z|Soweget|z−z−ˇm|2−|z−z|2πN−|m|2πN/2exp−≤ee√2m∈Z2,|m|≥42m∈Z2SoforeveryN≥1weget|z−z−ˇm|2−|z−z|2πNexp−≤Ce√2m∈Z2,|m|≥42Forthefinitesumwehaveeasily|z−z−ˇm|2−d(z,z)2πNexp−≤Ce√2m∈Z2,|m|≤42soweget(i).Concerningtheproofwithanyγ,γ,wehavetousetheinequality|z−z|2ϕγ,z,ϕγ,z≤C−1/2e−cwhereC>0,c>0dependonγ,γ,butnotinz,z. 1666QuantizationandCoherentStatesonthe2-TorusTheproofof(ii)usesthesamemethodwithz=z.Sowegetf≤ϕz,z(θ)−1z,ϕz+mm=(0,0)and√2f≤Ne−|m|πNz,z(θ)−1m=(0,0)Sowehaveproved(ii).6.5TheWeylandAnti-WickQuantizationsonthe2-TorusWewillshowhowaphase-spacefunction(classicalHamiltonian)H∈C∞(T2)canbequantizedasaselfadjointoperatorintheHilbertspaceHN(θ).Thesefunctionshavetobereal.6.5.1TheWeylQuantizationonthe2-TorusWeidentifythefunctionsHwiththefunctionsC∞onR2ofperiod(1,1)∈R2.ThenwehaveH(q,p)=H2iπσ((q,p),(m,n))m,ne(m,n)∈Z2Thenwedefine,following[104]and[64]:Definition16mnW(H)=HOpm,nTˆ,(6.32)NNm,nRecallthat=1,N∈N∗.2πNOnehasthefollowingproperty:Proposition62Letθ∈[0,2π)2and>0.ThenforanyfunctionH∈C∞(T2)onehasOpW(H)HN(θ)⊆HN(θ)ProofThisfollowsdirectlyfromthedefinitionofHN(θ)andfromT(m,n)ˆOpW(H)T(m,n)ˆ∗=OpW(H),ifm,n∈ZW(θ)InotherwordsOp(H)commuteswithΣ.N 6.5TheWeylandAnti-WickQuantizationsonthe2-Torus167ThuswedefinetheoperatorOpW(H)∈L(Hθ)astherestrictionof(6.32)to,θHN(θ).InthedecompositionofL2(R)asadirectintegral,OpW(H)isthefiberatθof,θOpW(H).WWdθOp(H)=Op(H)(6.33),θ2[0,2π[24πWehavealsothefollowingformulabyrestrictiontoHN(θ):nmOpW(H)=H(6.34)n,mTˆ,,θNNn,m∈ZInparticularweseethatthemapH→OpW(H)cannotbeinjective,sotheWeyl,θsymbolHofOpW(H)isnotunique.Itbecomesuniquebyrestrictingtotrigono-,θmetricpolynomialssymbols.LetusdenotebyTNthelinearspacespannedbyπn,m(q,p)=e2iπ(nq−mp),forn,m=0,...,N−1.ThenwehaveProposition63OpWisaunitarymapfromT22N(withthenormofL([0,1])onto,θL(HN(θ)),equippedwithitsHilbert–Schmidtnorm.Inparticularwehave,forev-eryH,K∈TN,TrOpW(H)OpW(K)∗=NH(z)K(z)dz(6.35),θ,θT2ProofLetusrecalltheformulaTˆk,e(θ)=eiπk/Nei(θ2+2πj)/Ne(θ)(6.36)NNjj+kUsingthatthediscreteFouriertransformisunitary,weget∗kkTrTˆ,Tˆ,=Nδk,kδ,(6.37)NNNNSothesystem{N−1/2T(ˆk,)}0≤k,≤N−1isanorthonormalbasisinL(HN(θ)),NNequippedwithitsHilbert–Schmidtnormandwegettheproposition.Corollary21EverylinearoperatorHˆinHN(θ)hasauniqueWeylsymbolH∈TN,H2iπσ(z,(m,n))H(z)=m,ne0≤m,n≤N−1where∗−1/2HˆTˆmnHm,n=NTr,(6.38)NN 1686QuantizationandCoherentStatesonthe2-TorusAsemi-classicalresultfortheWeylquantizationisthefollowing:Proposition64ForallH∈C∞(T2)onehas1limTrOpW(H)=dzH(z),θN→∞NT2(θ)ProofUsingtheorthonormalpositionbasiseonehasjN−1mnW(H)=HθθTrOp,θn,mej,Tˆ,ejNNj=0m,nNowweusetheproperty(6.36):1TrOpW(H)N,θ1N−1=Hn,N+kNj,k=0,n∈Zπn2πnθ2θθ×expi(N+k)−iθ1+i+j+kej,ej+kNN2πnθ2N−1n−iθ+i1i2πnj=Hn,N(−1)e1NeN(6.39)N,nj=0Thusweconclude1W(H)=Hiσ((,n),(θ1,θ2))TrOp0,0+HN,nNeN,θ,n∈Z∗Thelasttermtendsto0becauseoftheregularityofH,andthefirstoneisT2dzH(z),whichcompletestheproof.6.5.2TheAnti-WickQuantizationonthe2-TorusAsinthecontinuouscase(seeChap.2)theAnti-Wickquantizationisassociatedtothesystemofcoherentstates.Definition17LetH∈L∞(T2).Thenϕq,pθbeingthesystemofcoherentstatesdefinedintheprevioussection,wedefinedzOpAW(H):=H(z)ϕθϕθ,θzzT22π 6.5TheWeylandAnti-WickQuantizationsonthe2-Torus169Remark35NotethatOpAW(H)forH∈C∞(T2)issimplytherestrictionof,θAW(H)(consideredasanoperatoronS(R))toHOpN(θ).Letusrecallthatwealwaysassume2πN=1.AsforWeylquantization,Anti-WickquantizationonR2andonT2arerelatedwithadirectintegraldecompositionProposition65LetH∈C∞(T2).ThenwehavethedirectintegraldecompositionAWAWdθOp(H)=Op(H)(6.40),θ2[0,2π[2(2π)InparticularwehavetheuniformnormestimateOpAW(H)≤
H,θ∞(6.41)ProofUsingperiodicityofHanddirectintegraldecompositionofψ∈S(R)(θ)(ψ(θ)=Σψ),wegetNdθOpAW(H)ψ=H(z)ϕθ,ψ(θ)ϕz+nz+ndz2(6.42)2[0,2π[2T24πn=(n1,n2)∈ZUsingperiodicityinzofϕzandϕz+n=eiσ(n,z)/2T(n)ϕˆz,wegetdθAW(H)ψ=H(z)ϕθ,ψ(θ)ϕOpzz+ndz2(6.43)[0,2π[2T24πAW(H))AWSowehaveproved(Opθ=Op(H).,θNowweshowalinkbetweenAnti-WickquantizationandtheHusimifunction:Proposition66OnehasforanyH∈C∞(T2)andforanyψ∈Hθψ,OpAWHψ=NdzH(z)Hψ(z),θT2Andwehavethefollowingsemi-classicallimit:Proposition67Foranyz∈T2,anyθ∈[0;2π[2,andanyH∈C∞(T2)wehavelimϕθ,OpAWHϕθ=H(z)z,θzN→∞ProofWedenotez=(q,p)∈T2andBε(z)theballofcenterzandradiusεandbyBεc(z)itscomplementaryset.Takeε0.Wehaveϕθ,OpAW(H)ϕθ=dzHzϕθ,ϕθ2+dzHzϕθ,ϕθ2z,θzzzzzBεc(z)2πBε(z)2π 1706QuantizationandCoherentStatesonthe2-TorusItisclearthatthefirsttermintherighthandsidetendsto0as→0becauseofProposition61(i).Forthesecondtermwedenoteg(z,z)=N|ϕzθ,ϕθ|2.WehavezdzHzgz,z−H(z)Bε(z)≤dzHz−H(z)gz,z+H(z)gz,zdz−1Bε(z)Bε(z)≤ε
∇H∞gz,zdz+H(z)dzgz,z−1(6.44)Bε(z)Bε(z)Usingtheresolutionofidentitywehaveϕ(θ)2dzgz,z=z−dzgz,zBε(z)Bεc(z)UsingProposition61(ii)thefirsttermintherighthandsidetendsto1asN→∞,anditisclearthatthesecondissmallasN→∞.Thiscompletestheproof.AsinthecontinuouscasetheWeylandAnti-Wickquantizationsareequivalentinthesemi-classicalregime:Proposition68ForanyH∈C∞(T2)andanyθ∈[0,2π)2wehaveOpW(H)−OpAW(H)=ON−1,asN→∞(6.45),θ,θL(HN(θ))ProofThisresultfollowsfromthesimilaroneinthecontinuouscase(seeChap.2,Proposition27)usingtheestimateOpW(H)−OpAW(H)≤OpW(H)−OpAW(H)(6.46),θ,θL(HN(θ))L(L2(R))6.6QuantumDynamicsandExactEgorov’sTheorem6.6.1QuantizationofSL(2,Z)WenowconsideradynamicsinphasespaceinducedbysymplectictransformationsF∈SL(2,Z).ItcreatesadiscretetimeevolutioninT2andthen-stepevolutionisprovidedbyFn.OnewantsheretoquantizeFasanaturaloperatorinHN(θ).WehaveseeninChap.2(3.3)thatFisquantizedinL(L2(R))bythemetaplectictransformationR(F)ˆ.Letusrecallthefollowingproperty:R(F)ˆ∗T(z)ˆR(F)ˆ=TˆF−1z(6.47) 6.6QuantumDynamicsandExactEgorov’sTheorem171WeshallseenowhowtoassociatetoFanunitaryoperatorinHN(θ).Proposition69LetF∈SL(2,Z).Thenforanyθ∈[0,2π)2thereexistsθ∈[0,2π)2suchthatR(F)ˆHθ⊆HθFurthermoreθisdefinedasfollows:θθ2abmod2π2=F+πN,(6.48)θθ1cdmod2π1MoreoverwehaveR(F)Σˆ(θ)(θ)R(F)ˆ(6.49)=ΣNNProofWeusehereProposition57andformula(6.9).From(6.47)weget,ifψ∈HN(θ)andz=(z1,z2)∈Z2,T(z)ˆR(F)ψˆ=R(F)ˆTˆ−1−i(σ(z,(θ2,θ1))+πNzz)R(F)ψˆ(6.50)F(z)ψ=e12where−1zd−bz1z=Fz=1=z−caz22Wehaveσ(z,(θ2,θ1))=σ(z,F(θ2,θ1))andz=−cdz2+(ad+bc)z2z1211z2−abz2Butmodulo2wehavez2≡−z1,z2≡−z2,ad+bc≡1.Soweget,modulo2π,12,(θσz2,θ1)+πNz1z2≡z1(dθ1+cθ2+πNcd)−z2(bθ1+aθ2−πNab)+πNz1z2SowehaveR(F)ˆHθ⊂Hθwithθgivenby(6.48).Moreoveritiseasytocheckformula(6.49).Letusdenoteθ:=πF(θ).SoπFisasmoothmapfromthetorusR2/(2πZ)2intoitself.Remark36WecaneasilyseethatRˆθ(J)=F∗forθ=(0,0)inthebasisof(θ)N{e},uptoaphase.jj=1Definition18ForeveryF∈SL(2,Z),weshalldenoteRˆN,θ(F)therestrictionofR(F)ˆtoHRˆN(θ).ItisthequantizationofFinHN(θ).N,θ(F)isalinearoperatorfromHN(θ)inHN(θ). 1726QuantizationandCoherentStatesonthe2-TorusProposition70RˆN,θ(F)isaone-to-onelinearmapfromHN(θ)inHN(θ).FurthermorewehavethefollowingrelationshipbetweenRˆN,θ(F)andR(F)ˆ.⊕dθR(F)ˆ=VN,θRˆN,θ(F)(6.51)2[0,2π[2(2π)whereVN,θisthecanonicalisometryfromHN(θ)ontoHN(θ)definedby(θ)(θ)VN,θe=e.jjInparticularforeveryθ∈[0,2π[2,RˆN,θ(F)isaunitarytransformationfromHN(θ)ontoHN(θ).ProofWeknowthatR(F)ˆisanisomorphismfromS(R)ontoS(R)andfromS(R)ontoS(R).UsingthatHN(θ)isfinitedimensionalweseethatRˆN,θ(F)isaone-to-onelinearmapfromHN(θ)inHN(θ).WecaneasilycheckthatV(πF(θ))=Σ(θ)foreveryθ∈[0,2π[2.SousingN,θΣNNthatπFisanareapreservingtransformationweget,foreveryψ,η∈S(R),1η,R(F)ψˆ2=η(θ),VN,θRˆN,θψ(θ)dθL(R)4π2[0,2π[2HN(θ)Sothepropositionisprovedusingstandardpropertiesofdirectintegraldecomposi-tionsforoperators.Onehasthefollowingresultsconcerningtheinterestingcaseθ=θ.Theproofsarelefttothereaderorsee[29,30,104].Proposition71ConsiderF∈SL(2,Z)with|TrF|>2.Then∀N∈N∗thereexistsθ∈[0,2π)2sothatR(F)ˆHN(θ)⊆HN(θ)whereθcanbechosenindependentofNifandonlyifFisoftheformevenoddoddeven,oroddevenevenoddThecase|TrF|=3istheonlycasewherethechoiceofθisuniquewithθ=(π,π)forNoddandθ=(0,0)forNeven.MoreoverinthecaseevenoddF=oddeventhevalueθ=(0,0)isasolutionofthefixedpointequationθ=πF(θ).Remark37Ithasbeenshownin[64]thatforF∈SL(2,Z)ofthefollowingform:2g1F=22g−12g 6.6QuantumDynamicsandExactEgorov’sTheorem173(0)N−1theoperatorR(F)ˆhasmatrixelementsinthebasis{e}oftheformjj=0R(F)ˆCN2iπ22j,k=√expgj−jk+gkNNwhere|CN|=1,sothatitisrepresentedasaunitaryHadamardmatrix.WedonotknowatpresentwhetherthispropertyissharedbymoregeneralmapsF.6.6.2TheEgorovTheoremIsExactAsinthecontinuouscasetheEgorovtheoremisexactsinceR(F)ˆisthemetaplecticrepresentationofthelinearsymplecticmapF:Theorem38ForanyH∈C∞(T2)onehasRˆ∗OpW(H)RˆN,θ(F)=OpW(H◦F)(6.52)N,θ(F),θ,θProofBydenotingTˆθ(z)therestrictionofT(z)ˆtoHN(θ)onehasmnW(H)=HOpm,nTˆθ,,θNNm,n∈ZSomnRˆ∗OpW(H)RˆN,θ(F)=Hm,nRˆN,θ(F)∗Tˆ,RˆN,θ(F),θθN,θ(F)NNm,nButweknowthatRˆ∗TˆRˆTˆF−1z,∀z=(m/N,n/N)N,θ(F)θ(z)N,θ(F)=θWedothechangeofvariablesm−1m=FnnThenwegetmnRˆN,θ(F)∗OpW(H)RˆN,θ(F)=(H◦F)m,nTˆθ,=OpW(H◦F),θNN,θm,n∈ZThiscompletestheproof. 1746QuantizationandCoherentStatesonthe2-Torus6.6.3PropagationofCoherentStatesAsinthecontinuouscasethequantumpropagationofcoherentstatesisexplicitand“imitates”theclassicalevolutionofphase-spacepoints.Herethephasespaceisthe2-torusand∀z∈T2thetimeevolutionofthepointz=(q,p)isgivenbyqq1z==F,modpp1Weshalluse“generalizedcoherentstates”whichareactually“squeezedstates”.Takeγ∈Cwithγ>0.ThenormalizedGaussianϕγ∈L2(R)weredefinedinSect.6.4.1/42γiγxϕγ(x):=expπ2ThenthegeneralizedcoherentstatesinL2(R)areϕγγ,z:=T(z)ϕˆ(6.53)Thegeneralizedcoherentstatesonthe2-torusareasabove(θ)(θ)ϕγ,z=ΣNϕγ,z∈HN(θ)WetakesuchcoherentstateasinitialstateandapplytoitthequantumevolutionoperatorRˆθ(F).Onehasthefollowingresult:Proposition72LetF∈SL(2,Z)begivenbyabF=cdIfπF(θ)=θ,then1/2Rˆθ|bγ+a|θN,θ(F)ϕγ,z=ϕF·γ,Fzbγ+adγ+cwhereF·γ=.bγ+aProofWeknowthatΣ(θ)R(F)ˆ=R(F)Σˆ(θ).Letz=(q,p)∈T2.ThenwehaveNNRˆθ=RˆN,θ(F)Σ(θ)ϕγ,z=Σ(θ)R(F)ϕˆN,θ(F)ϕγ,zNNγ,zTheresultfollowsfromthepropagationofcoherentstatesbymetaplectictransfor-mationsintheplane(seeChap.3). 6.7EquipartitionoftheEigenfunctionsofQuantizedErgodicMaps1756.7EquipartitionoftheEigenfunctionsofQuantizedErgodicMapsonthe2-TorusOneofthesimplesttraceoftheergodicityofamapFonT2inthequantumworldistheequipartitionoftheeigenfunctionsofRˆN,θ(F)intheclassicallimitN→∞.Ithasbeenestablishedintheliteratureindifferentcontexts:forthegeodesicflowonacompactRiemannianmanifolditwasprovenby[45,174,206].ForHamiltonianflowsinRnitwasestablishedin[106],andforsmoothconvexergodicbilliardsin[83].Forthecaseofthed-torusthisproblemhasbeeninvestigatedin[28].Herewerestrictourselvesonthecaseofthe2-torus.Theorem39(Quantumergodicity)LetFbeanergodicareapreservingmaponT2,andRˆN,θ(F)∈U(Hθ)itsquantizationwhereθ=πF(θ).Denoteby{φN}j=1,...,NjtheeigenfunctionsofRˆN,θ(F).ThenthereexistsE(N)⊂{1,...,N}satisfying#E(N)lim=1N→∞Nsuchthat∀A∈C∞(T2)andallmapsj:N∈N−→j(N)∈E(N)wehave:limφN,OpW(A)φN=A(z)dz(6.54)j(N),θj(N)N→∞T2limφN,OpAW(A)φN=A(z)dz(6.55)j(N),θj(N)N→∞T2uniformlywithrespecttothemapj(N).Remark38ThisTheoremsaysthattheWignerdistributionandHusimidistribution(whendividedbyN)convergeinthesenseofdistributionstotheLiouvilledistribu-tionalongsubsequencesofdensityone.Webeginwithalemma:Lemma38LetusintroducethefollowingRadonprobabilitymeasuresμN,μ¯Nasjfollows:NNNAWN1Nμj(A)=φj,Op,θ(A)φj,μ¯N(A)=μj(A)Nj=1ThismeasuresareF-invariant.becauseφNareeigenstatesforRˆN,θ(F).jOnehas∀A∈C∞(T2):limμ¯N(A)=μ(A)N→∞μistheLiouvillemeasureonthe2-torusT2. 1766QuantizationandCoherentStatesonthe2-TorusProofWefindthatthemeasuresμNareF-invariant,moduloO(N−1),usingjEgorovtheoremandthatφNareeigenstatesforRˆN,θ(F).jClearlywehave1μ¯TrOpAW(A)N(A)=N,θSowehave1μ¯=TrOpAW(A)−μ(A)N(A)−μ(A),θN1≤OpAW(A)−OpW(A)+TrOpW(A)−μ(A),θ,θL(Hθ)N,θWededucetheresultusingPropositions64and68.Remark39TheLemmaisstilltruefortheSchwartzdistributionsνN(A)=jNWN1NNNφj,Op,θ(A)φjandν¯N(A)=Nj=1νj(A).TheνjareexactlyF-invariant.LetusprovenowProposition73ForeveryA∈C∞(T2)wehave1lim|μN(A)−μ(A)|2=0(6.56)jN→+∞N0≤j≤NProofWecanreplaceAbyA−μ(A)andassumethatμ(A)=0.Defineforn∈N∗the“time-average”ofA:k=n1kAn=A◦F.nk=1UsingtheRemarkafterLemma38,wecanreplaceμNbyνN.jjWehaveνj(A)=νj(An)foreveryn≥1.SowegetusingtheCauchy–SchwarzinequalityandFinvarianceofνN,jνN2WNN2OpWN2j(A)=Op(An)φj,φj≤(An)φj≤OpW(A∗WNN(6.57)n)Op(An)φj,φjButfromthecompositionruleforWeylquantization(Chap.2)wehave1W∗WW|A2Op(An)Op(An)=Opn|+O(6.58)N 6.8SpectralAnalysisofHamiltonianPerturbations177Foreveryn,considerthelimitN→+∞.UsingLemma38wegetμN2νN22limsupj(A)=limsupj(A)≤|An|dμN→+∞N→+∞T2UsingergodicityassumptionandtheLebesguedominatedconvergencetheoremwehavelim|A2n|dμ=0n→+∞T2Thelimit(6.56)followsifμ(A)=0.Now,theBienaymé–Tchebichevinequalitygivesthefollowingresultaccordingtowhich“almost-all”eigenstatesareequidistributedonthetorus.Proposition74ForanyH∈C∞(T2)and∀ε>0#{j:|μN(H)−μ(H)|<ε}jlim=1N→∞NAlongthesamelinesasin[106]onecanconcludesfortheexistenceofaH-independentsetE(N)suchthatthetheoremholdstrue.Remark40Anaturalquestionis“isthequantumergodictheoremtruewithE(N)=N”(uniquequantumergodicity)?Theanswerisnegative.In[58]thefollowingre-sultisproved.LetC={τ1,...,τK},KperiodicorbitsforF.Considertheprobabil-itymeasureμC,α=1≤j≤Kαjμτj,whereαj∈[0,1]and1≤j≤Kαj=1.ThenthereexistsasequenceNk→+∞suchthat11limφN,OpW(A)φN=A(z)dz+μNk,θNkC,α(A)(6.59)k→∞2T22Thisresultshowsthatsomeeigenstatescanconcentratealongperiodicorbits,thisphenomenonisnamedscarring.6.8SpectralAnalysisofHamiltonianPerturbationsThepreviousresultscanbeextendedtosomeperturbationsofautomorphismsofthetorusT2.LetHbearealperiodicHamiltonian,H∈C∞(T2)andF∈SL(2,Z).Letθ∈[0,2π[2besuchthatθ=πF(θ).WeconsiderherethefollowingunitaryoperatorinHN(θ),where2πN=1:εwRˆUε=exp−iOp,θ(H)N,θ(F) 1786QuantizationandCoherentStatesonthe2-TorusWeshallseethatifFishyperbolicandεsmallenoughthequantumergodictheoremisstilltrue.Topreparetheproofwebeginbysomeusefulpropertiesconcerningthepropa-−itHˆ−itHˆgatorsV(t)=e,VN,θ(t)=eN,θ.LetusintroducethefollowingHilbertspaces:Ks=Dom(Hˆosc+1)s/2,where2s/222d22s≥0,withthenormψs=
(Hˆosc+1)ψ(Hˆosc=−dx2+x).Lemma39Foreverys≥0andeveryT>0,thereexistsCs,T>0suchthatV(t)ψ≤Cs,Tψs,∀ψ∈Ks,∀t∈[−T,T],∀∈]0,2π]sProofItissufficienttoassumethats∈N(usingcomplexinterpolation).Fors=0weknowthatV(t)isunitary.LetusdenoteΛ=(Hˆosc+1)1/2.Letusassumes=1.ItisenoughtoprovethatΛV(t)Λ−1isboundedfromL2(R)intoL2(R).WehavedV(−t)ΛV(t)=V(−t)H,ΛˆV(t)idtUsingthesemi-classicalcalculus(Chap.2),andthatHisperiodic,weknowthati[Hˆ2,Λ]isboundedonL(R).Sothelemmaisprovedfors=1.Nowwewillprovetheresultforeverys∈Nbyinduction.Assumethelemmaisprovedfork≤s−1.ComputedV(−t)ΛsV(t)=V(t)H,ΛˆsV(t)idtButi[H,Λˆs]isanpseudodifferentialoperatoroforders−1fortheweightμ(x,ξ)=(1+x2+ξ2)1/2.Inparticulartheoperatori[H,Λˆs]Λ1−sisboundedonL2(R).Sowegettheresultforsusingtheinductionassumption.ThefollowingresultwillbeusefultotransformpropertiesfromthespaceL2(R)tothespacesHN(θ).Letψ∈S(R)andforθ∈[0,2π[2besuchthatψ(θ)˜=(ψ(θ,˜0),...,ψ(θ,N˜−1)∈CN(coefficientofψ(θ)inthecanonicalbasisofHN(θ)).LetusdenoteHs([0,2π[2)theperiodicSobolevspaceoforders≥0offunctionsfrom[0,2π[2NintoCNItsnormisdenoted
·
N,s.Lemma40Foreverys≥0thereexistsCssuchthatψ˜≤Csψs,∀ψ∈S(R)(6.60)N,sInparticularwehavethefollowingpointwiseestimate:foreverys>1thereexistsCssuchthatψ(θ)˜≤Csψs,∀ψ∈Ks(6.61) 6.8SpectralAnalysisofHamiltonianPerturbations179ProofLetusrecallthat,foreveryψ∈S(R),θ=(θ1,θ2)∈]0,2π]2,jθ2ψ(θ,j)˜=N−1/2eiθ1ψ(qj−),qj=+N2πN∈ZSowehave∂θψ(θ,j)˜=N−1/2eiθ1i(−qj)ψ(qj−)+iN−1/2qjψ(θ,j)˜(6.62)1∈Zd∂θψ(θ,j)˜=N−1/2eiθ1iψ(qj−)(6.63)2dx∈ZReasoningbyinductionon|m|=m1+m2,m=(m1,m2),weeasilyget∂m2(∂k2x2θψ(θ,j)˜dθ≤Cmx)ψ(x)+ψ(x)dx(6.64)[0,2π[2Rk+≤|m|Soestimate(6.60)follows.Estimate(6.61)isaconsequenceofSobolevestimateindimension2.(θ)Letusnowconsiderthepropagationofcoherentstatesϕγ,zunderthedynamicsVN,θ(t)inHN(θ).Weshallprovethatestimatescanbeobtainedfromthecorre-spondingevolutioninL2(R)(seeChap.4),usingthetwopreviouslemmas.Recalltheseresults.WehavecheckedapproximatesolutionsfortheSchrödingerequation:i∂tψt=Hψˆt,ψ0=ϕγ,z,γ>0(M)Wehavefoundψz,tsuchthat(M)(M)(N+3)/2(M)(M)i∂tψz,t=Hψˆz,t+Rz,t,ψz,0=ϕγ,z(6.65)(M)where,foreverys≥0,Rz,tKs=O(1)for→0.(M)ψz,thasthefollowingexpression:Miδtj/2x−qtΓtψz,t(x)=eπjt,√ϕz(x)(6.66)t0≤j≤Mwherezt=(qt,pt)istheclassicalpathinthephasespaceR2suchthatz0=zsatisfyingq˙∂Ht=(t,qt,pt)∂p(6.67)∂H(t,qp˙t=−t,pt),q0=q,p0=p∂qandϕΓt=T(zˆΓtztt)ϕ.(6.68) 1806QuantizationandCoherentStatesonthe2-TorusϕΓtistheGaussianstate:Γt−d/4iϕ(x)=(π)a(t)expΓtx.x(6.69)2Γtisacomplexnumberwithpositivenondegenerateimaginarypart,δtisarealfunction,a(t)isacomplexfunction,πj(t,x)isapolynomialinx(ofdegree≤3j)withtimedependentcoefficients.MorepreciselyΓtisgivenbytheJacobistabilitymatrixoftheHamiltonianflowz→Φtz:=zt.IfwedenoteH∂qt∂pt∂qt∂ptAt=,Bt=,Ct=,Dt=(6.70)∂q∂q∂p∂pthenwehaveΓ−1t=(Ct+γDt)(At+γBt),Γ0=γ,(6.71)tqtpt−q0p0δt(z)=psqs−H(zs)ds−,(6.72)02−1/2a(t)=det(At+γBt),(6.73)wherethecomplexsquarerootiscomputedbycontinuityfromt=0.UsingthetwolemmasandtheDuhamelformula,weget,usingthenotation(θ)(θ)ψ=Σψ,NProposition75Foreverym≥0andeveryθ∈[0,2π[2wehaveV(θ)−ψ(m,θ)=ON−(m+1)/2(6.74)N,θϕγ,zz,tHN(θ)(0,θ)(θ)Inparticularwehaveψz,t=ϕΓt,ztwiththenotationofSect.6.4.LetuscomebacktoHamiltonianperturbationsofhyperbolicautomorphismF.LetusdenoteFε=Φε◦F.FεissymplecticonT2(itpreservesthearea).BytheHHHC1stabilityofAnosovdynamicalsystems,forεsmallenough,FεisAnosov.TheHquantumanalogueofFεistheunitaryoperatorHRˆεwRˆN,θ,ε(F)=exp−iOp,θ(H)N,θ(F)Wehavethefollowingsemi-classicalcorrespondence.Proposition76Thefollowingestimatesholdtrueuniformlyinz∈T2andε∈[0,1]:Rˆ(θ)=Cεϕθ(6.75)N,θ,ε(F)ϕγ,zγε,Fε(z)H 6.8SpectralAnalysisofHamiltonianPerturbations181Cε+F·γDεwhereγε=andAε+F·γBε1/21/2iδε(F(z))/|γb+a||Aε+F·γBε|Cε=eγb+aAε+F·γBεInparticularCεisacomplexnumberofmodulusone.ProofThisisadirectconsequenceofpropagationofcoherentstates(Sect.4.3).NowweshallprovesomespectralpropertiesforRˆN,θ,ε(F)forεsmallenough.LetusdenoteηN,0≤j≤N−1theeigenvaluesofRˆN,θ,ε(F),sothatjRˆN,θ,ε(F)ψN=ηNψNwhere{ψN}0≤j≤N−1isanorthonormalbasisofHN(θ)jjjjandηN∈S1,theunitcircleofthecomplexplane.jTheorem40Forε>0smallenough,whenN→+∞,theeigenvalues{ηN}0≤j≤N−1areuniformlydistributedonS1i.e.foreveryintervalIonS1)wejhave{j;ηN∈I}jlim=μ(I)(6.76)N→+∞NwereμistheLebesgueprobabilitymeasureonS1.ProofInafirststepwewillprovethatforeveryf∈C1(S1)wehave1limTrfRˆN,θ,ε(F)=f(x)dμ(x)(6.77)N→+∞NS1UsingFourierdecompositionoffitisenoughtoprove(6.77)forf(z)=zk,k∈Z.Hencewehavetoprovethatforeveryk=0,1klimTrRˆN,θ,ε(F)=0(6.78)N→+∞NWeassumek≥1(fork≤−1thereareobviousmodifications).UsingthatthecoherentstatesareanovercompletesysteminHN(θ),wehaveRˆkθRˆkθTrN,θ,ε(F)=ϕz,N,θ,ε(F)ϕzT2UsingthepropagationofcoherentstateswegetθRˆkθ=θ+O−1/2ϕz,N,θ,ε(F)ϕzϕz,ϕγ(ε,k),(Fε)kεNHandusingProposition61wehave 1826QuantizationandCoherentStatesonthe2-Torusθkθ√−c(ε,k)d((Fε)kz,z)2N−1/2ϕz,RˆN,θ,ε(F)ϕz≤C(ε,k)NeH+OεNButweknowthatforεsmallenoughFεisAnosovsoitisergodicanditsperiodicHsethaszeromeasure.Soforeveryδ>0wehaveμ{z∈T2,d((Fε)kz,z)≥δ}=0.HUsingthatO(εN−1/2)isuniforminz∈T2weget(6.78)hence(6.77).NowwegeteasilytheTheoremconsideringf±∈C1(S1)suchthatf−≤1I≤f+andS1(f+−f−)dμ<δwithδ→0. Chapter7Spin-CoherentStatesAbstractInthischapterweconsiderthattheunitsphereS2oftheEuclideanspaceR3withitscanonicalsymplecticstructureisaphasespace.ThencoherentstatesarelabeledbypointsonS2andallowustobuildaquantizationofthetwosphereS2.Theyaredefinedineachfinite-dimensionalspaceofanirreducibleuni-taryrepresentationofthesymmetrygroupSO(3)(oritscoveringSU(2))ofS2andgiveasemi-classicalinterpretationforthespin.AsanapplicationwestatetheBerezin–Liebinequalitiesandcomputethether-modynamiclimitforlargespinsystems.7.1IntroductionUptonowwehaveconsideredGaussiancoherentstatesandtheirrelationshipwiththeHeisenberggroup,thesymplecticgroupandtheharmonicoscillator.Thesestatesareusedtodescribefieldcoherentstates(Glauber[90]).Forthedescriptionofassemblyoftwo-levelsatom,physicistshaveintroducedwhattheyhavecalled“atomiccoherentstates”[5].ThesestatesaredefinedinHilbertspaceirreduciblerepresentationsofsomesymmetryLiegroup.AsweshallseelateritispossibletoassociatecoherentstatestoanyLiegroupirreduciblerepresentation.ThisgeneralconstructionisduetoPerelomov[155].InthischapterweconsidertherotationgroupSO(3)oftheEuclideanspaceR3anditscompanionSU(2).IrreduciblerepresentationsofthesegroupsarerelatedwiththespinofparticlesaswasdiscoveredbyPauli[154].Inthischapter(andintherestofthebook)weshallusefreelysomebasicnotionsconcerningLiegroups,Liealgebraandtheirrepresentations.WehaverecalledmostoftheminanAppendicesA,BandC.7.2TheGroupsSO(3)andSU(2)LetusconsidertheEuclideanspaceR3equippedwiththeusualscalarproduct,x=(x1,x2,x3),y=(y1,y2,y3),x·y=x1y1+x2y2+x3y3andtheEuclideannormM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,183TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_7,©SpringerScience+BusinessMediaB.V.2012 1847Spin-CoherentStatesx=(x2+x2+x2)1/2.TheisometrygroupofR3isdenotedO(3).1A∈O(3)123meansthatAx=xforeveryx∈R3.SO(3)isthesubgroupofdirectisometriesi.e.A∈SO(3)meansthatA∈O(3)anddetA=1.ItiswellknownthatA∈SO(3)isarotationcharacterizedbyaunitaryvectorv∈R3(rotationaxis)andanangleθ∈[0,2π[.Moreprecisely,visan1-eigenvectorforA,Av=vandAisarotationofangleθintheplaneorthogonaltov.Sowehavethefollowingformula,foreveryx∈R3:Ax:=R(θ,v)x=(1−cosθ)(v·x)v+(cosθ)x+sinθ(v∧x)(7.1)Recallthatthewedgeproductv∧xistheuniquevectorinR3suchthatdet[v,x,w]=(v∧x)·w,∀w∈R3ItiseasytocomputetheLiealgebraso(3)ofSO(3),so(3)=A∈Mat(3,R),AT+A=0whereATisthetransposedmatrixofA.Consideringrotationsaroundvectorsofthecanonicalbasis{e1,e2,e3}ofR3wegetabasis{E1,E2,E3}ofso(3)wheredEk=R(θ,ek)dθθ=0Itsatisfiesthecommutationrelation[Ek,E]=Em(7.2)foreverycircularpermutation(k,,m)of(1,2,3).Itiswellknownthatanyrotationmatrixisanexponential.Proposition77Foreveryv∈R3,v=1andθ∈[0,2π[wehaveR(θ,v)=eθM(v)(7.3)whereM(v)=1≤k≤3vjEj.Proofθ→R(θ,v)andθ→eθM(v)areoneparametergroups,soitisenoughtoseethattheirderivativesatθ=0arethesame.Thisistruebecausewehavev∧x=M(v)x.AsSO(2)(identifiedtothecircleS1)SO(3)isconnectedbutnotsimplycon-nected.TocomputeirreduciblerepresentationsofSO(3)itisconvenienttoconsider1AnisometryinanEuclideanspaceisautomaticallylinearsothatO(3)isasubgroupofGL(R3). 7.2TheGroupsSO(3)andSU(2)185asimplyconnectedcoverofSO(3)whichcanberealizedasthecomplexLiegroupSU(2).SU(2)isthegroupofunitary2×2matricesAwithcomplexcoefficients,A=ab,suchthat|a|2+|b|2=1.−b¯a¯TheLiealgebrasu(2)istherealvectorspaceofdimension3of2×2complexanti-Hermitianmatricesofzerotrace:su(2)=X∈gl(2,C)|X∗+X=0,TrX=0Thethreelinearlyindependentmatrices:10i10−11i0A1=,A2=,A3=2i021020−iformabasisofsu(2)onRandsatisfythecommutationrelations[Ak,A]=Am(7.4)providedk,l,misacircularpermutationof1,2,3.LetusconsidertheadjointrepresentationofSU(2).Thisrepresentationisdefinedintherealvectorspacesu(2)bytheformulaρ−1,U∈SU(2),A∈su(2)(7.5)U(A)=UAUInphysicsthespinisdefinedbyconsideringthePaulimatrices,whicharehermitian2×2matricesgivenby010−i10σ1=,σ2=,σ3=(7.6)10i00−1whichsatisfythecommutationrelations[σk,σl]=2iσm(7.7)iσwhichisequivalentto(7.4)becausewehaveAk=k.{σ1,σ2,σ3}isanorthonor-2malbasisforthethree-dimensionalreallinearspaceH2,0ofHermitian2×2ma-triceswithtrace0,whichwillbeidentifiedwithR3.ThescalarproductinH2,0is1A,B =TrA∗B2LetusdenoteRUthe3×3matrixofρUinthisbasis.ThefollowingpropositiongivesthebasicrelationshipbetweenthegroupsSO(3)andSU(2).Proposition78ForeveryU∈SU(2)wehave:(i)RUhasrealcoefficients.(ii)RUisanisometryinH2,0.(iii)detRU=1. 1867Spin-CoherentStates(iv)ThemapU→RUisasurjectivegroupmorphismfromSU(2)ontoSO(3).(v)ThekernelofU→RUiskerR={1,−1}.Proof(i)FromR1−1Uσk,σ =Tr(UσkUσ)wegetRUσk,σ =RUσk,σ.2(ii)UsingcommutativityoftracewehaveR1∗UA,RUB =TrAA=A,A.2(iii)WehavedetRU=±1becauseRUisanisometry.ButdetR1=1andSU(2)isconnectedsodetRU=1.(iv)ItiseasytoseethatRisagroupmorphism.LetusconsiderthefollowinggeneratorsofSU(2):e−iϕ/20cos(θ/2)−sin(θ/2)U1(ϕ)=iϕ/2,U2(θ)=0esin(θ/2)cos(θ/2)Letusremarkthatwehave,foreveryϕ,ϕ,θ,cos(θ/2)e−i/2(ϕ+ϕ)−sin(θ/2)ei/2(ϕ−ϕ)U1(ϕ)U2(θ)U1ϕ=sin(θ/2)e−i/2(ϕ−ϕ)cos(θ/2)ei/2(ϕ+ϕ)(7.8)Thencomputetheimage:⎛⎞⎛⎞cosϕsinϕ0cosθ0−sinθRU1(ϕ)=⎝−sinϕcosϕ0⎠,RU2(θ)=⎝010⎠001sinθ0cosθRU1(ϕ)istherotationofangleϕwithaxise3,RU2(θ)istherotationofangleθwithaxise2.So,ifϕ,θ,η∈[0,2π[wehaveRU1(ϕ)U2(θ)U1(η)=RU1(ϕ)RU2(θ)RU1(η)where(ϕ,θ,η)aretheEuleranglesoftherotationR(ϕ,θ,η):=RU1(ϕ)RU2(θ)×RU1(η).ButanyrotationcanbedefinedwithitsEulerangles,soRissurjective.(v)LetU∈SU(2)besuchthatUAU−1=AforeveryA∈H2,0.ThenweeasilygetUAU−1=AforeveryA∈Mat(2,C)henceU=λ1withλ=±1.WeshallseenowthateveryirreduciblerepresentationofSO(3)comesfromanirreduciblerepresentationofitscompanionSU(2).Corollary22ρisanirreduciblerepresentationofSO(3)ifandonlyifρisanirreduciblerepresentationofSU(2)suchthatρRU=ρR−U,∀U∈SU(2)ProofLetρbearepresentationofSU(2)inafinite-dimensionallinearspaceEsuchthatρRU=ρR−U.Thenwedefinearepresentationρ˜ofSO(3)inEbythe 7.3TheIrreducibleRepresentationsofSU(2)187equalityρ(R˜U)=ρ(U).ConverselyeveryrepresentationofSO(3)comesfromarepresentationofSU(2)likeabove.Inotherwordsthefollowingdiagramiscom-mutative:RSU(2)SO(3)ρρ˜GL(E)whereGL(E)isthegroupofinvertiblelinearmapsinE.Corollary23TheLiealgebrasso(3)andsu(2)areisomorphthoughtheisomor-phismDR(1)(differentialofRattheunitoftheLiegroupSU(2)).InparticularwehaveDR(1)Ak=Ek,k=1,2,3.Remark41Thegenerators{Lk}1≤k≤3oftherotationswithaxisekgiveabasisoftheLiealgebraso(3)asareallinearspace.RecallthatL=(L1,L2,L3)istheangularmomentum.2LkbelongstothecomplexLiealgebraso(3)⊕iso(3)andissometimesdenotedJk,J=ix∧∇x.ForexampleL3=i(x2∂x−x1∂x).Wehave12thecommutationrelations,foreverycircularpermutation(k,,m)of(1,2,3),[Lk,L]=iLm.(7.9)Thisbasiscanbeidentifiedwiththematrixbasis(iE1,iE2,iE3)consideredbefore.7.3TheIrreducibleRepresentationsofSU(2)ThegroupSU(2)issimplyconnected(ithasthetopologyofthesphereS3),soweknowthatallitsrepresentationsaredeterminedbytherepresentationsofitsLiealgebraso(2)(seeAppendicesA,BandC).Moreover,SU(2)isacompactLiegroupsoallitsirreduciblerepresentationsarefinitedimensional.7.3.1TheIrreducibleRepresentationsofsu(2)WeshallfirstconsidertherepresentationoftheLiealgebrasu(2)anddetermineallitsirreduciblerepresentations.Recallthatsu(2)isarealLiealgebraanditismoreconvenienttoconsideritscomplexificationsu(2)+isu(2).Butanymatrixcanbedecomposedasasumofan2Multiplicationbyigivesself-adjointgeneratorsinsteadofanti-self-adjointoperators. 1887Spin-CoherentStatesHermitianandanti-Hermitianpart,sowehavesl(2,C)=su(2)+isu(2)(7.10)absl(2,C)isthespaceofmatricesA=,a,b,c∈C.ItistheLiealgebraofthec−agroupof2×2complexmatricesgsuchthatdetg=1.Itresultsfrom(7.10)thatir-reduciblerepresentationsoftherealLiealgebrasu(2)aredeterminedbyirreduciblerepresentationsofthecomplexLiealgebrasl(2,C).Oneconsiderssl(2,C)endowedwiththebasis{H,K+,K−}inwhichthecom-mutationrelationsare[H,K±]=±K±,[K+,K−]=2H(7.11)where1100100H=,K+=,K−=20−10010ItisalsoconvenienttointroduceHermitiangenerators(seefootnote2):σ3K++K−σ1K+−K−σ2K3=H=,K1==,K2==(7.12)2222i2Let(E,R)bea(finite-dimensional)irreduciblerepresentationofsl(2,C).Forcon-venienceletusdenoteHˆtheoperatorR(H).Hˆadmitsatleastaneigenvalueλandaneigenvectorv=0:Hvˆ=λvFromthecommutationrelations(7.11)wehaveHˆKˆ+v=Kˆ+Hˆ+Kˆ+v=(λ+1)Kˆ+vHˆKˆ−v=Kˆ−Hˆ−Kˆ−v=(λ−1)Kˆ−vSincetheremustbeonlyafinitenumberofdistincteigenvaluesofHˆ,thereexistsaneigenvalueλ0ofHˆandaneigenvectorv0suchthatHvˆKˆ0=λ0v0,−v0=0λ0isthesmallesteigenvalueofHˆ.Onedefinesthenkvk=Kˆ+v0ItmustobeyHvˆk=(λ0+k)vkOnecanshowbyinductiononkthatKˆ−vk=ckvk−1,whereck+1=ck−2(λ0+k),∀k∈N 7.3TheIrreducibleRepresentationsofSU(2)189Sowegetck=−k(2λ0+k−1),k∈NSincethevectorsvk=0arelinearlyindependentandthevectorspaceEisfinitedimensionalthereexistsanintegernsuchthatv0=0,v1=0,...,vn=0,vn+1=0ThusfromKˆ−vn+1=0onededucesthat2λ0+n=0.Then∀k∈NonehasKˆ+,Kˆ−vk=2Hvˆk,H,ˆKˆ±vk=±Kˆ±vk(7.13)Onededucesthatthevectors{vk}ngenerateasubspaceofEinvariantbythek=0representationRandsincetherepresentationwelookforisirreducible,theygen-eratethecomplexlinearspaceEwhichisthereforeoffinitedimensionn+1.Theelementsofthebasis{vk}narecalledDickestatesin[5].k=0Inconclusionwehavefoundnecessaryconditionstogetanirreduciblerepresen-tation(E(n),R(n))ofdimensionn+1ofsl(2,C)withabasis{vk}nofE(n)suchk=0thatnR(n)(H)vk=k−vk2(7.14)R(n)(K+)vk=vk+1R(n)(K−)vk=k(n−k+1)vk−1for0≤k≤nandv−1=vn+1=0.Wehavetocheckthattheseconditionscanberealizedinsomeconcretelinearspace.LetE(n)thecomplexlinearspacegeneratedbyhomogeneouspolynomialszkzn−kofdegreenin(z1,z2)∈Cwiththebasisvk=12and(n−k)!(n)∂(n)∂(n)1∂∂R(K−)=z2,R(K+)=z1,R(H)=z1−z2∂z1∂z22∂z1∂z2Sowehaveproved:Proposition79Everyirreduciblerepresentationofsl(2,C)offinitedimensionisequivalentto(E(n),R(n))forsomen∈N.Inthephysicsliteratureoneconsidersjsuchthatn=2j.jisthuseitherintegerorhalf-integerandrepresentstheangularmomentumoftheparticles.WeshallseelaterthatrepresentationsofSO(3)correspondtoj∈Nsoniseven.jisthegreatesteigenvalueofR(n)(H).Inquantummechanicstherepresentation(E(2j),R(2j))isdenoted(V(j),D(j))andoneconsidersthebasisofV(j)indexedbythenumberm,−j≤m≤j,wheremisintegerifjis,andhalf-integerifjis. 1907Spin-CoherentStatesStatesinV(j)representspinstatesandtheoperatorsinV(j)arespinobservables.SoweintroducethenotationSˆ=R(2j)(K)(itisthespinobservablealongtheaxis0x,1≤≤3)andSˆ±=R(2j)(K±).Thisbasisisusuallywritteninthe“ket”notationofDiracas|j,m.Thecorrespondenceisthefollowing:j+m(j−m)!|j,m =(−1)vj+m(j+m)!InthisbasistherepresentationDjoftheelementsK3,K+,K−(basisofthecom-plexLiealgebrasl(2,C)definedatthebeginning)actasfollows:Sˆ3|j,m =m|j,mSˆ+|j,m =(j−m)(j+m+1)|j,m+1Sˆ−|j,m =(j+m)(j−m+1)|j,m−1HenceSˆ(j−m)!Sˆj+m−|j,−j =0,|j,m =+|j,−j(j+m)!(2j)!WerecallthatthetwocomponentsL1,L2oftheangularmomentumarerelatedtotheoperatorsL±asfollows:L+=L1+iL2,L−=L1−iL2Intherepresentationspace(V(j),Dj)ofsl(2,C)onecanconsiderthespinoperatorS=(Sˆ1,Sˆ2,Sˆ3)andSˆ2+Sˆ2+Sˆ2=S2123S2canberewrittenasS2=Sˆ−Sˆ++Sˆ3Sˆ3+1ItisclearthatfortherepresentationDj,thevector|j,miseigenstateofS2:S2|j,m =j(j+1)|j,mThusS2actsasamultipleoftheidentityandiscalledtheCasimiroperatoroftherepresentationDj.OnedefinesascalarproductonE(2j)byimposingthatthebasis|j,misanorthonormalbasis.Intheorderedbasis:|j,−j,|j,−j+1,...,|j,j−1,|j,jtheoperatorsS3,S±arerepresentedbythefollowing(2j+1)×(2j+1)matrices: 7.3TheIrreducibleRepresentationsofSU(2)191Sˆ3=diag(−j,−j+1,...,j−1,j)(7.15)⎛⎞000...000√⎜2j00...000⎟⎜√⎟⎜02(2j−1)0...000⎟Sˆ⎜⎜⎟⎟(7.16)+=⎜...............⎟⎜...√...⎟⎝000...2(2j−1)00⎠√000...02j0⎛√⎞02j0...00√⎜002(2j−1)...00⎟⎜⎟⎜000...00⎟Sˆ⎜⎜⎟⎟(7.17)−=⎜.............⎟⎜....√.⎟⎝000...02j⎠000...00TheseformulasareconsequencesofthepolynomialsrepresentationoftheDickestates|j,minthespaceV(j)givenbyj+mj−mzz|j,m(z121,z2)=√(j+m)!(j−m)!7.3.2TheIrreducibleRepresentationsofSU(2)Weshallseenowthatforeveryj∈NwecangetarepresentationT(j)ofSU(2)2suchthatitsdifferentialdT(j)coincideswiththerepresentationsD(j)ofsu(2)thatwehavestudiedintheprevioussection.FurthermoretheyaretheonlyirreduciblerepresentationsofSU(2).Sinceeveryunitarymatrixisdiagonalizablewithunitarypassagematriceswehave∀A∈SU(2):eit0−1A=g−itg0eforsomet∈R.Thenweshowthattheexponentialmapfromsu(2)toSU(2)issurjective:Fromtherelationeit0−it=exp(itσ3)0ewededuceA=expitgσ−13gwithigσ3g−1∈su(2),hencetheresult. 1927Spin-CoherentStatesTakingthePaulimatricesasabasiswefindthateveryA∈su(2)canbewrittenasA=ia·σ,a=(a31,a2,a3)∈RWededuceeasilythatdetA=a2andA2=−(detA)1Thereforewehaveproventhefollowinglemma:Lemma41ForanyA∈su(2)onehasA2=−(detA)1Furthermoreonehasthefollowingresult:Proposition80ForanyA∈su(2)suchthatdetA=1,onehas,∀t∈R,exp(tA)=cost1+sintA(7.18)ProofBothmembersof(7.18)haveAasderivativeatt=0.Itisthereforeenoughtoshowthatthemapt∈R→1cost+AsintisaoneparametersubgroupofGL(2,C).Takes∈R.Onehas(1cost+Asint)(1coss+Asins)=1costcoss+A2sinssint+(sinscost+cosssint)A=1cos(s+t)+Asin(s+t)(7.19)Asaconsequenceweseethateveryelementg∈SU(2)canbewrittenasg=α11+α2I+α3J+α4Kwhere0i0−1i0I=,J=,K=i0100−ithevector(α1,α2,α3,α4)ofR4beingofnorm1.ThusSU(2)canbeidentifiedwiththegroupofquaternionsofnorm1.ThegroupSU(2)actsonC2bytheusualmatrixaction.abg=∈SU(2)(7.20)−b¯a¯ 7.3TheIrreducibleRepresentationsofSU(2)193with|a|2+|b|2=1.Thenz1az1+bz2g=z2−bz¯1+¯az2ginducesanactionρ(g)onfunctionsf:C2→C:ρ(g)f=f◦g−1Sinceghasdeterminantoneitsinverseg−1equals−1a¯−bg=ba¯thusρ(g)f(z1,z2)=f(az¯1−bz2,bz¯1+az2)OneconsidersV(j)asthevectorspaceofhomogeneouspolynomialsinz1,z2ofdegree2j(werecallthatj∈1N).ConsiderthefollowingbasisinV(j):22j2j−1j+mj−m2jz,z1z,...,zz,...,z,−j≤m≤j22121ItisclearthatV(j)isstablebyρ.OneequipsV(j)withtheSU(2)-invariantscalarproductwhichmakesthemono-mialsk2j−kzzp12k(z1,z2)=√k!(2j−k)!anorthonormalbasisofV(j).Definetheactionofg∈SU(2)onanhomogeneouspolynomialpinthefollowingway:Tj(g)p(z−11,z2)=p◦g(z1,z2)Letusprovethefollowing.Lemma42Considerthehomogeneouspolynomialp:2jl2j−lp(z1,z2)=clz1z2l=0Thenthemap2j2=l!(2j−l)!|c2p→pl|l=0definesanHilbertiannormofV(j)thatisinvariantbytheactionofSU(2). 1947Spin-CoherentStatesProofItisenoughtocheckthatp◦g∗2=p2whichimpliesthattherepresen-tation(V(j),Tj)isunitary.Takep(z2j1,z2)=pα,β(z1,z2)=(αz1+βz2),α,β∈CwhichgenerateV(j).Thenif2j2jl2j−ll2j−lp(z1,z2)=clz1z2,p(z1,z2)=clz1z2l=0l=02j2j−1l2j−l−1l2j−lp◦g(z1,z2)=dlz1z2,p◦g(z1,z2)=dlz1z2l=0l=0onehas2j,p=p◦g−1,p◦g−1=cplc¯ll!(2j−l)!l=02j2j=dld¯ll!(2j−l)!=(2j)!αα¯+ββ¯(7.21)l=0NamelytheHermitianscalarproductinC2of(α,β)with(α,β)isinvariantunderSU(2).WeshallstudytherepresentationT(j)ofSU(2)obtainedbyrestrictionofρtoV(j).Onedefinesjj+mj−mfm(z1,z2)=z1z2WefirstconsiderdiagonalmatricesinSU(2).Theyareoftheformeit0gt=exp(−2itK3)=−it0eThen(j)jj−itit−2imtjT(gt)fm(z1,z2)=fmz1e,z2e=efm(z1,z2)(7.22)ThuseveryfjiseigenstateofT(j)(g−2imtmt)witheigenvaluee.WeshallnowconsiderthedifferentialdT(j)(g)forthebasiselementsK3,K±ofsl(2,C):oneconsidersX∈sl(2,C),αβX=γ−α 7.3TheIrreducibleRepresentationsofSU(2)195anda(t)b(t)gt=exp(tX)=c(t)d(t)Theng(0)=1andg(0)=Xsothata(0)=d(0)=1,b(0)=c(0)=0,a(0)=α,b(0)=β,c(0)=γ,d(0)=−αForanypolynomialintwovariablesf(z1,z2)onehasdd(dρ)(−X)f(z1,z2)=ρ(gt)−1f(z1,z2)=(f◦gt)(z1,z2)dtdtt=0t=0=(αz1+βz2)∂1f(z1,z2)+(γz1+δz2)∂2f(z1,z2)Therefore1(dρ)(K3)=(z1∂1−z2∂2)2(dρ)(K+)=z1∂2,(dρ)(K−)=z2∂1j(j)Weshalldeterminetheactionofdρ(K3),dρ(K±)onthebasisvectorsfmofV:jjdρ(K3)fm=mfmFurthermoreusingjj+m−1j−mjj+mj−m−1∂1fm=(j+m)z1z2,∂2fm=(j−m)z1z2wegetjjjjdρ(K+)fm=(j−m)fm+1,dρ(K−)fm=(j+m)fm−1Denoting1j|j,m =√fm(j−m)!(j+m)!wegetdT(j)(K3)|j,m =m|j,m(7.23)(j)(KdT+)|j,m =j(j+1)−m(m+1)|j,m+1(7.24)dT(j)(K−)|j,m =j(j+1)−m(m−1)|j,m−1(7.25)Weuseheretheabuseofnotationρ=T(j).Sowededucethefollowingresult: 1967Spin-CoherentStatesProposition81ThedifferentialoftherepresentationT(j)ofSU(2)coincideswiththerepresentationDjofsu(2).Proposition82Foranyj∈1N,(V(j),T(j))isanirreduciblerepresentationof2SU(2).ItdefinesanirreduciblerepresentationofSO(3)ifandonlyifj∈N.Ifnisodd(j∈N,j/∈N)thenT(j)isaprojectiverepresentationofSO(3).2ProofItfollowsfromgeneralresultsaboutthedifferentialoftherepresentationsofLiegroupsthatthedifferentialofT(j)isanirreduciblerepresentation.Thesecondpartcomesfromthefollowingfact:T(j)(−g)=T(j)(g),∀g∈SU(2)ifandonlyifniseven.Theproofofthelastpartislefttothereader.Corollary24EveryirreduciblerepresentationofSU(2)isequivalenttooneoftherepresentations(V(j),T(j)),j∈1N.2ProofSinceSU(2)iscompactweknowthateveryirreduciblerepresentationofSU(2)isfinitedimensional.Wehaveseenthateveryirreduciblerepresentationoffinitedimensionofsu(2)isoneoftheDj,j∈1N.Thisimpliestheresultsince2SU(2)isconnectedandsimplyconnected.7.3.3IrreducibleRepresentationsofSO(3)andSphericalHarmonicsWehaveseenabovethatirreduciblerepresentationsofSO(3)aredescribedby(T(j),V(j))forj∈N.AmoreconcreteequivalentrepresentationcanbeobtainedwithaspectraldecompositionoftheLaplaceoperatoronS2.Recallthatinsphericalcoordinates(r,θ,ϕ)wehave∂2∂2∂2∂22∂1=2+2+2=∂r2+r∂r+r2S2∂x∂x∂x123whereS2isthesphericalLaplaceoperatoronS2,∂21∂1∂2S2:=∂θ2+tanθ∂θ+2∂ϕ2(7.26)sinθSO(3)hasanaturalrepresentationΣinthefunctionspaceL2(R3)(andinL2(S2))definedasfollows:Σ(g)f(x)=f(g−1x)whereg∈SO(3),f∈L2(R3)andtheLaplaceoperatorcommuteswithΣ.(j)LetusintroducethelinearspaceHofhomogeneouspolynomialsfin3(x1,x2,x3)oftotaldegreejandsatisfyingf=0andrestrictedtothesphere2(j)S;Histhespaceofsphericalharmonics.3 7.3TheIrreducibleRepresentationsofSU(2)197RecallthattheEuclideanmeasureonS2isdμ2(θ,ϕ)=sinθdθdϕ.(j)∞2Theorem41HisasubspaceofC(S)ofdimension2j+1,invariantforthe3(j)actionΣ.Therepresentation(Σ,H)isirreducibleandisunitaryequivalentto3therepresentation(T(j)V(j)).Recallthefollowingexpressionforthemeasuredμ2onthesphere:dμ2(θ,ϕ)=sinθdθdϕ.ProofWeshallprovesomepropertiesofsphericalharmonicswhichareprovedinmoredetailsforexamplein[130].ThespaceHjisinvariantbythegeneratorsL1,L2,L3ofrotations.Insphericalcoordinateswehave1∂L3=(7.27)i∂ϕ1∂sinϕ∂L2=cosϕ−(7.28)i∂θtanθ∂ϕ∂cosϕ∂L1=isinϕ+(7.29)∂θtanθ∂ϕWecancomputetheCasimiroperator:L2:=L2+L2+L2=−S2.Inparticular123wehave[L3,S2]=0.IfL±:=L1±iL2thenwehave[L3,L±]=±L±,[L+,L−]=2L3(7.30)L22+L−=L−L3(L3−1),L−L+=L−L3(L3+1)(7.31)Letf∈Hj.Inpolarcoordinateswehavef(r,θ,ϕ)=rjY(θ,ϕ).Sowegetf=0⇐⇒−S2Y=j(j+1)YL3canbediagonalizedinHjLimϕ3Y=λY⇐⇒Y(ϕ,θ)=ef(θ),m∈Z,−j≤m≤jSoadmittingthatHjhasdimension2j+1weseethattherepresentation(Σ,Hj)isunitaryequivalenttotherepresentation(Tj),V(j)).Remark42UsingthesamemethodasinSect.7.3.1,foreveryj∈Nwehavek(j)kanorthonormalbasis{Y}−j≤k≤jofHwhereYareeigenfunctionsofL3:j3jL3Yk=kYk,−j≤k≤j.InotherwordsYkareDickestates.Moreovertheyhavejjjthefollowingexpression:02j+1Yj(θ,ϕ)=Pj(cosθ)(7.32)4π 1987Spin-CoherentStateswherePjaretheLegendrepolynomials1djjP2j(u)=u−12jj!dujFork=0wecanusethefollowingformula:√Lk=j(j+1)−k(k+1)Yk+1+Yjj√(7.33)Lkk−1−Y=j(j+1)−k(k−1)YjjLetusprovenowtwousefulpropertiesofthesphericalharmonicsProposition83(j)(i)Foreveryj∈N,Hhasdimension2j+1.3(ii){Yk,−j≤k≤j,j∈N}isanorthonormalbasisofL2(S2)or,equivalently,jHj=L2S2j∈N(j)ProofLetusintroducethespacePofhomogeneouspolynomialsin(x1,x2,x3)3(j+1)(j+2)oftotaldegreej.ThedimensionofP3,jis.Iteasytoprovethatis2surjectivesoweget(i):(j+1)(j+2)(j−1)jdim(ker)=−=2j+122(j)Toprove(ii)letusintroduceonPascalarproductsuchthatwehaveanorthonor-3k1k2k3malbasis{√x1x2x3}(j)k!k!k!k1+k2+k3=j.LetusintroducetheHilbertspaceH:=⊕P3.123∂andxSothelinearoperatorskarehermitianconjugate.Henceisconjugateto∂xkr2=x2+x2+x2.1232(j)(j+2)(j)(j−2)WehaverP⊆PandP=P.Usingtheformula(Fredholm3333∗⊥(j)j2(j−2)property)kerA=(ImA),wegetP=H⊕rP.Stepbystepweget333(j)(j)2(j−2)2(j−2)P=H⊕rH⊕···⊕rH(7.34)3333wherej−1≤2≤j.NowwecanprovethatthesphericalharmonicsisatotalsysteminL2(S2).Letusremarkthatthealgebraj∈NP3,jisdenseinC(S2)forthesup-norm(conse-(j)quenceofStone–WeierstrassTheorem).Sousing(7.34)weseethatj∈NH3is2(j)22denseinC(S)forthesup-norm,soj∈NH3isdenseinL(S). 7.4TheCoherentStatesofSU(2)1997.4TheCoherentStatesofSU(2)7.4.1DefinitionandFirstPropertiesLetusstartwithareference(nonzero)vectorψ0∈V(j)andconsiderelementsoftheorbitofψ0inV(j)bytheactionoftherepresentationT(j).Wegetafamilyofstatesoftheform(j)(g)ψ|g =T0jwillbefixed,sowedenote|g =T(g)ψ0.Inamoreexplicitformwehaveg−1(z2T(g)ψ0(z1,z2)=ψ01,z2),(z1,z2)∈CInprincipleanyvectorψ0∈V(j)canbetakenasreferencestate.Howeverforthestatesψ0=|j,±jwecanseethatthedispersionofthetotalspinoperatorS=(Sˆ1,Sˆ2,Sˆ3),isminimal,sothatthestates|j,±jdeterminethesystemofcoherentstateswhich,insomesense,isclosesttotheclassicalstates.Inpracticewechooseinwhatfollowsψ0=|j,−jLetusrecallthedefinitionofthedispersionforanobservableAˆforastateψ,whereψisanormalizedstateinanHilbertspaceHandAˆaself-adjointoperatorinH.ForψinthedomainofAˆtheaverageisAˆψ:=ψ,Aψˆandthedispersionisdefinedlikethevarianceforarandomvariableinprobability:Aˆ−Aˆ2Aˆ2Aˆ2ψAˆ:=1=−ψψψψLetusrecallheretheHeisenberguncertaintyprinciple:ifAˆ,Bˆareself-adjointop-eratorsinHandψ∈H,ψ=1thenwehave12ψAˆψBˆ≥iA,ˆBˆ(7.35)4ψApplicationtothetotalspinobservablegives22ψS=ψSˆk=Sˆkψ−Sˆkψ,ψ1≤k≤31≤k≤3Forψ0=|j,mwegetS=j(j+1)−m2ψ0Sothedispersionisminimalform=±j.TheHeisenberginequalityforspinoperatorsreads(see(7.35))Sˆ2Sˆ21Sˆ212≥3(7.36)4Itiseasytoseethatthisinequalityisanequalityforψ0=ψn.0 2007Spin-CoherentStatesNowourgoalistostudythemainpropertiesofthecoherentstates|g.LetusfirstremarkthatthefullgroupG=SU(2)isnotagoodsettoparametrizethesecoherentstatesbecausethemapg→|gisnotinjective.Soweintroducetheso-calledisotropygroupHdefinedasfollows:H=g∈G,∃δ∈R,T(g)ψiδ0=eψ0WefindthatHisthesubgroupofdiagonalmatrices α0H=,α=exp(iψ)0α¯Nowthemapg˙→|gisabijectionfromthequotientspaceX:=G/Hontotheorbitofψ0,whereG/HisthesetofleftcosetgHofHinGandg→˙gisthecanonicalmap:G→G/H.01LetusdenoteX0=X.−10Lemma43X0isisomorphictothesetofelementsoftheform αβ222,α∈R,α=0,β=β1+iβ2∈C,α+β1+β2=1−βα¯abProofLetusdenoteg(a,b)agenericelementofSU(2),g=,a,b∈C,−b¯a¯|a|2+|b|2=1.Wefirstremarkthatforevery|b|=1,g(0,b)isinthecosetofg(0,1).Nowif|a|2+|b|2=1anda=0thenwehaveauniquedecompositiong(a,b)=ga,bg(α,0)witha>0,α=a,(a)2+|b|2=1.|a|Sowegetthelemma.Remark43Concerningtheorbitwithourchoiceofψ0theimageofX0doesnot2jcontainthemonomialz,whichareobtainedwithg(0,1).1Choosingtheparametrizationθθ−iϕα=cos,β=−sine,0≤θ<2π,0≤ϕ<2π22weseethatthespaceX0isjustarepresentationofthetwo-dimensionalsphereS2minusthenorthpole,namelythesetofunitthree-dimensionalvectorsn=(sinθcosϕ,sinθsinϕ,cosθ),0≤θ<π,0≤ϕ<2π 7.4TheCoherentStatesofSU(2)201andanyelementgn∈Xcanbewrittenas θgn=expi(sinϕσ1−cosϕσ2)(7.37)2whereσ1,σ2arethePaulimatrices.Thusgndescribesarotationbytheangleθaroundthevectorm=(sinϕ,−cosϕ,0)belongingtotheequatorialplaneofthesphereandperpendic-ularton(itiswelldefinedbecauseθ∈[0,π[).Definition19ThecoherentstatesofSU(2)arethefollowingstatesdefinedintherepresentationspaceV(j):|n =T(gn)ψ0:=D(n)ψ0(7.38)Inthephysicsliteraturetheyarecalledthespin-coherentstatesbecausethespinisclassifiedwiththeirreduciblerepresentationsofSU(2)(see[154]).Thesecoherentstateshaveseveralothernames:atomiccoherentstates,Blochcoherentstates.Choosingg=gnthecoherentstateoftheSU(2)groupcannowbewrittenas|n =T(gn)ψ0=exp(iθm·K)ψ0wherem=(sinϕ,−cosϕ,0),K=(K1,K2,K3)mistheunitvectororthogonaltobothnandn0=(0,0,1).NotethatthisdefinitionexcludesthesouthpolenS=(0,0,−1).ThusacoherentstateofSU(2)correspondstoapointofthetwo-dimensionalsphereS2whichmaybeconsideredasthephasespaceofaclassicaldynamicalsystem,the“classicalspin”.Thecoherentstatesassociatedwiththesouthpolewill2jbetheimageof(g(0,β)),|β|=1givingmonomialsz.1SowehaveparametrizedthespincoherentbythesphereS2.Anotherusefulparametrizationcanbeobtainedwiththecomplexplane,usingthestereographicprojectionfromthesouthpoleofthesphereS2ontothecomplexplaneC.Ifn=(n1,n2,n3)∈S2thenthestereographicprojectionofnfromthesouthpoleisthecomplexnumberζ(n)=n1+in2.Soinpolarcoordinateswehaveζ(n)=1+n3tan(θ/2)eiϕ.Aswehavealreadyremarked,thegroupSU(2)naturallyembedsintothecom-plexgroupSL(2,C)whichisthegroupofcomplexmatriceshavingdeterminantone.ThefollowingGaussiandecompositioninSL(2,C)willbeuseful.Theproofisaneasyexercise.Lemma44Foranyg∈SL(2,C)oftheformαβg=,withδ=0γδ 2027Spin-CoherentStatesonehasaunique(Gaussian)decompositiong=t+·d·t−wheredisdiagonalandt±aretriangularmatricesoftheform1ζ10t+=,t−=(7.39)01z1andε−10d=0εWehavetheformulasγβε=δ,z=,ζ=(7.40)δδMoreoverifg∈SU(2)thenwehave22−12−1|ε|=1+|ζ|=1+|z|(7.41)ThisallowstowriteasconsequenceofGaussdecomposition,(j)(g)=T(j)(t(j)(j)T+)T(d)T(t−)Letuswriteε=reiswithr>0ands∈R.Takingψ0=|j,−jasthereferencestateonegetsintherepresentation(T(j),V(j))(j)(tz+Sˆ−T−)ψ0=eψ0=ψ0(7.42)T(j)(d)ψ−2j(logr+is)0=eψ0(7.43)SowehaveT(g)ψiϕ0=eNT(t+)ψ0(7.44)From(7.41)wegetN=(1+|ζ|2)−j.ϕisarealnumber(argumentofδ).Ifg=gnthenδisrealsowegets=ϕ=0.Inconclusionwehaveobtainedanidentificationofthecoherentstates|nwiththestate|ζdefinedasfollows:2−j|ζ =1+|ζ|expζSˆ+|j,−jMorepreciselywedenote|ζ =|gnwiththefollowingcorrespondence:θ−iϕn=(sinθcosϕ,sinθsinϕ,cosθ),ζ=−tane2Thegeometricalinterpretationisthat−ζ¯isthestereographicprojectionofn. 7.4TheCoherentStatesofSU(2)203Recallthefollowingexpressionofgn: cosθ−sinθe−iϕg22n=(7.45)sinθeiϕcosθ22Anotherformfor|nisgivenbythefollowingequivalentdefinitionusingthatgn=eiθ(sinϕSˆ1−cosϕSˆ2),|n =D(ξ)ψ0whereD(ξ)=expξSˆ+−ξ¯Sˆ−andξ=tan(θ)eiϕ.2TheGaussiandecompositionalsoprovidesa“normalform”ofD(ξ):D(ξ)=expζSˆ+expηSˆ3expζSˆ−withη=−2log|ξ|,ζ=−ζ¯Sinceζ,ζ,ηdonotdependonjitisenoughtocheckthisformulaintherepresen-tationwherej=1,S=1R(σ),whereσisthethreecomponentPaulimatrix.22Foreachn∈S2thecoherentstateψnminimizesHeisenberginequalityobtainedbytranslationof(7.36)byD(n)i.e.puttingS˜k=D(n)SˆkD(n)−1insteadofSˆk,1≤k≤3.7.4.2SomeExplicitFormulasManyexplicitformulascanbeprovedforthespin-coherentstates.TheseformulashavemanysimilaritieswithformulasalreadyprovedfortheHeisenbergcoherentstatesandcanbewrittenaswellwiththecoordinatesnonthesphereS2orinthecoordinatesζinthecomplexplaneC.LetusfirstremarkthatS2andCcanbeidentifiedwithaclassicalphasespace.S2isequippedwiththesymplectictwoformσ=sinθdθ∧dϕ.Inthestereographicdζ∧dζ¯projectionitistransformedinσ=2i(1+|ζ|2)2.Thisisaneasycomputationusingζ=−tanθe−iϕ.2LetusconsiderfirstsomepropertiesofoperatorsD(n).Weshallalsousetheno-tationsD(ξ)orD(ζ)whereξandζaregivenbyξ=tan(θ)eiϕandζ=−tanθe−iϕ22usingpolarcoordinatesforn.ThemultiplicationlawfortheoperatorsD(n)isgivenbythefollowingformula:Proposition84(i)Foreveryn1,n2outsidethesouthpoleofS2wehaveD(n1)D(n2)=D(n3)exp−iΦ(n1,n2)J3(7.46) 2047Spin-CoherentStateswhereΦ(n1,n2)istheorientedareaofthegeodesictriangleonthespherewithverticesatthepoints[n0,n1,n2].n3isdeterminedbyn3=Rgn2(7.47)n1whereRgistherotationassociatedtog∈SU(2)asinProposition78andθgn=expi(σ1sinϕ−σ2cosϕ)(7.48)2(ii)Moregenerallyforeveryg∈SU(2)andeveryn∈S2suchthatnandg·nareoutsidethesouthpolewehave(j)(g)ψDn=exp−ijA(g,n)ψg·n(7.49)whereg·n=Rg(n)andA(g,n)istheareaofthesphericaltriangle[n0,n,g·n].ProofWeprovetheresultinthetwo-dimensionalrepresentationofSU(2).Thevec-torn3isdeterminedonlybygeometricalruleandisthusindependentoftherepre-sentation.WechoosetherepresentationinV1/2.DefineR(g)tobetherotationinSO(3)inducedbyanyg∈SU(2).Bythedefi-nition(7.48)ofgnwehaveR(gn)n0=n,∀n=(sinθcosϕ,sinθsinϕ,cosθ)Weneedtocomputeg=gngn.Weusethefollowinglemma:12Lemma45∀g∈SU(2)∃m∈S2andδ∈Rsuchthatg=gmr3(δ)δσwherer3(δ)=exp(i3).2Applyingthelemmatog=gngnweget12g=gmr3(δ)andweneedtoidentifymwithn3givenby(7.47).WehaveRr3(δ)n0=n0ThenR(g)=R(gn)R(gn)=R(gm)Rr3(δ)12Applyingthisidentitytothevectorn0wegetR(gn)R(gn)n0=R(gn)n2=R(gm)n0=m121Thuswehaveproventhatm=n3=R(gn)n2.1 7.4TheCoherentStatesofSU(2)205SowehaveseenthatthedisplacementoperatorD(n)transformsanyspin-coherentstate|n1intoanothercoherentstateofthesystemuptoaphase:D(n)|n1 =D(n)D(n1)ψ0=D(n2)expiSˆ3Φ(n,n1)ψ0=exp−ijΦ(n,n1)|n2wheren2=R(gn)n1.Thesecondfactorintherighthandsideof(8.65)doesdependontherepresentation.ThecomputationofΦ(n1,n2)willbedonelater.Intheproofof(7.49)thenontrivialpartistocomputethephaseA(g,n)whichalsowillbedonelater.Thefollowinglemmashowsthatthespinisindependentofthedirection.Lemma46OnehasD(n)Sˆ−13D(n)=n·SWeshallprovethelemmaintherepresentationofthePaulimatrices.FirstnotethatθθD(n)=cos+isin(sinϕσ1−cosϕσ2)22Then−1θθD(n)σ3D(n)=cos+isin(sinϕσ1−cosϕσ2)σ322θθ×cos−isin(sinϕσ1−cosϕσ2)(7.50)22WeusethepropertiesofthePaulimatrices:σ1σ3=−iσ2,σ2σ3=iσ1,σ2σ1=−iσ3tocomputetherighthandside.OnegetsD(n)σ−13D(n)=cosθσ3+sinθ(sinϕσ2+cosϕσ1)=n·σThefollowingconsequenceisthat|nisaneigenvectoroftheoperatorn·S:Proposition85Onehasn·S|n =−j|nProofDenoteby|n0thevectoriS3θψe0ThenSiθSˆ33|n0 =−jeψ0=−j|n0sincewetakeψ0=|j,−j. 2067Spin-CoherentStatesNowweshallusetheabovelemma:n·S|n =D(n)J3ψ0=−j|nThiscompletestheproofoftheproposition.AsintheHeisenbergsetting,thespin-coherentstatesfamily|nisnotanorthog-onalsystem.Onecancomputethescalarproductoftwocoherentstates|n,|n:Proposition86OnehasjijΦ(n,n)1+n·nn|n =e(7.51)2whereΦ(n,n)isarealnumber.Ifthesphericaltrianglewithvertices{n0,n,n}isanEulertrianglethenΦ(n,n)istheorientedareaofthistriangle.ProofToeachpointnonthesphereS2weassociateitssphericalcoordinatesθ∈[0,π),ϕ∈[0,2π)asusual:x=sinθcosϕ,y=sinθsinϕ,z=cosθThecorrespondingelementgn∈SU(2)isdefinedasθgn=expi(sinϕσ1−cosϕσ2)2Thematrixi(sinϕσ1−cosϕσ2)canbeviewedasapurequaternionthatwede-noteq.Using(7.18)wehaveθθgn=cos+qsin22Takingn∈S2withsphericalcoordinatesθ,ϕweget(usingquaternioncalculus)θθθθgngn=coscos−sinsincosϕ−ϕ2222θθθθθθ+σ3isinsinsinϕ−ϕ+qcossin+qsincos(7.52)222222Thereforegngnisoftheform(7.20)withθθθθi(ϕ−ϕ)a=coscos−sinsine2222TheelementgnofSU(2)canbewrittenas cosθ−sinθe−iϕg22n=sinθeiϕcosθ22 7.4TheCoherentStatesofSU(2)207NowweturntotherepresentationT(j)(g)inthespaceV(j)ofhomogeneouspoly-nomialsofdegree2jinz1,z2.Thecoherentstate|nisoftheformT(gn)ψ0for2j2√z2somen∈Sandψ0beingareferencestate.Wechooseψ0=inthehomoge-(2j)!neouspolynomialrepresentation.Notethatthisiscoherentwiththechoice|j,−j.Theoverlapbetweentwocoherentstatesisgivenbythescalarproduct:(j)(j)T(gn)ψ0,T(gn)ψ0!2j2j"1θθiϕθθiϕ=z2cos+z1sine,z2cos+z1sine(7.53)(2j)!2222Wemakeuseofthefollowingresult:Lemma47LetΠα,β(z1,z2)=(αz1+βz2)2j.ThenthescalarproductinV(j)oftwosuchpolynomialsequals2jΠα,β,Πα,β =(2j)!αα¯+ββ¯Thenweget2jθθθθi(ϕ−ϕ)n|n =coscos+sinsine2222Byaneasycalculusweobtain2cosθcosθ+sinθsinθei(ϕ−ϕ)=1+n·n22222Letusnowcomputethephaseoftheoverlapn|n.ItisanontrivialandinterestingcomputationrelatedwithBerryphaseasweshallsee.ItcanbeextendedtoamoregeneralsettingforcoherentstatesonKählermanifolds[32].Wefollowheretheelementaryproofofthepaper[4].Letusdenoteθθθθi(ϕ−ϕ)η=argcoscos+sinsine2222Usingclassicaltrigonometricformulawegetsinθsinθsin(ϕ−ϕ)tanη=(7.54)(1+cosθ)(1+cosθ)+sinθsinθcos(ϕ−ϕ)Wenowcomparethisformulawiththefollowingsphericalgeometricformulaal-readyknownbyEulerandLagrange(see[75]foradetailedproof).Let3pointsn1,n2,n3beontheunitsphereS2,notallonthesamegreatcircleandsuchthatthesphericaltrianglewithverticesn1,n2,n3isanEulertrianglei.e. 2087Spin-CoherentStatestheanglesandthesidesareallsmallerthanπ.Letωbetheareaofthistriangle.Thenwehaveω|det[n1,n2,n3]|tan=(7.55)21+n1·n2+n2·n3+n3·n1Noticethatthisformulatakesaccountoftheorientationofthepiecewisegeodesiccurvewithverticesn1,n2,n3.Theorientationispositiveiftheframe{On1,On2,On3}isdirect.From(7.54)and(7.55)wegetdirectlythattanη=tanω.Butω,η∈]−π,π[so22wecanconcludethatη=ω.Asisexpected,thespin-coherentstatesystemprovidesa“resolutionoftheiden-tity”intheHilbertspaceV(j):Proposition87Wehavetheformula#2j+1dn|nn|=1(7.56)4S2Orusingcomplexcoordinates|ζ,#dμj(ζ)|ζζ|=1(7.57)Cwherethemeasuredμjis2j+1d2ζdμj(ζ)=π(1+|ζ|2)2withd2ζ=|dζ∧dζ¯|.2ProofThetwoformulasareequivalentbythechangeofvariablesζ=−tanθe−iϕ.2Soitissufficienttoprovethecomplexversion.Letusrecalltheanalyticexpressionfor|nand|ζ.2j1θiϕθ|n =ψn(z1,z2)=√−sinez1+cosz2(7.58)(2j)!22112j|ζ =ψζ(z1,z2)=√2jζz¯1+z2(7.59)(2j)!(1+|ζ|)j(j)RecallthatwehaveanorthonormalbasisofDickestates{d}−j≤k≤jinVwherekj+kj−kjz1z2d(z1,z2)=√k(j+k)!(j−k)! 7.4TheCoherentStatesofSU(2)209Soweget1/2(2j)!2−jj+kψζ,dk =1+|ζ|ζ(7.60)(j+k)!(j−k)!AndusingtheParsevalformulaweget2−j2−j2jη|ζ =1+|ζ|1+|η|1+ηζ¯(7.61)Equation(7.61)isthecomplexversionfortheoverlapformulaoftwocoherentstates(7.53).ItisconvenienttointroducenowthespinBargmanntransform(seeChap.1fortheBargmanntransformintheHeisenbergsetting).Foreveryv∈V(j)wedefinethefollowingpolynomialinthecomplexvariableζ:j,2jv(ζ)=ψζ|v1+|ζ|Ifv=−j≤k≤jckekadirectcomputationgives#2vj,(ζ)2dζ=π|ck|2C(1+|ζ|2)2j+22j+1−j≤k≤jπ2=v(7.62)2j+1Orequivalently#2ψ2dζ=πv2(7.63)ζ|v(1+|ζ|2)22j+1CThisformulaisequivalenttotheovercompletenessformulabypolarisation.Remark44Fromtheproofwehavefoundthatthespin-Bargmanntransform:Bjv(ζ):=vj,(ζ)isanisometryfromV(j)ontothespaceP2jofpolynomialsofdegreeatmost2jequippedwiththescalarproduct#22j+1dζP,Q =P(ζ)Q(ζ)¯πC(1+|ζ|2)2j+2Inparticularwehavej,2−j2jψη(ζ)=1+|η|(1+¯ηζ)WenowextendthecomputationofthephaseΦ(n,n)inthegeneralcasebygivingforitadifferentexpressionrelatedwiththewellknowngeometricphase.Asimilarcomputationwasdonein[146]. 2107Spin-CoherentStatesProposition88Foranyn,n∈S2wehave$iΦn,n=−ψn,dψn(7.64)j[n,n]wheretheintegraliscomputedontheshortestgeodesicarcjoiningntonoftheonedifferentialformψn,dψn.ToexplaintheformulawerecallherethemainideabehindthegeometricphasediscoveredbyBerry[24]andPancharatnam[149](seealso[2]).Letusconsideraclosedloopn(t):[0,1]→S2whichiscontinuousbypart,n(0)=n(1).Wedefinethetime-dependentHamiltonianasH(t)ˆ=n(t)·SThesolutionofthetime-dependentSchrödingerequationwiththisHamiltonianisdenotedψ(t).Letusconsiderη(t)intheHilbertspaceandα(t)∈Rsuchthatψ(t)=eiα(t)η(t)Wechooseα(t)suchthatη(t),η(t)˙ =0.Inothertermsη(t)describesaparalleltransportalongthecurve.Thenthegeometricalphaseα(t)obeysiα(t)˙+ψ(t),ψ(t)˙=0Wethushave$α(1):=α(γ)=iψ,dψγIfγdelimitatesaportionΓofS2wehavebyStokestheorem#α(γ)=dψ,dψΓwheretheproductofthedifferentialsistheexternalproduct.Letψnbethecoherentstate|nobtainedatt=1fromψ(0)=ψn.Intheho-0mogeneouspolynomialrepresentationweget2j1θiϕθψn=√sinez1+cosz2(2j)!22Thus2j−1∂ψnjθiϕθθiϕθ=√z1cose−z2sinsinez1+cosz2∂θ(2j)!22222j−1∂ψn2jθiϕθiϕθ=√isinez1z1sine+z2cos∂ϕ(2j)!222 7.4TheCoherentStatesofSU(2)211UsingtheinvarianceofthescalarproductinV(j)underSU(2)transformationsweperformthechangeofvariablesθθZ1=z1cos−z2e−iϕsin22(7.65)θθZ2=z1eiϕsin+z2cos22OnethushaveusingtheorthogonalityrelationsinV(j):!"∂ψnjiϕ2j−12jψn,=eZ1Z,Z=0(7.66)∂θ(2j)!22∂ψn∂ψnOnecancalculatethescalarproductofand:∂θ∂ϕ!"!"∂ψn∂ψn2j2−iϕθ2j−1θiϕθ2j−1,=iesinZ1Z,Z1cose+Z2sinZ∂θ∂ϕ(2j)!222221=ijsinθ(7.67)2since2j−12j−1Z1Z,Z1Z=(2j−1)22Weshallnowcalculatethephaseofthescalarproductψn,ψnbycalculatingthegeometricphasealongthegeodesictriangleT=[n0,n,n,n0].DenotebyΩthedomainonS2delimitedbyT.Wehave$#α(T)=iψn,dψn =idψn,dψn(7.68)TΩThisyields#α(T)=−jsinθdθdϕ=−jArea(Ω)ΩWedenoteby[n1,n2]theportionofgreatcircleonS2betweenn1andn2.Wenowintegrate(7.68)successivelyalong[n0,n],[n,n]and[n,n0].Fromthefactthat[n0,n],[n,n0]lieinverticleplaneswehave##ψn,dψn =ψn,dψn =0[n0,n][n,n0]ThusforanEulertriangleweget#−jArea(Ω)=−jΦn,n=iψn,dψn[n,n] 2127Spin-CoherentStatesInthegeneralcasewecansubdividethetriangleinseveralEulertriangleswithvertexatn0byaddingverticesbetweennandn.Thenweget#iΦn,n=−ψn,dψnj[n,n]Asaconsequenceofourstudyofthegeometricphase,letusnowcomputethephaseinformula(7.46).WealreadyknowthatD(niα1)D(n2)ψ0=D(n3)eandwehavetocomputeα.Wehaveψiα0,D(n1)D(n2)ψ0=eψ0,D(n3)ψ0∗ψ(7.69)=D(n1)0,D(n2)ψ0FromLemma48weknowthatψ0,D(n3)ψ0 ≥0.ButD(n1)∗=D(n∗)wheren∗11isthesymmetricofn1onthegreatcircledeterminedbyn0andn1.Applyingthecomputationofthephasein(7.53)wegetα=argn∗,n12Withanelementarygeometricargumentwegetthephaseinformula(7.46).Asaconsequence,wecangetthephaseA(g,n)informula(7.49).Ifg=gmthenA(gm,n)=Φ(m,n).Foragenericg∈SU(2)wehaveg=gmr3(δ)forsomeδ∈R.Sowehaveggn=gmr3(δ)gn.Letn=(θ,ϕ)(polarcoordinates).Wehaver3(δ)gn=gnr3(δ)wheren=(θ,ϕ+δ).SousingcomputationofΦin(7.46)andelementarygeometrywefindthatA(g,n)isequaltotheareaofthesphericaltriangle[n0,n,g·n].Letuscloseourdiscussionconcerningthegeometricphaseforcoherentstatesbythefollowingresult:Lemma48Letψ1andψ2betwodifferentstatesonagreatcircleofSj(unitsphereonV(j)).Oneparametrizesthisbytheangleθinthefollowingway:ψ(θ)=x1(θ)ψ1+x2(θ)ψ2wherexi(θ)∈Randψ(0)=ψ1,ψ(θ0)=ψ2.Oneassumesthatψ(θ),ψ(θ)˙=0Thenψ1andψ2areinphase,namelyψ1,ψ2>0 7.5CoherentStatesontheRiemannSphere213ProofOnecanassumethata= ψ1,ψ2>0.Thenbyaneasycalculuswegeta1x1(θ)=cosθ−√sinθ,x2(θ)=√sinθ,a=cosθ01−a21−a21 ψ(θ),ψ(θ)˙=√ ψ1,ψ221−a2Onededucesthatψ1,ψ2isrealthereforepositive.Remark45Fortwocoherentstates|nand|ntherearetwonaturalgeodesicsjoiningthem:thegeodesiconthetwosphereS2andthegeodesiconthesphereSj(sphere(2j+1)dimensional).ItisaconsequenceofresultsprovedabovethatthegeometricphasesforthesetwocurvesinV(j)arethesame.Thiswasnotobviousbeforecomputations.7.5CoherentStatesontheRiemannSphereWehaveseenthatitisconvenienttocomputeonthesphereS2usingcomplexcoor-dinatesgivenbythestereographicprojection.Weshallgiveheremoredetailsaboutthis.Inparticularthisgivesasemi-classicalinterpretationforthespin-coherentstatesandaquantizationofthesphereS2.Thepictureisanalogoustothehar-monicoscillatorcoherentstatesandtheassociatedWickquantizationofthephasespaceR2d.ThestereographicprojectionofthesphereS2fromitssouthpoleisthetransfor-mationπs(n)=ζ,definedbyn1+in2ζ=1+n3wheren=(n1,n2,n3).πsisanhomeomorphismfromS2:=S2{(0,0,1)}onthecomplexplaneC.MoreoverπscanbeextendedinanhomeomorphismfromS2onC˜:=C∪{∞}suchthatπs(0,0,1)=∞.C˜isaone-dimensionalcomplexandcompactmanifoldcalledtheRiemannsphere.πs−1isdeterminedbytheformulaπs−1ζ=(n1,n2,n3)whereζ+ζ¯ζ−ζ¯1−|ζ|2n1=,n2=,n3=1+|ζ|2i(1+|ζ|2)1+|ζ|2ItisknownthatthegroupofautomorphismsofC˜(bijectiveandbiholomorphictransformations)istheMöbiusgroup,thegroupofhomographictransformationsh(z)=az+bwherea,b,c,d∈Csuchthatad−bc=1.Theconventionsare:ifcz+dc=0thenh(∞)=∞;ifc=0thenh(∞)=a,h(−d)=∞.ccabLetusdenoteg=,g∈SL(2,C)andf=hg.Thenisnotdifficulttoseecdthathg=1ifandonlyifg=±12. 2147Spin-CoherentStatesWehaveseenthatifg∈SU(2)thenRgdefinesarotationinS2.InC˜weseethatRgbecomesaMöbiustransformation:Lemma49LetR˜−1abg=πsRgπsandg∈SU(2),g=.Thenwehave−b¯a¯aζ+bR˜gζ=,∀ζ∈C˜(7.70)−bζ¯+¯aProofWeonlygiveasketch.Firstitisenoughtoconsiderg=g2(θ).AftersomecomputationswegettheresultusingthatMöbiustransformationspreservethecrossratioζ1−ζ3.ζ2−ζ3ForsimplicitywedenoteR˜gζ:=g·ζ.Nowouraimistorealizetherepresentation(T(j),Vj)inaspaceofholomorphicfunctionsontheRiemannsphereC˜.Thisisachievedeasilywiththespin-BargmanntransformBjintroducedabove.RecallthatBjv(ζ)=ψζ,v(1+|ζ|2)j.WegettheimagesoftheDickebasisandofthecoherentstates:1/2d˜j(2j)!(ζ):=B(dk)(ζ)=ζ,j+k=(7.71)!(2j−)!ψ˜2j2−j2jζ(z)=ψz,ψζ1+|z|=1+|ζ|1+ζz¯,z,ζ∈C(7.72)LetusremarkthattheHilbertspaceVjistransformedintheHilbertspaceP2j(polynomialsofdegree≤2j+1)andthatP2jcoincideswiththespaceofholo-morphicfunctionsPonCsuchthat#P(ζ)22−2j−221+|ζ|dζ<+∞CSotheexponentialweightoftheusualBargmannspaceisreplacedherebyapoly-nomialweight.WeseenowthattheactionofSU(2)intheBargmannspaceP2jissimpleandhasanicesemi-classicalinterpretation.LetusdenoteT˜j(g):=BjTj(g)Bj.OntheRiemannspheretherepresentationT˜jhasthefollowingexpression:Proposition89Foreveryψ∈P2jandg∈SU(2)wehaveT˜j(g)ψ(ζ)=μj(g,ζ)ψg−1(ζ)(7.73)whereg=ab,μ2jaζ¯−b−b¯a¯j(g,ζ)=(a+bζ)¯andR˜g−1(ζ)=bζ¯+a. 7.5CoherentStatesontheRiemannSphere215ProofItisenoughtoprovethepropositionforψ(ζ)=ζ.FromdefinitionofBjwegetjj(j+k)!ζz¯2jj+kbz¯j−kBT(g)dk(ζ)=(2j)!√1+z2,(az¯1−bz2)1+az2Vj(j−k)!UsingthatthescalarproductinVjisinvariantfortheSU(2)actionweobtain1/2jj(2j)!2jaζ¯−bBT(g)d(ζ)=a+bζ¯(7.74)!(2j−)!bζ¯+aHencewegettheproposition.Now,followingOnofri[148]weshallgiveaclassicallymechanicalinterpretationofthetermμ(g,ζ).Thisinterpretationcanbeextendedtoanysemi-simpleLiegroup,asweshallseelater.LetusintroduceK(ζ,ζ)¯=2log(1+ζζ)¯(Kählerpotential),dtheexteriordif-ferential,∂theexteriordifferentialinζ,∂¯theexteriordifferentialinζ¯.Wehaved=∂+∂¯andd∂=∂∂¯=−∂∂¯.Weintroducetheoneformθ=−i∂Kandthetwoformdζ∧dζ¯ω=dθ=2i(1+|ζ|2)2ωisclearlyanon-degenerateantisymmetrictwoform.So(C˜,ω)isasymplecticmanifold.MoreoveritisaKählerone-dimensionalcomplexmanifoldfortheHer-mitianmetric2dζdζ¯ds=4(1+|ζ|2)2ItisnotdifficulttoseethatωisinvariantbytheactionofSU(2):gω=ω.InotherwordsSU(2)actsinC˜bycanonicaltransformations.C˜isconnectedandsimplyconnected,sothereexistsasmoothfunctionS(g,ζ)suchthatdS(g,ζ)=θ−gθ.Nowletuscomputedμasfollows.From(7.73)withψ=ψ˜0=1andusingthatψζ,ψ0 =(1+|ζ|2)jwegetψ0,Tj(g−1gζ)ψ0μj(g,ζ)=ψ0,ψζThenwecomputedμ=(ij(θ−gθ))μ=(ijdS)μ.Hencewegettheclassicallymechanicalinterpretationforμ(g,ζ)%ζij(θ−gθ)μ(g,ζ)=μ(g,0)e0(7.75) 2167Spin-CoherentStates7.6ApplicationtoHighSpinInequalitiesOneofthefirstsuccessfuluseofspin-coherentstateswasthethermodynamiclimitofspinsystemsasanapplicationofBerezin–Liebinequalities.Berezin–Liebin-equalityholdstrueforgeneralcoherentstates.7.6.1Berezin–LiebInequalitiesWeshallfollowherethenotationsofSect.2.6concerningWickquantization.WeassumeherethattheHilbertspaceHisfinitedimensional(thisisenoughforourapplication).LetAˆ∈L(H)withacovariantsymbolAcandcontravariantsymbolAcdefinedonsomemetricspaceMwithaprobabilityRadonmeasuredμ(m).Itisnotdifficulttoseethatthesetwosymbolssatisfythefollowingdualityformulas:#TrAˆBˆ=Ac(m)Bc(m)dμ(m)(7.76)M#e2cAcm=m|emA(m)dμ(m)(7.77)MInparticularwehave#TrAˆ=Ac(m)dμ(m)(7.78)MLetusremarkthatifAciswelldefined,Acisnotuniquelydefinedingeneral.Theorem42LetHˆbeaself-adjointoperatorinHandχaconvexfunctiononR.Thenwehavetheinequalities##χHc(m)dμ(m)≤TrχHˆ≤χHc(m)dμ(m)(7.79)MMProofLetusrecalltheJenseninequality[170],whichisthemaintoolforprovingtheBerezin–Liebinequalities.ForanyprobabilitymeasureνonM,anyconvexfunctionχonRandanyf∈L1(M,dν)wehave##χfdν(m)≤χf(m)dν(m)(7.80)MMWestartwiththeformula#TrχHˆ=em,χHˆemdμ(m)MUsingthespectraldecompositionforself-adjointoperators,wehave#em,χHˆem=χ(λ)dνm(λ)R 7.6ApplicationtoHighSpinInequalities217whereνmisthespectralmeasureofthestateem.Itisaprobability(discrete)mea-sure.SotheJenseninequalitygivesem,χHˆem≥χAc(m)IntegratinginmwegetthefirstBerezin–Liebinequality#χHc(m)dμ(m)≤TrχHˆMForthesecondinequalityweintroduceanorthonormalbasisofH:{vn}1≤n≤NofeigenfunctionsofAˆ.Sowehave#χHˆHvˆcv2vnvn=χvnn=χH(m)n|emdμ(m)MFromJenseninequalityappliedwiththeprobabilitymeasure|vn|em |2dμ(m)weget#χHˆcv2vnvn≤χH(m)n|emdμ(m)MSumminginnwegetthesecondBerezin–Liebinequality.7.6.2HighSpinEstimatesWeconsideraone-dimensionalHeisenbergchainofNspinSn=(Sn,Sn,Sn),1≤123n≤N.TheHamiltonianofthissystemisHˆ=−SnSn+11≤n≤N−1HˆisanHermitianoperatorinthefinite-dimensionalHilbertspaceHNHnN=⊗whereHn=V(j)foreveryn.FromthecoherentstatessystemsinV(j)wegetinastandardwayacoherentsysteminHNparametrizedon(S2)N(orCNusingthecomplexparametrisation).Inthesphererepresentation,ifΩ=(n1,...,nN)∈SNwedefinethecoherentstateψΩ=ψn1⊗ψn2···ψnN.Weget,asforN=1,anovercompletesystemwitharesolutionofidentity.WedenotedΩNtheprobabilitymeasure(4π)−Ndn1⊗···⊗dnN.Wickandanti-WickquantizationarealsowelldefinedasexplainedinChap.1.InparticulartheBerezin–Liebinequalitiesaretrueinthissetting.Asusualthefollowingconventionisused:ifAˆ∈L(Hm)andBˆ∈L(Hm)thenAˆBˆ=1⊗1···⊗Aˆ⊗···⊗Bˆ⊗1···⊗1withAˆatpositionm,Bˆatposi-tionm. 2187Spin-CoherentStatesInparticularforthecovariantandcontravariantsymbolsofAˆBˆwehavethefollowingobviousproperties:mm(AB)c(Ω)=AcnBcn(7.81)(AB)c(Ω)=AcnmBcnm(7.82)Weshallprovethefollowingresultsconcerningthesymbolsofonespinoperators(S1,S2,S3):Proposition90ThecovariantsymbolsareS1,c(n)=−jsinθcosϕ(7.83)S2,c(n)=−jsinθsinϕ(7.84)S3,c(n)=−jcosθ(7.85)ThecontravariantsymbolsareSc(n)=−(j+1)sinθcosϕ(7.86)1Sc(n)=−(j+1)sinθsinϕ(7.87)2Sc(n)=−(j+1)cosθ(7.88)3ProofItseemsmoreconvenienttocomputeinthecomplexrepresentationψζforthecoherentstates.InthespaceV(j)thespinoperatorsarerepresentedbydifferen-tialoperators1S1=(z1∂z+z2∂z)(7.89)2121S2=(z1∂z−z2∂z)(7.90)212i1S3=(z1∂z−z2∂z)(7.91)122jThecoherentstateψhastheexpressionζ12−j2jψζ(z1,z2)=√1+|ζ|(ζz1+z2)(2j)!Soadirectcomputationgives ζζ|S1|ζ =−2j=−jsinθcosϕ(7.92)1+|ζ|2InthesamewaywecancomputeS2,c(n)andS3,c(n). 7.6ApplicationtoHighSpinInequalities219Computingcontravariantsymbolsismoredifficultbecausewehavenodirectformula.Thetrickistostartwithalargeenoughsetoffunctionsincomplexvariables(ζ,ζ)¯andtocomputetheiranti-Wickquantizations.LetusdenoteAα,β,τ(ζ,ζ)¯=(1+|ζ|2)−τζαζ¯βandAˆα,β,τ(k,)thematrixele-mentsoftheoperatorAˆα,β,τ,−j≤k,≤j,inthecanonicalbasisofV(j).Weshallusethefollowingformula:#Ac(ζ)vv,Avˆk=,Πζvkdμj(ζ)C(7.93)v,Πζvk =v,ψζψζ,vk =ckcζ¯j+ζj+kwherec(2j)!1/2k=().(j+k)!(j−k)!Soweget#Aˆ2j+k+αζ¯j++β2−2j−τ−2α,β,τ(k,)=ckcdζζ1+|ζ|CUsingpolarcoordinatesζ=reiγweget#∞r2j+2α+2k+1Aˆα,β,τ(k,)=ckcδα+k,β+2(2j+1)dr22j+τ+20(1+r)Wecomputetheonevariableintegralusingbetaandgammafunctions#∞sΓ(s+1)Γ(t−s+1)r1s+1s+1122dr=B,t−=0(1+r2)t2222Γ(t)FinallywehavetheformulaΓ(j+k+α+1)Γ(j−α−k+τ+1)Aˆα,β,τ(k,)=(2j+1)ckcδα+k,β+Γ(2j+τ+2)(7.94)Forexampleifτ=1,α=0,1,β=0,1,2wefind1Aˆ1,0,1(k,k+1)=(j+k+1)(j−k)(7.95)2j+21Aˆ0,1,1(k,k+1)=(j+k)(j−k+1)(7.96)2j+21Aˆ0,0,1(k,k+1)=(j−k+1)(7.97)2j+21Aˆ1,1,1(k,k+1)=(j+k+1)(7.98)2j+2UsingtheseresultsweeasilygetacontravariantsymbolforSasin(7.86). 2207Spin-CoherentStatesNowwecancomputethecovariantandcontravariantsymbolsoftheHamilto-nianHˆ.Applyingtheaboveresults,translatedinthespherevariables,wegetH2nmnm+1(7.99)c(Ω)=j1≤m≤N−1Hc(Ω)=(j+1)2nmnm+1(7.100)1≤m≤N−1Wearereadynowtoprovethemainresultwhichisaparticularcaseofmuchmoregeneralresultsprovedin[136,177].Letusintroducethequantumpartitionfunctionq1−βHˆZ(β,j)=Tre,β∈R(2j+1)NThecorrespondingclassicalpartitionfunctionisobtainedtakingtheaverageofspinoperatorsoncoherentstates.Sowedefine#Zc(β,j)=e−βHc(Ω)dμ(Ω)(S2)NPuttingtogetherallthenecessaryresultswehaveprovedthefollowingLieb’sin-equalities:Proposition91Wehavethefollowinginequalities:Zc(β,j)≤Zq(β,j)≤Zc(β,j+1),∀j,integerorhalf-integer(7.101)Thisresultcanbeusedtostudythethermodynamiclimitoflargespinsystems(see[136]).Corollary25WiththenotationsofthepreviouspropositionwehaveZq(β,j)lim=1(7.102)j→+∞Zc(β,j)7.7MoreonHighSpinLimit:FromSpin-CoherentStatestoHarmonic-OscillatorCoherentStatesWewanttogiveheremoredetailsconcerningthetransitionbetweenspin-coherentstatesandHeisenbergcoherentstates.Letusbeginbythefollowingeasyconnectionbetweenspin-coherentstatesandharmonicoscillatorcoherentstate.WecomputeontheBargmannside.Westartfromj,2−j2jψη(ζ)=1+|η|1+¯ηζ 7.7MoreonHighSpinLimit:FromSpin-CoherentStates221ηζIfwereplaceηandζby√and√andletj→+∞thenweget2j2jj,ζηζ¯−|η|2/2√limψη√=e=2πϕη(ζ)j→+∞√2j2jIntherightsideϕη(ζ)istheBargmanntransformoftheGaussiancoherentstatelocatedinη.Inthissensethespin-coherentstatesconverge,inthehighspinlimit,tothe“classical”coherentstates.InthesamewayweshallprovethattheDickestatesconvergetotheHermitebasisoftheharmonicoscillator.Letusdenotenowdj,ktheDickebasisofV(j):Hdˆj,k=kdj,k,−j≤k≤j.Thespin-Bargmanntransformofdj,kiseasilycomputed:(2j)!j+kd(ζ)=ζj,k(j+k)!(j−k)!Recallthat(seeChap.1)theBargmanntransformoftheHermitefunctionψ(∈N)isζψ(ζ)=√2π!ThenwehaveProposition92Forevery∈Nandeveryr>0wehaveζ√limd√=2πψ(ζ)(7.103)j,kj→+∞,k→−∞2jj−k→uniformlyin|ζ|≤r.ProofTheresultfollowsfromthefollowingapproximations.Forj,klargeenoughsuchthatj−k≈wehave(2j)!(2j)!(2j)≈≈(j+k)!(j−k)!!(2j−)!!Adifferentapproachconcerningthehighspinlimitofspin-coherentstatesistoconsidercontractionsoftheLiegroupSU(2)intheHeisenbergLiegroupH1(see[121]).WearenotgoingtoconsiderherethetheoryofLiegroupcontractionsingeneralbuttocomputeontheexampleofSU(2)anditsrepresentations.LetusconsidertheirreduciblerepresentationofindexjofSU(2)intheBargmannspaceP2j.P2jisclearlyafinite-dimensionalsubspaceoftheBarg-mann–FockspaceF(C). 2227Spin-CoherentStatesLetuscomputeinthisrepresentationtheimagesL±,L3ofthegeneratorsK±,K3oftheLiealgebrasl(C).Wegeteasily∂L=ζ−j(7.104)3∂ζ2∂L+=2jζ−ζ(7.105)∂ζ∂L−=(7.106)∂ζLetusintroduceasmallparameterε>0anddenoteεL+=εL+(7.107)εL−=εL−(7.108)ε1L3=L3+21(7.109)2εWehavethefollowingcommutationrelations:Lε,Lε=2ε2Lε−1,Lε,Lε=±Lε(7.110)+−33±±Asε→0(7.107)defineafamilyofsingulartransformationsoftheLiealgebrasu(2)andforε=0weget(formally)L0,L0=−1,L0,L0=±L0(7.111)+−3±±ThesecommutationrelationsarethosesatisfiedbytheharmonicoscillatorLiealge-bra:L0+≡a†,L0−≡a,L0≡N:=a†a.3Wecangiveamathematicalproofofthisanalogybycomputingcovariantsym-bols.Proposition93Assumethatε→0andj→+∞suchthatlim2jε2=1.ThenwehavejLεψj2†limψ√3√=|ζ|=ϕζ|aa|ϕζ(7.112)ζ/2jζ/2jjLεψj†limψ√+√=ζ¯=ϕζ|a|ϕζ(7.113)ζ/2jζ/2jjLεψjlimψ√−√=ζ=ϕζ|a|ϕζ(7.114)ζ/2jζ/2jProofFromthecomputationsofSect.7.5wehavejj1−|ζ|2ψLψ=−jζ3ζ1+|ζ|2 7.7MoreonHighSpinLimit:FromSpin-CoherentStates223SowehavejLεψjζ2+O1ψ√√=+ζ/2j3ζ/2jjandweget(7.112).Theotherformulasareprovedinthesamewayusingthefollowingrelations:jLψj2jζ¯jLψj2jζψζ+ζ=1+|ζ|2,ψζ−ζ=1+|ζ|2 Chapter8Pseudo-Spin-CoherentStatesAbstractWehaveseenbeforethatspin-coherentstatesarestronglylinkedwiththealgebraicandgeometricpropertiesoftheEuclidean2-sphereS2.Weshallnowconsideranaloguesettingwhenthesphereisreplacebythe2-pseudo-spherei.e.theEuclideanmetricinR3isreplacedbytheMinkowskimetricandthegroupSO(3)bythesymmetrygroupSO(2,1).ThemainbigdifferencewiththeEuclideancaseisthatintheMinkowskicasethepseudo-sphereinnoncompactaswellitssym-metrygroupSO(2,1)andthatallirreducibleunitaryrepresentationsofSO(2,1)areinfinitedimensional.8.1IntroductiontotheGeometryofthePseudo-Sphere,SO(2,1)andSU(1,1)Inthissectionweshallgiveabriefintroductiontopseudo-Euclideangeometry(of-tennamedhyperbolicgeometry).Thereexistsahugeliteratureonthissubject.Formoredetailswerefertoin[11]ortoanystandardtextbookonhyperbolicgeometrylike[36].8.1.1MinkowskiModelOnthelinearspaceR3weconsidertheMinkowskimetricdefinedbythesym-metricbilinearformx,yM:=xy:=x1y1+x2y2−x0y0wherex,y∈R3,x=(x0,x1,x2),y=(y0,y1,y2).Sowegetthethree-dimensionalMinkowskispace(R3,·,·M)withitscanonicalorthonormalbasis{e0,e1,e2}.LetusremarkthattheMinkowskimetricistherestrictionoftheLorentzrelativis-ticmetric,definedbythequadraticformx2+x2+x2−x2,tothethree-dimensional1230subspaceofR4definedbyx3=0.ThesurfaceinR3definedbytheequation{x∈R3,x,xM=−1}isahyper-boloidwithtwosymmetricsheets.Thepseudo-spherePS2isoneofthissheet.Sowecanchoosetheuppersheet:PS2=x=(x2220,x1,x2),x1+x2−x0=−1,x0>0M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,225TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_8,©SpringerScience+BusinessMediaB.V.2012 2268Pseudo-Spin-CoherentStatesPS2isasurfacewhichcanbeparametrizedwiththepseudopolarcoordinates(τ,ϕ):x0=coshτ,x1=sinhτcosϕ,x2=sinhτsinϕ,τ∈[0,+∞[,ϕ∈[0,2π[PS2isaRiemannsurfaceforthemetricinducedonPS2bytheMinkowskimetric.Incoordinates(x0,x1,x2)PS2isdefinedbytheequationx0=1+x2+x2.So,12incoordinates{x1,x2}onPS2,themetricds2=−dx2+dx2+dx2isgivenbythe012followingsymmetricmatrix:⎛⎞x2xx1−112⎜1+x2+x21+x2+x2⎟G=1212⎝2⎠x1x21−x21+x2+x21+x2+x21212Henceweseethatds2ispositive-definiteonPS2.Inpolarcoordinateswehaveasimplerexpression:ds2=dτ2+sinhτ2dϕ2.ThecurvatureofPS2is−1every-where(comparewiththesphereS2withcurvatureis+1everywhere).ByanalogywiththeEuclideansphereweshalldenotenthegenericpointonS2.TheRiemanniansurfacemeasureinpseudopolarcoordinatesisgivenbycomput-√ingthedensitydetG,whereGisthematrixofthemetricincoordinates(τ,ϕ).SothesurfacemeasureonPS2isd2n=sinhτdτdϕWecanseethatthegeodesicsonthepseudo-spherePS2aredeterminedbytheirintersectionwithplanesthroughtheorigin0.WeconsidernowthesymmetriesofPS2.Letusdenote⎛⎞−100L=⎝010⎠001thematrixofthequadraticform•,•M:x,yM=Lx·y(recallthatthe·denotestheusualscalarproductinR3).Theinvariancegroupof•,•MisdenotedO(2,1).SoA∈O(2,1)meansAxAx=xxforeveryx∈R3orequivalently,ATLA=L(ATisthetransposedmatrixofA).InparticularifA∈O(2,1)thendetA=±1.ThedirectinvariancegroupisthesubgroupSO(2,1)definedbyA∈O(2,1)anddetA=1.ThisgroupisnotconnectedsoweintroduceSO0(2,1)thecomponentof1inSO(2,1),itisaclosedsubgroupofSO(2,1).Thepseudo-spherePS2isclearlyinvariantunderSO0(2,1).Itisnotdifficulttoseethat1isalwaysaneigenvalueforeveryA∈SO0(2,1).Letv∈R2besuchthatv=1,Av=vandv,vM=0.TheorthogonalcomplementofRvfortheMinkowskiform.,.Misatwo-dimensionalplaneinvariantbyA.SoAlookslikearotationinEuclideangeometry. 8.1IntroductiontotheGeometryofthePseudo-Sphere,SO(2,1)andSU(1,1)227LetusgivethefollowingexamplesoftransformationsinSO0(2,1):⎛⎞100Rϕ=⎝0cosϕ−sinϕ⎠:rotationintheplane{e1,e2}(8.1)0sinϕcosϕ⎛⎞coshτsinhτ0B1,τ=⎝sinhτcoshτ0⎠:boostinthedirectione1(8.2)001⎛⎞coshτ0sinhτB2,τ=⎝010⎠:boostinthedirectione2(8.3)sinhτ0coshτThesethreetransformationsgenerateallthegroupSO0(2,1).Thiscanbeeasilyprovedusingthefollowingremark:ifAv=vandifUisatransformationthenAU(Uv)=UvwhereAU=UAU−1.SO0(2,1)isaLiegroup.Inparticularitisathree-dimensionalmanifold.8.1.2LieAlgebraWecangetabasisfortheLiealgebraso(2,1)ofSO(2,1)bycomputingthegener-atorsofthethreeone-parametersubgroupsdefinedin(8.1).Weget⎛⎞000dE0:=Rϕ|ϕ=0=⎝00−1⎠(8.4)dϕ010⎛⎞010dE1:=B1,τ|τ=0=⎝100⎠(8.5)dτ000⎛⎞001dE2:=B2,τ|τ=0=⎝000⎠(8.6)dτ100ThecommutationrelationsoftheLiealgebraarethefollowing:[E0,E1]=E2,[E2,E0]=E1,[E1,E2]=−E0(8.7)Ifwecomparewithgeneratorsofso(3)weremarktheminussigninthelastrelation.Letusconsidertheexponentialmapexp:so(2,1)→SO(2,1).IfA=x0E0+x1E1+x2E2wherex2+x2+x2=1,wehavetheone-parametersubgroupof012SO(2,1),R(θ)=eθA.R(θ)isapseudo-rotationwithaxisv=(x0,−x2,x1).Itsgeometricalpropertiesareclassifiedbythesignofv,vM. 2288Pseudo-Spin-CoherentStates•v,vM>0(“time-like”axis):R(1)hasauniquefixedpointonPS2,eachorbitisbounded.R(1)issaidtobeelliptic•v,vM<0(“space-like”axis):thereexistsauniquegeodesiconPS2invariantbyU(1),R(1)issaidhyperbolic•v,vM=0(“light-like”axis):thegeodesicsasymptoticallygoingtoRvarein-variantbyR(1).R(1)issaidparabolicTheseclassificationwillbemoreexplicitonothermodelsofPS2asweshallsee.AsintheEuclideancasewecanrealizetheLiealgebrarelations(8.7)inaLiealgebraofcomplex2×2matrices.WereplacethegroupSU(2)bythegroupSU(1,1)ofpseudo-unitaryunimodularmatricesoftheform:αβ22g=,|α|−|β|=1β¯α¯SU(1,1)isaLiegroupofrealdimension3.LetusintroducenowaconvenientparametrizationofSU(1,1):ti(ϕ+ψ)/2ti(ϕ−ψ)/2α=coshe,β=sinhe22wherethetriple(ϕ,t,ψ)runsthroughthedomain0≤ϕ<2π,00}bythehomographyH0ζ=ζ+1orH−1z=−z+i,whereζ∈D,z∈H.Explicitly,ifζ=reiϕandz=u+iv,we0z+ihave2rsinϕ1−r2u=,v=1+r2+2rcosϕ1+r2+2rcosϕInHthemetricandthesurfaceelementofthepseudo-spherehavethefollowingexpression:du2+dv2dudvds2=,d2z=v2vSowegetanothermodelofthepseudo-spherenamedthePoincaréhalf-planemodel.Inthismodeltheboundaryofthediscistransformedintotherealaxis.InHthegeodesicsareverticallinesandhalfcirclesorthogonaltotherealaxis.AsforthediscwecanseethatthedirectsymmetrygroupofHisthegroupαz+βSL(2,R)withthehomographicactionHg(z)=,α,β,γ,δ∈R,αδ−γβ=1γz+δandHg=1Hifandonlyifg=±1.WerecoverthefactthatSU(1,1)andSL(2,R)areisomorphgroups.Finallythereisanotherrealizationofthepseudo-spherewhichisimportanttodefinecoherentstates:PS2canbeseenasaquotientofSU(1,1)byitscompactmaximalsubgroupU(1). 8.2UnitaryRepresentationsofSU(1,1)231Lemma50Foreveryn=(coshτ,sinhτsinϕ,sinhτcosϕ)definecosh(τ/2)sinh(τ/2)e−iϕgn=iϕsinh(τ/2)ecosh(τ/2)Thenthemapn →{gnω0(t),t∈[−2π,2π[}isabijectionfromS2intherightcosetsofSU(1,1)moduloU(1)whereU(1)isidentifiedwiththediagonalmatriceseit/200e−it/2,t∈[0,4π[.ProofLetusdenoteg(α,β)=αβ,α,β∈C,|α|2−|β|2=1.Thelemmaisaβ¯α¯directconsequenceofthefollowingdecomposition:thereexistα>0,β∈C,t∈[0,4π[,uniquesuchthatωg(α,β)=gα,β0(t)Morepreciselywehaveα=|α|,t=2argα,β=eit/2β.Thisprovesthelemma.Somodulocompositionbyarotationofaxise0ontheright,everyg∈SU(1,1)isequivalenttoauniquegn,n∈PS2.Wecanseethatgnisapseudo-rotationwithaxisv=(0,sinϕ,cosϕ)(remarkachangeofsign).Togetthat,remark−1d1gngn=(cosϕσ1+sinϕσ2)=(cosϕb1−sinϕb2)dτ2sogn=eτ(cosϕb1−sinϕb2),thisisindeedapseudo-rotationofaxisdirection(0,sinϕ,cosϕ).8.2UnitaryRepresentationsofSU(1,1)LetusbeginbyintroducingausefulandimportantinvariantoperatorforLiegrouprepresentation:theCasimiroperator.WefirstdefinetheKillingformonaLiealge-brag,1X,Y0=Trad(X)ad(Y)(8.9)2whereadistheLieadjointrepresentation:ad(X)Y=[X,Y],∀Y∈g.ad(X)istheinfinitesimalgeneratoroftheone-parametergrouptransformationing:GθX−θXθ(Y)=eYeWehavetheantisymmetricproperty:ad(X)Y,Z=−Y,ad(X)Z,∀X,Y,Z∈g00 2328Pseudo-Spin-CoherentStatesConsiderabasis{Xj}ofg.TheKillingforminthisbasishasthematrixgj,k=tr(ad(Xj)ad(Xk)).Wedenotegj,ktheinverseofthematrixgj,k.LetusnowconsiderarepresentationRofaLiegroupGinthelinearspaceVandρ=dRthecorrespondingrepresentationofitsLiealgebraginL(V).TheρCasimiroperatorCasisdefinedasfollows:Cρ=gj,kρ(Xasj)ρ(Xk)LetusremarkthatifVisinfinitedimensionalsomecareisnecessarytochecktheρdomainofCas.Neverthelessastandardcomputationgivesthefollowingimportantproperty.ρLemma51TheCasimiroperatorCascommuteswiththerepresentationρ,ρρρ(X)Cas=Casρ(X),∀X∈g.ρρInparticulariftherepresentationisirreduciblethenCas=cas1,cas∈R.LetusnowremarkthateveryirreducibleunitaryrepresentationofSU(1,1)isinfinitedimensional,thisisabigdifferencewithSU(2).Proposition95LetρbeaunitaryrepresentationofSU(1,1)inafinite-dimensionalHilbertspaceH.Thenρistrivial,i.e.ρ(g)=1H,∀g∈SU(1,1)ProofUsingtheisomorphismfromSL(2,R)ontoSU(1,1),itisenoughtoprovethepropositionforaunitaryrepresentationofSL(2,R).Foreveryx∈R,a∈R,a=0,wehave−12a01xa01ax−101−1=0a0a011xThenweseethatρareallconjugateforx>0.Bycontinuitytheyareconjugate0110to1H=ρ01becausetherepresentationisfinitedimensional.Thesameproperty10holdstrueforx<0andformatrices.ButthegroupSL(2,R)isgeneratedbyy11x10thetwomatrices01andy1.Sowecanconcludethatρ(A)=1HforeveryA∈SL(2,R).UnitaryirreduciblerepresentationsofSU(1,1)havebeencomputedindepen-dentlybyGelfand–Neumark[81]andbyBargmann[15].WeshallfollowherethepresentationbyBargmannwithsomeminormodifications. 8.2UnitaryRepresentationsofSU(1,1)2338.2.1ClassificationofthePossibleRepresentationsofSU(1,1)LetρbeanirreducibleunitaryrepresentationofSU(1,1)insomeHilbertspaceH.ItcanbeprovedthatHisinfinitedimensionalexceptifρistrivial(ρ(g)=1,∀g∈SU(1,1)).SoHwillbeinfinitedimensional.IntherepresentationspacethegeneratorsbjdefinetheoperatorsdBj:=iρωj(t),j=0,1,2dtt=0withthecommutationrelations[B1,B2]=−iB0,[B2,B0]=iB1,[B0,B1]=iB2(8.10)OrwiththecomplexnotationB±=B2±iB1,wehave[B−,B+]=2B0,[B0,B±]=±B±(8.11)Wehaveρ(ω0(t))=e−itB0hencee−i4πB0=1.SothespectrumofB0isasubsetofk,k∈Z}.Thereexistsψk0{0∈H,ψ0=1andB0ψ0=λψ0,λ=,k0∈Z.22UsingthecommutationrelationwehaveB0B+ψ0=(λ+1)B+ψ0(8.12)B0B−ψ0=(λ−1)B−ψ0(8.13)Reasoningbyinduction,wegetforeveryk∈N,Bkk0B+ψ0=(λ+k)(B+)ψ0(8.14)kψkB0(B−)0=(λ−k)(B−)ψ0(8.15)IntroducenowtheCasimiroperatorCasoftherepresentationwhichissupposedtobeirreducible,soCas=cas1where21Cas=B0−(B−B++B+B−)2Hencewehavetheequations(B2−B++B+B−)ψ0=2λ−casψ0(8.16)(B−B+−B+B−)ψ0=2λψ0(8.17)SowegetB−B+ψ0=λ(λ+1)−casψ0(8.18)B+B−ψ0=λ(λ−1)−casψ0(8.19) 2348Pseudo-Spin-CoherentStatesUsingnowB±kψ0insteadofψ0wehaveprovedforeveryk∈N,kψB−B+ψ0=(λ+k−1)(λ+k)−cas0(8.20)kψB+B−ψ0=(λ−k+1)(λ−k)−cas0(8.21)+=((λ+k−1)(λ+k)−c−Letusdenoteνas)andν=((λ−k+1)(λ−k)−cas).kkUsingthatB+=B−∗andB−=B+∗wegetfrom(8.20),Bk+12+Bk2+ψ0=νk+1+ψ0(8.22)Bk+12−Bk2−ψ0=νk+1−ψ0(8.23)From(8.22)wecanstartthediscussion.(I)Supposethatforallk∈N,B+kψ0=0andB−kψ0=0.Thenforeveryk∈N,λ±kisaneigenvalueforB0.(I-1)If0isinthisfamily(i.e.λ∈Z)thenwecansupposethatB0ψ0=0sowecanchooseλ=0.From(8.22)wefindthenecessaryconditioncas<0.1+kk(I-2)If0isnotinthefamilyλ±k,λ=20,k0∈ZandusingB±wecanassumethatλ=1.From(8.22)wegetcas<1/4.2Inthesetwocaseswegetanorthonormalbasis{ϕm}m∈ZforH,suchthatε)ϕB0ϕm=(m+m,whereε=0inthefirstcaseandε=1inthesecondcase.2Thisisreallyabasisbecausethelinearspacespanbytheϕmisinvariantbytherepresentationwhichisirreducible.k0+1k0(II)Supposenowthatthereexistsk0∈NsuchthatB+ψ0=0andB+ψ0=0.ψk0Using(8.20)weseethatforevery∈N,B−0isproportionaltoB−B+ψ0,sok0wecanreplaceψ0byB+ψ0.SowehaveB0ψ0=λψ0,B+ψ0=0.Hencethis+=0andcgivesνas=λ(λ+1).Ifλ=0wegetB−ψ0=0andtheHilbert1spaceisunidimensional.Sowehaveλ>0.AsabovewegetanorthonormalbasisofH{ϕm}m∈NsuchthatB0ϕm=(λ−m)ϕm.k0+1k0Ifthereexistsk0∈NsuchthatB−ψ0=0andB−ψ0=0wehaveasimilarresultwitheigenvaluesλ+mforB0.Incase(I)theCasimirparametercasvariesinanintervalandwesaidtherepresen-tationbelongstothecontinuousseries;incase(II)theCasimirparametervariesinadiscretesetandwesaythattherepresentationbelongstothediscreteseries.8.2.2DiscreteSeriesRepresentationsofSU(1,1)InthelastsectionwehavefoundnecessaryconditionssatisfiedbyanyirreduciblerepresentationofSU(1,1).Nowwehavetoprovethattheseconditionscanbereal-izedinsomeconcreteHilbertspaces. 8.2UnitaryRepresentationsofSU(1,1)2358.2.2.1TheHilbertSpacesHn(D)Letnbearealnumber,n≥2,Hn(D)istheHilbertspaceofholomorphicfunctionsfontheunitdiscDofthecomplexplaneCsatisfying2n−1f(z)22n−2f1−|z|dxdy<+∞,z=x+iy(8.24)Hn(D):=πDThemeasuredνn−12n−2n(z):=(1−|z|)dxdyisaprobabilitymeasureonDandπHn(D)isacompletespacewiththeobviousHilbertnormisaconsequenceofstan-dardpropertiesofholomorphicfunctions.ItisusefultoproducethefollowingcharacterizationofHn(D)usingtheseriesexpansionoff:ckf(z)=k(f)zk≥0whichisabsolutelyconvergentinsidethediscD.In(8.24)letuscomputetheintegralinpolarcoordinatesz=reiθ.FromtheParsevalformulaforFourierseriesweget1f=2(n−1)πc2r2k+11−r2n−2dr(8.25)Hn2(D)k(f)0k≥0Sowehavef2c2Γ(n)Γ(k+1)(8.26)Hn(D)=k(f)Γ(n+k)k≥0ThisgivesaunitaryequivalentdefinitionofHn(D)asaHilbertspaceoffunctionsontheunitcircle.InparticularthescalarproductinHn(D)off1andf2hasthefollowingexpression:f1,f2Hn(D)=ck(f1)ck(f2)γn,k(8.27)k≥0whereγΓ(n)Γ(+1)n,=.Γ(n+)zFromformula(8.26)and(8.27)weeasilygetthate(z):={√}≥0isanor-γn,thonormalbasisofHn(D).8.2.2.2DiscreteSeriesRealizationofSU(1,1)inHn(D)TheserepresentationscanbeintroducedusingGaussdecompositioninthecomplexLiegroupSL(2,C),αβ10α01β=γ1α(8.28)γδ1001αα 2368Pseudo-Spin-CoherentStateswhereα,β,γ,δarecomplexnumberssuchthatαδ−βδ=1,α=0.Moreoverthisdecompositionasaproductlike10u01wz10101uisunique.SothisallowsustodefinenaturalactionsofSU(1,1)inthediscD.101zLetusdenotebyt−(z)thematrices,t+(z)thematricesandd(u)=z101u0−1.0uLetusdenotegagenericelementofSU(1,1),αβg=β¯α¯Considerthet−matrixintheGaussiandecompositionofgt−(z).Wehavet−=β¯+¯αzt−(z)˜wherethecomplexnumberz˜isz˜=M−(g)(z)=.Wegeteasilythatα+βz|M−(g)(z)|=1if|z|=1andfromthemaximumprinciple,|M−(g)(z)|<1if|z|<1.SowehavedefinedarightactionofSU(1,1)inD.Inthesamewaywegetaleftactionconsideringthet1z−1+matrixinthedecompositionofg.Sowe01−β+αzgettheactionM+(g)(z)=.α¯−βz¯NowwewanttodefineunitaryactionsofSU(1,1)inthespaceHn(D)asfol-lows:−(g)f(z)=m−(z)fMg−1z(8.29)Dng−wherethemultiplierm−g(z)ischosensuchthatDn−definesanunitaryrepresentationofSU(1,1).Weprovenowthatitistruewiththechoicem−g(z)=(α¯−βz)−n.Theorem43Foreveryintegern≥2wehavethefollowingunitaryrepresentationofSU(1,1)intheHilbertspaceH(D):−−n−β¯+αzDn(g)f(z)=(α¯−βz)f(8.30)α¯−βz+−nβ+¯αzDn(g)f(z)=(α+βz)¯f(8.31)α+βz¯ProofItisnotdifficulttoseethatDn±areSU(1,1)actionsinthelinearspaceHn(D).LetusprovethatDn−isunitary.Thisfollowswiththeholomorphic−β¯+αzβ¯+¯αZchangeofvariablesZ=,Z=X+iY,z=x+iy.Wegetz=,α¯−βzα+βZdz=(βZ+α)−2.WeconcludeusingthatforaholomorphicchangeofvariabledZintheplane,wehavefromtheCauchyconditions,2∂(x,y)dzdet=∂(X,Y)dZ 8.2UnitaryRepresentationsofSU(1,1)237Sowegetforanyf∈Hn(D),2−2n−β¯+αz2n−2|¯α−βz|f1−|z|dxdyDα¯−βzf(Z)22n−2=1−|Z|dXdYDwhichsaysthatD−isunitary.Letusremarkherethattheabovecomputationsshownthatthemultiplierm−gisnecessarytoproveunitarity.Thenextstepistoprovethattheserepresentationsareirreducible.TodothatwefirstcomputethecorrespondingLiealgebrarepresentation.LetuscomputetheimageofthisbasisbytherepresentationD−.Straightforwardncomputationsgived−1dDnω0(t)f(z)=n+2zf(z)(8.32)dtt=02idzd1dD−ωnz+z2f(z)(8.33)n1(t)f(z)=−1dtt=02dzd1dD−ωnz+1+z2f(z)(8.34)n2(t)f(z)=dtt=02idzSowegetthethreeself-adjointgeneratorsB0,B1,B2,Bj=idDn−(bj),whereddenotesthedifferentialonthegroupat1,ndB0=+z(8.35)2dzidB21=nz+z−1(8.36)2dz1dB22=nz+z+1(8.37)2dzwiththecommutationrelations[B1,B2]=−iB0,[B2,B0]=−iB1,[B0,B1]=iB2(8.38)UsingthenotationB±=B2∓iB1,wehavedB−=(8.39)dz2dB+=nz+z(8.40)dzand[B−,B+]=2B0,[B0,B±]=±B±(8.41) 2388Pseudo-Spin-CoherentStatesRemark46OperatorsBafora=0,1,2,±arenon-boundedoperatorsintheHilbertspaceHn(D)sowehavetodefinetheirdomains.Hereweknowthattherepresen-tationDn−isunitary.SoStone’stheoremgivesthatB0isessentiallyself-adjointandthelinearspaceP∞ofallpolynomialsinzisacoreforB0.MoreoverthespectrumofB0isdiscrete,withsimpleeigenvalues{n/2+k,k∈N}.B1andB2alsohaveauniqueclosedextension.Moreoveritcouldbepossibletocharacterizetheirdomains(lefttothereader!).OperatorsB±areclosableinHn(D).WekeepthesamenotationB±fortheirclosures.Wehavethefollowingusefulproperty.Lemma52B±areadjointofeachother:B±∗=B∓.Inparticularforeveryζ∈Ctheoperatori(ζB−−ζB¯+)isself-adjoint.ProofItisenoughtoproveB±∗=B∓.ThisisformallyobviousbecauseweknowthatB1,B2areself-adjoint.Welefttothereadertocheckthatthedomainsarethesame.LetuscomputetheCasimiroperatorCasnfortherepresentationDn−.Adirectcomputationofthecoefficientsgj,koftheKillingformshowsthatgj,k=0ifj=kandg0,0=−1,g1,1=g2,2=1.Sowegetn22221Cas=B0−B1−B2=B0−(B+B−+B−B+)(8.42)2LetusassumeforthemomentthatD−isirreducible.ThenbySchurlemmawenknowthatCasnisanumber;thisnumbercanbecomputedusingthemonomialz0.WefindeasilynnnnCas=−1=k(k−1),k=:(8.43)222Letusremarkthatk:=nisthelowesteigenvalueofB0andiscalledBargmann2index.8.2.3IrreducibilityofDiscreteSeriesHereweprovethatforeveryintegern≥2,therepresentationDn−isirreducible.LetEbeaclosedinvariantsubspaceinHn(D).TherestrictionofDn−tothecompactcommutativesubgroupg(θ,0,0)isasumofone-dimensionalunitaryrep-resentations.Sothereexistu∈E,ν∈RsuchthatD−g(θ,0,0)u=eiνθu,∀θ∈RnButuhasaseriesexpansionu(z)=ajzj.Sobyidentificationwehaveeiνθaj=e−i(n+2j)θaj.Fromthiswefindthatthereexistsj0suchthataj=0sowefind0 8.2UnitaryRepresentationsofSU(1,1)239ν=n+2j0.Butthisentailsaj=0ifj=j0.Henceweconcludethatthemonomialzj0belongstoE.NowplayingwithB±weconcludeeasilythatEcontainsallthemonomialszj,j∈N.WehaveprovedabovethatthemonomialsisatotalsysteminHn(D).SoE=Hn(D).Thediscreteseriesrepresentationshaveanimportantproperty:theyaresquareintegrable(seeAppendicesA,BandC).OnthegroupSU(1,1)wehavealeftandrightinvariantHaarmeasureμ.μispositiveoneachnonemptyopensetanduniqueuptoapositiveconstant(see[128]).Thefollowingresultisprovedin[129].Proposition96Foreveryf∈Hn(D)wehave−2Dn(g)f,fHn(D)dμ(g)<+∞SU(1,1)wheredgistheHaarmeasureonSU(1,1).ThereexistotherunitaryirreduciblerepresentationsforSU(1,1):theprincipalseriesandthecomplementaryseries(seealsothebookofKnapp[127]formoredetails).Theserepresentationsarenotsquareintegrable.Uptoequivalence,discreteseries,principalseriesandcomplementaryseriesaretheonlyirreducibleunitaryrepresentations.Letusexplainnowwhatprincipalseriesare.8.2.4PrincipalSeriesTheserepresentationscanalsoberealizedinHilbertspacesoffunctionsontheunitcircle.Theyaredefinedinthefollowingway:takeanonnegativenumberλandapointzontheunitcircle.Thenthehomographictransformationαz+β¯z →βz+¯αobviouslymapstheunitcircleintoitself.Onedefines−1+2iλαz+β¯Piλ(g)f(z)=|βz+¯α|fβz+¯αOneconsiderstheHilbertspaceL2(S1)withthescalarproduct2π1f1,f2=dθf¯1(θ)f2(θ)2π0 2408Pseudo-Spin-CoherentStatesToprovetheunitarityoftherepresentationPiλ(g)intheHilbertspaceweperformthechangeofvariableθ →θwhereiθαeiθ+β¯e=βeiθ+¯αTheJacobiansatisfiesdθ−2=βeiθ+¯αdθThuswehaveforanyrealλ:2π2πiθ2fiθ2dθPiλ(g)fe=dθe00Nowwecanprove:Proposition97Foranyλ∈R,PiλisaunitaryirreduciblerepresentationofSU(1,1)intheHilbertspaceL2(S1).MoreoveritsCasimiroperatorisCiλ=−(1+λ2).4Forthegeneratorω0(t)oftheLiegroupsu(1,1)onegetsPiθi(θ+t)iλω0(t)fe=feThusthecorrespondinggeneratorofSU(1,1)issimplydL0=dθTofindthegeneratorassociatedtoω1weneedtocalculate2ttsinheiθ+cosh=cosht+cosθsinht22Thustiθtcoshe+sinhPiθ−1/2+iλ22iλω1(t)fe=(cosht+sinhtcosθ)fsinhteiθ+cosht22ThusthegeneratorL1ofSU(1,1)isgivenby1dL1=−+iλcosθ−sinθ2dθSimilarlyusingthethirdgeneratorω2onefinds1dL2=−−+iλsinθ−cosθ2dθ 8.2UnitaryRepresentationsofSU(1,1)241ThereforetheusuallinearcombinationsB±=±L1+iL2satisfy1−iθ−iθdB+=−+iλe−ie2dθ1iθiθdB−=−−+iλe−ie2dθWedefinedB0=iL0=idθUsingthesamemethodasfordiscreteseries,wecanprovethattheserepresentationsareirreducible(startwithB0anduseB±).TocalculatetheCasimiroperatorB2−1(B−B++B+B−)itisenoughtoapply02ittotheconstantfunction.ThusB1+iλ)e∓iθ.Onefinds01=0,B±1=±(−22112C:=B0−(B−B++B+B−)=−−λ1248.2.5ComplementarySeriesWhentheparameterλoftheprincipalseriesisimaginarytherepresentationisnotunitaryinthespaceL2(S1).So,followingBargmann[17]weintroduceadifferentHilbertspace.Letusintroducethesesquilinearformdependingoftherealparameterσ∈]0,1[,2σ−1/2f1,f2σ=c1−cos(θ1−θ2)f1(θ1)f2(θ2)dθ1dθ2[0,2π]2f1,f2σiswelldefinediff1,f2arecontinuousonS1.Theconstantciscomputedsuchthat1,1σ=1,c=21/2−σπB(σ,1/2)−12πσ−1/2Theintegral(1−cosθ)dθiscomputedusingthechangeofvariable0x=cos(θ)sowegetc.ThefollowingpropertiesareusefultobuildtheHilbertspaceHσ.Proposition98(i)Foreveryf1,f2∈C(S1)wehaveσ−1/2ffdθ1−cos(θ1−θ2)1(θ1)2(θ2)1dθ2≤f1f2[0,2π]2 2428Pseudo-Spin-CoherentStates(ii)ek,eσ=0,ifk=whereek(θ)=eikθ.(iii)ek,ekσ=λk(σ)whereΓ(1/2+σ)Γ(|k|+1/2−σ)λk(σ)=(8.44)Γ(1/2−σ)Γ(|k|+1/2+σ)Inparticularλ0=1andλk(σ)>0foreveryk∈Zandσ∈]0,1/2[.Proof(i)isprovedusingthechangeofvariableu=θ1−θ2andCauchy–Schwarzin-equality.(ii)Itisaconsequenceofthefollowingequality,fork=,σ−1/2−ikθi(θ+u)ek,eσ=1−cos(u)eedθdu=0[0,2π]2(iii)Wehaveλ−k=λk,soitisenoughtoconsiderthecasek≥0.Hencewehaveπσ−1/2ek,ekσ=2c1−cos(u)cos(ku)du0Wecomputetheintegralusingthechangeofvariablex=cosu,socos(ku)=Tk(x),whereTkistheTchebichevpolynomialoforderk.1σ−1/22−1/2ek,ekσ=2c(1−x)1−xTk(x)dx−1Butwehavethefollowingexpression,knownastheRodriguesformula[56]:kk−1(k−1)!21/2dk2k−1/2Tk(x)=(−1)21−x1−x(2k)!dxkHencewegettheresultbyintegrationsbypartsandwellknownformulasforgammaandbetaspecialfunctions.1jSoiff1,f2∈C(S),fj=k∈ZckekistheFourierdecompositionoffj.Thenasaresultwehavef121,f2σ=λk(σ)ckckk∈Z 8.2UnitaryRepresentationsofSU(1,1)243Thisshowsthat(f1,f2) →f1,f2σisapositive-definitesesquilinearformonC(S1).NowwecandefinethecomplementaryseriesCσ,0<σ<1/2,asfollows.ItisrealizedintheHilbertspaceHσoffunctionsfonS1suchthatk∈Zλk(σ)|ck|2<+∞equippedwiththescalarproductf1,f2σ(seeproposition(iii)).Sowecandefine,forf∈Hσ,−1+2σαz+β¯Cσ(g)f(z)=|βz+¯α|fβz+¯αProposition99Forall0<σ<1/2,CσisaunitaryirreduciblerepresentationofSU(1,1)inH21σ.MoreoveritsCasimiroperatorisCσ=σ−.4ProofWeusethesamemethodsasforthediscreteandcontinuousseries.Inpartic-ularthecomputationsarethesameasforthecontinuousserieswithσinplaceofiλ.ThemaindifferencehereisinthedefinitionoftheHilbertspacewhichisnecessarytogetaunitaryrepresentation.8.2.6BosonsSystemsRealizationsLetusstartwithaonebosonsystem.Weconsidertheusualannihilationandcre-ationoperatorsa,a†inL2(R)(seeChap.1).Thefollowingoperatorssatisfythecommutationrelations(8.41)oftheLiealgebrasu(1,1):1†2121††B+=a,B−=a,B0=aa+aa(8.45)224WehaveseeninChap.3thatthemetaplecticrepresentationisaprojectiverepre-sentationofthegroupSp(1)=SL(2,R)anditisdecomposedintotwoirreduciblerepresentationintheHilbertsubspacesofL2(R),L2ev(R)ofevenstatesandL2(R)odofoddstates.ButthegroupSL(2,R)isisomorphictothegroupSU(1,1)bytheexplicitmapg →F−11ig:=M0gM,g∈SU(1,1),Fg∈SL(2,R)withM0=.0i1SothemetaplecticrepresentationdefinesarepresentationofthegroupSU(1,1)inthespaceL2(R)withtwoirreduciblecomponentsRˆev,odinthespaceL2(R).ev,odInquantummechanicsitisnaturaltoconsiderray-representations(orprojectiverepresentations)insteadofgenuinerepresentations.Forexamplethemetaplecticrepresentationisaray-representation.LetuscomputetheCasimiroperatorsCev,odforeachcomponents.WecomputeCevusingtheboundstateψ0fortheharmonicoscillator(ψ0∈L2ev(R)).1(aa†+a†a)=aa†−1wegetC3UsingthatHˆosc=evψ0=−ψ0and22163ψ13Codψ1=−1(B0ψ0=ψ0andB0ψ1=ψ1).1644Weseethatthecommutationrelations(8.45)definetwoirreducible“representa-tions”ofSU(1,1)whichareneitherinthediscreteseriesneitherinthecontinuous 2448Pseudo-Spin-CoherentStatesseries.Thereasonisthattheyareray-representationscorrespondingwiththeevenpartandoddpartofthemetaplecticrepresentation.Moredetailsconcerningray-representationscanbefoundin[16,197].Inpar-ticulartheseray-representationsaregenuinerepresentationsofthecoveringgroupSU(1,1)(whichissimplyconnectedbutSU(1,1)isnot).A“doublevaluedrepre-sentation”ρinalinearspacesatisfiesρ(gh)=C(g,h)ρ(g)ρ(h),withC(g,h)=±1Letusnowconsiderthetwobosonssystem.Weconsidertwoannihilationandcreationoperatorsa†221,2,ainL(R)(seeChap.1).Thefollowingoperatorssatis-1,2fiesthecommutationrelations(8.41)oftheLiealgebrasu(1,1):††1††B+=aa,B−=a1a2,B0=aa1+aa2+1(8.46)12212TheCasimiroperatoris11††2Cas=−+aa2−aa14421WeknowfromChap.1thatwehaveanorthonormalbasisofL2(R2),{φm}=(a†)m1(a†)m2φ0,0suchthat1,m2(m1,m2)∈N2,whereφm1,m212m1+m2+1B0φm1,m2=φm1,m2(8.47)2B+φm1,m2=φm1+1,m2+1(8.48)B−φm1,m2=φm1−1,m2−1(8.49)Adirectcomputationgives12Casφm1,m2=−+(m1−m2)φm1,m2(8.50)4Soifweintroducek=1(1+|n0|),wegeteasily(assumingn0≥0)thefollowing2lemma.Lemma53Foreverypositivehalfintegerk,theHilbertspacespannedby{φm2+2k−1,m2,m2∈N},isanirreduciblespacefortherepresentationoftheLiealgebrawithgenerators(8.45).WeknownowthatthisLiealgebrarepresentationdefinesaunitaryrepresentationofSU(1,1)butonlyaprojectiverepresentationofSU(1,1). 8.3Pseudo-CoherentStatesforDiscreteSeries2458.3Pseudo-CoherentStatesforDiscreteSeriesWecannowproceedtotheconstructionofcoherentstatesbyanalogywiththeharmonicoscillatorcase(Glauberstates)andthespin-coherentstates.Weconsiderherethediscreteseriesrepresentation.8.3.1DefinitionofCoherentStatesforDiscreteSeriesLetusconsidertherepresentation(Dn−,H(D)).ItcouldbepossibletoworkwithDn+aswell.Everyg∈SU(1,1)canbedecomposedasg=gnhwhereh∈U(1)andn∈PS2.Itisconvenienttostartwithψ0∈H(D)suchthathψ0=ψ0sowe−1/20−takeasafiducialstateψ0(ζ)=γn,0ζandwedefineψn=Dn(gn)ψ0.MostofpropertiesofψnwillfollowfromsuitableformulafortheoperatorfamilyD(n)=Dn−(gn).Therearemanysimilaritieswiththespinsetting.Weshallexplainnowthesesimilaritiesinmoredetail.Usingpolarcoordinatesfornwehavecosh(τ/2)sinh(τ/2)e−iϕgn=iϕsinh(τ/2)ecosh(τ/2)SousingthedefinitionoftherepresentationD−wehavethestraightforwardfor-nmulaforthepseudo-spin-coherentstates.2n/2−niϕψζ(z)=1−|ζ|1−ζz¯,whereζ=sinh(τ/2)e(8.51)NowweshallgiveaLiegroupinterpretationofthecoherentstates.LetusrecallthatBm=idDn−(1)bm,m=0,1,2,andB+=B2+iB1,B−=B2−iB−.ThenwehaveD(n)=exp−iτ(cosϕB1−sinϕB2)(8.52)iϕ−B−iϕ(8.53)=expτ/2B−e+eThesecondformulareadsτD(n)=D(ξ)=expξB¯,withξ=e−iϕ(8.54)−ξB+2WecangetasimplerformulausingaheuristicfollowingfromGaussdecomposition:cosh(τ/2)sinh(τ/2)e−iϕgn=iϕsinh(τ/2)ecosh(τ/2)10cosh(τ/2)01tanh(τ/2)e−iϕ=iϕ1(8.55)tanh(τ/2)e1001cosh(τ/2) 2468Pseudo-Spin-CoherentStatesRecallthatBn0ψ0=ψ0and|ζ|=tanh(τ/2).Moreoverifb∈su(1,1)thenwe2haveD−etb=e−itB,withB=idD−(1)b(8.56)nnSupposethat(8.56)canbeusedforb1±b2(whicharenotintheLiealgebrasu(1,1).Thenwegetthefollowingrepresentationofpseudo-coherentstatesinthePoincarédiscD,whereζandnrepresentsthesamepointonthepseudo-spherePS2,2n/2ζ¯B+ψn=ψζ=1−|ζ|eψ0(8.57)Letusremarkthatinthespincasethisheuristicisrigorousbecausetherepresen-tationDjiswelldefinedonSL(2,C)whichisnottrueforDn−.Neverthelessitispossibletogivearigorousmeaningtoformula(8.57)asweshallseeinthenextsection.8.3.2SomeExplicitFormulaWefollowmoreorlessthecomputationsdoneinthespincase.Weshallgivedetailsonlywhentheproofsarereallydifferent.Itisconvenienttocomputeinthecanonicalbasis{e}∈NoftherepresentationspaceHn(D)(analogueofDickestatesorHermitebasis).Weeasilygettheformu-lasB+e=(n+)(+1)e+1(8.58)B−e=(n+−1)e−1,B−e0=0(8.59)nB0e=+e.(8.60)2LetusremarkthatthelinearspacePjofpolynomialsofdegree≤jisstableforB0andB−butnotforB+.Forevery∈Nwehave1/2B+e0=n(n+1)···(n+−1)!eζB¯Followingourheuristicargumentweexpandtheexponente+asaTaylorseries(whichisnotallowedbecauseB+isunbounded)andwerecovertheformula:2k−2kψζ(z)=1−|ζ|1−ζz¯(8.61)Letusgivenowarigorousproofforthis.ItisenoughtoexplainwhatisetB+eforeveryt∈Dandevery∈N.Forsimplicityweassumet∈]−1,1[.Proposition100Foreverym∈N,thedifferentialequationφ˙mt=B+φt,φ0(z)=z 8.3Pseudo-CoherentStatesforDiscreteSeries247hasauniquesolutionholomorphicin(t,z)∈D×Dgivenbythefollowingformulas.Form=0φ−nt(z)=(1−tz)(8.62)form≥1,φ−nmt(z)=(1−tz)−1+z(1−tz)−n−m(8.63)ProofWecheckφt(z)=∈Nx(t)z.Sowecancomputex(t)usingtheinduc-tionformulatx+1(t)=x+1(0)+(n+)x(s)ds0Theresultfollowseasily.FromourcomputationswegettheexpansionofψζinthecanonicalbasisΓ(2k+)1/2ψζ=1−|ζ|2kζe¯(8.64)Γ(+1)Γ(2k)∈NProposition101Foreveryn1,n2∈PS2wehaveD(n1)D(n2)=D(n3)exp−iΦ(n1,n2)B0(8.65)whereΦ(n1,n2)istheorientedareaofthegeodesictriangleonthepseudo-spherewithverticesatthepoints[n0,n1,n2].n3isdeterminedbyn3=R(gn)n2(8.66)1whereR(g)istherotationassociatedtog∈SU(1,1)andτgn=exp−i(σ1sinϕ+σ2cosϕ)(8.67)2ProofComputationofn3iseasyusingthefollowinglemma.Thephasewillbedetailedlater.Lemma54Forallg∈SU(1,1)thereexistm∈PS2andδ∈Rsuchthatg=gmr3(δ)whererδ3(δ)=exp(iB0).2Thefollowinglemmashowsthatthepseudo-spinisalsoindependentofthedi-rection. 2488Pseudo-Spin-CoherentStatesLemma55OnehasD(n)Bˆ−10D(n)=−nB(8.68)ProofLet(τ,ϕ)bethepseudopolarcoordinatesofn,n=n(τ,ϕ).WehaveD(n(τ))=exp(−iτ(cosϕB1−sinϕB2)).LetususethenotationA(τ):=D(n(τ))AD(n(τ))−1whereAisanyoperatorinH(D).ThenwehavetheequalitiesdB0(τ)=−cosϕB2(τ)−sinϕB1(τ)(8.69)dτdB1(τ)=−sinϕB0(τ)(8.70)dτdB2(τ)=−cosϕB0(τ)(8.71)dτWehavethefollowingconsequences:d2dB0(τ)=B0(τ),B0(0)=−cosϕB2−sinϕB1(8.72)dτ2dτhencewegetB0(τ)=−n(τ)B.Thefollowingconsequenceisthat|nisaneigenvectoroftheoperatornB:Proposition102OnehasnB|n=−k|n(8.73)wherek=n.2AsintheHeisenbergandspinsettings,thepseudo-spin-coherentstatesfamily|nisnotanorthogonalsysteminH(D).Onecancomputethescalarproductoftwocoherentstates|n,|n:Proposition103Onehas−knn=e−ikΦ(n,n)1−nn(8.74)2whereΦ(n,n)istheorientedareaofthehyperbolictriangle{n0,n,n}.ProofWeusethecomplexrepresentationofcoherentstates,startingfromthedefi-nition,weget2k−2kψζ(z)=1−|ζ|1−ζz¯.(8.75) 8.3Pseudo-CoherentStatesforDiscreteSeries249Nextweusetheseriesexpansion−2k(2k+−1)!ζz¯1−ζz¯=(2k−1)!!≥0tocomputetheFouriercoefficientofthecoherentstate|ζinthebasise1/2Γ(2k+)2ke|ζ=1−|ζ|ζ(8.76)Γ(+1)Γ(2k)TheParsevalidentitygivesn2k2k−2kn=1−|ζ|1−|ζ|1−ζ¯ζ(8.77)Wecantranslatethisequalityinpseudopolarcoordinates(ζ=tanh(τ/2)e−iϕ)andwegetni(ϕ−ϕ)−2kn=cosh(τ/2)coshτ/2−sinh(τ/2)sinhτ/2e(8.78)Aneasycomputationnowgivesthefollowinglemma:Lemma56−2knn2=1−nn(8.79)2ThecomputationofthephaseΦinformula(8.74)canbedoneasforthespincaseusingthegeometricphasemethod.Asisexpected,thepseudo-spin-coherentstatesystemprovidesa“resolutionoftheidentity”intheHilbertspaceH(D):Proposition104Wehavetheformula2k−1dn|nn|=1(8.80)4PS2Orusingcomplexcoordinates|ζ,dν2k(ζ)|ζζ|=1(8.81)Dwherethemeasuredνnisn−1d2ζdνn(ζ)=π(1−|ζ|2)2withd2ζ=|dζ∧dζ¯|.2 2508Pseudo-Spin-CoherentStatesProofThetwoformulasareequivalentbythechangeofvariablesζ=tanhτe−iϕ.2Soitissufficienttoprovethecomplexversion.Weintroducef(ζ)˜=ζ|fHn(D).DecomposefinthebasisofHn(D),f=≥0ce,wehavef2Γ(2k+)22k22(ζ)=1−|ζ||ζ||c|Γ(+1)Γ(2k)≥0Afterintegrationinζwehave22k−1f(ζ)2dζ=f(z)2d2zπD(1−|ζ|2)2DSowegettheresolutionofidentitybyapolarisationargument.8.3.3BargmannTransformandLargekLimitHereweintroducethe(pseudo-spin)Bargmanntransformandprovethatask→+∞therepresentationD−contractstotheHarmonicoscillatorrepresenta-2ktionorHeisenberg–Schrödinger–Weylrepresentation.Letusdenotek,2−kϕ(ζ)=ζ|ϕHn(D)1−|ζ|,ϕ∈H2k(D),ζ∈D.Infactthistransformationistrivial,hereitisidentity!ButitisconvenienttoseethisasaBargmanntransform.UsingtheParsevalformulaweeasilygetk,−1/2ϕ(ζ)=ζγ2k,e,ϕHn(D)=e(ζ)e,ϕHn(D)∈N∈NHereweshallnotethedependenceintheBargmannindexk,sowedenotethepseudo-spin-coherentstateψk(z).ζProposition105Thepseudo-coherentstatesψkconvergetotheGlaubercoherentζstateϕζ(seeChap.1)ask→+∞inthefollowingBargmannsenseandtheDickestateskconvergetotheHermitefunctionψ,forevery∈N:√k,limψ√ζ/2k=ϕζ(ζ),∀ζ,ζ∈C(8.82)k→+∞ζ/2k 8.3Pseudo-CoherentStatesforDiscreteSeries251ProofItisaneasyexercise,knowingthat|ζ|2ϕζ=expζζ¯−ζ2andthatζψ(ζ)=√2π!AsforthespincasewehaveanalogousresultsforthegeneratorsoftheLiealgebras.Letusintroduceasmallparameterε>0anddenoteBε=εB±±(8.83)ε1B0=B0−21(8.84)2εWehavethefollowingcommutationrelations: Bε,Bε=2ε2Bε−1,Bε,Bε=±Bε(8.85)+−33±±Asε→0equations(8.83)defineafamilyofsingulartransformationsoftheLiealgebrasu(1,1)andforε=0weget(formally) B0,B0=−1,B0,B0=±B0(8.86)+−3±±ThesecommutationrelationsarethosesatisfiedbytheharmonicoscillatorLiealge-bra:B+0≡a†,B−0≡a,B0≡N:=a†a.0Wecangiveamathematicalproofofthisanalogybycomputingtheaverages.Proposition106Assumethatε→0andk→+∞suchthatlim2kε2=1.Thenwehavelimψk√Bεψk√=|ζ|2=ϕζ|a†a|ϕζ(8.87)ζ/2k0ζ/2klimψk√Bεψk√=ζ¯=ϕζ|a†|ϕζ(8.88)ζ/2k+ζ/2klimψk√Bεψk√=ζ=ϕζ|a|ϕζ(8.89)ζ/2k−ζ/2kProofFromtheproofofLemma55wecancomputethefollowingaverages:ψk,Bψk=knnnUsingtheζparametrizationwegettheresultasinthespincase. 2528Pseudo-Spin-CoherentStates8.4CoherentStatesforthePrincipalSeriesAsfordiscreteserieswecanconsidercoherentstatesfortheprincipalandcom-plementaryseries.TheprincipalseriesisrealizedintheHilbertspaceL2(S1)withtheHaarprobabilitymeasureonthecircleS1andwithitsorthonormalbasise(θ)=eiθ,∈Z.e0beinginvariantbytherotationssubgroupofS(1,1)wedefinethecoherentstatesψλ(z),z∈S1,asn,ζiλcosh(τ/2)+sinh(τ/2)e−iϕ2iλ−1ψn(z)=z(8.90)1−iλ2iλ−1ψiλ(θ)=1−|ζ|221−ζz¯(8.91)ζInthefirstformulacoherentstatesareparametrizedbythepseudo-sphereandinthesecondformulatheyareparametrizedbythecomplexplane.Propertiesofthesecoherentstatesareanalyzedinthebook[156](pp.77–83).8.5GeneratorofSqueezedStates.ApplicationWeshallproveherethattheSU(1,1)generalizedcoherentstatesconsideredabove(introducedbyPerelomov[156])arenothingbuttheone-dimensionalsqueezedstatesintroducedinChap.3.WeconsidertherealizationoftheLiealgebrasu(1,1)definedbythegenerators1B†+a†a0=aa41†2B+=a212B−=a2ThesegeneratorsaredefinedasclosedoperatorsintheHilbertspaceL2(R).Theyobeythecommutationrules(8.41).FurthermorefortheCasimiroperatorwehaveC31.as=−16WehavealreadyremarkedinSect.8.2.6thatthisrepresentationoftheLieal-gebrasu(1,1)givesaprojectiverepresentationofSU(1,1)(notagenuinegrouprepresentation). 8.5GeneratorofSqueezedStates.Application2538.5.1TheGeneratorofSqueezedStatesConsidernowone-dimensionalsqueezedstates.Recallthefollowingdefinition.Takeacomplexnumberωsuchthat|ω|<1.Wedefineωβ(ω)=argtanh|ω||ω|D(β)=expβBˆ+−β¯Bˆ−D(β)isalsoknownasthe“Bogoliubovtransformation”andgeneratessqueez-ing.For|0=ϕ0beingthegroundstateofB0,letthegeneralizedcoherentstateψβbedefinedasψβ=D(β)|0Remark47ItisnotdifficulttoshowthatintermsoftheoperatorsQˆ,PˆofquantummechanicsonehasiiD(β)=expβQˆ2−Pˆ2−βQˆPˆ+PˆQˆ22WefirstrecallthefundamentalpropertyofD(β)provedinChap.3inanydi-mension.Lemma57OnD(Q)ˆ∩D(P)ˆthefollowingidentitiesholdstrue:(i)D(β)isunitaryandsatisfiesD(β)−1=D(−β)(ii)2−1/2†D(β)aD(−β)=1−|ω|a−ωa(iii)sinh(2r)iθ−iθD(β)B0D(−β)=cosh(2r)B0−B+e+B−e2withβ=reiθbeingthepolardecompositionofβintomodulusandphase.ThefollowingresultsaredirectconsequencesofChap.3:Proposition107(i)Defineδ=1−ω.Onehas1+ωδ>0 2548Pseudo-Spin-CoherentStatesand1/41/22δ1+ωxψβ(x)=exp−δπ|1+ω|2(ii)MoregenerallyifφkisthekthnormalizedeigenstateofB0(Hermitefunction)onehas−k/21/4k+1/2√22δ1+ωδxD(β)φk=√Hkxδexp−n!π|1+ω|2whereHkisthenormalizedkthHermitepolynomial.Nowweaddressthefollowingquestion:whatistheWignerfunctionofasqueezedstate?ItwillappearthatitisaGaussianinq,pbutwithsqueezinginsomedirectionanddilatationintheotherdirection.Onehasthefollowingresult:Proposition108TheWignerfunctionWψ(q,p)isgivenbyβq2δ1W(q,p)=2exp−−(p+qδ)2ψβδRemark48(i)Forβ=0,δ=1,andthuswerecovertheWignerfunctionofϕ0.(ii)Itisclearthat1dqdpWψ(q,p)=1β2πTheproofisaneasycomputationofGaussianintegrals.8.5.2ApplicationtoQuantumDynamicsConsiderthetimedependentquadraticHamiltonianHˆλ(t)B¯2(t)=λ(t)B++−+μ(t)B0(8.92)whereλandμareC1functionsoft,λiscomplexandμisreal.ItspropagatorisdenotedU2(t,s).ThisisaparticularcaseofgeneralquadraticHamiltonianstudiedinChap.1andinChap.4.WerevisitherethecomputationofU2(t,s)usingthesu(1,1)Liealgebrarela-tionssatisfiedby{B0,B+,B−}.Itisconvenienttoformulatetheresultinanabstractsetting. 8.5GeneratorofSqueezedStates.Application255Proposition109AssumethatB0,B±areclosedoperatorswithadensedomaininaHilbertspaceHsuchthatB∗=B0,B+∗=B−andsatisfyingthecommutation0relations:[B−,B+]=2B0,[B0,B±]=±B±.ThenHˆ2(t)definedby(8.92)hasapropagatorgivenbyU2(t,s)=D(βt)expi(γt−γs)B0D(−βt)(8.93)wherethecomplexfunctionβtandtherealfunctionγtsatisfythedifferentialequa-tions2+μωiω˙t=λω¯tt+λ,ω0=0(8.94)γ˙=−λω¯−λω¯−μ,γ0=0(8.95)ProofThefirststepistocomputethefollowingderivatives:diD(βt)=(αtB++¯αtB−+ρtB0)D(βt)(8.96)dtwhereω˙tαt=i(8.97)1−|ω|2ωtω˙¯t−˙ωtω¯tρt=i(8.98)1−|ωt|2Using(8.96)wecancomputediU2(t,s)=αtB++¯αtB−+ρtB0−˙γD(βt)B0D(−βt)dtandwedirectlyget(8.94).Letusprovenow(8.96).Themethodisthefollowing.DenoteL(t)=βtB+−β¯tB−.WehavedL(t+δt)=L(t)+δL(t)≈L(t)+δL(t)dtApplyingtheDuhamelformulaweget1eL(t+δt)−eL(t)=dsesL(t+δt)δL(t)e(1−s)L(t)0Thenasδt→0wehave1dL(t)sL(t)L(t)˙e(1−s)L(t)e=dsedt0 2568Pseudo-Spin-CoherentStates˙¯NowwehaveL(t)˙=β˙tB+−βtB−anddsL(t)−sL(t)sL(t)−sL(t)eB+e=−2β¯eB0edsanddsL(t)−sL(t)sL(t)−sL(t)eB−e=−2βeB0edsUsingLemma57wegetformula(8.96).Remark49ThedifferentialequationsatisfiedbyωtisaRicattiequation.WehaveseeninChap.4thatthisequationcomesfromaclassicalflow.Inparticularωtisdefinedforeverytimetandsatisfied|ωt|<1(ω0=0).LetusnowconsiderthetimedependentHamiltonian1g2HˆPˆ2+f(t)Qˆ2+g(t)=22Qˆ2wheregisacouplingconstantandfafunctionoftimet.PropertiesofthisHamilto-nianhavebeenconsideredby[156]andusedin[48]tostudythequantumdynamicsforionsinaPaultrap.Thesu(1,1)LiealgebrarelationsaresatisfiedbyPˆ2+Qˆ2g2Qˆ2−Pˆ2g2QP+PQB0=+,B±=−∓(8.99)44Qˆ244Qˆ24Soweget11Hˆg(t)=f(t)−1B++f(t)−1B−+1+f(t)B022g2Thealgebraisthesameasabovebutherethepotentialhasanonintegrable2Qˆ2singularityandwehavetotakecareofthedomainofdefinitionforoperatorsB0,B±.LetusconsidertheHilbertspaceL2(R+).RecalltheHardyinequality+∞2+∞dx|u(x)|≤4dxu(x)2,∀u∈H1(R+)|x|2000RecallthatH1(R+)istheSobolevspaceH1(R+)withtheconditionu(0)=0.0DenotebyL21(R+)thespace{u∈L2(R+),xu∈L2(R+)}.Thesesquilinearform(u,v) →u,B0viswelldefinedonV:=H1(R+)∩L2(R+)andisHermitian,non01negative.SoB0hasaself-adjointextensionasaunboundedoperatorinL2(R+).FurthermoreweseethatB±arealsodefinedasformsonVandhaveclosedex-tensionsinL2(R+)suchthatB+∗=B−.Theseextensionsalsosatisfythesu(1,1) 8.5GeneratorofSqueezedStates.Application257Liealgebrarelations.HencetheunitaryoperatorsD(β),eiγB0arewelldefinedinL2(R+)withβcomplexandγreal.1SowecanapplyProposition109tothepropagatorUg(t,s)ofHg(t).Corollary26TheHamiltonianHg(t)hastimedependentpropagatorUg(t,s)givenby(8.93)Ug(t,s)=D(βt)expi(γt−γs)B0D(−βt)(8.100)whereB0,B±aredefinedby(8.99)andβt,γtaredeterminedby(8.94).MoreovertheyarerelatedtocomplexsolutionsoftheNewtonequationξ¨ξ¨t=f(t)ξt,0=ig,ξ+iξ˙1(8.101)ωt=,γt=−argξ−iξ˙ξ−iξ˙2Remark50NotethatthesolutionoftheclassicalequationofmotionforH0(t)solvesthequantumevolutionproblemforHˆgforeveryg∈R.Thesu(1,1)LiealgebracanalsobeusedtosolvethestationarySchrödingerequationsforthehydrogenatom.ItisnothingelsethanagrouptheoreticapproachofamethodalreadyusedbySchrödingerhimself[176].LetusconsidertheradialHamiltonianforthehydrogenatomwithmass1,=1,chargee,energyE>0.d22d2e2(+1)++2−+2ER(r)=0(8.102)dr2rdr2rr2Wetransformthisequationbythechangeofvariabler=x2andoffunctionR(r)=x−3/2f(x).Thenwegetd224(+1)+3/42+8Ex−+8ef=0dx2x2Anotherchangeofvariablex=λuwithλ=(−1)1/4gives8Ed224(+1)+3/422+u+−+8λef=0(8.103)du2u2ThisequationistheeigenvalueequationforthegeneratorB0withg2=4(+1)+3/4.SothenegativeenergiesoftheradialSchrödingerequation(8.102)aredeterminedbytheeigenvaluesofB0. 2588Pseudo-Spin-CoherentStatesLemma58Theself-adjointoperatorB0hasacompactresolvent.Itsspectrumisadiscretesetofsimpleeigenvaluesgivenby1/22s+1112λk=+k,k∈N,wheres=++g(8.104)424ProofThedomainofB0isincludedinH1(R+)∩L2(R+)sowededucethatits01resolventiscompact.Thecomputationofthespectrumisstandard,usingB−andB+asannihilationandcreationoperatorsonthegroundstateψ0.UsingtheresultsofSect.8.2.1.WecomputethegroundstatebysolvingequationB−ψ0=0.Thisasingulardifferentialequation.Weputψ0(x)=xsϕ(x).Wecaneliminatethesingularityby111/2−x2choosings=+(+g).Thentheequationissatisfiedifϕ(x)=exp().So242s−x22s+1wehaveψ0(x)=C0xexp()wheresislikein(8.104)andB0ψ0=ψ0.24ThenwegetallthespectrumofB0andalltheboundedstatesψk=CkB+kψ0wheretheconstantsCKarechosentohaveanorthonormalbasisinL2(R+).Applyingthislemmaandformula(8.103)weseethat(8.102)hasnontriviale4solutionsforE=En=−2,n≥1,thewellknownenergylevelsofthehydrogen2natom.Morepropertieswillbegiveninthenextchapter.Remark51TheCasimiroperatorishereCas=cas1.casiscomputedby21Casψ0=B0−(B+B−+B+B−)ψ0=k(k−1)ψ02withk=2s+1.ThisisnotcompatiblewithadiscreterepresentationofSU(1,1)4exceptifsishalfaninteger.Whatwehaveconsideredhereisanirreduciblerepre-sentationoftheuniversalcoverSU(1,1).Itcouldbepossibletostudycoherentstatesϕβ=D(β)ψ0inthisrepresentationtooaswehavedoneforthediscreterepresentations.8.6WaveletsandPseudo-Spin-CoherentStatesAsiswellknownwaveletsareassociatedwiththeaffinegroupoftransformationsoftherealaxisR:t →at+bwherea>0andb∈R.Wedenoteg(a,b)thisaffinetransformationandAFthegroupofallaffinetransformations.WaveletsarerealfunctionsdefinedbytheactionofthegroupAFonagivenfunctionψsowehave1t−bψa,b(t)=√ψaa 8.6WaveletsandPseudo-Spin-CoherentStates259Ifϕa,bistheFouriertransformofψa,bandϕtheFouriertransformofψ,wehave√ϕ−ibsa,b(s)=aeϕ(as)Nowweshallsee,followingideastakenfromthepaper[25],thatwaveletsandpseudo-spin-coherentstatesarecloselyrelated.Thisisnotsurprisingusingthefollowingfacts:theaffinegroupisisomorphtoasubgroupoftheSL(2,R)group1whichisisomorphtoSL(2,R)/SO(2)andthisoneisisomorphtoSU(1,1)/U(1).2RecallthatthegroupsSU(1,1)andSL(2,R)areconjugate−111iCSU(1,1)C=SL(2,R),whereC=√2i1Consideringconsequencesofthesefacts,weshallfindthatthediscreteseriesrep-resentationsofSU(1,1)haverealizationswithstrongconnectionswiththeaffinegrouphencerelationshipbetweenpseudo-coherentstatesandgeneralizedwaveletswillfollow.RemarkthattheaffinegroupcanbeidentifytoR∗+×Rwiththegrouplaw:(a,b)×(a,b)=(aa,ab+b).Letusconsiderthemapping√a√baM:(a,b) →0√1aItiseasytoseethatMisagroupisomorphismfromAFintoSL(2,R).ItsimageisdenotedAsl.SU(2)isacompactsubgroupofSL(2,R)anditisnotdifficulttoprovethatthequotientspaceSL(2,R)/SO(2)canbeidentifiedtoAsl:cosθsinθLemma59ForeveryA∈SL(2,R)thereexitsarotationR(θ)=and−sinθcosθauniqueaffinetransformation(a,b)∈R∗+×RsuchthatA=M(a,b)R(θ)InparticulartheleftcosetssetSL(2,R)/SO(2)isisomorphtoAsl.LetusconsiderthediscreteseriesDn+whichwillbenowdenotedDn(n≥2isaninteger,n=2kwherekistheBargmannindex).Usingtheisomorphismζ →ζ+iz(ζ)=fromtheunitdiscDontothePoincaréhalf-planeH,Dncanberealized1+iζintheHilbertspaceHn(H).Hn(H)isthespaceofholomorphicfunctionsfinH1RecallthatSL(2,R)isthegroupof2×2realmatricesofdeterminantone.2e−iθ0RecallthatU(1)isidentifiedherewiththegroupofmatricesiθ,θ∈R.0e 2608Pseudo-Spin-CoherentStatessuchthat2n−1f(X+iY)2n−2fYdXdY<+∞Hn(H)=πHwiththenaturalnorm.Sowehaveaunitarymapf →FfromHn(H)ontoHn(D)whereF(ζ)=2(1+iζ)−nf(z(ζ)).InHn(H)thediscreteseriesDngivesnaturallyaunitaryrepresentationofSL(2,R)ab−naz−cDnf(z)=(d−bz)f(8.105)cd−bz+dToestablishaconnectionwiththeaffinegroupitisconvenienttorealizetherep-resentationDninthespaceHˇn(H)ofanti-holomorphicfunctionsonH,sothatf∈Hˇn(H)meansthatf(z)ˇ=f(z)¯withf∈Hn(H).f →fˇisaunitarymap.WedenoteIˇnF=fˇ.ThenDnisunitaryequivalenttotherepresentationDˇnDˇab−ndz−bnf(z)=(a−cz)f(8.106)cd−cz+aNotethattheunitaryequivalencebetweenDˇnandDnisimplementedbythegroup,isomorphisminSL(2,R),abdc →cdbaTherestrictionofDˇntotheaffinegrouphasthefollowingexpression:Dˇ−n/2z−bnM(a,b)f(z)=af(8.107)aWaveletsarefunctionsofarealvariable,sothelaststepistofindarealizationofDninthespace +∞ L2(R+)=ϕt1−nϕ(t)2dt<+∞n0withthenaturalnorm.Letusintroducethe(anti-holomorphic)Fourier–Laplacetransform:+∞(Lϕ(t)e−itz¯dt,z∈C,(z)>0nϕ)(z)=cn0cnisanormalizationconstant.UsingtheFourierinverseformulaandthePlancherelformulawehave+∞1tyitxϕ(t)=eLnϕ(x−iy)edx,y>0,t>0(8.108)2πcn−∞ 8.6WaveletsandPseudo-Spin-CoherentStates261+∞1L2yn−2dxdy=Γ(n−1)21−nt1−nϕ(t)2−ndtnϕ(x−iy)2πcnH0(8.109)SowegetisometriesbetweenthespacesL2n(R+),H(H)andHˇ(H)choosing2n−2cn=.Wehaveobtainedthefollowingirreducibleunitaryrepresentationofπ(n−2)!theaffinegroupinthespaceL2n(R+):Wn(a,b):=L−1DˇM(a,b)Lnnnϕ1−n/2−ibta,b(t):=Wn(a,b)ϕ(t)=aeϕ(at)whichrepresentwaveletsontheFourierside.CoherentstatesforSU(1,1)wheredefinedintheHilbertspaceHn(D)byanactionofSU(1,1),startingfromafiducialstatesψ0invariantbytheactionoftheunitcircleU(1)(isomorphtoSO(2)).ThenSU(1,1)coherentstatesareparametrizedbythequotientSU(1,1)/U(1)ButfromLemma59andtheisomorphismbetweenSU(1,1)andSL(2,R)weseethatSU(1,1)/U(1)canalsobeparametrizedbytheaffinegroup:(a,b) →ga,bwherewechooseoneelementga,bineachleftcoset.WehaveseenaboveintheconstructionofcoherentstatesthatSU(1,1)/U(1)canbeparametrizedbyC(orbypseudo-sphere):ξ →gξ.Ifga,bandgξareinthesamecosetthenwehavega,b=gξhwhereh∈U(1).Wehavechosenψ0rotationinvariantsotheactionsofga,bandgξdefinethesamecoherentstate.LetusmovethisconstructioninHn(H)andinL2n(R+).Weget,respectively,fiducialstatesf0(z)=dn(1−iz)nandϕ0(t)=entn−1e−twherednandenaresuit-ableconstants.ThenusingpropertiesoftherepresentationDnwegetabijectivecorrespon-dencebetweenSU(1,1)coherentstatesdefinedinHn(D)forDn+andwaveletsinL2n(R+).Morepreciselywehaveobtainedϕ1−n/2−ibta,b(t):=Wn(a,b)ϕ0(t)=aeϕ(at)whichrepresentwaveletsontheFourierside.TheirrelationshipwiththeSU(1,1)coherentstatesisgivenbyL−1Iˇnnψξ=ϕa(ξ),b(ξ),∀ξ∈C(8.110)Iˇ−1Lnϕa,b=ψ,∀(a,b)∈R∗×R(8.111)nξ(a,b)+whereξ →(a(ξ),b(ξ))isabijectionfromContoR∗+×Rand(a,b) →ξ(a,b)isabijectionfromR∗+×RontoC.Inparticularwealsohavearesolutionofidentityforwaveletswhichcanbeobtainedfrom(8.81)orbyadirectcomputation.n−1dadb2f=2ϕab,fϕa,b,∀f∈Ln(R+)(8.112)4πR∗+×Ra 2628Pseudo-Spin-CoherentStatesThereadercanfindin[25]severalexplicitformulasconcerningthethreerealiza-tionsofDn±inHn(D),Hn(H)andL2n(R+).FinallyremarkthatWnisarepresentationinL2n(R+)ofasubgroupofS(1,1)conjugatedtotherestrictionofDn+.ButalltherepresentationsWnareequivalentcontrarytotherepresentationsD+whicharenon-equivalent.nIfMnistheunitarymapMnϕ(t)=t1−n/2ϕ(t)fromL2n(R+)ontoL2(R+)thenwehaveclearlyMnWn=W2MnsoWnandW2areconjugateforeveryn≥2. Chapter9TheCoherentStatesoftheHydrogenAtomAbstractTheaimofthischapteristopresentaconstructionofasetofcoherentstatesforthehydrogenatomproposedbyC.Villegas-Blas(ThomasandVillegas-BlasinCommun.Math.Phys.187:623645,1997;Villegas-BlasinPh.D.thesis,1996).WeshowthatinasemiclassicalsensetheyconcentrateessentiallyaroundtheKeplerorbits(inconfigurationspace)oftheclassicalmotion.Asuitableunitarytransformation(theFockoperator)mapsthepure-pointsubspaceofthehydrogenatomHamiltonianontotheHilbertspacefortheS3sphere.WestudythecoherentstatesfortheS3sphere(asintroducedbyA.Uribe(J.Funct.Anal.59:535556,1984))andshowthattheactionofthegroupSO(4)isirreducibleinthespacegen-eratedbythesphericalharmonicsofagivendegree.NotethatcoherentstatesforthehydrogenatomhavebeenextensivelystudiedbyJ.Klauderandhisschool.WehavechosennottopresentthemhereandrefertheinterestedreadertoKlauderandSkagerstam(CoherentStates,1985).9.1TheS3SphereandtheGroupSO(4)9.1.1IntroductionItiswellknownthatthenon-relativisticquantummodelforthehydrogenatomis|p|213thequantizationHˆoftheKeplerHamiltonianH(x,p)=−,p,x∈R.2|x|ThenaturalsymmetrygroupforHseemstobetherotationgroupSO(3).WeshallseeinSect.9.2thatthehydrogenatomhashiddensymmetriesanditssym-metrygroupisthelargergroupSO(4)whichexplainthelargedegeneraciesoftheenergylevelsofHˆ.ThisiswhywestartbystudyingthegroupSO(4),itsirreduciblerepresentationsandhypersphericalharmonics.RecallthatSO(4)isthegroupofdirectisometriesoftheEuclideanspaceR4oritsunitsphereS3,S3=x=(x22221,...,x4)|x1+x2+x3+x4=1LetusintroducetheLaplacianΔS3forthesphereS3,astherestrictiontotheunit2sphereS3oftheLaplaceoperatorΔ∂.MoreexplicitlycomputingR4:=1≤j≤4∂x2jM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,263TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_9,©SpringerScience+BusinessMediaB.V.2012 2649TheCoherentStatesoftheHydrogenAtomΔR4inhypersphericalcoordinates:x1=rsinχsinθcosϕx2=rsinχsinθsinϕ(9.1)x3=rsinχcosθx4=rcosχ,whereχ,θ∈[0,π[,ϕ∈[0,2π[weget1∂3∂1ΔR4=r3∂rr∂r+r2ΔS3(9.2)where∂22∂1ΔS3=∂χ2+tanχ∂χ+2ΔS2,where(9.3)sinχ1∂∂21∂2ΔS2=tanθ∂θ+∂θ2+2∂ϕ2(9.4)sinθEquation(9.2)definestheoperatorΔS3.ItisaselfadjointoperatorintheHilbertspaceL2(S3)fortheEuclideanmeasuredμ23(θ,ϕ,χ)=sinχsinθdθdϕdχ.Themeasuredμ3andtheoperatorΔS3areinvariantbythegroupSO(4).ThespectrumofΔS3canbedescribedhasfollows:thereexistsanexplicitconstantc∈RsuchthatifΔ3:=−ΔS3+cthenΔ3hasthediscretespectrumλ2|k∈Nk=(k+1)Eachλkisknowntohavemultiplicity(k+1)2(see[147]orwhatfollows),whichcoincideswithmultiplicitiesofboundstatesofhydrogenatomasweshallseelater.9.1.2IrreducibleRepresentationsofSO(4)SO(4)isacompactLiegroupsoweknowthatallitsirreduciblerepresentationsarefinitedimensional.TheLiealgebraso(4)ofSO(4)isthealgebraofantisymmetric4×4realmatri-ces;so(4)hasdimension6sotheLiegroupSO(4)hasdimension6.WeshallseethatitsirreduciblerepresentationscanbededucedfromirreduciblerepresentationsofSU(2)(computedinthechapterSpinCoherentStateswhichwillbedenoted(SCS)).ItisconvenienttousethequaternionfieldHanditsgenerators{1,I,J,K}.Hisa4-dimensionalreallinearspacewhichcanberepresentedasthespaceofab2×2matricesq=wherea,b∈C.ThebasisisrelatedwithPaulimatrices:−b¯a¯1=σ0,I=iσ3,J=iσ2,K=iσ1.Thefollowingpropertiesareeasytoprove. 9.1TheS3SphereandtheGroupSO(4)2651.{1,I,J,K}isanorthonormalbasisforthescalarproductq,q:=1tr(q·q).2SoHcanbeidentifiedwiththeEuclideanspaceR4.2.q·q=q·q=(|a|2+|b|2)1.Inparticularifq=0,qisinvertibleandq−1=q2222where|q|=|a|+|b|.|q|3.SU(2)={q∈H,|q|=1}.4.su(2)={q∈H,q+q=0}.Aquaternionqissaidpure(orimaginary)ifq+q=0andrealifq=q.5.g∈SU(2)⇐⇒g=eθA=cosθ+(sinθ)A,θ∈R,Aapurequaternion.NowwecanidentifySO(4)withthedirectisometriesgroupoftheEuclideanspaceH.Inparticularifforanyg−11,g2∈SU(2)wedefineτ(g1,g2)q=g1·q·g2thenτ(g1,g2)∈SO(4).FurthermorewehaveProposition110τisagroupmorphismfromSU(2)×SU(2)inSO(4).Thekernelofτiskerτ={(1,1),(−1,−1)}andτissurjective.InparticularthegroupSO(4)isisomorphictothequotientgroupSU(2)×SU(2)/{(1,1),(−1,−1)}anditsLiealgebraso(4)isisomorphictotheLiealgebrasu(2)⊕su(2).ProofItisclearthatτisagroupmorphism.(g1,g2)∈kerτmeansthatg1·q=q·g2,∀q∈H.Sowegetsuccessively:g1=g2,g1=λ1,λ∈C,λ=±1becauseg1∈SU(2).Toprovethatτissurjective,weusethatτ(SU(2)×SU(2))isasubgroupofSO(4),itsactiononHistransitiveandthatwehaveasurjectivegrouphomomor-phismg →RgfromSU(2)inSO(3)(actinginpurequaternions).LetA∈SO(4).IfA1=1thenthereexistsg∈SU(2)suchthatA=τ(g,g).IfA1=qthenthereexistg1,g2∈SU(2)suchthatq=τ(g1,g2)1sowecanwriteA=τ(g1·g,g2·g).NowweusethefollowingclassicalresulttodeduceirreduciblerepresentationsofSO(4)(foraproofsee[34]).Proposition111LetG1,G2betwocompactLiegroups,(ρ1,V1)and(ρ2,V2)twoirreduciblerepresentationsofG1andG2,respectively.Then(ρ1⊗ρ2,V1⊗V2)isanirreduciblerepresentationofG1×G2.ConverselyeveryirreduciblerepresentationofG1×G2islikethis.Corollary27Let(ρ,V)beanirreducibleunitaryrepresentationofSO(4).ThenNsuchthatjthereexistj1,j2∈1+j2∈Nandsuchthat(ρ,V)isunitarilyequiv-2alentto(T(j1)⊗T(j2),V(j1)⊗V(j2)).ProofUsingProposition110wecanassumethat(ρ,V)isanirreduciblerepre-sentationofSU(2)×SU(2)/{(1,1),(−1,−1)}.So(ρ,V)isanirreduciblerepre-sentationofSU(2)×SU(2)andfromProposition111wehave(ρ,V)≡(T(j1)⊗ 2669TheCoherentStatesoftheHydrogenAtomT(j2),V(j1)⊗V(j2)).TogetarepresentationofSU(2)×SU(2)/{(1,1),(−1,−1)}itisnecessaryandsufficientthatT(j1)(−1)⊗T(j2)(−1)=1V(1)⊗V(2)Sowefindtheconditionj1+j2∈N.9.1.3HypersphericalHarmonicsandSpectralDecompositionofΔS3AswehavealreadyseeninChap.7forSO(3)weshallseenowthatirreduciblerepresentationsofSO(4)arecloselyrelatedwiththehypersphericalharmonicsandspectraldecompositionofΔS3.(k)LetusintroducethespacePofhomogeneouspolynomialsf(x1,x2,x3,x4)4(k)inthefourvariables(x1,x2,x3,x4)oftotaldegreek∈NandH˜thespaceof4(k)f∈P4suchthatΔR4f=0.H˜(k)isdeterminedbyitsrestrictionH(k)tothesphereS3whichisbydefinition44thespaceofhypersphericalharmonicsofdegreek.Letf∈P(k).Forx∈R4wehavex=rω,r>0,ω∈S3andf(x)=rkψ(ω).4So,using(9.2)wehave(k)ΔS3ψ=−k(k+2)ψ,∀ψ∈H4(9.5)(k)LetusintroducethemodifiedLaplacianΔ3=−ΔS3+1.ThenH4isaneigen-spaceforΔ3witheigenvalueλk=(k+1)2.ThegroupSO(4)hasanaturalunitaryrepresentationinL2(S3)definedbytheformulaρ−13gψ(ω)=ψgω,g∈SO(4),ω∈S(k)ρcommuteswithΔS3andeachH4isinvariantbyρ.Themaingoalofthissub-sectionistoprovethefollowingresults.Theorem44(i)TheHilbertspaceL2(S3)isthedirectHilbertiansumofhypersphericalsub-spaces:23(k)(0)LS=H,H=C(9.6)44k∈NMoreoverwehave(k)2dimH=(k+1)(9.7)4 9.1TheS3SphereandtheGroupSO(4)267(k)(ii)Foreveryk∈Ntherepresentation(ρ,H)isirreducibleandequivalentto4therepresentation(T(k/2)⊗T(k/2),V(k/2)⊗V(k/2))where(T(j),V(j))istheirreduciblerepresentationofSU(2)definedinChap.7.ProofWeshallfollowmoreorlesstheproofofProposition83ofChap.7.WefirstgetthefollowingdecompositionforhomogeneouspolynomialspacesinR4:(k)(k)2(k−2)P=H˜⊕rP(9.8)444P(k)=H˜(k)⊕r2H˜(k−2)⊕···⊕r2H˜(k−2)(9.9)4444wherek−1≤2≤kandr2ismultiplicationbyr2:=x2+x2+x2+x2.1234(k)UsingtheStoneWeierstrasstheoremitfollowsthatk∈NH4isdenseinL2(S3)hencewegetthedecompositionformula(9.6).Furthermorewehave(k)(k)(k−2)(k)k+3dimP=dimH+dimP,anddimP=4444ksowegetformula(9.7)bythebinomialNewtonformula.(k)LetusprovenowthatHisirreduciblefortherepresentationρ.4DenotebyΣ(k)therestrictiontoS3ofP(k).LetusconsiderafixedpointonS3,44forexamplen0=(0,0,0,1).WecanidentifywithSO(3)thesubgroupofSO(4)(k)(k)fixingn0.LetΣbethesubspaceofψ∈Σsuchthatρgψ=ψ,∀g∈SO(3).4,34(k)IrreducibilityofHwillfollowfromthe4Lemma60(k)k(i)ThedimensionofΣis[]+1([λ]isthegreatestintegernsuchthatn≤λ).4,32(ii)LetE={0}beafinite-dimensionalSO(4)-invariantsubspaceofthespaceC(S3).Thenthereexistsψ0=0,ψ0∈Esuchthatρgψ0=ψ0∀g∈SO(3).Admittingthislemmaforamomentletusfinishtheproofofthetheorem.(k)ProofForeverykwehavethefollowingdecompositionofΣintoirreducible4factors:(k)Σ=E41≤≤LwhereeachE(j)(j)isequivalenttosomeV⊗V.Letusapplythelemma(ii)toeachE.Weget(k)kL≤dimΣ=+1(9.10)4,32 2689TheCoherentStatesoftheHydrogenAtomButfrom(9.8)weget(k)(k−2j)Σ=H442j≤k(k−2j)(k)kSoifoneoftheHisreduciblethenΣwouldhaveatleast[]+2irreducible442(k)factorswhichisnotpossiblebecauseof(9.10).SoallthespacesHareirreducible4forρ.Letusprovenowthat(ρ,H(k))isequivalenttotherepresentation(T(k/2)⊗4T(k/2),V(k/2)⊗V(k/2)).ThisisaconsequenceofthePeterWeyltheoremwhosestatementisTheorem45[34]LetGbeacompactLiegroupwithHaarmeasuredμandlet(λ)(ρλ,V)bethesetofallitsirreducibleunitaryrepresentations(uptounitaryλ∈Λ(λ)(λ)equivalence).Foreachλ∈ΛconsideranorthonormalbasisofV:{e}1≤k≤kkλ(λ)√(λ)(λ)andthematrixelementsa(g)=kλe,ρλ(g)(e),g∈G.k,k(λ)2Then{a,1≤k,≤kλ,λ∈Λ}isanorthonormalbasisofL(G).k,InthequaternionmodelwecanseethatS3isisomorphictotheLiegroupSU(2)anduptoanormalizationconstanttheHilbertspacesL2(S3)andL2(SU(2))coin-cide.InV(j)considertheorthonormalbasisvk,0≤k≤2janddefinethelinearmapbyv(j)(j)(j)2(j)k⊗v →afromV⊗VinL(SU(2)).WegetaunitarymapUk,fromV(j)⊗V(j)onasubspaceE(j)ofL2(SU(2)).NowapplyingthePeterWeyltheoremwehavethefollowingunitaryequivalences:L2S3∼L2SU(2)∼E(j)∼V(j)⊗V(j)j∈N/2j∈N/2Usinguniquenessforthedecompositionintoirreduciblerepresentationswecancon-cludethatH(k)∼V(k/2)⊗V(k/2).49.1.4TheCoherentStatesforS3WewanttointroducecoherentstatesdefinedonS3.FollowingUribe[189]wecon-siderapairofunitvectorsa,binS3suchthata·b=0(·denotestheusualbilinearforminC4).Letα=a+ib∈C4 9.1TheS3SphereandtheGroupSO(4)269Fig.9.1CoherentstatesonthesphereThegeometricalinterpretationisthatαrepresentsatangentvectorbatthepointa∈S3.SothesetA={α=a+ib|a·b=0,|a|=|b|=1}canbeidentifiedwiththeunittangentbundleS(S3)overS3.MoreovertheintersectionofS3andtherealplanecontainingthevectorsa,bisageodesiconS3whichwillbedenotedα˚.Wehaveclearlyα˚={ω∈S3,|ω·α|=1}.Foreveryα∈A,ω∈S3andk∈NwedefineΨkkα,k(ω)=(α·ω)=(a·ω+ib·ω)Uribe[189]hasdefinedthesecoherentstatesforsphereSninanydimensionn.Weshallcallthemsphericalcoherentstates.UsingdefinitionofAweseethatΨα,kisasphericalharmonicsofdegreeknamelyitisaneigenfunctionofΔ3witheigenvalue(k+1)2.Letusremarkthatifωisnotonthegeodesic˚αthenwehave|α·ω|<1henceΨα,k(ω)isexponentiallysmallask→+∞,soΨα,k(ω)livesclosetothegeodesicα˚whenkislarge.TheL2-normofΨα,kinL2(S3)isgivenbyΨ22kα,k=|α·ω|dμ3(ω)S3UsinghypersphericalcoordinateswegetππΨ2=2πsin2k+2χdχsin2k+1θdθα,k00π/2nUsingwellknownexpressionfortheWallisintegralswn=sin(θ)dθ,weget02π2Ψ2α,k=(9.11)k+1LetuschecknowthecompletenessofthecoherentstatesΨα,k. 2709TheCoherentStatesoftheHydrogenAtomProposition112ThecoherentstatesΨα,kformacompletesetontheirreducible(k)eigenspaceHofthemodifiedLaplacianΔ3associatedwiththeeigenvalueλk:4Pk=C(k)|Ψα,kΨα,k|dμ(α)˚(9.12)α˚∈Γ(k)wherePkistheprojectorontoHofΔ3belongingtotheeigenvalueλk,C(k)is4aconstantofnormalization,Γisthespaceofgeodesicsα˚,anddμ(α)˚istheSO(4)invariantprobabilitymeasureonΓ.MoreoverwecancomputeC(k):(k+1)3C(k)=2π2ProofLetg∈SO(4).Onedefinesgα=ga+igbItisnotdifficulttoseethatSO(4)actsonAtransitivelysothatA=gα0|g∈SO(4)whereα0∈Aisfixed.ThentheHaarprobabilitymeasuredμHonSO(4)inducesapushedforwardmeasure(orimagemeasure)dμ(α)onA.Theintegraloperatorintherighthandsideof(9.12)commuteswithanyrotationoperatorρggivenbyarotationginSO(4).ThereforebySchurslemmathisintegral(k)operatormustbeamultipleoftheidentityoneachirreduciblesubspaceH.4ThecomputationofC(k)isstraightforwardtakingthetracein(9.12)andusing(9.11).AswehavealreadyremarkedforspincoherentstatesinChap.7,theΨα,karenotmutuallyorthogonal.HereitismoredifficulttocomputeΨα,k,Ψα,kforα=αbutitispossibletocomputethisoverlapasymptoticallyfork→+∞asisshownin[191]and[185].Hereweonlystatetheresult.Proposition113Foranyδ>0wehavefork→+∞2k2πα·αδ−1−∞Ψα,k,Ψα,k=1+Ok2+Ok(9.13)k2Remark52αandαdefinethesamegeodesicifandonlyif|α·α|=1.So(9.13)showsthattheoverlapisexponentiallysmallifandonlyif˚α=α˚.NoticethatonAthegeodesicflowismultiplicationbyeit,t∈R. 9.2TheHydrogenAtom2719.2TheHydrogenAtom9.2.1GeneralitiesWeconsiderthehydrogenatomHamiltonianPˆ2111Hˆ:=−=−ΔR3−(9.14)2|Qˆ|2|x|HˆisthequantizationoftheKeplerHamiltonianH(q,p)=p21−,(q,p)∈2|q|R3{0}×R3.ThenotationsarethesameasinChap.1.Thefirstexpressionismoreoftenusedinphysicsthesecondinmathematics.Hˆisaself-adjointoperatorinL2(R3)withdomain1DHˆ=DPˆ2∩D=H2R3(Sobolevspace,Katosresult)|Qˆ|Forsimplicitywehavetakenm=e==1.ItiseasytoprovethatHˆcommuteswiththeangularmomentumoperatorLˆ=Qˆ∧PˆwhichisaconsequenceofitsSO(3)symmetryproperty:HˆcommuteswiththethreegeneratorsLˆ=(Lˆ1,Lˆ2,Lˆ3)ofSO(3).OthersymmetrieswerediscoveredalongtimeagofortheKeplerproblem(Laplace,Runge,Lenz)andgive,afterquantization,symmetriesforthehydro-genatom.LetusconsiderfirsttheclassicalHamiltoniansettingandintroducetheLaplaceRungeLenzvector:qM:=p∧L−=(M1,M2,M3)(9.15)|q|wheretheclassicalangularmomentumisL=q∧p=(L1,L2,L3).Wehavethefollowingproperties.1.{Mk,H}=0,k=1,2,32.L·M=03.{Mj,Lk}=εj,k,M4.{Mj,Mk}=−2Hεj,k,Mwhereεj,k,istheusualantisymmetrictensor.InparticularwecandeduceformthesepropertiesthatiftheenergyofHisfixedandnegative(H=E<0)thenthesixintegralsL,MspanaLiealgebra(forthePoissonbracket)isomorphictotheLiealgebraso(4)(seeSect.9.1.2ofthischapter).InthesesensetheKeplerproblemhashiddensymmetriescontainedintheLaplaceRungeLenzvectorM.ForanhistoricalpointofviewaboutMwereferto[94]. 2729TheCoherentStatesoftheHydrogenAtomAfterquantizationwegetaLaplaceRungeLenzoperator:1XMˆ=P∧Lˆ−Lˆ∧P−2|X|Onehasthefollowingcommutationrules,correspondingtotheaboveclassicalone(see[182]fordetailedcomputations).1.[Lˆj,Lˆk]=iεjklLˆl2.[Lˆj,Mˆk]=iεjklMˆl3.[Mˆj,Mˆk]=−2iεjklMˆlHˆ4.Lˆ·Mˆ=Mˆ·Lˆ=05.Mˆ2−1=2H(ˆLˆ2+1)ItisknownthatHˆhasapurelyabsolutelyspectrumon[0,+∞)andanegativepointspectrumoftheform1∗En=−,n∈N2n2ThedegeneracyoftheeigenvalueEnisn2.WeshallseethatthisisduetothehiddensymmetriescontainedintheLaplaceRungeLenzoperatorMcommutingwithHˆ(LˆisalsocommutingwithHˆbutitgeneratesapparentsphericalsymmetriesofHˆ).Letusnowrecalltheusualproofforthefollowingresult.Lemma61TheeigenvalueE1ofHˆhasdegeneracyn2.n=−22nProofWegivehereasketchofproof,forthedetailswerefertoanytextbookinquantummechanics.Insphericalcoordinateswehave∂211Hˆ=−rr−ΔS2−∂r2r2rRecallthatthesphericalharmonicsYmsatisfyLˆ2Ym=(+1)Ym(seeChap.7).Eigenvaluesareobtainedbysolvingtheradialequation∂2(+1)1−rr+−f(r)=Ef(r)∂r2r2rSowegettheeigenvaluesEnforn≥1andabasisoftheeigenspace:En:={ψn,,m,−≤m≤,0≤≤n}ThedegeneracyofEnequalstothedimensionofEn:n−1dim(Hn)=(2p+1)p=0 9.2TheHydrogenAtom273Tothissumofoddnumbersupto2n−1weaddandsubtractsthesumofevennumbersupto2n−2.Thisyieldsdim(H2n)=n(2n−1)−n(n−1)=nInthefollowingsectionweshallrecoverthisresultusingtheSO(4)symmetryinatransparentway.9.2.2TheFockTransformation:AMapfromL2(S3)tothePure-PointSubspaceofHˆWefollowapresentationofBanderItzykson[14].Considertheeigenvalueproblemofthehydrogenatom:P21Hψˆ=−ψ=Eψ2|X|InFouriervariableponegetsp21ψ(˜q)−Eψ(˜p)=dq(9.16)2222πR3|p−q|wherewehaveset=1forsimplicityandψ(˜p)=(2π)−3/23dqe−iq·pψ(q)dq.RSincetheboundstatesofthehydrogenatomhavenegativeeigenenergiesEwedefinep0>0suchthat2E=−p20ThenwedefinethestereographicprojectionfromthemomentumspaceR3ontoS30(thesphereS3withthenorthpole(0,0,0,1)removed):considerS3dividedintotwohemispheresbythemomentumspaceR3.Givenavectorp/p0∈R3(homogeneouscoordinates),takethelinefromthenorthpoletothispoint.ItwillintersectthesphereS3atapointw∈S3.Wehave:0w=(w1,w2,w3,w4)(9.17)2p0wi=pi,i=1,2,3(9.18)p2+p20p2−p2w0(9.19)4=p2+p20w:=F(p)definesanewparametrizationofthesphereS3.Theinversetransforma-0tionissimply−1p0wiF(w)=pi(w)=,i=1,2,31−w4 2749TheCoherentStatesoftheHydrogenAtomInthisparametrizationtheEuclideanmeasuredμ3ofS3canbecomputedusingtheformula∂F∂F3dμ3(w)=det,dp∂pk∂pSoweget232pdμ24033(w)=2δw−1dw=dpp2+p20Whenp0=kwedenotebywk(p)thecorrespondingstereographictransformation(9.17).TothechangeofvariablesFisassociatedthefollowingunitarytransformUFfromL2(R3)intoL2(S3):2221p(w)+pUF(ψ)(˜w):=Φ(w)=√0ψ˜p(w)(9.20)p2p00NowwecanshowthattheL2normsofψˆandΦarethesame:22Φ(w)2Φ(w)2p+p0ψ(˜p)2L2(S3)=dμ3(w)=2dpS3R32p0Duetothevirialtheorem1onehas2Eψ(˜p)2dp=−pψ(˜p)2dp(9.21)R32ThusΦ(w)=ψ(ˆp)L2(S3)L2(R3)Onehasthefollowingremarkableproperty(justcompute).Lemma62Givenq∈R3wedefinethepointv∈S3bythestereographicequations0(9.17)withqinsteadofpandthesamep0.Onehas(p2+p2)(q2+p2)|p−q|2=00|w−v|2(2p0)2Thereforefrom(9.16)onefindsthattheequationobeyedbyΦissimply1Φ(v)Φ(w)=dμ3(v)(9.22)222πp0S3|v−w|1Recallthatthevirialtheoremsays:ifψisaboundstateofHˆandAˆ=x·∇x+∇x·xthen2i[H,ˆAˆ]ψ,ψ=0.Forthehydrogenatomwehavei−1[H,ˆAˆ]=−Δ−1.Sowehave(9.21).|x| 9.2TheHydrogenAtom275ConsidertheoperatorTinL2(S3)definedbyΦ(v)(TΦ)(w)=dμ3(v)2S3|w−v|NotethatTcommuteswithrotationsρRdefinedby(ρR−1vRΦ)(v)=Φ∗(n−1)withR∈SO(4).Forn∈N,Histhefinite-dimensionalspacegeneratedby4theharmonicpolynomialsofdegreen−1inthevariables(v1,v2,v3,v4)restricted3(n−1)toS.WehaveshownthattheoperatorsρRrestrictedtothespaceH4giveanirreduciblerepresentationofSO(4).ThusduetoSchurslemma,theoperatorTacts(n−1)asamultipleoftheidentityinH:4T|(n−1)=λn1H4Weshallnowcalculateλn.Proposition114Onehas2π2λn=n(n−1)ItisenoughtotakeaparticularfunctioninHsay4G(v)=(vn−13+iv4)solutionofG(v)G(w)=dμ3(v)=λnG(w)2S3|w−v|Weintroducethesphericalcoordinates(χ,θ,φ)inS3:v1=sinχsinθcosφv2=sinχsinθsinφv3=sinχcosθv4=cosχandchoosew=(0,0,0,1).Wegetthefollowingequation:2πππ(sinχcosθ+icosχ)n−1dχdθdφsin2χsinθ=λn−1ni0002(1−cosχ) 2769TheCoherentStatesoftheHydrogenAtomDoingtheintegrationwithrespecttoθ,φweget2π2πsinχdχsin(nχ)=λnn01−cosχwhichfinallyyields2π2λn=nComparingwith(9.22)weobtainp10=andthusn1En=−2n2Thuswerecoverthepointspectrumofthehydrogenatomwithitsdegeneracy:n2=(n−1)dim(H).4Weshallnowuse(9.20)toshowthattheeigenfunctionsofHˆmapontothesphericalharmonicsdefinedonS3.RecallthatLˆisthequantumangularmomentumoperator,andLˆ2=Lˆ2+Lˆ2+Lˆ2123ConsiderthenormalizedeigenfunctionsofthehydrogenatomΨn,,msatisfying:1HΨˆn,,m=−Ψn,,m(9.23)n2Lˆ2Ψn,,m=(+1)Ψn,,m,=0,1,...,n−1(9.24)Lˆ3Ψn,,m=mΨn,,m,m=−,−+1,...,−1,(9.25)RecallthatEnisthen2-dimensionalvectorspacespannedbythefunctions{Ψn,,m|0≤l≤n−1,−l≤m≤l}.LetHppbethesubspaceofL2(R3)spannedbytheeigenfunctionsofHˆ.WedefinetheoperatorU:Hpp →L2(S3)using(9.20):2221p(w)+pUΨˆn,,m(w)=√0Ψˆp(w)n,,mp2p00andextendUlinearlytoallthespaceHpp.SinceUΨˆn,,msatisfies(9.22)UΨˆn,,m(n−1)mustbelongtothespaceH,thusonehas42UΨˆΔ3(UΨˆn,,m)=nn,,mwhereΔ3isthemodifiedLaplacianonS3witheigenvaluen2. 9.3TheCoherentStatesoftheHydrogenAtom2772(n−1)SinceUpreservestheLnormandthespacesEnandHhavethesame4(n−1)finitedimensionUisanunitaryoperatorfromEnontoH.Furthermoreitisa4unitaryoperatorfromH23(n−1)pp=n∈NEnontoL(S)=n∈NH4.LetΠppbetheorthogonalprojectorontothepure-pointspectrumspaceofHˆ.Onehas∞1Hˆ−12−Πpp=nΠn2n=1whereΠnistheprojectorontoEn.OnehasProposition115UHˆ−1ΠppU−1=−2Δ3(n−1)ProofLetΠnbetheprojectorontoH4.Onehas∞Δ23=nΠn.n=1SinceUΠ−1nU=Πnthisyieldstheresult.9.3TheCoherentStatesoftheHydrogenAtomUsingtheunitaryoperatorintroducedinSect.9.2,U:L2(R3) →L2(S3),wedefinethecoherentstatesofthehydrogenatomasΨˆ−1Ψα,k(9.26)α,k=UFromnowonwedefineΨα,k(w)=ck(α·w)kwheretheconstantckischosensothatΨ2k+1α,kL2(S3)=1.ItwascomputedinSect.9.1:ck=2π2.Equation(9.26)readsusing(9.20):22√2p0√2p0kΨˆα,k(p)=p02Ψα,kwk(p)=ckp02α·wk(p)p2+pp2+p00(werecallthatwk(p)isthestereographicprojection(9.17)forp0=k).LetusdefinethedilationoperatorDkinL2(R3)as(D3/2Ψ(kx)kΨ)(x)=k 2789TheCoherentStatesoftheHydrogenAtomorinmomentumspacepDkΨˆ(p)=k−3/2ΨˆkDefiningJtobethemultiplicationoperatorby(2)2wegetp2+122Ψˆα·w1(p)k(9.27)α,k(p)=(DkJΨα,k)(p)=ckDk2p+1TakingtheFouriertransformweseethatthecoherentstateforthehydrogenatominconfigurationspaceequals2ckip·x2kΨα,k(x)=expα·w1(p)dp3/22(2πk)R3kp+1NowweshallconsiderthestateΨα,k(x)dilatedbyk2:Φα,k(x):=(Dk2Ψα,k)(x)3/22k2=ckexpikp·x+klogα·w1(p)dp22πR3p+1Notethatlog(α·w1(p))iswelldefined:ifα4=0then|α·w1(p)|→0as|p|→∞andonethusgetsadecreaseoutsideacompactKofR3.OnKthelogarithmisdefinedlocally.Ifα4=0then1α·w1(p)=α4+O|p|whereα4∈C.ThenthereexistsR>0suchthatfor|p|>Rthelogarithmiswelldefined.TheaimisnowtoshowthatΦα,k(x)concentratesintheneighborhoodofaKe-plerorbitwhenkbecomeslarge.Thisisasemiclassicalresultsincekplaystheroleof1.Fordoingthisweusecomplexstationaryphaseestimatesappliedtotheintegral:3/22k2Φα,k(x)=ckdpexpkf(x,p)22πR3p+1withf(x,p)=ix·p+logα·w1(p)(9.28)Thestationaryphaseconditionreadsf(x·p)=0(9.29)∇pf(x,p)=0(9.30) 9.3TheCoherentStatesoftheHydrogenAtom279Butf(x,p)=logα·w1(p)Wehave|α·w1(p)|≤1withequalityonlywhenw1(p)isintheplanegeneratedbya=α,b=α.Takeforsimplicityα=ˆe1+i(eˆ2cosγ+ˆe4sinγ)Weassumeγ=π,3π.Thevectorseˆiareunitvectorsinthedirectionofthecompo-22nentswi.Thusthefirststationaryphasecondition(9.29)imposesthatw1(p)mustsatisfyaparametricequationoftheform:w1(p)=ˆe1cosβ+sinβ(eˆ2sinγ+ˆe4cosγ)Thecorrespondingconditionsforthemomentumcomponentsarecosβp1=1−sinβsinγsinβcosγp2=1−sinβsinγp3=0SopdescribesthecircleC221γintheplanep3=0:p1+(p2−tanγ)=cos2γ.Oneseeseasilythatα·w1(p)=eiβwhichisacomplexnumberofmodulus1asrequired.Nowweconsidercondition(9.30):itreads2ixj+αj+α4−α·w1(p)pj=0(9.31)(p2+1)(α·w1(p))Thusxmustsatisfytheparametricequations:x1(β)=sinβ−sinγx2(β)=−cosβcosγx3=0ThusxmustbelongtotheellipseE(γ)ofequation22x2(x1+sinγ)+=1(9.32)cosγofenergy−1/2:p211−=−2|x|2 2809TheCoherentStatesoftheHydrogenAtomFig.9.2KeplerellipseToapplythesaddlepointmethod(seeSect.A.4)oneneedstoshowthattheHessianmatrixisnonsingularatthecriticalpoint(x(β),p(x(β))).OnecalculatestheHessianmatrixHβH2β=∂p,pfx(β),px(β)FirstofallweeasilyseethatifxisnotontheellipseE(γ)thenwehaveΦ−∞α,k(x)=OkAtediouscalculationsketchedinSect.A.3showsthatonE(γ)wehave|detHβ|=(1−sinβsinγ)4sin2γ+2sinβsinγ+1sin2γ−2sinβsinγ+1(9.33)SowehavedetHβ=0.FromnowonitisenoughtoconsideraneighborhoodVofafixedpointx0:=x(β)onE(γ).f(x,p)beingholomorphicinacomplexneighborhoodof(x0,p(x0))inC3×C3,thesaddlepointmethod(seeSect.A.4)canbeappliedandgivesforeveryx∈V,221Φ=Cstk1/2expkfx,p(x)1+Ok−1/2α,k(x)1+p2(x)|detHβ|(9.34)Letusremarkthattheexponentin(9.34)isfastdecreasinginkforx∈/E(γ). 9.3TheCoherentStatesoftheHydrogenAtom281Toanalyzemorecarefullythebehaviorof|Φα,k(x)|nearbyx0wechoosecon-venientcoordinates.Letx(β,t,s)=x(β)+tνβ+seˆ3whereβ∈[0,2π[,νβisthenormalvectortoE(γ)atx(β):νβ=sinβcosγeˆ1−cosβeˆ2.(t,s)aresuchthatt2+s2<δ2withδ>0smallenough.Weusetheshorternotationsx0=x(β),xt,s=x(β,t,s),p0=p(x0).UsingtheTaylorexpansionwegetfxt,s,p(xt,s)−f(x0,p0)12=∂xf(x0,p0)·(xt,s−x0)+∂x,xf(x0,p0)(xt,s−x0)·(xt,s−x0)212,pf(x+∂x0,p0)(xt,s−x0)·p(xt,s)−p0212·p(x3+∂p,pf(x0,p0)p(xt,s)−p0t,s)−p0+O|xt,s−x0|(9.35)2and223/2p(xt,s)−p0=∂xp(x0)(xt,s−x0)+Ot+s(9.36)ButwehaveH−1andweget,takingtherealpartintheβ=−i∂pxso∂xp(x0)=iHβTaylorexpansion,1−1223/2fxt,s,p(xt,s)=H(tνβ+seˆ3)·(tνβ+seˆ3)+Ot+s(9.37)2βHerewehaveusedthat∂xfisimaginaryand∂x,x2(x0,p0)=0,∂x,p2(x0,p0)=i.ThematrixH−1(realpartofH−1)hasthefollowingform(seecomputationsinSect.A.3):⎛⎞sin2βcos2γ−sinβcosβcosγ0−11⎜−sinβcosβcosγcos2β0⎟H=−⎝⎠h(β,γ)h(β,γ)002sinγ−2sinβsinγ+1whereh(β,γ)=(1−sinβsinγ)2sin2γ+2sinβsinγ+1TheeigenvectorsofH−1arev1=(cosβ,sinβcosγ,0)v2=νβ=(sinβcosγ,−cosβ,0)(9.38)v3=ˆe3=(0,0,1) 2829TheCoherentStatesoftheHydrogenAtomwithcorrespondingeigenvalues:λ1=0sin2βcos2γ+cos2βλ2=−h(β,γ)(9.39)1λ3=−sin2γ−2sinβsinγ+1Sinceλ2,λ3<0weseethatforklarge|Φα,k(x)|2behaveslikeaGaussianhighlyconcentratedaroundtheellipseatthepointx0.Furthermoreitdecreasesinthedi-rectionoftheeigenvectorsv2,v3namelyinthedirectionperpendiculartotheplaneoftheellipseandintheplaneoftheellipseinthedirectionnormaltotheellipse(notethatp(x0)·v2=0).Morepreciselywehave4Φ221kQ(t,s)−1α,k(xt,s)=Cst2|detHek+Ok(9.40)p(xt,s)+1β,t,s|where1−1223/2Q(t,s)=H(tνβ+seˆ3)·(tνβ+seˆ3)+Ot+s2βThequadraticformQ1−10(t,s):=H(tνβ+seˆ3)·(tνβ+seˆ3)isdefinite-negative2βintheplane(t,s).Nowweshallseethatask→+∞thedensityprobability|Φα,k|2convergestoaprobabilitymeasuresupportedintheellipseE(γ).Thefollowingstatementiscloseto[191](Thesis,Proposition4.1).Proposition116ForeverycontinuousandboundedfunctionψinR3wehave2limΦα,k(x)ψ(x)dx=Cstψ(e)d(e,O)d(e)(9.41)k→+∞R3E(γ)whered(e)isthelengthmeasureontheellipseE(γ),d(e,O)isthedistanceofe∈E(γ)toitsfocusO,Cstisanormalizationconstant.ProofFromcomputationsalreadydonewehave421p(x0)2+1|detHβ|2−1/22−1/2=sinγ+2sinγsinβ+1sinγ−2sinγsinβ+1anddetQ0=λ2λ31−sin2γsin2β=(1−sinγsinβ)2(sin2γ+2sinγsinβ+1)(sin2γ−2sinγsinβ+1) 9.3TheCoherentStatesoftheHydrogenAtom283Sowecanapplythestationaryphasetheoreminvariables(t,s)toget2limΦα,k(x)ψ(x)dxk→+∞R32π221/2=Cst(1−sinγsinβ)1−sinγsinβψx(β)dβ0Butfore=x(β)wehave(1−sin2γsin2β)1/2ψ(x(β))dβ=d(e)and(1−sinγsinβ)isthedistancebetweenx(β)andO.Letusremarkthatthespeedv(β)ofaclassicalparticletravellingonE(γ)is1−sin2γsin2βv(β)=1−sinγsinβ Chapter10BosonicCoherentStatesAbstractInafirstpartwegiveabriefpresentationofgeneralFockspacesettingtodescribequantumfieldtheory.Bosonsarequantumparticlewithintegerspinandhavesymmetricwavefunctions;fermionsarequantumparticlewithhalf-integerspinandarerepresentedwithanti-symmetricwavefunctions.Thefunctionalset-tingisgivenbysymmetricoranti-symmetrictensorproductofHilbertspaces.Wedescribethesespacesandtransformationsbetweenthesespaces.Weshallfollowthereferences(BerezininTheMethodofSecondQuantization,1966;BratteliandRobinsoninOperatorAlgebraandQuantumStatisticalMechanicsII,1981).Coher-entstatesaredefinedbytranslatingthevacuumstateswiththeWeyloperators.Thisiseasilydonehereforbosons.Weshallseeinthenextchapterhowtodealwithfermions.Inasecondpartwegiveaninterestingapplicationofbosoniccoherentstatestothestudyoftheclassicallimitas0ofnon-relativisticbosonsystemswithtwobodyinteractionintheneighborhoodofasolutionoftheclassicalsystem(heretheHartreeequation).Theclassicallimitcorrespondsheretothemean-fieldlimitasthenumberofparticlesgoestoinfinity.Aswehavedoneforfinitesystems,wehereuseHepp’smethod,whichisalinearizationprocedureofthequantumHamiltonianaroundtheclassicalfield.ThefluctuationsaroundthissolutionarecontrolledbyapurelyquadraticHamiltonian.Inaseriesofseveralimportantpapers(GinibreandVeloinCommun.Math.Phys.68:45–68,1979;Ann.Phys.128(2):243–285,1980;Ann.Inst.HenriPoincaré,Phys.Théor.33:363–394,1980)GinibreandVelohaveprovenanasymptoticexpansionandremainderestimatesforthesequantumfluctuations.Finally,followingthepaper(RodnianskiandSchleininCommun.Math.Phys.291:31–61,2009)onecanshowthat,inthelimit0,themarginaldistributionofthetime-evolvedcoherentstatestendsintrace-normtotheprojectorontothesolutionoftheclassicalfieldequation(Hartreeequation)withauniformremainderestimatesintime.10.1IntroductionThischapterisverydifferentfromtheothersinthisbook.Untilnowwehavecon-sideredcoherentstatesdependingonaparameterlivinginafinitedimensionalspaceM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,285TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_10,©SpringerScience+BusinessMediaB.V.2012 28610BosonicCoherentStates(typicallyaphasespaceforaclassicalmechanicalsystemormoregenerallyaLiegroup).Butcoherentstatesmayalsobeausefultooltoanalyzequantumsystemswithaninfinitenumberofparticles(thiswasthemainmotivationforthefounderofcoherentstates,R.J.Glauber).Largenumberofparticlessystemsarestudiedinmanydomainsofphysics:statisticalmechanics,quantumfieldtheory,quantumop-ticsforexample.Therearemanybooksandpapersinthephysicalliterature(Wein-berg[194]).Thereexistalsobooksmorerigorousfromthemathematicalpointofview[33]andforadiscussionconcerningphysicalandmathematicalaspectsseethebook[78].10.2FockSpaces10.2.1BosonsandFermionsLetusstartwithaquantumsystemofidenticalparticles.EachparticlehasitsstatesintheHilbertspaceh.ThestatesofsystemsofkparticlesareintheHilbertspaceh⊗k=h⊗···⊗handifthenumberofparticlesisnotfixed(likeinquantumfieldtheory)thetotalHilbertspaceistheFockspaceF(h):=h⊗k(10.1)k≥0whereh⊗0=C(“no-particle”space).Letusrecallherethatif{ej}j∈Jisanorthonormalbasisofhthen{ei⊗1ei⊗···⊗ei,|i1,...,ik∈J}isanorthonormalbasisofh⊗k.Soifwedenote2k(k)(k)ψ=ei⊗ei⊗···⊗ei,ψthenwehavei1,i2,...,ik12kψ2=ψ(k)2(10.2)i1,i2,...,ikk≥0,i1,...,ik∈JRecallthatthedifferencesbetweenbosonsandfermionsaredeterminedbytheirbehaviorunderpermutations(Pauliexclusionprincipleforfermions).LetusdenoteSkthegroupofpermutationsof{1,2,...,k}andbyεπthesignatureofπ∈Sk.ThefollowingequalitiescanbeextendedintwoprojectionsinF(h):1ΠB(ψ1⊗···⊗ψk)=ψπ1⊗ψπ2⊗···⊗ψπkk!π∈Sk1(10.3)ΠF(ψ1⊗···⊗ψk)=επψπ1⊗ψπ2⊗···⊗ψπkk!π∈Skwhereψ1,...,ψk∈h.Thefollowingnotationswillbeused:ψ1∧ψ2∧···∧ψk=ΠF(ψ1⊗···⊗ψk)ψ1∨ψ2···∨ψk=ΠB(ψ1⊗···⊗ψk) 10.2FockSpaces287Definition20ThesubspaceFB(h):=ΠBF(h)istheFockspaceofbosonsandthesubspaceFF(h):=ΠFF(h)istheFockspaceoffermions.⊗k=Π⊗khB,F(h)arethek-particlessubspacesforbosons(B)orfermions(F).B,FThenumberoperatorNisdefinedasfollows:Nψ(k)=kψ(k),ψ(k)∈h⊗k(10.4)Ncanbeextendedasaself-adjointoperatorinF(h)withdomainD(N)=ψ∈F(h),k2ψ(k)2<+∞(10.5)k≥0MoreoverNcommuteswithΠB,FsoNisaself-adjointoperatorinthespacesFF,B(h).WecandefineoperatorsinFB,F(h)startingfromanHamiltonianHinhbyamethodknownassecondquantizationasfollows.DefineH(0)=0andfork≥1,(k)ΠHB,F(ψ1⊗···⊗ψk)=ΠB,Fψ1⊗···⊗ψj−1⊗Hψj⊗ψj+1⊗···⊗ψk(10.6)1≤j≤kBylinearitythedirectsumofH(k)definesanoperatorH:=kH(k)inFB,F(h).HisthesecondquantizationofH.ItisconvenienttointroducethedensesubspacedefinedasFψ∈F(h)|ψ(k)=0ifklargeenough0(h)=kH(k)iswelldefinedinF0(h).Morepreciselywehavethefollowingeasytoprovelemma.Lemma63IfHisaself-adjointoperatorinhthenHcanbeextendedasauniqueself-adjointoperator(withdensedomain)inFB,F(h).ThisoperatorisalsodenoteddΓ(H)orH.IfUisaunitaryoperatorthenkU(k)canbeextendedinauniqueunitaryoperatorinFB,F(h).ThisoperatorisdenotedΓ(U)orU.Remark53IfH=1thenweseethatdΓ(1)=N,thenumberoperator.IfUt=e−itHwithHself-adjointinhthenwehaveΓ(Ut)=e−itdΓ(H)inotherwordstheinfinitesimalgeneratorofΓ(Ut)isthesecondquantizationofthegener-atorofUt.Inquantumfieldtheorythenumberofparticlesofthesystemisnotconstantsowehavetodefinetwokindsofobservable:annihilationoperatorsandcreationoperators(othernamesareabsorptionandemissionoperators). 28810BosonicCoherentStatesDefinition21Foreveryf∈hwedefinetheoperatorsa(f)anda∗(f)bythefol-lowingconditions:a(f)ψ(0)=0,a∗(f)ψ(0)=f1/2f,ψa(f)(ψ1⊗···⊗ψk)=(k+1)1ψ2⊗···⊗ψka∗(f)(ψ−1/21⊗···⊗ψk)=kf⊗ψ1⊗ψ2⊗···⊗ψkRemarkthatf→a(f)isantilinearandf→a∗(f)islinearonh.Lemma64Foreveryψ(k)∈h⊗k,f∈h,wehavea(f)ψ(k)≤k1/2fΨ(k),a∗(f)ψ(k)≤(k+1)1/2fΨ(k)(10.7)a(f)anda∗(f)aredefinedonthelinearspaceD(N1/2)andsatisfy,a(f)ψ≤f(N+1)1/2ψ(10.8)a∗(f)ψ≤f(N+1)1/2ψ,∀ψ∈DN1/2(10.9)a(f)anda∗(f)leavethesubspacesFB,F(h)invariant.Sotheannihilationandcreationoperatorsforbosons(B)andfermions(F)aredefinedasfollows:aB,F(f)=a(f)ΠB,F=ΠB,Fa(f)(10.10)a∗(f)=a∗(f)ΠB,F=ΠB,Fa∗(f)B,FRemark54Startingfromthevacuumstate:Ω=(1,0,...,0,...)wecreateaparti-clewithstatea∗(f)Ω=(0,f,0,...).Moregenerallyifh1,...,hk∈hwegetkB,Fparticlesinthestatea∗(h1)a∗(h2)···a∗(hk)Ω.ItisnotdifficulttoprovethatΩiscyclic,whichmeansthatthefamily{a∗(h1)a∗(h2)···a∗(hk)Ω,|,hj∈h,k∈N}isdenseinF(h)andthesamepropertyholdstrueforbosonsandfermions.IntheFockspacesFB,F(h)wehavethecanonicalcommutationrelations(CCR)forbosonsandanticommutationrelations(CAR)forfermions.Moreexplicitly,ifH,Karetwooperators,wedenotethecommutator[H,K]:=HK−KHandtheanticommutator[H,K]+:=HK+KH.InwhatfollowsoperatorsaredefinedonFB,F(h)∩F0(h),h1,h2∈h.Wehaveforbosons(CCR)aB(h1),a∗(h2)=h1,h21BaB(h1),aB(h2)=a∗(h1),a∗(h2)=0BBandforfermions(CAR)aF(h1),a∗(h2)=h1,h21F+aF(h1),aF(h2)=a∗(h1),a∗(h2)=0+FF+ 10.2FockSpaces289Remark55Ifanorthonormalbasis{ϕj}i∈Iofhisgiventheannihilation/creationoperatorsaredeterminedbya(∗):=a(∗)(ϕi)(thesubscriptF,Biserasedwhentheicontextisclear).Inparticularthenumberoperatorcanbewrittenasa∗aN=ii(10.11)i∈IWeshallnowdetailsomeconsequencesofrelations(CCR)and(CAR).10.2.2BosonsFirstofallweremarkthattheBargmann–FockrealizationofquantummechanicsfornparticlesisisomorphictothebosonicFockrealizationwithh=Cn.RecallthatwehaveseeninChap.1thatintheBargmannspaceF(Cn)wehave∂ζj,=δj,k∂ζkandanorthonormalbasisφ#(ζ)=(2π)−n/2(α!)−1/2ζααIf{ej}1≤j≤nisthecanonicalbasisofCn,wegetaunitarymapΦBfromF(Cn)ontoFB(Cn)bythepropertyφ#=ΦΦBαB(eα1⊗eα2⊗···⊗eαk)NotethatΦB(eα⊗eα⊗···⊗eα)isthesymmetrictensorproductoftheeα.We12kjhaveeasily∂−1∗−1aj=ΦBΦB,aj=ΦBζˆjΦB(10.12)∂ζjwhichprovedthatbosonicFockrealizationandBargmann–Fockrealizationofquantummechanicsareequivalent.InquantumfieldtheorytheoneparticlespaceisusuallytheinfinitedimensionalHilbertspaceh=L2(Rn).Inphysicalapplicationsitisconvenienttoconsiderfieldoperatorsdependingonapointx∈Rn(eachparticlehasndegreeoffreedom).TheyareoperatorvalueddistributionsonL2(Rn).LetFnbethebosonicFockspace:kF2nn=∨LRk≥0withk0k∨L2Rn=C,∨L2Rn=L2Rns 29010BosonicCoherentStatesk2nThesubscriptsindicatesthesymmetrictensorproduct.sL(R)isthesubspaceofkL2(Rn)ofsymmetricfunctionsonRn×Rn···Rn.So∨kL2(Rn)isthek-ktimesparticlesspaceforbosons.Recallthatavectorψ∈Fnisasequenceψ=ψ(k)k≥0kofk-particlewavefunctionsψ(k)∈∨L2(Rn).ThescalarproductinFnoftwofunc-tionsψα,ψβisgivenby(k)(k)ψα,ψβ=ψα,ψβL2(Rnk)k≥0RecallthatthestateΩ={1,0,...,0,...}iscalledthevacuum.Thecreationandannihilationoperatorsa∗(x),a(x)aredefinedasoperator-distributionbyk∗(k)1(k−1)a(x)ψ(x1,...,xk)=√δ(x−xj)ψ(x1,...,xˆj,...,xk)(10.13)kj=1(k)√(k+1)a(x)ψ(x1,...,xk)=k+1ψ(x,x1,...,xk)(10.14)wherexˆjmeansthatxjisabsent.Thecanonicalcommutationrelationsassumetheforma(x),a∗(y)=δ(x−y),a(x),a(y)=a∗(x),a∗(y)=0Forf∈L2(Rn)werecoverthedefinitions:a∗(f)=dxf(x)a∗(x)(10.15)a(f)=dxf(x)a(x)¯(10.16)ThenumberoperatorNhastheformN=dxa∗(x)a(x)(10.17)LaterweshallconsidertheHamiltonianofabosonssystemwithpairwiseinter-actionsdescribedbyapotentialV=V(x−y).Vissupposedtobeaneven,realfunctiononRn.ThisHamiltoniancanbewrittenasfollowsintheFockspaceF,whereweassumeherethat=1:1∗1∗∗H=dx∇a(x)·∇a(x)+dxdyV(x−y)a(x)a(y)a(y)a(x)(10.18)22 10.2FockSpaces291Formula(10.18)needstobeinterpretedinthedistributionsense.Adirectcomputa-tionshowsthatrestrictionH(k)ofHtothek-particlesspaceis,asexpected:1H(k)=−Δj+V(xi−xj)(10.19)21≤j≤k1≤i0wehavernΦ(f)nψ(k)<+∞n!n≥0TheWeyloperatorsandtheircompanioncoherentstatesaredefinedasfollows: 10.3TheBosonsCoherentStates293Definition22Forf∈hwedefinetheWeyltranslationoperators:T(f)=expa∗(f)−a(f)=expdxf(x)a∗(x)−f(x)a(x)¯(10.23)ThecoherentstateΨ(f)foreveryf∈histhendefinedasΨ(f)=T(f)ΩThebosoniccoherentstateshavethefollowingexpression(analogueofanex-pressionalreadygiveninChap.1forfinitesystemsofbosons):Proposition117Foreveryf∈hwehavef21−⊗kΨ(f)=e2√fk!k≥0Inparticulartheprobabilitytohavekparticlesinψ(f)isequaltoe−f2f2k/k!,wherewerecognizethePoissonlawwithmeanf2.ProofWegiveformalargumentfromwhichitisnotdifficulttosupplyrigorousproofs.Onehastheusefulformula:f2−∗T(f)=e2expa(f)exp−a(f)(10.24)sincethecommutator[a(f),a∗(f)]=f2commuteswitha(f),a∗(f).Wede-duce−f2(a∗(f))k−f2f⊗kΨ(f)=e2Ω=e2√k!k!k≥0k≥0wheref⊗kistheFock-vector{0,0,...,f⊗k,0,...}.Letπkbetheorthogonalprojectorontothek-particlespace⊗ksh.Wehavefoundf2f⊗k−πkΨ(f)=e2√k!sotheprobabilitytohavekparticlesinψ(f)isequaltoπk(Ψ(f))2=e−f2f2k/k!.ThemainpropertiesofWeyloperatorsandcoherentstatesaregiveninthefol-lowingproposition.Proposition118Letf,g∈h. 29410BosonicCoherentStates(i)T(f)isaunitaryoperatorandonehasT(f)∗=T(f)−1=T(−f)(ii)TheWeyloperatorsatisfiesthecommutationrelations:T(f)T(g)=T(g)T(f)exp−2if,g=T(f+g)exp−if,gInparticularwehaveT(g)Ψ(f)=e−ig,fΨ(g+f)(iii)WehaveT∗(f)a(g)T(f)=a(g)+g,f1,T∗(f)a∗(g)T(f)=a∗(g)+f,g1(iv)Thecoherentstatesareeigenfunctionsoftheannihilationoperators:a(g)Ψ(f)=g,fΨ(f)(v)TheexpectationofthenumberoperatorNinthecoherentstateΨ(f)isΨ(f),NΨ(f)=f2alsowehaveforthevariance:222Ψ(f),NΨ(f)−Ψ(f),NΨ(f)=f(vi)Thecoherentstatesarenormalizedbutnotorthogonaltoeachother:1Ψ(f),Ψ(g)=exp−f−g2−if,g2whichimpliesthat1Ψ(f),Ψ(g)=exp−f−g22(vii)Thesetofoperators{T(f),f∈h}isirreducibleonF(h):theonlyboundedoperatorsBinF(h)commutingwithT(f)forallf∈F(h)arethescalarB=λ1,λ∈C.Inparticularthesetofcoherentstates{Ψ(f),f∈h}istotalinF(h).ProofProperties(i)to(ii)arelefttothereader.For(iii)wecomputedT∗(tf)a(g)T(tf)=T∗(tf)a(g),a∗(f)T(tf)=T∗(tf)g,fT(tf)dtandweget(iii)integratingintbetween0and1.(iv)are(v)areconsequencesof(iii). 10.3TheBosonsCoherentStates295−1fa∗(f)For(vi)wewrite,using(10.24):T(f)Ω=e2eΩ.SowegetT(f)Ω,T(g)Ω=Ω,T∗(f)T(g)Ω=Ω,T(g−f)Ωeif,g−1g−f2if,g=e2e(10.25)Letusnowprove(vii).RemarkfirstthatBcommuteswithΦ(f)foreveryf∈hhencewitha(f)anda∗(f).InparticularBcommuteswiththenumberopera-torN.Let{ei}i∈Ibeanorthonormalbasisforhandai=a(ei).Denoteψk1,...,kn=(k1!···kn!)−1/2(a∗)k1···(an∗)knΩ.ThisisanorthonormalbasisfortheFockspace1FB(h)withtheobviousindexset.Computeψk1,...,kn,Bψj1,...,jm.Thisis0ifthesets{k1,...,kn},{j1,...,jm}arenotequal.Finallywehave,usingCCR,···a∗∗ψi1,...,in,Bψi1,...,in=Ω,ai1inai1···ainΩ=Ω,BΩhenceB=Ω,BΩ1.Thetwofollowinglemmaswillbeusefullater.LetusintroducetheoperatorfamilyNr=(1+N2)r/2forr∈R.Lemma66Foreveryr∈Randeveryf∈h,NrT(f)N−rextendsinaboundedoperatorintheFockspaceFB(h).Inotherwordsforeveryr≥0,T(f)isboundedontheHilbertspaceD(Nr)forthenormψr:=NrψF(h).BProofFromthepreviousproposition(iii)wehaveT∗(f)NT(f)=N+a(f)+a∗(f)+f21(10.26)Weprovethelemmaforr=1.Itisnotdifficultbyiterationandinterpolationto2provetheresultforeveryr.WehaveN1/2−1/22−1/2∗−1/2T(f)NΨ=Ψ,NT(f)NT(f)NΨWecanreplaceNbyN,use(10.26)andCauchy–SchwarzinequalitytogetN1/2−1/222T(f)N≤CfLemma67Letα∈C1(R,h),t→α(t).ThenT(α(t))isstronglydifferentiableintfromD(N1/2)toF(h).ThederivativeisgivenbydTα(t)=Tα(t)a∗α(t)˙−aα˙¯+i(α¯·˙α)(10.27)dtwhereα˙=dα.dt 29610BosonicCoherentStatesProofItisenoughtocomputethederivativefort=0.UsingLemma10.3thecom-putationscanbedoneforψ∈D(N1/2).Inwhatfollowsψwillbeomitted.WehaveT(α(t))−T(α(0))=T(α(0))T∗(α(0))(T(α(t))−1)andT∗α(0)Tα(t)=Tα(t)−α(0)eiα(0),α(t)UsingDuhamelformulaonD(N1/2)wegetTα(t)−α(0)=1+ta∗α(˙0)−aα(˙0)+Ot2hencedT∗α(0)Tα(t)=a∗α(˙0)−aα(˙0)+iα(0),α(˙0)dtt=0Theformula(10.27)follows.Inthefollowingsectionweshallstudythemean-fieldbehavioroflargesystemsofbosonswithweaktwoparticlesinteractions.Forthatpurposeweintroduceone(1)particledensityoperatorΓforeveryΨ∈F(h),asfollows.ItisdefinedasaΨsesquilinearforminh:1∗(1)(f,g)→Ψ,a(f)a(g)Ψ:=f,Γg,f,g∈hΨ,NΨΨIfh=L2(R3)theSchwartzkernelofΓ(1)satisfiesΨ(1)1∗Γ(x,y)=Ψ,a(x)a(y)Ψ(10.28)Ψψ,NΨMoreoverifΨisak-particlestatethenΓ(1)istherelativetraceinh=L2(Rn)ofΨtheprojector|ΨΨ|andwehave(1)Ψ¯Γ(x,y)=dxdyΨx,xy,yΨRn(k−1)×Rn(k−1)WeshallbeinterestedtoconsideringtheoneparticledensityforΨ(t)beingatimeevolutionofacoherentstateΨ(ϕ(t)),whereϕ(t)=−1/2ϕ(t),dependingona(1)small(semi-classical)parameter.WecallitΓ.Weshallseethat,undersome,t(1)conditions,Γconvergesintrace-normoperatorto|ϕ(t)ϕ(t)|when0where,tϕ(t)followsaclassicalevolution.10.4TheClassicalLimitforLargeSystemsofBosons10.4.1IntroductionWehavealreadyconsideredtheclassicallimitproblemforsystemswithafinitenumberofbosonsinChap.1.Ithasbeenanaturalquestionsincetheearlydaysof 10.4TheClassicalLimitforLargeSystemsofBosons297quantummechanicstocomparetheclassicalandquantummechanicaldescriptionsofphysicalsystems.OneoftheoldestandbynowbestknownrelationbetweenthetwotheoriesgoesbacktoEhrenfest[74].Thishasbeenputonafirmmathemat-icalbasesbyHepp[113].HeprovedthatinthelimitwherethePlanckconstant1tendstozero,thematrixelementsofquantumobservablesbetweensuitable-dependentcoherentstatestendtotheclassicalvaluesevolvingaccordingtotheap-propriateequation.Moreoverheprovedthatthequantummechanicalfluctuationsevolveaccordingtotheequationobtainedbylinearizingthequantummechanicalevolutionaroundtheclassicalsolution.Heppapproachcoversthecaseofquantummechanics(thatwehavestudiedindetailinChap.4)ofbosonfieldtheories,bothrelativisticandnonrelativistic,andmoregenerallyofallquantumtheorieswhichcanbeexpressedintermsofobservablessatisfyingtheCanonicalCommutationRelations(CCR).OneisledtostudyaperturbationproblemfortheevolutionofasetofoperatorssatisfyingtheCCRinasuitablerepresentation.Thesmallpa-rameterwhichcharacterizestheperturbationtheoryis1/2.InthemostfavorablecasesthisevolutionisimplementedbyaunitarygroupofoperatorsW(t,s).ThesolutionoftheunperturbedproblemisgivenbyaunitarygroupU2(t,s),thein-finitesimalgeneratorofwhichisquadraticinthefieldoperatorsanddependsontheclassicalsolutionaroundwhichoneisconsideringtheclassicallimit.Theop-eratorU2(t,s)describestheevolutionofthequantumfluctuations.Hepp’sresultconsistsofprovingstrongconvergenceofW(t,s)towardsU2(t,s)whengoestozero.WeshallexplaintheresultsobtainedbyGinibre–Velo[86–88]toestimatetheerrorterminHepp’sresults.WealsoexplainresultsobtainedmorerecentlybyRodnianski–Schlein[168]usingHepp’sapproachtogettheconvergenceoftheoneparticlemarginalforevolvedcoherentstatestowardsaclassicalfield.Notethatthisresultissomehowanextensiontothequantumfieldcontextofthesemi-classicalex-pansionconsideredbeforeinChap.4fortimeevolutionofGaussiancoherentstatesforafixednumberofbosons.10.4.2Hepp’sMethodWefollowherethepresentationgivenin[87].WestartwiththeabstractsettingofageneralFockspaceF(h)andanorthonormalbasis{ei}i∈Iinh.I={1,2,...,ν},ν≤+∞.Recallthatai:=a(ei).Sothesystemisdescribedbythefamilyofquantumoperatorsa=(ai)i∈Isatis-fyingthecanonicalcommutationrules:[a∗i,aj]=0,ai,aj=δi,j1Inphysicsisaconstantequalto1.055×10−34Js.AsisusualinquantummechanicsweconsiderhereasaneffectivePlanckconstantobtainedbyscaling,forexample→√,where2mmisthemassand0meansm+∞. 29810BosonicCoherentStatesThevariablesaexpectedtohaveaclassicallimitarerelatedtoabya=1/2aora(f)=1/2a(f),f∈hora=1/2aii.Consideraself-adjointHamiltonianHinFB(h),suitablyregular(forinstancepoly-nomialina∗,a),withnoexplicit-dependence:∗n1∗nkm1mH=C(n1,...,nk|m1,...,m)a1···a1a1···a,C(•|•)∈CIntheHeisenbergpicturethetimeevolutiona(t)ofaisgivenbythefollowingequation:dia(t)=a(t),H(a),a(0)=a(10.29)dtWewanttorelatethetimedependentoperatorsa(t)withafamilyof-independentc-numbervariables:2ϕ(t)=ϕi(t)i∈Iwhichwillappeartobetheclassicallimits.ϕ(t)canbeidentifiedwithaclassical trajectoryintheHilbertspacehwritingϕ(t)=i∈Iϕi(t)ei.WethusexpandHinpowerseriesofa−ϕi,(a)∗−¯ϕiinaneighborhoodofiiϕ,ϕ¯:H(a)=H(ϕ)+H1(a−ϕ)+H2(a−ϕ)+H≥3(a−ϕ)(10.30)wherethefunctionsH1,H2,H≥3arepolynomialsina−ϕ,a∗−¯ϕwithtotaldegree1,2and≥3,respectively,withtimedependentcoefficients.DefiningH(a)=a,Hkk(a),k=1,2,≥3weseethatHare-independent,thatH(a)isac-numberandthatHislinearink12a,a∗.Equation(10.29)canberewrittenasddiϕ+i(a−ϕ)=H1(a)+H2(a−ϕ)+H≥3(a−ϕ)(10.31)dtdtButwehaveH(a)=H(a).Sowechooseϕtobeasolutionoftheclassical11evolutionequationassociatedwithHamiltonianH,namelydiϕ=H1(a)(10.32)dtWedefineϕ=−1/2ϕ2Ac-numberhereisafamilyoftimedependentcomplexnumbersindexedbyI.Itbecanidentified(1)withavectorinh.Inthelanguageofquantummechanicsc-numbersaretheoppositeofΓ,,toperatorsinanHilbertspace. 10.4TheClassicalLimitforLargeSystemsofBosons299Then(10.31)becomesdi(a−ϕ)=H(a−ϕ)+−1/2H1/2(a−ϕ)2≥3dtLetusnowintroducetheWeyloperatorforany(αi)i∈Iα∗∗∗T(α)=expiai−αiai=expa(α)−a(α),whereα=αieii∈Ii∈I(10.33)wherewehavechoseninitialtimes=0forsolvingequation(10.29).RecallthatT(α)areunitaryandobeyT(α)∗aT(α)=a+α Notethatherewehavechosencoordinatesinh.Iff=i∈IαieiwehaveT(f)=T(α).Wedefineanewvariableb(t)as∗b(t)=Tϕ(s)a(t)−ϕ(t)Tϕ(s)Theinitialvalueproblemfor(10.29)thenreducestofindingafamilyb(t)ofoper-atorssatisfyingthe(CCR)andtherelationsb(0)=ad(10.34)ib=H(b)+−1/2H1/2bdt2≥3Notethatthesecondtermintherighthandsideof(10.34)isO(1/2)sinceHhas≥3degreeatleasttwo.Thereforeb(t)isexpectedtoconvergetowardsthesolutionofthelinearizedequationdib=Hb,b(0)=a(10.35)dt2whichgovernsthequantumfluctuationsaroundtheclassicalequation.WeintroducethepropagatorU2(t,s)definedbythequadraticHamiltonianHˆ2(t):diU2(t,s)=Hˆ2(t)U2(t,s),U(s,s)=1dtSowehaveb(t)=U∗2(t,0)aU2(t,0)Inthesameway,thetimeevolutionofoperatorsb(t)willbeimplementedbyaunitarygroupW(t,s)suchthatb(t)=W(t,0)∗aW(t,0) 30010BosonicCoherentStateswhereW(t,s)obeysthedifferentialequationdi−1/2H1/2W(t,s)=H2(a)+≥3aW(t,s),W(s,s)=1(10.36)dtOnehasthefollowingresult,whichiseasilyprovedusingLemma67.Proposition119OnehasW(t,s)=expiω(t,s)Tˆϕ(t)U(t−s)Tˆϕ(s)withU(t−s)=exp−i−1(t−s)H(a)andtω(t,s)=−1dτHϕ(τ)−ϕ(τ),Hϕ(τ)1sThereforewehaveprovenatleastformallythefollowingresult:Proposition120∗∗∗Tϕ(s)U(t−s)a(s)−ϕ(t)U(t−s)Tϕ(s)=W(t,s)a(s)W(t,s)withU(t)andW(t,s)givenbyProposition119.ThedifficultmathematicalproblemistoanalyzetheunitarypropagatorsU2(t,s)andW(t,s).Onehasthefollowingresult(see[113]):Letϕ(t,x)beasolutionoftheclassicalequation(10.32)withinitialdataϕatt=s.Forf∈L2(Rn)letusdefineϕ(f,t)=dxf(x)ϕ(t,x)Similarlyconsiderthesolutionsb(t)ofthelinearizedproblem(10.35).Letb(t,f)=f¯ibi(t),iff=fieii∈Ii∈IAsusualdefinetheoperatorvalueddistributions(b)(t,x)suchthatb(t,f)=dxf(x)b(t,x)Heredenoteseithernothingor∗. 10.4TheClassicalLimitforLargeSystemsofBosons301Itispossibletoapplythestrategydescribedabovetoprovesemi-classicallimitresultsforbosonsystemswhen0.TheHamiltonianHisdefinedas2∗∗∗H=dx∇a(x)·∇a(x)+dxdyV(x−y)a(x)a(y)a(y)a(x)(10.37)22Notethatthelimit0isequivalenttothemean-fieldlimitN+∞consideredin[168](N=−1)fortheHamiltonian1∗1∗∗HN=dx∇a(x)·∇a(x)+dxdyV(x−y)a(x)a(y)a(y)a(x)22N(10.38)Herewehave1ϕ(y)2H1=−Δϕ(x)+ϕ(x)dyV(x−y)(10.39)2sothatonerequiresthattheclassicalevolutionϕt(x)=ϕ(t,x)isasolutionoftheHartreeequation:d1V∗|ϕ2iϕt=−Δϕt+ϕtt|(10.40)dt2TostaterigorousresultssometechnicalassumptionsareneededforthepotentialV.Recallthefollowingdefinition(see[115]formoredetails).Forsimplicityweonlyconsiderthecaseh=L2(R3).Definition23ApotentialV(x),x∈R3iscalledaHardypotentialifVisrealandthereexistsC>0suchthatVϕ≤Cϕ,∀ϕ∈H1R3(10.41)L2(R3)H1(R3)ItiswellknownthatifV(x)=c,c∈RthenVisaHardypotential(bythe|x|usualHardyinequality).HardyclasspotentialsincludedtheKatoclasspotentials.(see[115]fordetails).ThefollowingpropositionisaparticularcaseofmoregeneralonesconcerningHartreeequation[88].Thefollowingpropositionissketchedin[168],Remark1.3.In[86–88]morerefinedresultsaregivenconcerningsolutionsforHartreeequationwithsingularpotentials.Proposition121LetV(x)beaHardypotential.Letϕ0∈H1(R3).Thenequation(10.40)hasauniquesolutionϕ∈C(R,H1(R3)).FurthermoreonehasthepropertyofconservationoftheL2-normandoftheenergy:ϕt=ϕ0,∀t∈RE(ϕt)=E(ϕ0),∀t∈R 30210BosonicCoherentStateswhere121ϕ(x)2ϕ(y)2E(ϕ)=∇ϕ+dxdyV(x−y)22Proposition122LetV(x)beanevenHardypotential.AssumethattheinitialdatafortheHartreeequationsatisfyϕs∈H1(R3).Thenonehasfor|t−s|0andeveryT>0thereexistsC(δ,T)suchthatNδU2(t,s)N−δ≤C(δ,T),for|t−s|≤T(10.46)TheDuhamelformulagivestW(t,s)=U1/22(t,s)−iW(t,τ)H3(τ,a)+H4(a)U2(τ,s)dτ(10.47)s 10.4TheClassicalLimitforLargeSystemsofBosons303whereH3(t,a)=A3(t)+A3(t)∗anddydxV(x−y)ϕ¯∗A3(t)=t(x)a(y)a(y)a(x)(10.48)andHdydxV(x−y)a∗(x)a∗(y)a(y)a(x)(10.49)4=UsingthatF0,0isdenseinF(L2(R3))andthatW(t,s)isunitaryinF(L2(R3)),itisenoughtoprovethatlimW(t,s)Ψ=U2(t,s)Ψ,foreveryΨ∈F0,0(10.50)→0Thisresultisprovedusing(10.47),(10.48),(10.49),assumptionsonVandϕandstandardestimates.(iii)isprovenusingProposition120and(i).Corollary28LetAbeasmoothandboundedfunctionofaanda(see[19]).Thenwehavethefollowingsemi-classicallimitevolutionforquantumexpectationsincoherentstates:limU(t)Ψ(ϕ),U(t)Aa−ϕ,a∗−¯ϕU(t)Ψ(ϕ)→0=Ω,Ab(t),b(t)∗Ω(10.51)whereb(t)=U2(t,0)∗aU2(t,0)isthelinearevolutionattheclassicalevolutionϕ(t).Remark56In[88]theauthorsprovedafullasymptoticexpansionin1/2foradensesubsetofstatesψ.WeshallseelaterthatwehaveabetterresultifVisbounded.10.4.3RemainderEstimatesintheHeppMethodHeppmethodwasrevisitedbyGinibre–Velo[88,89]toextendittosingularpoten-tialandtogetquantumcorrectioninatanyorder.Thespiritoftheworksof[88,89]istoexploitthedifferentialequation(10.36)(andtheformula(10.47))towriteaDysonseriesexpansionforW(t,s).Theauthorsobtainapowerseriesin1/2andstudyitsanalyticitypropertiesinκ:=1/2.ForboundedpotentialsVthisseriescanbeshowntobeBorelsummableinvectornormwhenappliedtofixed-independentvectorstakenfromasuitabledensesetincludingcoherentstates.ThepotentialVisalsoassumedtobestable,whichmeansthatthereexistsaconstantB≥0suchthatH4+BN≥0(10.52) 30410BosonicCoherentStates(Nisthenumberoperator(10.17)).Roughlyspeakingitmeansthatthepotentialissufficientlyrepulsiveneartheoriginifattractivesomewhereelse.Moreexplicitlywecanfindin[171]asufficientconditionofstabilityforapotentialV.H4hasthefollowingexpression:(k)(x(k)(H4Ψ)1,...,xk)=V(xi−xj)Ψ(x1,...,xk)i3wehaveϕν−1ν−11(t)tdt=+∞,ϕ2(t)tdt<+∞ThenVisstable([171],Proposition3.2.8).RemarkthatthestabilityconditionisarestrictiononthenegativepartofV.Beforetostatethesummabilityresultletusrecallsomedefinitionsconcerningasymptoticseries. Consideraformalpowercomplexseriesf(κ)=j∈Nαjκj.Definition24(i)fisaGevreyseriesoforder1/s,s>0,ifthereexistC>0,ρ>0suchthat|α|≤Cρj(j!)1/s,∀j∈N(10.54)(ii)Thes-BoreltransformoftheGevreyseriesfisdefinedasαjBjsf(τ)=τ(10.55)jΓ(1+)j∈Nswheretheseriesconvergesforτ∈C,|τ|<ρ−1.(iii)TheGevreyseriesfissaidBorels-summableif(iii)1itss-BoreltransformBsfhasananalyticextensiontoaneighborhoodofthepositiverealaxis,(iii)2thefollowingintegral:∞s−1−(u)sduBsf(u)ueκ0convergesfor0<κ<κ0.Ifthisholdsthenwesaythatfhasas-Borelsumf(κ)definedby∞−ss−1−(u)sf(κ)=sκduBsf(u)ueκ(10.56)0 10.4TheClassicalLimitforLargeSystemsofBosons305Remark57(i)Fors=1theabovedefinitioncorrespondstotheusualBorelsummability.(ii)Attheformallevel,formula(10.56)iseasytocheckusingdefinitionofΓfunctionandchangesofvariables.(iii)(j!)j/sandΓ(1+j)havethesameorderasj→asaconsequenceofthesStirlingformulau√uΓ(1+u)=2πu1+o(1)asu→+∞.eWestatenowasufficientconditionforBorelsummabilityduetoWatson,Nevan-linnaandSokal(see[161,180]andreferenceswithextensiontoanys>0).Theorem46LetfbeanholomorphicfunctioninthecomplexdomainDs,R:={κ∈C,κ−s>R−s}forsomes>0.AssumethatthereexistC>0,ρ>0suchthatinthisdomainwehavef(κ)−αjκj≤CρN(N!)1/s|κ|N(10.57)0≤j0.ThisiseasilyseenbystoppingtheseriesattheorderN≈δwithδsmallenough.κsAddingananalyticconditioninasuitabledomainasintheabovetheoremerasethiserrortermsothatf(κ)isuniquelydetermined.ThefollowingresultgivesanaccurateasymptoticdescriptionforthequantumfluctuationoperatorW(t,s),improvingProposition122.Theorem47LetVbestableandboundedpotential.Letϕ∈C1(R,L2(R3))beasolutionofHartreeequation(10.40).Letβ>0andΦ∈D(exp(βN)).Thenthereexistsθ>0suchthatforalls,t∈R,t≥ssuchthatt−s≤θ,W(t,s)Φisanalyticinκinthesector−π0(Proposition3.1in[88]).InathirdstepitisprovedthatW(t,s)Φhasananalyticexpansioninκ:=1/2inadomainlikeD2,R.Herethestabilityconditionisused.Thelaststepconsistsofestimatingtheremaindertermoftheasymptoticexpan-sionofW(t,s)Ψinκsmall.TodothattheDysonseriesexpansionisperformedintwosteps,introducinganauxiliaryevolutionoperatorU4(t,s)obeyingdiU4(t,s)=H2(t)+H4U4(t,s)dtThenifU2(t,s)isthepropagatorforH2(t)(whichdescribestheevolutionofthequantumfluctuations),onehastU4(t,s)=U2(t,s)−idτU2(t,τ)H4U4(τ,s)stW(t,s)=U1/24(t,s)−idτU4(t,τ)H3W(τ,s)s Thenitcanbeprovedthattheseriesj∈Nj/2Wj(t,s)Φis2-Borelsummableandits2-BorelsumisW(t,s)Φ.Usingsimilarmethodsthecaseofunboundedpotentialshasbeenconsideredin[89].Theauthorsobtainthesameanalyticitydomainasin[88]ofW(t,s)withrespecttoκ=1/2andseethattheseriesisstillasymptotic(inthePoincarésense)andGevreyoforder2.Itisnotknownthattheseriesisstill2-Borelsummableforsingularpotentials.NotethatD(exp(βN))containsthecoherentstatesΨ(f),f∈L2(R3).10.4.4TimeEvolutionofCoherentStatesWehaveseenthatHepp’smethodconcernsthequantumfluctuationsinaneighbor-hoodofaclassicaltrajectory.Itdoesnotgiveinformationonthequantummotion3NotionsofBorelsummabilityhavebeendefinedbeforeforcomplexvaluedseries,extensiontovectorvaluedseriesisstraightforward. 10.4TheClassicalLimitforLargeSystemsofBosons307itself,inparticularwhentheinitialstateisacoherentstate.InChap.4thisproblemwasconsideredforfinitebosonsystems.Ourgoalhereistoexplainanextensionofthisresultforlargesystemsofbosons,followingthepaper[168].AccordingtotheprevioussectiononehastostudytheevolutionoperatorU(t)=e−iHtassociatedwiththeHamiltonian1∗∗∗H=dx∇a(x)·∇a(x)+dxdyV(x−y)a(x)a(y)a(y)a(x)22bytheevolutionequationdiU(t)=HU(t)dtandapplyittothecoherentstateΨ(−1/2ϕ)whereϕ=−1/2ϕ,ϕisasolutionoftheHartreeequation(10.40).BydefinitiontheHamiltonianHleavessectorsk∨L2(R3)withfixednumberofparticlesinvariant.OnethushaveU(t)∗NU(t)=N(10.59)(1)Onehasthefollowingresult.Γ(definedattheendofSect.10.3)isthemarginal,toperatorintheoneparticlespaceL2(R3)deducedfromthequantumevolutionU(t)Ψ(ϕ)ofthecoherentstateΨ(ϕ).RecallthatinanHilbertspace√hthetrace-normofanoperatorAisdefinedasATr=Tr(A∗A)(seeChap.1).Theorem48SupposethatVisaHardypotential(see(10.41).ThenthereexistconstantsC,K>0(onlydependingontheH1(R3)normofϕandonC)suchthatΓ(1)−|ϕtϕt|≤CeKt,t∈R(10.60),tTrϕtisthesolutionofHartreeequation(10.40)attimetwithϕ0=1.ProofWeshallgiveherethemainideasoftheproof.Thedetailsareinthepaper[168].(1)WenowwritethefunctionΓ(x,y)using(10.28).Onehastocalculatethe,tdenominator:U(t)ψ(ϕ)Ω,NU(t)ψ(ϕ)ΩUsing(10.59)weareleftwithψ(ϕ)Ω,Nψ(ϕ)ΩWenowusethetranslationpropertyoftheWeyloperator:T(ϕ)∗NT(ϕ)=dxa∗(x)−¯ϕ(x)a(x)−ϕ(x)(10.61) 30810BosonicCoherentStatesButusingthata(x)Ω=0weseethattheexpectationvalueof(10.61)inthevacuumsimplyequals−1ϕ2=−1(1)ThereforeΓ(x,y)hasthefollowingdecomposition:,t(1)Γ(x,y),t−1/2∗∗∗−1/2=Ω,TϕU(t)a(y)a(x)U(t)TϕΩ=¯ϕ1/2∗∗t(y)ϕt(x)+ϕ(y)¯Ω,T(ϕ)U(t)a(x)−ϕt,(x)U(t)T(ϕ)Ω+1/2ϕ∗∗a∗t(x)Ω,T(ϕ)U(t)(x)−ϕt,(y)T(ϕ)Ω+Ω,T(ϕ)∗U(t)∗∗(y)−¯ϕ(y)a(x)−ϕat,t,(x)U(t)T(ϕ)ΩNowweusethefactdemonstratedintheprevioussectionthatT(ϕ)∗U(t)∗a(x)−ϕ(x)U(t)T(ϕ)=W(t,s)∗a(x)W(t,s)s,t,s,Thuswegettheequalitybetweenthetwokernels:(1)∗∗Γ(x,y)−ϕt(x)ϕ¯t(y)=Ω,W(t,0)a(y)a(x)W(t,0)Ω,t1/2ϕ∗∗+t(x)Ω,W(t,0)a(y)W(t,0)Ω+1/2ϕ¯∗t(y)Ω,W(t,0)a(x)W(t,0)Ω(10.62)Itisremarkedin[168]thathereitisenoughtoestimatetheHilbert–Schmidtnorm(1)ofΓ−|ϕtϕt|insteadofitstrace-norm.Sothemaintechnicalpartofthepaper,t[168]istoshowthattheL2normin(x,y)oftherighthandsideof(10.62)isboundedabovebyCeKt,usingsuitableapproximationofthedynamicsW(t,s).Asin[88,89]W(t,s)iscomparedwiththedynamicsU4(t,s)generatedbyH2(t)+H4(withouttheH3(t)term),namelydiU4(t,s)=H2(t)+H4U4(t,s)dtIn[168]thefollowinglemmasareproven.Lemma69OnehasW(t,0)−U4(t,0)Ω≤C1/2eKtLemma70W(t,0)Ω,NW(t,0)Ω≤CeKt 10.4TheClassicalLimitforLargeSystemsofBosons309Now,introducingU4(t,s),therighthandsideof(10.62)isrewrittenas(1)Γ(x,y)−ϕt(x)ϕ¯t(y),t=Ω,W(t,0)∗a∗(y)a(x)W(t,0)Ω+1/2ϕ∗∗t(x)Ω,W(t,0)a(y)W(t,0)−U4(t,0)Ω∗−U∗a∗∗+Ω,W(t,0)4(t,0)(y)U4(t,0)Ω+1/2ϕ¯∗t(y)Ω,W(t,s)a(x)W(t,s)−U4(t,s)Ω∗∗a(x)U+Ω,W(t,s)−U4(t,s)4(t,s)Ω(10.63)whereweusethatU4(t,s)preservestheparityofthenumberofparticles:∗a∗(y)U∗Ω,U4(t,s)4(t,s)Ω=Ω,U4(t,s)a(x)U4(t,s)Ω=0ThenwegetfromLemmas69and70theHilbert–SchmidtestimateΓ(1)222Ktdxdy(x,y)−ϕt(x)ϕ¯t(y)≤Ce,∀t≥0,tR3×R3Remark59In[43]theauthorshaveextendedthepreviousresulttothecaseofarbitraryfactorizedinitialdata. Chapter11FermionicCoherentStatesAbstractThischapterisanintroductiontosomecomputationtechniquesforfermionicstates.AfterdefiningGrassmannalgebrasitispossibletogetaclassi-calanalogueforthefermionicdegreesoffreedominaquantumsystem.FollowingthebasicworkofBerezin(TheMethodofSecondQuantization,1966;IntroductiontoSuperanalysis,1987),weshowthatwecancomputewithGrassmannnumbersaswedowithcomplexnumbers:derivation,integration,Fouriertransform.Afterthatweshowthatwehavequantizationformulaforfermionicobservables.InparticularthereexistsaMoyalproductformula.Asanapplicationweconsiderexplicitcom-putationsforpropagatorswithquadraticHamiltoniansinannihilationandcreationoperators.11.1IntroductionWehaveseenintheChapteronBosonsthatrelations(CCR)canberealizedwithrealorcomplexnumbers.Weseeherethatanti-commutationrelations(CAR)needtointroduceanewkindofnumber,nilpotent,calledaGrassmannnumber.Insomesense,nilpotenceisclassicallyequivalenttothePauliexclusionprinciplewhichcharacterizefermions.Thisappearstobestrangefromaphysicistpointofviewbe-causePauliexclusionprincipleisapurelyquantumproperty,withoutclassicalana-logue.Neverthelesssuchamathematicalmodelexists.Evenif“classicalfermions”donotexistinNature,theyareaconvenientmathematicaltoolforcomputationsandallowtoputonthesamefootingbosonsandfermions.Thisisimportanttoelabo-ratesupersymmetricmodelswhichwillbeconsideredinmoredetailsinthenextChapter.Ourmaingoalhereistointroducefermioniccoherentstatesandtodescribetheirproperties.ThemaindifferencefromthebosoniccaseconsideredinChap.1isthatwehavetoreplacecomplexnumbersbyGrassmannnumbers.Sotheconstructionshavetoberevisitedwithsomecare.SoweshallstudyinmoredetailsGrassmannal-gebrasandweshallseethatmanypropertiesandconstructionswellknownforusualnumberscanbeextendedtoGrassmannalgebras.Ourconstructionoffermionicco-herentstatesmainlyfollowsthepaper[37].WealsoconsiderinmoredetailsthepropagationofFermionsforquadraticevolutions.M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,311TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0_11,©SpringerScience+BusinessMediaB.V.2012 31211FermionicCoherentStatesIn[21](seethebook[156],Chap.9)theauthorintroducedFermioniccoherentstatesfromanotherpointofview,withoutGrassmannalgebras.HehaveconsideredanirreduciblerepresentationoftherotationgroupSO(2n,R)(forasystemofnfermions)calledthespinrepresentation.Weshallseethatthetwopointofviewsaremathematicallyequivalent.11.2FromFermionicFockSpacestoGrassmannAlgebrasItfollowsfromtheChapteronbosonsthatthefermionicFockspaceistheHilbertspaceFF(h)=k∈N∧kh,wherehistheonefermionspaceand∧khistheanti-symmetrictensorspannedbythestates1επψπ1⊗ψπ2⊗···×ψπk,ψj∈h.k!π∈SkTheannihilationandcreationoperatorsa(f),a∗(f)areboundedoperatorsinFF(h).ThisisaconsequenceoftheCanonicalAnti-commutationRelation∗(f∗a(f1)a2)+a(f1)a(f2)=f1,f21.(11.1)LetusbeginwithanexplicitmodeltorealizeCARrelations(11.1)whichiscalledthespinmodel.WestartwithHF=C2andthematrices0100σ+=,σ−=.0010Sowehave(CAR)fora=σ−anda∗=σ+:[σ22+,σ−]+=12,σ+=σ−=0.Thisisamodelforonestatespin.WehavethenumberoperatorN=a∗a=10000andthegroundstateise0=.1WegetamodelfornspinstatesintheHilbertspaceH2⊗n2nH=(C)=C.Theannihilationandcreationoperatorsareak=σ3⊗···σ3⊗a⊗12⊗···⊗12,k−1n−k∗⊗a∗⊗1ak=σ3⊗···σ32⊗···⊗12.n−kThegroundstateishereΩ0=e0⊗···⊗e0.n 11.2FromFermionicFockSpacestoGrassmannAlgebras313ThismodelisunitaryequivalenttothefermionicFockmodelwithh=Cn.WehaveHn2nnnF=C⊕C⊕∧C⊕···⊕∧C.Noticethatdim(HF)=2n.LetusgivenowsomespecificpropertiesforthegeneralfermionicFockmodel(fordetailedproofssee[33]).Proposition123(1)Foreveryf∈h,a(f)anda∗(f)areboundedoperatorsinFF(h):a(f)=a∗(f)=f,∀f∈h.(11.2)(2)Let{ϕj}j∈Jbeanorthonormalbasisforh.Thefamilyofstatesdefinedas=a∗(ϕ∗ψj1,j2,...,jkj1)···a(ϕjn)Ω,jk∈J,n≥0,isanorthonormalbasisforFF(h).(3)IfTisaboundedoperatorinFF(h)commutingwithalltheoperatorsa(f)anda∗(g),f,g∈h,thenT=λ1forsomeλ∈C.Property(3)ofthepropositionmeansthattheFockrepresentationforFermionisirreducible.Whenthesystemhasnidenticalparticlesthecreation/annihilationoperatorsare(∗):=a(∗)(ϕdenotedaj),1≤j≤n.jAnaturalproblemistorepresentthecommutationrelations(CAR)withderiva-tiveandmultiplicationoperatorsascanbedonefor(CCR)withqanddorindqtheFock–Bargmannrepresentation(seeChap.1).Inotherwordswewouldliketorepresentanti-commutationrelationslike∂∂θ+θ=1.(11.3)∂θ∂θThisisnotpossibleinthenaivesense.ThisproblemisequivalenttobuildaclassicalanalogueforFermionicobservables.Inordertosatisfy(11.3)itisnecessarytore-placetheusualrealorcomplexnumbersbyGrassmannvariablesaswasdiscoveredbyBerezin[22].LetusdefinetheGrassmannalgebras.Definition25TheGrassmannalgebraGnwithngenerators{θ1,...,θn}isanalge-bra,withunit1,aproductandaK-linearspace(K=RorK=C)suchthatθjθk+θkθj=0,∀j,k=1,...,nandeveryg∈Gncanbewrittenasg=c0(g)+cj1,...,jk(g)θj1···θjk,(11.4)k≥1j1,...,jkwherec0(g)andcj1,...,jk(g)areK-numbers. 31411FermionicCoherentStatesThisdefinitionhassomedirectalgebraicconsequencesgivenbelowandeasytoprove1.Inequality(11.4)thedecompositionisnotunique.Wegetauniquedecomposi-tionifweaddinthesumtheconditionj1n.Inparticularifθ=(θ1,...,θn),γ=(γ1,...,γn)are2nGrassmanngeneratorstheneθ·γiswelldefinedwhereθ·γ=1≤k≤nθkγk.Lemma72Ifk≥2wehaveμ(k)θθ1θ2···θk=(−1)kθk−1···θ1,(11.13)where1ifk≡2,3mod(4)μ(k)=(11.14)0ifk≡0,1mod(4)eθ·γ=1+θ·γ+(−1)ν(ε)θεγε(11.15)ε∈E[n],|ε|≥2wheretheintegerν(ε)isdefinedasfollows:1if|ε|≡2,3mod(4)ν(ε)=(11.16)0if|ε|≡0,1mod(4)ProofEquation(11.13)istruefork=1,2.WegetthegeneralcasebyinductiononkusingLemma71.(11.15)isprovedbyinductiononnusingtheidentityeθ·γ=eθkγk=(1+θkγk).1≤k≤n1≤k≤n11.3.2CalculuswithGrassmannNumbersWehavealreadydefinedderivativesinGrassmannalgebrasintheprevioussection.TopreserveanalogywithbosonsitisusefultodefineintegrationinGrassmannvariables.Definition27ConsideraGrassmannalgebraGnwithgenerators{θ1,...,θn}.Letψ∈Gnand1≤j1,...,jk≤n.ψdθj···dθj∈Gnisdefinedbythefollowingproperties.1k 31811FermionicCoherentStates(i)ψ→∫ψdθj···dθj∈Gnislinear1k(ii)∫dθj=0,∀j=1,...,n.(iii)∫θjdθk=δj,k,∀j,k.(iv)∫dθjdθk=0,∀j,k.(v)∫∫ψ(θj)ϕ(θk)dθjdθk=ψ(θj)dθjϕ(θk)dθk,∀j,kWeseethatforfermionsintegrationcoincideswithdifferentiation;weeasilyfind∂∂ψdθj···dθj=···ψ(θ1,...,θn).(11.17)1k∂θj∂θjk1Thisformulawillbeusedtocomputeintegrals.Lemma73(Changeofvariables)LetAbearealinvertiblen×nmatrix(AmayhaveitscoefficientinaGrassmannalgebrasuchthatAisevenandinvertible).Thenwehaveψ(Aθ)dθ=det(A)ψ(θ)dθ.(11.18)ProofIfθ=θAwehave=θθkjAj,k.1≤j≤nUsingdefinitionofdeterminantweget···θ=θθn1n···θ1(detA).Butwehaveψdθ=ψ(1,...,1)whereψ(1,...,1)isthecomponentofψinitsexpan-sionψ(θ)=ε∈E[n]θεψε.Hencewededuce(11.18).FromtheLeibnitzrulewededuceanintegrationbypartstheformula:∂ψ∗∂ϕ∗ϕdθdθ=Pψdθdθ.(11.19)∂θk∂θkRemarkonnotation:heredθ∗dθmeans1≤j≤ndθ∗dθj.Sometimesitwillbede-jnotedd2θ.11.3.3GaussianIntegralsInordertopreserveanalogywiththeBargmann–Fockrealizationwithholomorphicstates(seeChap.1)weneedtohaveacomplexstructureonourGrassmannalgebra.SoweconsidertheGrassmannalgebraG2nwithgenerators{θ1,...,θn;θ∗,...,θn∗}.1InG2nwecandefineacomplexanti-linearinvolutionsuchthatθk→θ∗.Weimposekthatψ→ψ∗isR-linearandsatisfies 11.3IntegrationonGrassmannAlgebra319(i)(zα)∗=¯zα∗ifz∈Candαisagenerator(ii)(αβ)∗=β∗α∗foreveryα,β∈G2nLetusdenoteθ∗=(θ∗,...,θn∗).TheGrassmannalgebraG2nwithitscomplexstruc-1turewillbedenotedGc.nIntegrationonGcisdefinedasfollows:n∗=ψdθ∗∗ψdθdθ1dθ1···dθndθn.Wehavethefollowingproperty.Lemma74Theintegralψdθdθ∗isinvariantunderunitarychangeofvariable:θ=Uζ,U∈SU(n).Sowehaveψdθdθ∗=ψUζ,Uζ¯dζdζ∗.ProofFromtheproofofthechangeofvariablelemmawehaveθn···θ1=(detU)ζn···ζ1.Thenwegetθ∗θ∗∗∗∗∗nn···θ1θ1=ζnζn···ζ1ζ1detUdetU=ζnζn···ζ1ζ1.Thelemmafollows.Proposition125LetBbean×nHermitianmatrix.Thenwehave−θ∗·Bθ∗edθ·dθ=detB.(11.20)ProofThereexistsaunitarymatrixUsuchthatUBU∗=DwhereDisthediago-nalmatrixD=(b1,...,bn).Usingthechangeofvariablesη=Uθweget−θ∗·Bθ∗−η∗·Dη∗−bjη∗ηj∗edθ·dθ=edη·dη=ejdηjdηj=detB.1≤j≤nWecanextendtheabovepropositiontomoregeneralGaussianintegrals.Proposition126LetC[γ,γ∗]beanothercopyofthecomplexGrassmannalgebraGncandBan×nHermitianmatrix.Thenwehave−θ∗·Bθγ∗·θ+θ∗·γ∗γ∗·B−1γeedθ·dθ=(detB)e.(11.21)ProofAsintheproofofProposition125webeginbyaunitarychangeofvariabletodiagonalizeB.Thenwegettheresultaftersomecomputations(reductiontothecasen=1)lefttothereader.AnotherusefulGaussianintegralcomputationis 32011FermionicCoherentStatesProposition127Letusconsideraquadraticformin(γ,γ∗):1Φγ,γ∗=γ·Kγ+γ∗·Lγ∗+2γ∗·Mγ,2whereK,Lareanti-symmetricmatrices.SothefollowingmatrixΛofΦisanti-symmetric:K−MTΛ=.MLAssumeΛisnon-degenerate.ThenwehavetheFouriertransformresultΦ(γ,γ∗)γ∗·ξ−ξ∗·γ∗Φ(−1)(ξ∗,ξ)eedγdγ=PfΛe,(11.22)wherePfΛisthePfaffianofΛandΦ(−1)isthequadraticformwiththematrixΛ−1.ProofForξ=0itiswellknownthattheintegralisequaltoPfΛwhichisthePfaffianofΛ(see[207]).RecallthatPfΛ2=detΛ.Thenbyausualtrickwecaneliminatethelineartermsintheintegral.WedothatbyperformingachangeofGrassmannvariablesγ=θ−αξ,γ∗=ζ−βξ.θandζarenewGrassmannintegrationvariables(notnecessarilyconjugated)andαξ,βξarecomputedsuchthatthelineartermsareeliminated.Sowefindαξ−1ξ∗=Λ.βξξTheformula(11.22)follows.11.4Super-HilbertSpacesandOperatorsAsinthebosoniccasewewanttofindaspacetorepresentfermionicstatesasfunctionsinsomeL2-space.11.4.1ASpaceforFermionicStatesLetusdefineH(n)thesubspaceofG2nofholomorphicfunctionsinθ=(θ(n)means∂ψ=0for1≤k≤n.1,...,θn).ψ∈H∗∂θkH(n)isacomplexvectorspaceofdimensionnwiththebasis{θε,ε∈E[n]}.MoreoverH(n)isaHilbertspaceforthescalarproductθ∗·θ∗∗ψ,ϕ:=eψ(θ)ϕ(θ)dθdθ 11.4Super-HilbertSpacesandOperators321and{θε,ε∈E[n]}isanorthonormalbasis.Weshalldenoteeε(θ)=θεforε∈E[n].ThespaceH(n)hasdimension2noverCandisisomorphictotheFock(n)spaceH.FRemark60Thenotationψ∗forthecomplexinvolutionissometimesreplacedbythemoresuggestivenotationψ¯whichmaybesometimesconfusing.Thecreation/annihilationoperatorsinH(n)are∂a∗ψ(θ)=θjjψ(θ),ajψ(θ)=ψ(θ).∂θjTheysatisfyanti-commutationrules(CAR)anda∗isthehermitianadjointofakforkevery1≤k≤n.TheHilbertspaceH(n)isnotlargeenoughtorepresentFermionicstatesandtocomputewiththem.AsusualinGrassmanncalculusweneedtoaddnewGrassmanngenerators(γ,γ∗),γ=(γ1,...,γn).LetusdenoteΓnc:=C[γ,γ∗].AsaboveweintroduceH˜(n)thesub-GrassmannalgebraofGnc⊗Γncofholo-morphicelementsinθ.H˜(n)willbeseenasamodule1overthealgebraΓncwithbasis{θε,ε∈E[n]}.H˜(n)isakindoflinearspacewherethecomplexfieldnumberisreplacedbytheGrassmannalgebranumbers:Γc.ThefollowingoperationsmakesensebecausentheyarewelldefinedinGnc⊗Γnc,withusualpropertieseasytostate:ifψ,ϕ∈H˜(n),λ∈Γnc,thenψ+ϕ∈H˜(n)andλψ,ψλ∈H˜(n).Inparticulareveryψ˜∈H˜(n)canbedecomposedasθεcψ=ε(ψ),ε∈E[n]wherecε(ψ)∈Γnc.ThescalarproductinH(n)canbeextendedasasesquilinearformtoH˜(n)bytheformula∗θ∗·θ∗ψ,ϕ:=ψ(θ)ϕ(θ)edθdθbutnowψ,ϕisaGrassmannnumberinΓncandnotalwaysacomplexnumber.Themap(ψ,ϕ)→ψ,ϕhasusualproperties:itissesquilinearandnonnegative.ButitisdegenerateandonlyitsrestrictiontotheHilbertspaceH(n)isnon-degenerate.Herewedonotusethegeneraltheoryofsuper-space,onourexamplesdirectcom-putationscanbedone(seethebook[62]formoredetailsconcerningsuper-Hilbertspaces).LetusremarkthateverylinearoperatorinH(n)canbeextendedasalinearoperatorinH˜(n).InparticulartheparityoperatorPinvariablesθiswelldefined.1AmoduleoversomeringoralgebraislikealinearspacewherethefieldnumberRorCisreplacedbyaringoranalgebra(seeanytextbookinadvancedalgebra). 32211FermionicCoherentStatesAsafirstapplicationletusconsidertheFermionicDiracdistributionatθ.Itisnotdifficulttoestablishthefollowingidentityforeveryψ∈H(n):−(θ−γ)·γ∗∗ψ(θ)=ψ(γ)edγdγ.(11.23)Inotherwordstheidentityisanintegraloperatorwithakernel∗E(θ,γ):=e−(θ−γ)·γ=1−(θ∗k−γk)γk.(11.24)1≤k≤nItisusualtodenoteE(θ,γ)=δ(θ−γ).Wehavealsothemoresymmetricformδ(θ−γ)=(θ∗∗.k−γk)θ−γk1≤k≤nThiscouldbeusedtoprovethateverylinearoperatorinH(n)hasakernel.11.4.2IntegralKernelsProposition128ForeverylinearoperatorHˆ:H(n)→H(n)2thereexistsKH∈C[θ,γ∗]suchthat∗Hψ(θ)ˆ=KHθ,γ∗ψ(γ)eγ·γdγdγ∗,∀ψ∈H(n).(11.25)MoreoverKH∈C[θ,γ∗]canbecomputedasfollows:KHθ,γ∗=eε,Heˆεeε(θ)eε(γ)∗.(11.26)ε,ε∈E[n]ProofUsingthat{eε}isanorthonormalsystemitisnotdifficulttoprovethat(11.25)issatisfiedforψ=eεwithKHgivenby(11.26).Sowegettheresultforeveryψ.Corollary29EverylinearoperatorHˆ:H(n)→H(n)hasauniquedecomposi-tionlikeHˆ=Hε,εa∗εaε,(11.27)ε,ε∈E[n]whereHε,ε∈C.2Asforbosons,whenweconsiderquantizationofobservables,itisconvenienttodenoteoperatorswithahataccenttomakeadifferencebetweenclassicalandquantumobservables.Sometimesthisruleisnotappliedwhenthecontextisclear. 11.4Super-HilbertSpacesandOperators323Theseresultscanbeeasilyextendedtolinearoperatorsinthesuper-HilbertspaceH˜(n).Proposition129ForeverylinearoperatorHˆ:H˜(n)→H˜(n)thereexistsKH∈C[θ,γ,γ∗]suchthat∗Hψ(θ)ˆ=KHθ,γ∗ψ(γ)eγ·γdγdγ∗,∀ψ∈H˜(n).(11.28)MoreoverK∈C[θ,γ,γ∗]canbecomputedasfollows:Kθ,γ∗=eε(θ)eε,Heˆεeε(γ)∗,(11.29)ε,ε∈E[n]andeverylinearoperatorHˆinH˜(n)hasauniquedecompositionHˆ=∗εεHε,εaa(11.30)ε,ε∈E[n]cwhereHε,ε∈Γn.Thisrepresentation,withannihilationoperators,firstiscalledthenormalrepresentationofHˆ.WedenotebyEnd(H˜(n))thespaceoflinearoperatorsinH˜(n).Corollary30EverylinearoperatorHˆinH˜(n)hasanhermitianconjugateHˆ∗:ψ,Hϕˆ=Hˆ∗ψ,ϕ,∀ψ,ϕ∈H˜(n).(11.31)MoreoverHˆ→Hˆ∗isalinearcomplexinvolutioninEnd(H˜(n)),satisfyingforeverylinearoperatorH,ˆFˆinH˜(n)andλ∈Γnc,•(HˆF)ˆ∗=Fˆ∗Hˆ∗.•(λH)ˆ∗=Hˆ∗λ∗,(Hλ)ˆ∗=λ∗Hˆ∗.•a∗isthehermitianconjugateofakfor1≤k≤n.kItisconvenienttodefineafermionicFouriertransformwhichisakindofFourier-symplectictransform.Thiswillbeusedforfermionicquantization.11.4.3AFourierTransformDefinition28Foranyψ∈H˜(n)theFouriertransformisdefinedasFξ∗·α−α∗·ξ∗ψ(α)=eψ(ξ)dξdξ. 32411FermionicCoherentStatesThefollowingpropertiesareeasytoprove:∗∗1.ψ(ξ)=e(α·ξ−ξ·α)ψF(α)dα∗dα(inverseformula).TheFouriertransformisidempotent:(ψF)F=ψ.2.1F(α)=α∗·α,ξF(α)=α,ξ∗F(α)=−α∗.3.ψF(α)(ϕF(α))∗dα∗dα=ψ(ξ)(ϕ(ξ))∗dξ∗dξ(Parseval’srelation1).4.ψF(α)ϕF(−α)dα∗dα=ψ(ξ)ϕ(ξ)dξ∗dξ(Parseval’srelation2).5.(ψϕ)F(α)=ψF(α−β)ϕF(β)dβ∗dβ(Fourier-convolution).F(α)=ψF(α)eζ∗·α−α∗·ζ(translation-modulation),whereτ6.(τζψ)ζψ(ξ)=ψ(ξ−ζ).Letusremarkthatthesepropertiesarealsosatisfiedifψ,ϕarereplacedbylinearoperatorsinEnd(H˜(n))dependingonGrassmannvariables.11.5CoherentStatesforFermionsAsintheprevioussectionweconsidertwocomplexGrassmannalgebrasGc=nC[θ,θ∗]andΓnc=C[γ,γ∗].H(n)(andH˜(n))arespacesofholomorphicstatesinvariablesθ.11.5.1WeylTranslationsTheWeyltranslationsaredefinedasusual:T(γ)ˆ=ea∗·γ−γ∗·a,γ=(γ1,...,γn).(11.32)RemarkthatT(γ)ˆdependsonthe2nindependentGrassmannvariables(γ,γ∗).NeverthelesswesimplynoteT(γ)ˆandsometimesT(γ,γˆ∗)ifnecessary.TheyaretranslationsinthephasespaceandinparticularwehaveTˆ0,γ∗ψ(θ)=ψθ−γ∗,T(γ,ˆ0)ψ(θ)=eθ·γψ(θ).Recallthatγ·a=1≤k≤nγkakanda∗·γ=1≤k≤na∗γk(bewareoftheorder!).kUsinganti-commutationsrelations(CAR)wehavethecommutationrelations:α∗jaj=−ajαjforαj=γj,γj,a∗∗=γ∗jγj,γkakjγkδj,k,(11.33)a∗j,akγk=γkδj,k,[aj,akγk]=0. 11.5CoherentStatesforFermions325Wehaveasimilarrelationinvertingaanda∗andusing[A,B]∗=−[A∗,B∗].Fromtheserelationsweget∗∗1T(γ)ˆ=eakγk−γkak=1+a∗γk−γ∗ak+a∗ak−γ∗γk.kkk2k1≤k≤n1≤k≤n(11.34)Thefollowingpropertiesareeasilyobtainedwithlittlealgebraiccomputationssim-ilartothebosoniccase(seeChap.1).InparticularwealsohaveaBaker–Campbell–Hausdorffformula:Lemma75LetA,B,beΓnc-linearoperatorsinH˜(n)suchthat[A,B]commuteswithA,B.ThenAB−1[A,B]A+Beee2=e.ProofLetusremarkfirstthateAiswelldefinedbytheTaylorseriesbecausedim[H˜(n)]<+∞andeAisaΓnc-linearoperator.HencetheresultfollowsasinLemma1ofChap.1.NowwestatemainspropertiesofthetranslationoperatorsT(γ)ˆ.1.T(γ)ˆisΓnc-linearinH˜(n)(onrightandleft).2.T(αˆ+γ)=T(α)ˆT(γ)ˆexp(1(α∗·γ+α·γ∗)).23.(T(γ))ˆ−1=T(ˆ−γ)=T(γ)ˆ∗.InparticularT(γ)ˆisaunitaryoperatorinthesuper-Hilbertspace,whichmeansthatthesuper-innerproductispreserved:T(γ)ψ,ˆT(γ)ϕˆ=ψ,ϕ.4.Translationproperty:T(γ)ˆ∗aT(γ)ˆ=a+γ,(11.35)T(γ)ˆ∗a∗T(γ)ˆ=a∗+γ∗.(11.36)11.5.2FermionicCoherentStatesNowwecangiveadefinitionforFermioniccoherentstates.Definition29ForeveryGrassmanngeneratorγweassociateastateinthesuper-HilbertspaceH˜(n)denotedψγ=|γbytheformulaψγ=T(γ)ψˆ∅,(11.37)whereψ∅(θ)=e0,...,0(θ)isthevacuumstate,usuallydenoted|0. 32611FermionicCoherentStatesFirstpropertiesofcoherentstatesψγareeasilydeducedfrompropertiesofthetranslationT(γ)ˆ.1.ψγareeigenvectorsofannihilationoperatorsa:aψγ=γψγwherea=(a1,...,an)andγ=(γ1,...,γn).2.Theinnerproductoftwocoherentstatessatisfies1ψγ∗·α−γ∗·γ+α∗·α.(11.38)γ,ψα=exp23.1T(γ)ψˆα∗·γ−γ∗·αψα+γ.(11.39)α=exp2WecangiveamoreexplicitformulaforψγwhichisverysimilartothatobtainedintheBargmannrepresentationforbosons(Chap.1).Fromtheabovecomputationswehave∗−γ·γ/21+a∗γψψγ(θ)=ekk∅1≤k≤n−γ∗·γ/2∗∗=eak1γk1···akjγkjψ∅k10|detAt=0}andαtαAtBtαα∗=Ftα∗=B¯A¯α∗tttisthepseudo-classicalflowandΓt=BtT(ATt)−1.Moreovertheformula(11.147)extendsholomorphicallyfort∈CZwhereZ:={t∈C,detAt=0}isadiscretesubsetofC.Prooft→ψα,tisclearlyholomorphicinC.Soitisenoughtoprovetheformula(11.147)for00suchthatdet(Ft−1)=0.Thenfor00theeigenstatesforEarepair(ψB,ψF)whereψF=QψˆB.IfE=0thenHψˆ=0ifandonlyifQˆ1ψ=Qˆ2ψ=0thenitissaidthatsuper-symmetryispreserved. 12.2QuantumSupersymmetry355Butifthereexistsψ∈D(Q)ˆ∩D(Qˆ∗)suchthatQψˆ=0andQˆ∗ψ=0thenwesaythatsupersymmetryisbroken.ThenthevacuumenergyEsatisfiesE>0.LetusnowconsiderthewellknownWittenexampleofaquantummechanicalsystemwithsupersymmetry.Itisaspin1exchangemodelinonedegreeoffreedom.22(R,C2)withtheunitaryinvolutionσ10.ThesuperchargeisSoH=L3=0−1Qˆ=1d00+V(x)σ−,σ−=(12.2)idx10Qˆ∗1d01=−V(x)σ+,σ+=(12.3)idx00AneasycomputationgivesthefollowingHamiltonian:1d2σ3Hˆ=−+V(x)212−V(x)(12.4)2dx22AssumeforsimplicitythatV(x)isapolynomialofevendegree:V(x)=c0x2k+···cmx2k+mwithc0>0.ThenHˆisaself-adjointoperatorwithadiscretespectrum.ItiseasytosolveequationsQψˆ=0andQˆ∗ψ=0.WefindQψˆ=0⇐⇒ψ(x)=ψ(xV(x0)−V(x)(12.5)0)eQˆ∗ψ=0⇐⇒ψ(x)=ψ(x0)eV(x)−V(x0)(12.6)SoweseethatQψˆ=0hasnon-zeroL2solutionandQˆ∗ψ=0hasnonon-zeroL2solution,hencethesupersymmetryisbroken.ωx2AsupersymmetricharmonicoscillatorisobtainedwithV(x)=,ω>0.So2wegettheHamiltonian1d2ωHˆ−+ω2x212−σ3sos=22dx2ItiseasytogetthespectraldecompositionforHˆsosusingtheHermitebasis{ψk}k∈Nψk(x)0(seeChap.1).Denoteψk(1/2,x)=andψk(−1/2,x)=,with0ψk−1(x)ψ−1=0.Fork≥1,{ψk(1/2,·),ψk(−1/2,·)}isabasisofeigenvectorsfortheeigenvaluekofmultiplicitytwoand0isanondegenerateeigenvaluewitheigenvectorψ0(x).0Letusremarkthatbesidesthepositionxwehavehereanotherdegreeoffreedoms=1/2,−1/2whichisdiscreteandrepresentsthespinofafermion.Wecanrewritethesupersymmetricharmonicoscillatorwithcreationandanni-hilationoperatorsforbosonsandfermions:1d1aB=√ωBq+,aF=√σ+(12.7)2ωBdq2ωF∗1d∗1aB=√ωBq−,aF=√σ−(12.8)2ωBdq2ωF 35612SupercoherentStates—AnIntroductionWegetthesuper-harmonicoscillator,Hˆ=ω∗,aB]++ωF[a∗,aF](12.9)B[aBFwhere(a∗,aB)satisfies(CCR)orWeyl–Heisenbergalgebrarelationsand(a∗,aF)BFsatisfies(CAR)orCliffordalgebrarelations.WeshallseethatHˆissupersymmetricifωB=ωF.Itispossibletoconsidersystemswithnbosonsgeneratorsandmfermionsgen-eratorsaswell,writingaB=(aB,1,...,aB,n)andaF=(aF,1,...,aF,m)satisfying(CCR)relationsforaB,(CAR)relationsforaF.Thegoalofasupersymmetrytheoryistoputbosonsandfermionsonthesamefooting.Forexamplewewouldliketoconsidertheparametersforthespinasaclassicalvariableonthesamefootingasxfortheposition.Inotherwordsthequestionistofindakindofclassicalanalogueforthespin.Thequestionseemsstrangebecausethespinispurelyquantalanddisappearsintheusualsemi-classicallimit.NeverthelessBerezinhasinventednewspaceswherethisispossibleafterquan-tization;hecalledthemsuperspaceswithdifferentsuper-structures(linearspaces,algebras,groupsandmanifolds(see[20])).Weshallgivehereanintroductiontothiswidesubject,explainingonlyenoughdetailstounderstandsomesemi-classicalpropertiesofsupercoherentstates.Concerningamorecompletepresentationwere-fertothewealthofliterature[20,65,132,190].12.3ClassicalSuperspacesTheproblemconsideredhereistofindaclassicalanalogforthealgebradefinedin(12.1).Inotherwordswewanttoconstructclassicalspacesmixingbosonsandfermions.Wealreadyknowclassicalspacesforbosons:realorcomplexvec-torspacesormanifoldsandclassicalspacesforfermions:GrassmannalgebrasK[θ1,...,θn],K=RorC.Itisnotobvioustounderstandingeometricaltermswhatisaclassicalspaceforfermions.Weshallcomebacktothispointlater.FirstnoticeherethatitisnotthealgebraK[θ]butinsteadamysteriousspaceBsuchthatthespaceof“smoothfunctions”onBwithvaluesintheGrassmannalgebraKisK[θ].InparticularBisasetofdimension0.Thiswasmoreorlessimplicitintheprevioussections.12.3.1MorphismsandSpacesTheconfigurationspaceofaclassicalsystemwithnbosonsandmfermionswillbeasymbolicspacedenotedRn|msuchthatC∞Rn|m:=C∞Rn[θ∞n1,...,θn]:=CR⊗Gm 12.3ClassicalSuperspaces357whereGmistheGrassmannalgebrawithmgenerators.InotherwordsC∞(Rn|m)isaC∞(Rn)modulewithbasis{θ1,...,θn}.(x,θ)isinterpretedasacoordinatessystemofapointinthesymbolicspaceRn|m;x=(x1,...,xn)aretheevencoordinatesandθ=(θ1,...,θm)aretheoddcoordinates.AsforGrassmannalgebras,C∞(Rn|m)isasuper-module,withaparityoperatorP,C∞(Rn)-linear.SoC∞Rn|m=C∞Rn|m⊕C∞Rn|m+−where[•]+istheevenpartand[•]−istheoddpart.ForapplicationsweneedtounderstandtransformationmapbetweendifferentsuperspacesRn|m,Rr|s.AfirstapproachtodefineamapΦ:Rn|m→Rr|sistousecoordinatessystem:Φ(y,η)=(x,θ)wherex,yareevencoordinates,θ,ηareoddcoordinates.ButthesymbolicspacesaredefinedimplicitlybytheC∞functionsdefinedonthem.SoweinterpretΦasachangeofvariablesinfunctionsf∈C∞(Rr|s),Φ(f)(x,θ)=fΦ(x,θ)(12.10)wheref→Φ(f)isalinearmap.Itisimportanttounderlineherethatthemeaningoftherighthandsidein(12.10)isgivenbythelefthandsidebecausesuperspacesareonlysymbolicandaredefinedbytheiralgebraoffunctionsandnotbyawellidentifiedgeometricalobjectintheusualsense.TodefinesupermanifoldsitisnecessarytodefinelocalsuperspacesinanopensetUofRn.ThenUn|misthesuperspacedefinedbyitsalgebraofC∞functions:C∞(Un|m):=C∞(U)⊗Gm.SowehaveanaturaldefinitionformorphismsfromUn|minVs|rwhereVisanopensetofRs.Letusgivenowsomemoreformaldefinitionsfollowing[20,132,190].12.3.2SuperalgebraNotionsDefinition34(Superlinearspaces)AsuperlinearspaceVisavectorspaceonK=RorCwithadecomposition(Z2-grading)V=V0⊕V1.V0isthesetofevenvector,V1isthesetofoddvectors.ThenonVthereexistsaparity(linear)operatorP,definedbyPv=vforv∈V0,Pv=−vforv∈V1.ElementsinV0∪V1arecalledhomogeneous.AlinearmapFfromthesuperlinearspaceVintothesuperlinearspaceWissaidsuperlinear,oreven,ifitpreservesparities,soFsendsVjinWj,j=0,1.AlinearmapfromVintoWissaidoddifF(Vj)⊆Wj+1(j∈Z2).Theparitynumberπ(v)isdefinedforhomogeneouselements:π(v)=0ifv∈V0;π(v)=1ifv∈V1.LetusremarkthateverylinearmapfromVintoWisthesumofanevenmapandanoddmap.Thefollowingnotationswillbeused: 35812SupercoherentStates—AnIntroductionSHom(V,W)isthelinearspaceofevenmapsVintoWandHom(V,W)isthespaceofalllinearmapsfromVintoW.ThenHom(V,W)isagainasuperlinearspaceanditevenpartisthespaceofevenmapsfromVintoW.Hom(V,W)=SHom(V,W)0Definition35(Superalgebras)AsuperalgebraisalinearspaceAwithanassocia-tivemultiplicationwithunit1(fg)→f·g,suchthatforeveryf,g∈A+∪A−wehavetheparitycondition:π(f·g)=π(f)+π(g)ThesuperalgebraAissaid(super)commutativeiff·g=(−1)π(f)π(g)g·fExamples:GrassmannalgebrasK[θ1,...,θn]aresuper-commutativealgebras.Definition36(Superspace)IfBisasuperspace(forexampledefinedbyaGrass-mannalgebra)thesetofB-pointsofthesuperspaceRn|misthesetRn|m(B)definedbythefollowingequality:Rn|m(B)=HomC∞Rn|m,C∞(B)TheinterpretationisthatRn|m(B)=Hom(B,Rn|m)soRn|m(B)canbeseenasthe“pointsofRn|m”parametrizedbyB.Anotherimportanttrickforacorrectinter-pretationofsuperspaceconcernstheproductofsuperspaces.WeshouldliketohaveforexampleRn|m=Rn|0×R0|m(ofcourseRn|0istheusualspaceRnandR0|misthespacewhoseC∞-functionsareR[θ1,...,θm]).Letusconsider3classicalsuperspacesX,Y1,Y2.Wecanunderstandidentifi-cationbetweenHom(X,Y1×Y2)andHom(X,Y1)×Hom(X,Y2)throughiden-tificationbetweenHom(C∞(Y1×Y2),C∞(X))andHom(C∞(Y1),C∞(X))×Hom(C∞(Y2),C∞(X)).Thisisdoneasfollows.IfΦ∗isamorphismfromC∞(Yk)intoC∞(X)thenwegetauniquemorphismkΠ∗fromC∞(Y1×Y2)intoC∞(X)satisfyingΦ∗(f∗∗∞1⊗f2)=Φ1(f1)Φ2(f2),forfk∈C(Yk)(12.11)ConverselyifΦ∗isgivenwegetΦ∗(f1)=Φ∗(f1⊗1)andΦ∗(f2)=Φ∗(1⊗f2).12Sotheidentificationisdeterminedby(12.11).12.3.3ExamplesofMorphismsWecomputeheremorphismbetweensomesuperspaces.(1)ΦisamorphismfromRninRr.ItisknownthatwehaveΦ(f)(x)=f(Φ(x))withΦsmoothfunctionRn→Rr. 12.4Super-LieAlgebrasandGroups359(2)ΦisamorphismfromR0|1intoR.UsingparitywehaveΦ(f)=b0(f)whereb0(f)isreal,linearandmultiplicativeinf.Sothereexistsx0∈RsuchthatΦ(f)=f(x0),whichmeansthatΦisconstant,asitshould.(3)ΦisamorphismfromR1|1intoR.Usingparity,wehaveΦ(f)=c0(f)wherec0isamorphismfromC∞(R)inC∞(R).HencethereexistsasmoothfunctionΦ:R→RsuchthatΦ(f)=f◦Φ.(4)ΦisamorphismfromR1|1intoR0|1.Usingparity,wehaveΦ(1)=1andΦ(θ)=c1θ.(1istheconstantfunction1).SowehaveΦ(b0+b1θ)=b0+b1c1θ.From(3)and(4)weclearlyseethat(x,θ)arecoordinatesforR1|1.(5)ΦisamorphismfromR1|2intoR1.WehaveΦ(f)=b0(f)+b1(f)θ1θ2Thesuperalgebraevenmorphismconditionsgiveb∞0(fg)=b0(f)b0(g),f,g∈C(R),b0(1)=1(12.12)b1(fg)=b0(f)b1(g)+b1(f)b0(g)(12.13)Fromthefirstcondition,thereexistsϕ:R→Rsuchthatb0(f)=f◦ϕ.Thenthesecondconditionmeansthatb1isaderivationatϕ(x).Sothereexistsχ(x,t),smoothinaneighborhoodofthegraphofϕsuchthatdfb1(f)(x)=χ(x,t)(12.14)dtt=ϕ(x)12.4Super-LieAlgebrasandGroupsStandardsymmetries(attheclassicalorquantumlevelaswell)aredescribedthroughthegroupaction.Supersymmetrieswillbedescribedthroughtheactionofsuper-groups.Moreoverthesesuper-groupsarederivedbyexponentiatingsuper-LiealgebrasasintheusualcaseofLiegroups.12.4.1Super-LieAlgebrasDefinition37Asuper-LiealgebrasisasuperlinearspaceonK=R,Cwithabi-linearbracket(A,B)→[A,B]satisfyingthefollowingidentitiesforhomogeneouselementsA,B,C: 36012SupercoherentStates—AnIntroduction[A,B]=(−1)1+π(A)π(B)[B,A](12.15)π[A,B]=π(A)+π(B)(12.16)(−1)π(A)π(C)A,[B,C]+(−1)π(B)π(A)B,[C,A]+(−1)π(C)π(B)C,[A,B]=0(12.17)Remark69Thefirstequalitymeansthat[A,B]isanticommutativeexceptedifAandBareodd.Inthiscasethebracketiscommutative.Thesecondidentitymeansthatthebracketoftwoevenortwooddelementsisevenandthebracketofanevenandoddelementisodd.Thethirdidentityisthesuper-Jacobiidentity.AsinthestandardcaseofLiealgebras,manyexamplesofasuper-Liealgebraarerealizedasmatrixsuperalgebras.LetusconsiderasupervectorspaceonthefieldK=R,C,V=V0⊕V1wheretheevenspaceV0hasdimensionnandtheoddspaceV1hasdimensionm.AnysuperlinearoperatorAinVhasamatrixrepresentationA++A+−A=A−+A−−DenoteEnd(V)thespaceofallsuperlinearoperatorsinV.WehaveseenthatEnd(V)isasupervectorspacewithparityπ.Itisasuper-Liealgebraforthesuper-bracket:[A,B]=AB−(−1)π(A)π(B)BAChoosingbasisinV0andV1thesuperspaceVisisomorphictoKn⊕Km.ThisspaceisdenotedKn,m(notthesamemeaningasKn|m)andthesuper-Liealgebraisdenotedgl(n|m).Wedenotesl(n,m)thesub-super-Liealgebraofgl(n|m)definedbythesuper-tracelesscondition:StrA=0.su(n,m)isthesub-super-LiealgebraofmatrixAsuchthatA∗∗++=−A++,A−+=−iA+−Form=0werecovertheclassicalLiealgebrasgl(n),sl(n),su(n).Definition38LetAbeasuperalgebra(notnecessarilyassociative).Alinearopera-torDinAiscalledasuperderivationifitsatisfies,foreveryhomogeneouselementsf,ginEnd(A),D(f·g)=D(f)·g+(−1)π(D)π(f)f·D(g)(12.18)ThebasicexampleisD=adAwhereAisasuper-LiealgebraandadA(B)=[A,B](likeinstandardLiealgebras). 12.4Super-LieAlgebrasandGroups361InthesuperalgebraC∞(Rn|m),X∂∂j=andΘk=aresuperderivations(the∂x∂θjkfirstiseven,thesecondisodd).Andifh∈C∞(Rn|m),hXjandhΘkareagainsuperderivations.12.4.2Supermanifolds,aVeryBriefPresentationTodefinesupermanifoldsitisnecessarytodefinelocalsuperspacesinanopensetUofRn.ThenwedenoteUn|mthesuperspacedefinedbyitsC∞functions:C∞(Un|m):=C∞(U)⊗Gm.SowehaveanaturaldefinitionformorphismsfromUn|minVs|rwhereVisanopensetofRs.Un|misasuperspaceofdimensionn|m.Asforastandardmanifold,asupermanifoldMofdimensionn|misatopologicalspaceM0(calledtheunderlyingspace)suchthatforeachpointm∈M0wehaveanopenneighborhoodUandasuperalgebraRUisomorphictoC∞(Vn|m),whereVisanopensetofRn.Moreoveragluingconditionhastobesatisfied(see[190],p.135).InparticularM0isastandardmanifoldofdimensionn.Wehavediscussedherethesuper-analogueofrealC∞manifolds.Itispossibletodefinerealanalyticorholomorphicsupermanifolds.ForthatwehavetoreplaceC∞(U)bythespaceCω(U)ofanalyticfunctionsinU.InthecomplexcaseUisanopensetofCn(recallthatcomplexanalytic=holomorphic).Nowadaysthereexistessentiallytwokindsof(almost)equivalentdefinitionsofsupermanifolds.Onedefinessupermanifoldbyitsmorphisms[22,132,190],itiscalledthealgebro-geometricalapproach;theotherisclosertothestandarddefinitionwhereamanifoldisasetofpointsbutthefieldofrealnumbersisreplacedbyanon-commutativeandinfinitedimensionalBanachalgebra[62,169].Hereweshallusetheterminologyofthealgebro-geometricalapproach.See[169]foradiscussionaboutcomparisonofthesetwodefinitions.Weonlygivehereaverybriefandsketchyintroductiontothisrichsubject.Ourgoalisonlytohaveabetterintuitionofwhatisgoingonsomefewexamples,inparticularconcerningtheFermi–Bose(orsuper)oscillator.Definition39Roughlyspeaking,aC∞-supermanifoldofdimensionn|misasu-perspaceMdefinedbyanunderlyingmanifoldofdimensionnsuchthatMislocallyisomorphtoUn|m,whereUisachart(opensetinRn)oftheunderlyingmanifoldM0.TherearegluingcompatibilityconditionsbetweenthechartsasformanifoldswhereC∞(Un|m)replacesC∞(U).IfUisanopensetofRn,thesupermanifoldUn|misdefinedbytheunderlyingspaceUandwithC∞(Un|m)=C∞(U)[θ1,...,θm],θ1,...,θmaregeneratorsofarealGrassmannalgebra.Definition40AvectorfieldinthesuperdomainUn|misaderivationinthesuper-algebraC∞(Un|m). 36212SupercoherentStates—AnIntroductionAsinthestandardcasethesetV(Un|m)ofvectorfieldsinUn|misasuper-Liealgebra.NotethatitisamoduleoverC∞(Un|m)withthefollowingbasisincoordi-nates(x,θ):∂∂∂∂···;,...,∂x1∂xn∂θ1∂θmInasupermanifoldMthetangentsuperspaceat“P∈M”ofcoordinates(x,θ)isdefinedlocallyinachartUn|masthespaceV(Un|m)restrictedat(x,θ)(see[190]foramoreprecisedefinition).12.4.3Super-LieGroupsInshort,asuper-Liegroupisasupermanifoldandasuper-groupwherethegroupoperationsaremorphims.Itisnottheplaceheretodefinerigorouslyalltheseterms;wereferto[62,132,169,190]foradetailedstudyofsupermanifoldsandsuper-groups.AsuperLie-groupisdefinedasasupermanifoldwherethegroupstructureisdefinedbymorphismsμ:G×G→G,ι=G→G,ν0|0→G0:RThesemorphismsdefine,respectively,themultiplication,theinverseelementandtheunitelement.Ofcourseweneedconditionstotranslatethegrouppropertiesonmorphisms(see[190]fordetails).WehavedefinedbeforethesuperspaceRn|m.ItisacommutativeLiegroupforthe“natural”additionwiththefollowingmorphisms:(x,θ)+(y,ζ)=(x+y,θ+ζ)(12.19)Additionrulehastobedefinedintermsofmorphismsasfollows.RecallthatthespaceRn|misdefinedbyitsmorphismsandforeverysuperspaceB,Rn|m(B)isthesetofBpointsofRn|m,Rn|m(B)=Hom(C∞(Rn|m),C∞(B)).AdditioninRn|m(B)isdefinedasfollows.Itisfirstnoticedthatamorphismψ∈Rn|m(B)isdeterminedbyitsimagesonthefollowinggeneratorsofRn|m:fj(x,θ)=xj,gk(x,θ)=θk,1≤j≤n,1≤k≤mψ(fj)andψ(gk)determineψ.Hencetheadditivegroup{Rn|m,+}isdefinedbytherule,ψ,ψ∈Rn|m(B),12+ψ(f)=ψ(f)+ψ(f),forf=fψ1212j,gk,foranyψ,ψ∈Rn|m(B)andanysuperspaceB.12 12.4Super-LieAlgebrasandGroups363Forsimplicity,assumen=m=1.Practicallyiff∈C∞(R1|1)wehavef(x,θ)=f0(x)+f1(x)θandthemeaningof(12.19)isμ(f)(t,s,θ,ζ)=f0(t+s)+f1(t+s)θ+f1(t+s)ζItiseasytoexplicitlywriteιandν.0Asinthestandardcase,itisveryimportanttohaveconnectionsbetweensuper-Liealgebrasandsuper-groups.Withoutgoingintodetailsofthetheoryofsuper-Liegroups,(see[22,190]),letusrecallhereadefinition(uptosuper-isomorphism)ofthesuper-Liealgebraofasuper-groupSGofdimensionn|m.Definition41Thesuper-Liealgebrasgofthesuper-groupSGisthesuperlinearspaceofleftinvariantderivationsinthealgebraC∞(Un|m)whereUisaneighbor-hood1oftheunderlyingLiegroupG0.TheeasiestexampleisthesuperlineargroupGL(n,m)correspondingtothesuper-Liealgebragl(n,m).Thedetailscanbefoundin[22].ElementsgofGL(n,m)aresuperlinearisomorphismsinRn|m.TheyareparametrizedbymatricesA++A+−A(g)=A−+A−−wherethematricesA++,A−−haveevenelementsandA+−,A−+haveoddele-ments.TherealcomponentofA(g)isA++0A0(g)=0A−−A0(g)isinvertibleifandonlyifdet(A++)det(A−−)=0.DenoteG0=GL(n)×GL(m).G0isastandardmanifoldofdimensionn2+m2.Soweseethatthesuper-manifoldGL(n,m)isdefinedbythesheafofsmoothfunction∞GL(n,m):=C∞(G2nmC0)⊗RItcanbeprovedthatitisasuperLiegroup[22].Remark70OneveryusefultooltocomputeinsuperanalysisiswhatBerezincalled“theGrassmannanalyticcontinuationprinciple”[22].Itwassometimesappliedim-plicitlyaboveandweshalloftenapplyitlaterwithoutmorejustifications.Thereex-istseveralwaystoexplainthisprincipleinamoremathematicalrigorousapproach(see[65,105]).Thepracticalruleisthefollowing.Letfbearealfunctionofnrealvari-ablesx1,...,xnandnnilpotentGrassmannnumbersξ1,...,ξn.Thatmeansthatjεξj∈Gnandξj=|ε|>0aεθwhereθ1,...,θnaregeneratorsofGn.Denotex= 36412SupercoherentStates—AnIntroduction(x1,...,xn)andξ=(ξ1,...,ξn)(usingtheusualnotationinseveralvariablefunc-tions),theGrassmannanalyticcontinuationprinciplesaysξα∂αf(x+ξ)=f(x)(12.20)α!∂xααRecallthatα=(αnαα1αnα1,...αn)∈N,ξ=ξ1···ξn,hencewehaveξ=0for|α|≥2nsotheTaylorexpansion(12.20)isfinite.Weconsidernowmoreinterestingexamples,whereevenandoddcoordinatesaremixed,whichgivessupersymmetriesafterquantization.LetusbeginbythetoymodelR1|1withthelawsuper-group:(t,θ)·(s,ζ)=(t+s−iθζ,θ+ζ)(12.21)LetusdenoteSG(1|1)thissuper-group(itiseasytoprovethatitisasuper-groupwithanon-commutativeproduct)andsg(1|1)itsLiealgebra.Iff∈C∞(R1|1),f(t,θ)=f0(t)+θf1(t),and(s,η)thecoordinatesofasuper-vectorvinR1|1,thelefttranslationoffbyvisdefinedbyτf(t,θ)=f(t−s+iηθ,θ−η)(s,η)=f0(t−s)+iηθf0(t−s)+(θ−η)f1(t−s)(12.22)HerewehaveusedtheGrassmannanalyticextensionprinciple.Butwehavethegroupproperty:τ(s,η)=τ(s,0)◦τ(0,η)andτf(t,θ)=f(t−s,θ),(usualtranslationbyarealnumbers)(12.23)(s,0)∂∂τ(0,ζ)f(t,θ)=f(t,θ)+ζiθ−f(t,θ)(12.24)∂t∂θSowecanconcludethatthesuper-Liealgebrasg(1|1),hasthetwogenerators{∂∂∂t,Dθ},where∂t=iseven,∂θ=,andDθ=iθ∂t−∂θareodd.Wehave∂t∂θthecommutationrule[Dθ,Dθ]+=−2i∂tByanalogouscomputationswegetabasisforrightinvariantvectorfields.Onlytheoddpartsismodified.WegetQθ=iθ∂t+∂θOnlythelastcommutationruleischanged:[Qθ,Qθ]+=2i∂t(12.25)Moreoverwehavethefollowingcommutation:[Q,D]=0(12.26) 12.4Super-LieAlgebrasandGroups365InmechanicsthegeneratorofthetimetranslationsistheHamiltonianH.Soequa-tion(12.25)isaclassicalanalogueforasupersymmetricsystem.ConsidernowamorephysicalexampleleadingtoaclassicalanaloguefortheWittenmodel[200].Thiskindofmodelwasconsideredbeforebyseveralphysicistsinthe1970s(Wess,Zumino,Salam).OnR1|2definethefollowingmultiplication(non-commutative)rule:(t,θ,θ∗)(s,ζ,ζ∗)=t+s−i(θζ∗−ζθ∗),θ+ζ,θ∗+ζ∗(12.27)Forf∈C∞(R1|2)wehavef(t,θ,θ∗)=f0(t)+f1(t)θ+f2(t)θ∗+f3(t)θθ∗.Themultiplicationmorphismμ∗isdeterminedbyμ∗(fj)for1≤j≤3(recallthatμ∗isasuperalgebramorphism).Foreveryf∈C∞(R)themeaningof(12.27)isμ(f)(t,s,θ,θ∗,ζ,ζ∗)=f(t+s−i(θζ∗−ζθ∗),θ+ζ,θ∗+ζ∗)=f(t+s)−i(θζ∗−ζθ∗)f(t+s)−θθ∗ζζ∗f(t+s)ιandνareeasilycomputedsuchthat{R1|2,μ}isasuperLiegroupthatweshall0denoteSG(1|2).AswehavedoneforSG(1|1)wecancomputegeneratorsforthesuper-Liealge-brasg(1|2)byidentifyingleftinvariantvectorfields.Denoteτs,η,η∗thelefttranslationby(s,η,η∗),τηthelefttranslationby(0,η,0)andτη∗thetranslationby(0,0,η∗).Asabovewegeteasilyforeveryf∈C∞(R1|2),τ∗ηf(t,θ)=f(t,η)+η(iθ∂t−∂θ)f(t,θ)(12.28)τη∗f(t,θ)=f(t,η)+η(iθ∂t−∂θ∗)f(t,θ)(12.29)Sowehavegotthefollowingbasisforsg(1|2):∗∂∂t,Dθ=iθt−∂θ,Dθ∗=iθ∂t−∂θ∗withoneevenandtwooddgenerators.Thecommutationrulesofthisalgebraare2=D2Dθθ∗=0(12.30)[Dθ,∂t]=[Dθ∗,∂t]=0(12.31)[Dθ,Dθ∗]+=−2i∂t(12.32)Byanalogouscomputationswegetabasisforrightinvariantvectorfields.Onlytheoddpartsismodified.Weget∗∂Qθ=iθt+∂θ,Qθ∗=iθ∂t−∂θ∗Onlythelastcommutationruleischanged:[Qθ,Qθ∗]+=2i∂t(12.33) 36612SupercoherentStates—AnIntroductionMoreoverwehavethecommutation[Q,D]=0(12.34)So(12.33)isaclassicalanalogueforthequantumsupersymmetricsystemofWitten.Thiswillbecomemoreexplicitlater.12.5ClassicalSupersymmetry12.5.1AShortOverviewofClassicalMechanicsTherearemanybooksconcerningthisverywellknownsubject.ForourpurposewerecallheresomebasicfactsconcerningLagrangiansandHamiltonians.Wereferformoredetailstothefollowingbooks[7,92,182].InphysicsclassicaldynamicalsystemsareusuallyintroducedwiththeirLa-grangianLandtheiractionintegralS=dxL(x)ifxisacoordinatesystemforclassicalpaths.ForapointparticlemovinginRnwithcoordinatesq=(q1,...,qn)t1wehaveS=dtL(q,q)˙,q˙isthetimederivativeofq.Theequationsofmotiont0(Euler–Lagrangeequations)arededucedfromtheleastactionprinciple,∂Ld∂L−=0(12.35)∂qdt∂q˙Euler–Lagrangeequationmaybeverydifficulttosolve.Muchinformationofitssolutionscanbeobtainedstudyingitssymmetries.AccordingtheNoetherfamoustheorem,symmetriesgiveintegralofmotionsI,functionsofq,q˙conservedalongthemotiondI(q,q)˙=0.InparticulartheHamiltonianenergyfunctionHiscon-dtserved:∂LH(q,q)˙=·˙q−L(q,q)˙(12.36)∂q˙Thecanonicalconjugatemomentumpisdefinedas∂Lp=∂q˙pisacotangentvectorin(Rn)∗.WeconsiderhereforsimplicitysystemswithoutconstraintsandthattheconfigurationspaceistheEuclideanspaceRn(see[7]formanifolds).LetusrecallnowastatementfortheNoethertheoremconcerningsymmetriesandintegralsofmotion.Theorem51LetV(q)=v∂s1≤j≤nj(q)∂qjbeavectorfieldandφVitsflow:ds(qφVs)=v(qs),q0=q,wherev(q)=v1(q),...,vn(q)(12.37)ds 12.5ClassicalSupersymmetry367(NI)AssumethatLisinvariantunderφsforscloseto0.ThenV∂LIV(q,q)˙=v(q)·∂q˙isanintegralofmotion:dIV=0.dt(NII)AssumethatthereexistsasmoothfunctionKonRn×Rnsuchthat∂L∂Ldv(q)·+˙v(q)=K(q,q)˙(12.38)∂q˙∂qdtthenI=v(q)·∂L−K(q,q)˙isanintegralofmotion.∂q˙Remark71ItisconvenienttostateNoether’stheoremwithfinitesmalldeforma-tionsofsizeε.Denoteδεq=εv(q),δεL=L(q+δεq,q˙+δεq)˙−L(q,q)˙.Theinvarianceassumptions(12.38)meansthatdδK(q,q)˙+Oε2(12.39)εL=εdtTheenergyHamiltonianHwasdefinedasafunctionof(q,q)˙.Itisconvenienttoreplacethevelocityq˙bythemomentumpusingtheLegendretransforminq˙→p.H(q,p)=p·˙q(q,p)−Lq,q(q,p)˙(12.40)∂2LIfthematrixisnon-degenerate,Hisasmoothfunctionof(q,p)only(atleast∂q∂˙q˙locally).InordertoquantizetheclassicalsystemwithLagrangianLitisassumedthatHisdefinedglobally.WeshallseethatforfermionsitisnottrueforinterestingexampleswheretheLagrangianisdegenerate:q˙→pisnotonto.ButinordertoquantizecanonicallyaclassicalsystemwehavetocomputeanHamiltonianandaPoissonbracket.Dirachasproposedamethod[69]todothatinthedegeneratecase.LetusdescriberoughlytheDiracmethodwhichwillapplytofermions.Wefol-lowhere[112]wherethereadercanfindmanydetails.Whenq˙→pisnotontotherangeofthemapping(q,q)˙→(q,p)(p=∂L)is∂q˙anon-geometricallytrivialpartofthephasespace.Inparticularitisnotpossibletorecoverq˙fromp.InthissituationwesaythattheLagrangianissingularordegenerate.Following[69],itwillbeassumedthatthisisasmoothmanifoldCdescribedbyequationsχj(q,p)=0,1≤j≤mwherethedifferentialdχjareeverywhereindependent.TheenergyHamiltonianHisalwaysafunctionofqandp(asintheregularcase)butisdefinedhereonlyundertheconstraint(q,p)∈C.DiracassumedthatHhasanextensiontothewholephasespaceRn×Rn(Rnisidentifiedwith(Rn)∗) 36812SupercoherentStates—AnIntroductionandhecomputedamodifiedPoissonbracket{•,•}DisuchthatforanyobservableF∈C∞(Rn×Rn)theequationofmotionisF˙={F,H}Di,inparticularq˙={q,H}Di,p˙={p,H}Di(12.41)OnC,usingdefinitionofpwehave∂L∂H∂HdH=˙qdp−dq=dq+dp(12.42)∂q∂q∂pThenthereexistsuj(q,q)˙,1≤j≤m,suchthat∂H∂χjq˙=+uj(12.43)∂p∂p1≤j≤m∂H∂χjp˙=−−uj(12.44)∂q∂q1≤j≤mSowehaveforeveryobservableFF˙={F,H}+uj{F,χj}(12.45)1≤j≤mTheconstraintsχjhavetobepreservedduringthetimeevolution,whichgivestheconditions{χk,H}+uj{χk,χj}=0,for1≤k≤m(12.46)1≤j≤mForsimplicityassumenowm=2and{χ1,χ2}=λ=0.Fromconditions(12.46)wecancomputeu1,u2:{χ2,H}{χ1,H}u1=,u2=−.λλThentheDiracbrackethasthefollowingexpression:1{F,G}Di={F,G}+{F,χ1}{χ2,G}−{F,χ2}{χ1,G}(12.47)λItiseasytoseethat(F,G)→{F,G}DiisaLiebracketonthelinearspaceC∞(Rn×Rn).LetusconsiderthefollowingexampleintheconfigurationspaceR2:LM=xy˙−yx˙−V(x,y),Visasmoothpotential.ThisLagrangianisdegenerate.Wehave∂LM∂LMpx:==−y,py:==x(12.48)∂x˙∂y˙ 12.5ClassicalSupersymmetry369TheenergyHamiltonianisH=V(x,y).Wehavethetwoconstraintsχ1=px+y,χ2=py−xand{χ1,χ2}=2.ThentheDiracbracketis11∂F∂G∂F∂G1∂F∂G∂F∂G{F,G}Di={F,G}−−−−22∂x∂y∂y∂x2∂px∂py∂py∂px(12.49)WecancheckthattheHamiltonianHandtheDiracbracket(12.49)givetheEuler–Lagrangeequationforthemotion.Wefind1∂V1∂Vx˙={x,V}Di=−;˙y={y,V}Di=(12.50)2∂y2∂xToprepareacanonicalquantizationoftheLagrangianLMwewritethecommuta-tionrelations1{x,y}Di={px,py}Di=−(12.51)21{x,px}Di={y,py}Di=(12.52)2{y,px}Di={x,py}Di=0(12.53)ADiracquantizationF→Fˆsatisfies{F,G}Di→−iF,ˆGˆSowegetarepresentationoftheLiealgebra(12.51)inL2(R2)satisfying1∂1∂xˆ=x+,yˆ=y−2i∂y2i∂x(12.54)1∂1∂pˆx=−y−,pˆy=x+2i∂x2i∂yNoticethatthisrepresentationinL2(R2)isnotirreduciblebecausewehavetheconstraintspˆx+ˆy=0,pˆy−ˆx=0andtherelations(12.54)defineanHeisenbergLiealgebraofdimension3.SowecangetaquantizationequivalenttotheWeylquantizationinL2(R).12.5.2SupersymmetricMechanicsSupermechanicsisanextensionofclassicalmechanics.Insupermechanicsapointhasreal(oreven)coordinatesx=(x1,...,xn)representingthebosonicdegreesoffreedomandGrassmanncoordinates,ξ=(ξ1,...,ξM),representingthefermionicdegreesoffreedom.Toencodethetwokindsofdegreeoffreedominthesameobjectoneintroducesa“realsupervariables”Xlivingintheconfigurationspace 37012SupercoherentStates—AnIntroductionofallthesystem.AsuperLagrangianisafunctionLSofXanditsderivativeDX,withvaluesinaGrassmannalgebra.IncoordinateswecanwriteX=(x,ξ)wherexhasrealdependentcomponents,ξhavefermionic(real)components.Incoordinateswegetapseudo-classicalLagrangianL(x,x,ξ,˙ξ)˙.Theconjugatemomentaare∂L∂Lp=,π=(12.55)∂x˙∂ξ˙AssumeforamomentthatwecandefinetheHamiltonianHastheLegendretrans-formofLin(x,˙ξ)˙.LetFbeanobservableonthephasespacedefinedbyitscoor-dinates(x,ξ,p,π).AlongthemotionwewanttohaveasusualF˙={F,H}whereFisanextensionoftheusualPoissonbracket.Adirectcomputation,usingequationsofmotion,gives∂H∂F∂H∂F∂H∂F∂H∂FF˙=−−+(12.56)∂p∂x∂x∂p∂π∂ξ∂ξ∂πRecallthatHandFareinthesuperalgebraC∞(Rn+n|m+m).Therightsidein(12.56)isthesuperPoissonbracket{F,H},thefirsttermistheusualantisym-metricPoissonbracketinbosonicvariables,insidethesecondparentheseswehaveasymmetricformforthecontributionoffermionicvariables.ForanyhomogeneousobservablesF,GthePoissonbracketisasuperLieprod-uctinC∞(R2n|2m)satisfying∂F∂G∂F∂Gπ(F)∂F∂G∂F∂G{F,G}=−+(−1)+(12.57)∂x∂p∂p∂x∂ξ∂π∂π∂ξFirstconsiderasimplesystemwithonebosonicstateandonefermionicstatewith-outinteraction.Itsstatesarerepresentedbythesuperfield:X=x+iθξ(12.58)xisarealnumberdependingontimet,ξisanoddnumberlivinginaGrassmannalgebra.WewantthatXisevenand“real”.WeintroducetwoGrassmanncomplexconjugategenerators{η,η∗}andchooseξ(t)=¯c(t)η+c(t)η∗wherec(t)isacom-plexnumber.Thenwehave(iθξ)∗=iθξsoXisevenandreal.RemarkthatXisnotaC∞functiononR1|1becausethecoefficientξisnotacomplexnumberbutaGrassmannvariable.AswehaveseenthesuperfieldXhastobeunderstoodasamorphismfromC∞(R)inC∞(R1|1);usingtheGrassmannanalyticextensionprinciplewehaveX∗(f)(x,θ)=f(x)+if(x)θξ 12.5ClassicalSupersymmetry371LetusintroducethesuperLagrangian1LS=DX·D(DX)2whereD=Dθ.Incoordinateswegetthepseudo-classicalLagrangian1L2pcl=dθLS=x˙+iξξ˙2WecaneasilysolvetheEuler–Lagrangeequations.Wegetx¨=0,ξ˙=0WecanconsiderthesamemodelwiththreesuperfieldsXj=xj+iθξj,1≤j≤3,1Lpcl=x˙·˙x−iξ˙·ξ2Herewehavethreeconstraintsiχj=πj+ξj2Butwehave{χj,χk}=−iδj,ksotheconstraintsaresecondorder.ThissystemisinvariantbyanyrotationinR3.ItsDiraccanonicalquantizationgivesaspinsystem[182].Thesuper-groupSG(1|1)transformsXbyrighttranslations,sowehaveX(t−iθη,θ+η)=X(t,θ)+ηQX(t,θ)whereQ=Qθ.SoavariationofXisgivenbyδηX=ηQX,whereδηisaderiva-tioninC∞(R)[θ,η,η∗]consideredasaR(η,η∗]module(coefficientsofXareinR[η,η∗]),δηF=[ηQ,F].IncoordinateswehaveδηX=δηx+iθδηξwhereδηx=iηξ,δηξ=ηx˙SowecomputethevariationoftheLagrangianandthevariationofthecorrespond-ingactionS=dtdθL.Usingthatδηisaderivation,wegetδηLS=ηQLS=η(iθ∂t+∂θ)L(12.59)Henceweget∂ηS=dt∂tdθiθLS(12.60)ByextensionoftheNoethertheoremtothefermioniccaseweseethatthevec-torfieldQisthegeneratorofasymmetryfortheLagrangianLS.Thissymmetry 37212SupercoherentStates—AnIntroductioniscalledsupersymmetrybecauseitconcernssupervariablesmixingusualnumbers(realorcomplex)andGrassmannnumbers.ForthecorrespondingvariationsofthepseudoclassicalLagrangianLpclwehave3iη∂δηLpcl=ξx˙2∂tAccordingtoNoether’sresultswehavetheconservedcharge∂L∂L3i∂IQ=iξ+˙x−ξx˙=−2iξx˙∂x˙∂ξ˙2∂tWecancomputetheHamiltonianHbyLegendretransform.ThephasespacehereisthesuperlinearspaceR2|1.Wegetfortheconjugatemomentum∂Liπ==−ξ∂ξ˙2ip2SothisLagrangianisdegenerate,withtheconstraintχ=π+ξ.WegetH=22wherepistheconjugatemomentumtox.Nowweshallconsideramoreinterestingexampleinvolvingthesuper-harmonicoscillator.Weconsiderthesuperfields:X=x+θψ∗+ψθ∗+yθθ∗(12.61)Thisfieldmaydependontimet,sox,y,ψ,ψaretimedependent,wherex,yarerealnumbers,θ,θareconjugateGrassmannvariables,ψ,ψ∗arenecessarilyoddnumbers.Asabove,tocomputethesenumberswehavetointroduceanotherpairofGrassmannvariables{ζ,ζ∗},anticommutingwiththepair{θ,θ∗}.Soweconsiderthatψ=cζ+dζ∗wherec,darecomplex(timedependent)numbers.Aswehavealreadyremarked,toavoidmathematicaldifficultiesitisnecessarytoaddextraGrassmannvariables.RecallthatthesuperfieldXdefinedintheformula(12.61)isamorphismfromR1|4inRwhichmeansthatitisdefinedbyamorphismXfromC∞(R)inC∞(R)[θ,θ∗,ζ,ζ∗]whichcanbecomputedbytheGrassmannanalyticcontinu-ationprinciple:Xf(x,θ)=f(x)+f(x)ψθ∗+θψ∗+f(x)yθθ∗−f(x)θθ∗ψψ∗(12.62)Thereexistotherrigorousinterpretationsofthisformula.Foradetaileddiscussionwerefertothepaper[105].LetusintroduceasuperpotentialV(X)whereV∈C∞(R)andthesuperLa-grangian1LS=Dθ∗XDθX+V(X)(12.63)2AssumethatVispolynomialforsimplicity.ThentheLagrangianhasthesupersym-metrydefinedbytheoddgeneratorsQθandQθ∗definedbefore.Theinfinitesimal 12.5ClassicalSupersymmetry373transformationsonfieldsareδηX=η∗Qθ+ηQθ∗Xandasaboveδηisaderivation.η,η∗areGrassmannvariables,independentoftheotherGrassmannvariables.WealsohaveδηV(X)=η∗Qθ+ηQθ∗V(X)and∗QδηLS=ηθ+ηQθ∗LSAsfortheabovetoymodel,itresultsthattheLagrangianLSisagainsupersymmet-ricwithgeneratorsQθ,Qθ∗.Nowweshallcomputeincoordinateswiththepseudo-classicalLagrangian.ComputeQθX=ψ∗+θ∗(ix˙+y)−iθθ∗ψ˙∗(12.64)Qθ∗X=−ψ+θ(ix˙−y)−iθθ∗ψ˙(12.65)SowegetδηX=η∗ψ∗−ηψ+θη(y−ix)+η∗(y+ix)θ˙∗−iη∗ψ˙∗+ηψ˙θθ∗(12.66)Henceincoordinatesthesupersymmetryhasthefollowinginfinitesimalrepresenta-tion:δ∗ψ∗−ηψ(12.67)ηx=ηδ∗ηψ=η(y+ix)˙(12.68)δ∗ηψ=η(y−ix)(12.69)δηy=−iη∗ψ˙∗+ηψ˙(12.70)Nowwecomputethepseudo-classicLagrangian:Lpcl:=dθdθ∗LS.UsingthecomputationrulesforBerezinintegralweget122i∗∗1∗Lpcl=x˙+y−V(x)y+ψψ˙−ψψ˙+V(x)ψψ(12.71)222Therealvariablecanbeeliminatedbecausewehave∂Lpcl=0.FromEuler–∂y˙∂LLagrangeequationwegetpcl=0soy=V(x).Wenowgetthefollowing∂yLagrangian:x˙2V(x)2i∗∗1∗Lw=−+ψψ˙−ψψ˙+V(x)ψψ(12.72)2222WecancomputethetwoNoetherchargesassociatedwiththegeneratorsQθ,Qθ∗.TheresultsareQ=x˙−iV(x)ψ,Q∗=x˙+iV(x)ψ∗(12.73) 37412SupercoherentStates—AnIntroductionAneasyexerciseistocomputethedynamicsfortheFermioscillator,wellknownfortheBose(orharmonic)oscillator.TheLagrangianisi1L∗∗∗F=ψψ˙−ψψ˙+ωψψ(12.74)22TheEuler–Lagrangeequationgivesψ˙=−iωψ;hencewegetψ(t)=ηe−iωt,whereηisanyGrassmanncomplexvariable.Itismoresuggestivetowritedownthedynamicsinrealcoordinates:ψ(t)+ψ∗(t)1−iωt∗iωtξ1(t)=√=√ηe+ηe(12.75)22ψ∗(t)−ψ(t)1∗iωt−iωtξ2(t)=√=√ηe−ηe(12.76)i2i212.5.3SupersymmetricQuantizationThefirststepistocomputetheHamiltonianHwfortheLagrangianLw.Themomentaaredefinedasusual∂Lwp==˙x(12.77)∂x˙∂Lwi∗π==−ψ(12.78)∂ψ˙2∗∂Lwiπ==−ψ(12.79)∂ψ˙∗2HencetheLegendretransformisnotsurjective:theLagrangianisdegenerate.WecanextendtheDiracmethod(seemoredetailsin[112])tofermionicvariableswiththetwoconstraintsi∗∗iχ1=π+ψ,χ2=π+ψ22Recallthathere{·}isthesuper-Poisson-bracket.Wehave{χ1,χ2}=−i,sotheconstraintisofsecondorder.Denote1H=˙xp+ψπ˙+ψ˙∗π∗−L22−V∗w=p+V(x)(x)ψψ2AccordingtoDirac’smethodwecomputemultipliersu1,u2suchthattheevolutionofanyobservableFobeystheequationF˙={F,H}+u1{F,χ1}+u2{F,χ2} 12.5ClassicalSupersymmetry375u1,u2arecomputedwiththecompatibilityconditionsχ˙k=0fork=1,2.Weget(x)ψ,u∗u1=−iV2=iV(x)ψThetotalDiracHamiltonianishere,aftercomputation,1Hp2+V(x)2+iV(x)(ψ∗π∗−ψπ)Di=H+u1χ1+u2χ2=2Toachievethequantizationofoursystemwedefine,asinthebosoniccase,theDiracbracket.LetF,Gdependingonlyonfermionicvariables.ThentheDiracbracketis1{F,G}Di={F,G}+{F,χ1}{χ2,G}+{F,χ2}{χ1,G}{χ1,χ2}Sowehave∗∗i∗∗1{ψ,ψ}Di=−i,{π,π}Di=,{ψ,π}={ψ,π}=(12.80)42AquantizationF→Fˆhastofollowthecorrespondenceprinciple.ForF=x,pweconsidertheusualWeyl–Heisenbergquantizationwiththecommutatorrule[ˆx,pˆ]=i.IfFandGarefermionicvariablestheni{F,G}=F,ˆGˆ(12.81)wherethebracketsaresymmetric(anticommutators).Inparticularwehaveψ,ˆψˆ∗∗−ψ,ˆπˆψˆ∗∗−i=,[ˆπ,πˆ]=,=,πˆ=(12.82)42TheDiracquantumHamiltonianis1Hˆpˆ2+V(x)2+iV(x)ψˆ∗πˆ∗−ψˆπˆ(12.83)Di=2ArealizationofthecommutationrelationsisobtainedwiththePaulimatrices:√√ψˆ=σ−,ψˆ∗=σ+(12.84)i∗iπˆ=−σ+,πˆ=−σ−(12.85)22For=1wegettheWittensupersymmetricHamiltonianconsideredatthebegin-ningofthischapter:1d2σ3Hˆ=−+V(x)212−V(x)(12.86)2dx22 37612SupercoherentStates—AnIntroductionWecanalsoremarkthatthesuperchargesQˆandQˆ∗areobtainedbyquantizationoftheNoetherchargesQ,Q∗.ωx2InparticularwecanconsideranharmonicpotentialV(x)=.Choosinga2quantizationsuchthatπˆ=−iψˆ∗andπˆ∗=−iψˆwegetthesuper-harmonicoscil-22lator1d2Hˆ−+ω2x2+ωψˆ∗ψˆ(12.87)sos=22dxWecanrealizethisHamiltonianwithψˆ=θandψˆ∗=∂θinthesuperHilbertspaceHS:=L2(R)⊗H˜(2).ItisnothingbutaGrassmannalgebrainterpretationoftheWittenmodelintroducedbeforewithH˜(2)inplaceofC2.ThescalarproductintheHilbertspaceHSisdefinedas∗θ∗θ∗F,G=dxdθdθeF(x,θ)G(x,θ),(12.88)wherexisarealvariable,θisanholomorphicGrassmannvariable.12.6SupercoherentStatesAscanonicalcoherentstatesarebuiltontheHeisenberg–WeylLiegroup,super-coherentstatesarebuiltonthesuperHeisenberg–WeylsuperLiegroup.Wefirstconsiderthesimplestcasewithonebosonandonefermion.Sowehavecreationandannihilationoperatorsa∗,a∗,aB,aFforbosonsandfermions.TheysatisfytheBFsuper-Liealgebracommutationrelations:∗]=1,[a∗[aB,aBF,aF]+=1(12.89)Allotherrelationsaretrivial.WehaveasupersymmetricharmonicoscillatorHˆsoswithsupersymmetrygener-atorsQ∗,Q,Hˆ∗aB+a∗aF,Qˆ=aBa∗,Qˆ∗=aFa∗(12.90)sos=aBFFBThisdefinitionofHˆsosmaydifferfromothersbyaconstant.Wehave[Hˆsos,Qˆ]=0so(Q,ˆQˆ∗)generatesaglobalsupersymmetrydefinedbythefollowingunitaryoperators:ηQˆ∗+η∗QˆUη=ewhere(η,η∗)areanycomplexconjugateGrassmannnumbers.Super-translationsT(z,γ)ˆareparametrizedby(z,γ),zisacomplexnumber,γisaGrassmann(complex)number,T(z,γ)ˆ=exp(za∗−¯zaB+a∗γ−γ∗aF)(12.91)BF 12.6SupercoherentStates377UsingtheBaker–Campbell–Hausdorffformulawehavethefollowingusefulprop-erties:1∗2∗∗∗T(z,γ)=expγγ−|z|exp(zaB)exp(aFγ)exp(−¯zaB)exp(−aFγ)2(12.92)1∗∗T(z,γ)T(u,δ)=exp(zu¯−¯zu+δγ+δγ)T(z+u,γ+δ)(12.93)2InparticularT(z,γ)−1=T(−z,−γ)isunitaryandT(z,γ)−1aBT(z,γ)=aB+z(12.94)T(z,γ)−1aFT(z,γ)=aF+γ(12.95)ThegroundstateofHˆsosisthestateψ0,0:=ϕ0⊗ψ0whereϕ0andψ0arethenormalizedgroundstatesofthebosonicandfermionicoscillators.Sowedefinesuper-coherentstatesbydisplacementofthegroundstateofthesuper-harmonicoscillatorbysuper-translations:ψz,γ=T(z,γ)ψ0,0(12.96)Using(12.92)wehaveγγ∗ψz,γ=1+|z,0+|z,1γ(12.97)2whereweusethenotation|z,ε=ϕz⊗θε,ε=0,1.Thecoherentstatesfamilyψz,γhasthefollowingexpectedproperties.Theproofsfolloweasilyfromresultsalreadyestablishedforbosonicandfermionicco-herentstates.1.(Normalization)2=ψψz,γz,γ,ψz,γ=1(12.98)2.(Non-orthogonality)11ψγ∗δ−(γ∗γ+δ∗δ)exp−|z|2+|u|2+¯zu(12.99)z,γ,ψu,δ=exp223.(Over-completeness)Foreveryψ∈HSwehaveψ(x,θ)=ψ∗dγ(12.100)z,γ,ψψz,γ(x,θ)dzdγ4.(Translationproperty)1∗∗T(z,γ)ψu,δ=exp(zu¯−¯zu+δγ+δγ)ψz+u,γ+δ(12.101)2 37812SupercoherentStates—AnIntroduction5.(Eigenfunctionsofannihilationoperators)aBψz,γ=zψz,γ,aFψz,γ=γψz,γ(12.102)Inparticularwehavetheaveragesforenergyandsuperchargesψ2+γ∗γ(12.103)z,γ,Hˆsosψz,γ=|z|ψ∗∗z,γ,Qψz,γ=zγ,ψz,γ,Qψz,γ=¯zγ(12.104)Remark72Definitionandpropertiesofsupercoherentstateswithonebosonandonefermioncaneasilybeextendedforsystemswithnbosonsandmfermions.Thenz=(z1,...,zn)∈Cn,γ=(γ1,...,γm)representasystemofmcomplexGrass-mannvariables,aB=(aB,1,...,aB,n),aF=(aF,1,...,aF,m),etc.Theformu-lasarethesameusingthemultidimensionalnotations:γ∗·aF=1≤j≤mγ∗aF,j.jThentheenergyHamiltoniancanbeHˆ=ωB·a∗aB+ωFa∗·aF,ωB,FarerealBFnumbers.TheHilbertspaceofthissystemisHn,m=L2(Rn)⊗H˜(m).Tohaveasupersym-metricsystemweneedthatn=mandωB=ωF.12.7PhaseSpaceRepresentationsofSuperOperatorsAsiswellknownforbosonsandaswasstudiedbeforeforfermionswecanextendtomixedsystemsbosons+fermionsrepresentationsformulasforoperatorsinthesuper-HilbertspaceH.PhasespacerepresentationmeansthatwearelookingforacorrespondencebetweenfunctionsonthephasespaceandoperatorsinsomeHilbertspace.Thiscorrespondenceiscalledquantizationinquantummechanics.OurgoalhereistorevisitWeylquantizationsforobservablesmixingbosonsandfermions.Itcouldbepossibletodoitalsoforanti-Wickquantization.Tohaveabetteranalogybetweenthebosonicandfermionicvariablesitisnicerq+ipntowritebosonicWeylquantizationincomplexcoordinatesζ:=√,ζ∈C,2q,p∈Rn.TheLebesguemeasureinCnisd2ζ=|dζ∧dζ∗|.AssumeforsimplicitythatHˆ,GˆhavesmoothSchwartzkernelsinS(R2n).WeconsidertheirWeylsymbolsHww,Gwwincomplexcoordinates.UsingthesameproofasinthefermioniccasewehavethefollowingMoyalformulasfortheWeylsymbolsofGˆHˆ:d2ηG1/2(ζ¯·η−ζ·¯η)(GwHw)(ζ)=w(ζ−η)Hw(η)e(12.105)Cn←−−→←−−→GwHw(ζ)=Gw(ζ)e1/2(∂ζ·∂ζ¯−∂ζ¯·∂ζ)Hw(ζ)(12.106)Thefirstformulagivesthecovariantsymbol,thesecondformulathecontravariantsymbol.Wehavethefollowingrelations:w2ζ¯·η−ζ·¯ηG(η)=dζeGw(ζ)(12.107) 12.8ApplicationtotheDickeModel379G−2nd2ζ·¯η−ζ¯·ηw(η)=(2π)ζeGw(ζ)(12.108)ItisnotdifficulttogetaMoyalformulaforclassesofoperatorsinthesuperspaceHn,m=L2(Rn)⊗H˜(m).Asusualitisenoughtoestablishformulasforsmoothingoperatorsthentheformulasareextendedtosuitableclassesofsymbols.SoweshallassumethatG,ˆHˆarelinearcontinuousoperatorsfromS(Rn)⊗H˜(m)intoS(Rn)⊗H˜(m)(thatmeansthattheirSchwartzkernelsareinS(R2n⊗Gc)).ThecovariantmsymbolHwisdefinedsuchthatHˆ=d2ζd2γHw(ζ,γ)T(ˆ−ζ,−γ)(12.109)Hw(ζ,γ)=StrHˆT(ζ,γ)ˆ(12.110)whereStrisdefinedastheusualtraceinbosonicvariableζandthesuper-traceinthefermionicvariableγ.Moreprecisely,foranytrace-classoperatorHˆinHn,mwehaveStrHˆ=TrH(ˆ1⊗χ)whereχisthechiralityoperatorinH˜(m)definedinChap.11.ThecontravariantsymbolisthesymplecticFouriertransformofthecovariantsymbol:w22η¯·ζ−η·ζ¯+α∗·γ+α·γ∗H(ζ,γ)=dηdαHw(η,α)e(12.111)SowegetthefollowingMoyalformulaforsupersymbols:←−−→←−−→←−−→←−−→GwHw(ζ,α)=Gw(ζ,α)e1/2(∂ζ·∂ζ¯−∂ζ¯·∂ζ+∂α∗··∂α+∂α·∂α∗)Hw(ζ,α)(12.112)Manyresultsexplainedbeforeforbosonsandfermionscouldbeextendedtomixedsystems.Insteadtodothatwenowdiscussasimpleapplication.12.8ApplicationtotheDickeModelThismodelwasstudiedin[1]asasupersymmetricsystem.TheHamiltonianforthismodelisHˆ=Ωa∗1∗a+ωσ3+g(a+a)σ1(12.113)2Itconcernsatwo-levelatominteractingwithamonochromaticradiationfield.a∗andaareoneparticlecreation/annihilationoperators,thePaulimatricesσ=(σ1,σ2,σ3)denotetheradiationfieldwithfrequencyΩ,ωistheleveldistanceofthestatesoftheatom,gisacouplingconstant. 38012SupercoherentStates—AnIntroductionThematricesσ±satisfythe(CAR)relation,sowecanidentifythemwithfermioniccreation/annihilationoperators.SowecanintroduceapairofcomplexconjugateGrassmannnumbers(β,β∗)suchthatσ+≡βˆ∗=∂β,σ−≡βˆismulti-plicationbyβ(seeChap.11)andσ3≡2βˆ∗βˆ−1.ButwiththischoiceHˆisnotanevenoperator.ToovercomethisproblemweaddanewGrassmannpairofcomplexvariables(θ,θ∗)andtherealGrassmannnumberη=θ+θ∗.Wehaveηˆ2=1(in[1]ηˆisdenotedcandiscalledaCliffordnumber,seeChap.11formoreexplanations).HencewehavethetwopairsofGrassmannvariables(β,β∗),(θ,θ∗)torepresentthePaulimatrices:σ∗+←→ˆηβˆ,σ−←→ˆηβˆwherewedenoteηˆ=θ+∂θ,b=βˆ,b∗=∂β.Wehavetoremarkthat(bη)ˆ∗bηˆ=b∗b.WiththissubstitutiontheHamiltonianHˆistransformedintoanevenHamilto-niandefinedinthespaceL2(R)⊗H˜(2)Hˆ∗1∗∗∗S=Ωaa+ω(bb−1)+g(a+a)(ηˆb+bη)ˆ(12.114)2Its(contravariant)Weylsymbolis1H|ζ|2−+ωβ∗β+gζ+ζ¯(ηβ∗+βη)S(ζ,β,θ)=Ω2OuraimistostudythedynamicsfortheHamiltonianHˆS.ItisdeterminedbythevonNeumannequation∂ρˆti=HˆS,ρˆt,ρˆt=0=ˆρ0(12.115)∂twhereρˆisadensityoperator(apositiveoperatoroftrace1),ρtisthecontravariantWeylsymbolofρˆt(alsocalledtheWeyl–WignerDistributionFunction).LetusremarkthatHˆSdependsonlyonaandγˆwhereγ=βη,becausewehaveb∗b=ˆγ∗γˆ.Moreoverwehave[ˆγ,γˆ∗]+=1.ρtsatisfiesaFokker–PlancktypeequationwhichcanbecomputedusingtheMoyalproductformulaappliedtoHSρt−ρtHS.Soweget∂ρt−→←−∗−=Lρt+ρtL(12.116)∂twherewefind(seealso[1])−→∗1−→1−→∗1−→1−→1−iL=Ωζ−∂ζζ+∂ζ∗+ωβ+∂ββ+∂β∗−22222∗1−→−→+gζ+ζ+∂ζ∗−∂ζ2−→∗1−→−→+θ+∂θβ−β+∂ζ−∂ζ.(12.117)2 12.8ApplicationtotheDickeModel381HˆSbeingtimeindependentwealsohaveρˆ−itHˆSitHˆSt=eρˆ0eSowestarttoconsiderthepropagationofthedynamicalvariablesaandγˆ.WefirstcomputethecommutatorsHˆ=−ˆγ+g(a+a∗)(2γˆ∗γˆ−1)(12.118)S,γˆHˆ∗=ˆγ∗−g(a+a∗)(2γˆ∗γˆ−1)(12.119)S,γˆHˆ=−g(γˆ∗+ˆγ)−Ωa(12.120)S,aHˆ∗=g(γˆ∗+ˆγ)+Ωa∗(12.121)S,aSowefindthenon-lineardifferentialsystem∂∗∗iγˆt=−ˆγt+g(a+a)(2γˆtγˆt−1)∂t∂∗∗∗∗iγˆt=ˆγt−g(a+a)(2γˆtγˆt−1)∂t(12.122)∂∗iat=−g(γˆt+ˆγt)−Ωat∂t∂∗∗∗iat=−g(γˆ+ˆγ)+Ωat∂tAssumenowthattheinitialWeyl–Wignerdistributionρˆdependsonlyona,a∗,γ,ˆγˆ∗.Thenρˆthasthefollowingshape:ρˆ0∗1∗1,∗∗∗2∗∗t=ˆρt(at,at)+ˆρt(at,at)γˆt+ˆρt(at,at)γˆt+ˆρt(at,at)γˆtγˆtUsing(12.122)wefindthatat,a∗t,γˆt∗γˆtarefunctionsoftanda,a∗,γ,ˆγˆ∗.Soρˆtcanbewrittenasfollows:(0)∗(1)∗(1)∗∗∗(2)∗∗ρˆt=ˆρt(a,a)+ˆρt(a,a)γˆ+ˆρt(a,a)γˆ+ˆρt(a,a)γˆγˆ(12.123)(•)∗Nowwecangivethephysicalmeaningofthebosonicoperatorsρˆt(at,at)bytakingtheaverageonthefermionicvariables(β,θ).WeshallseenowthatwerecoverthedynamicsgeneratedbytheHamiltonianHˆtakingthebasis{12,σ+,σ−,σ3}.WehavetocomputetherelativetraceTrH˜(ρˆt(1⊗G))ˆ,whereGˆisafermionic2operatorinH˜2,usingtheformula1ρˆ∗22TrH˜t1⊗Gˆ=ρt(a,a,β,θ)Gw(β,θ)dβdθ24WecomputethecovariantWeylsymbolsof12,γ,ˆγˆ∗,γˆ∗γˆ:1∗∗w(β,θ)=ββ+θθ 38212SupercoherentStates—AnIntroductionγ∗w(β,θ)=β(θ−θ)γ∗(β,θ)=(θ∗−θ)β∗w(γ∗γ∗w)(β,θ)=θθThenwegeteffectivedistributionfunctionsρeff(forsimplicitythetimetisnotwrittenexplicitlynorthecontravariantbosonicvariableζ)(0)∗∗22(0)ρ=ρ×(θθ+ββ)dθdβ=ρeff(1)∗∗∗22(1)ρ=ρ×β(θ+θ)(θ−θ)βdθdβ=−2ρeff(1)∗∗∗∗22(1)∗ρ=ρ×(θ+θ)ββ(θ−θ)dθdβ=−2ρeff(2)∗22(2)ρ=2ρ×θθdθdβ=2ρeffFinallyafteradirectcomputationwerecoverthedynamicsfortheHamiltonianHˆdefinedby(12.113)fromthedynamicsforthesuperHamiltonianHˆSdefinedby(12.114)[99,100]:(0)(1)(1),∗(2)ρeff=ρ12+ρσ+ρσ−+ρσ3effeffeffeff AppendixAToolsforIntegralComputationsA.1FourierTransformofGaussianFunctionsThisresultisthestartingpointforthestationaryphasetheorem.LetMbeacomplexmatrixsuchthatMispositive-definite.Wedefine[detM]1/2theanalyticbranchof(detM)1/2suchthat(detM)1/2>0whenMis∗real.Theorem52LetAbeasymmetriccomplexsymmetricmatrix,m×m.WeassumethatAisnonnegativeandAisnondegenerate.ThenwehavetheFouriertrans-formformulafortheGaussianeiAx·x/2iAx·x/2−ix·ξm/2−1/2(iA)−1ξ·ξ/2eedξ=(2π)det(−iA)e.(A.1)∗RmProofForAtherealformula(A.1)iswellknown:firstweproveitform=1thenform≥2bydiagonalizingAandusingalinearchangeofvariables.ForAcomplex(A.1)isobtainedbyanalyticextensionofleftandrighthandside.A.2SketchofProofforTheorem29RecallthatcriticalsetMofthephasefisM=x∈O,f(x)=0,f(x)=0.NotethatifaissupportedoutsidethissetthenJ(ω)isO(ω−∞).Usingapartitionofunity,wecanassumethatOissmallenoughthatwehavenormal,geodesiccoordinatesinaneighborhoodofM.Sowehaveadiffeomor-phism,χ:U→O,M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,383TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0,©SpringerScience+BusinessMediaB.V.2012 384AToolsforIntegralComputationswhereUisanopenneighborhoodof(0,0)inRk×Rd−k,suchthatχx,x∈M⇐⇒x=0andifm=χ(x,0)∈Mwehaveχxk=T,0RxmM,χxd−k=N,0RxmM,(normalspaceatm∈M).Sothechangeofvariablesx=χ(x,x)givestheintegralJ(ω)=eiωf(χ(x,x))ax,x|detχx,xdxdx.(A.2)RdThephasef˜x,x:=fχx,xclearlysatisfiesf˜x,x=0,f˜x,x=0⇐⇒x=0.xHence,wecanapplythestationaryphaseTheorem7.7.5of[117]inthevariablex,totheintegral(A.2),wherexisaparameter(theassumptionsof[117]aresatisfied,uniformlyforxcloseto0).WeremarkthatallthecoefficientscjoftheexpansioncanbecomputedusingtheabovelocalcoordinatesandTheorem7.7.5.A.3ADeterminantComputationHerewegivethedetailsconcerningcomputationsofthedeterminant(9.33)inChap.9.Wewriteα=(α,α4)whereα=ˆe1+icosγeˆ2.Thegradientofthephase(9.28)isix+G(p)whereG(p)isgivenby2G(p):=α+pα4−α·w1(p)=K(p)M(p),(p2+1)α·w1(p)where2K(p)=2α·p+α4(p2−1)and2pαM(p)=α+4−α·p1+p2K:R3→C,M:R3→C3. A.3ADeterminantComputation385Sincexisaconstantthehessianof(9.28)issimplyDG(p)whereDG(resp.DK,DM)isthefirstdifferentialofG(resp.K,M).Wewanttocalculateatthecriticalpointpcwithccosβcosγsinβp=,,0.1−sinβsinγ1−sinβsinγLetδpbeanarbitraryincreaseofp.OnehasDG(p)(δp)=DK(p)·δp·M(p)+K(p)DM(p)·δp.Wecanwrite∗DK(p)·δpM(p)=M(p)⊗DK(p)·δp,whereDK(p)∗∈(R3)∗+i(R3)∗C3.Usingthedualstructuretheidentificationof(R3)∗+i(R3)∗withC3isperformedviatheisomorphism:u→(v→u·v).WehaveKpc=e−iβ(1−sinβsinγ),Mpc=α+isinγ−eiβpc.Thusc−iβccc∗DGp=e(1−sinβsinγ)DMp+Mp⊗DKp.Itisconvenienttochooseasabasisofvectors(pc,qc,eˆ3)whereqc=α+isinγ−eiβpc.Thevectorspc,qcareC-linearlyindependentforγ=π+kπ.Simplecalculus2yieldsciβcc∗DMp=isinγ−e1R3−(1−sinβsinγ)p⊗q,DKpc=−e−2iβ(1−sinβsinγ)2α+αc.4pThuswegetDGpc=e−iβisinγ−eiβ(1−sinβsinγ)1R3−iβ2cc∗−e(1−sinβsinγ)p⊗q−2iβ2cciβc∗−e(1−sinβsinγ)q⊗q+ep.(A.3)LetH(p)betheHessianmatrixinthebasis(pc,qc,eˆ3)anddenoteH(p)H1(p)=.1−sinβsinγ 386AToolsforIntegralComputationsThesecondlineof(A.3)yieldsamatrixintheplanegeneratedby(pc,qc)oftheformd1d2d3d4thatwecalculateusingcc∗qc·pc(qc)2p⊗q=,00cc∗00q⊗q=qcc2,q·p(q)cc∗00q⊗p=c2cc.(p)p·qWehavec21+sinβsinγp=,1−sinβsinγcci−eiβsinβq·p=sinγ,1−sinβsinγc22iβq=−e.Wegetd1=−(1−sinβsinγ),d−iβ(1−sinβsinγ),2=−ed−iβ1+isinγe−iβ,3=−ed4=0.Thusthethree-dimensionalreducedHessianmatrixH1equals⎛⎞d1d20H1=⎝d300⎠00d5withd5=e−iβ(isinγ−eiβ).ItsdeterminantequalsdetH1=−d2d3d5.Onehas|detH22221|=(1−sinβsinγ)1−2sinβsinγ+sinγ1+2sinβsinγ+sinγ. A.4TheSaddlePointMethod387Finallywegettheresult:detHpc=(1−sinβsinγ)4sin2γ+2sinβsinγ+1sin2γ−2sinβsinγ+1.Weseethatitdoesnotvanishprovidedsinβsinγ=1.A.4TheSaddlePointMethodA.4.1TheOneRealVariableCaseThisresultiselementaryandveryexplicit.LetusconsidertheLaplaceintegralaI(λ)=e−λφ(r)F(r)dr0wherea>0,φandFaresmoothfunctionson[0,1]suchthatφ(0)=φ(0)=0,φ(0)>0,φ(r)>0ifr∈]0,1].Undertheseconditionswecanperformthechange√ofvariabless=φ(r)wherewedenoter=r(s)andG(s)=F(r(s))r(s).SowehaveProposition142ForeveryN≥1wehavetheasymptoticexpansionforλ→+∞,I(λ)=C−(j+1)/2−(N+1)/2,(A.4)jλ+Oλ0≤j≤N−1wherejG(j)(0)Cj=Γ+1.22j!InparticularC0=F(0)(φ(0))−1/2.ProofThisisadirectconsequenceofTaylorexpansionappliedtoGat0andusingthat,foreveryb>0andε>0,b−λs2j1j−(j+1)/2−ελesds=Γ+1λ+Oe.022A.4.2TheComplexVariablesCaseThisisanoldsubjectforonecomplexvariablebuttherearenotsomanyrefer-encesforseveralcomplexvariables.HerewerecallapresentationgivenbySjös-trand[179]or[178]. 388AToolsforIntegralComputationsLetusconsideracomplexholomorphicphasefunctioninaopenneighborhoodA×Uof(0,0)inCk×Cn,A×U(a,u)→ϕ(a,u)∈C.Assumethat•ϕ(0,0)=0,∂uϕ(0,0)=0.•det∂2ϕ(0,0)=0.(u,u)•ϕ≥0∀u∈U,ϕ>0forallu∈∂URwhereUR:=U∩Rnand∂URistheboundaryinRnofUR.BytheimplicitfunctiontheoremwecanchooseA×Usmallenoughsuchthattheequation∂uϕ(a,z)=0hasauniquesolutionz(a)∈U,a→z(a)beingholomor-phicinA.Thenwehavethefollowingasymptoticresult.Theorem53ForeveryholomorphicandboundedfunctionginUwehave,fork→+∞,ekϕ(a,z(a))e−kϕ(a,r)g(r)drURn/22π2−1/2−n/2−1=det∂(u,u)ϕa,z(a)g(a)+Ok.(A.5)kA.5KählerGeometryLetMbeacomplexmanifoldandhanHermitianformonM:h=hj,kdzj⊗dzk,hj,k(z)=hk,j(z).j,khisaKählerformiftheimaginarypartω=hisclosed(dω=0)anditsrealpartg=hispositive-definite.Wehaveω=ihj,k(z)dzj∧dz¯k.j,k(M,h)issaidaKählermanifoldifhisaKählerformonM.ThenonMexistsaRiemannmetricg=handsymplectictwoformω.Locallythereexistsareal-valuedfunctionK,calledKählerpotential,suchthat∂2hj,k=K.∂zj∂zkThePoissonbracketoftwosmoothfunctionsφ,ψonMisdefinedasfollows:{φ,ψ}(m)=ω(Xφ,Xψ), A.5KählerGeometry389whereXψistheHamiltonianvectorfieldatmdefinedsuchthatdψ(v)=ω(Xψ,v),forallv∈Tm(M).Moreexplicitly:∂ψ∂φ∂φ∂ψ{ψ,φ}(z)=ihj,k(z)−,∂zj∂z¯k∂zk∂z¯jj,kwherehj,k(z)istheinversematrixofhj,k(z).TheLaplace–Beltramioperatorcorrespondingtothemetricgis∂∂=hj,k(z).∂zj∂z¯kj,kInparticularfortheRiemannspherewehave2dζdζ¯ds=4,(A.6)(1+|ζ|2)222∂ψ∂φ∂φ∂ψ{ψ,φ}(z)=i1+|z|−,(A.7)∂z∂z¯∂z∂z¯222∂=1+|z|.(A.8)∂z∂z¯Forthepseudo-spherewehave2dζdζ¯ds=4,(A.9)(1−|ζ|2)222∂ψ∂φ∂φ∂ψ{ψ,φ}(z)=i1−|z|−,(A.10)∂z∂z¯∂z∂z¯222∂=1−|z|.(A.11)∂z∂z¯ AppendixBLieGroupsandCoherentStatesB.1LieGroupsandCoherentStatesInthisappendixwestartwithashortreviewofsomebasicpropertiesofLiegroupsandLiealgebras.ThenweexplainsomeusefulpointsconcerningrepresentationtheoryofLiegroupsandLiealgebrasandhowtheyareusedtobuildageneraltheoryofCoherentStatesystemsaccordingtoPerelomov[155,156].ThistheoryisanextensionoftheexamplesalreadyconsideredinChap.7andChap.8.B.2OnLieGroupsandLieAlgebrasWerecallheresomebasicdefinitionsandproperties.Moredetailscanbefoundin[72,105]orinmanyothertextbooks.B.2.1LieAlgebrasALiealgebragisavectorspaceequippedwithananti-symmetricbilinearproduct:(X,Y)→[X,Y]satisfying[X,Y]=−[Y,X]andtheJacobiidentity[X,Y],Z+[Y,Z],X+[Z,X],Y=0.Themap(adX)Y=[X,Y]isaderivation:(adX)[Y,Z]=[(adX)Y,Z]+[Y,(adX)Z].IfgandhareLiealgebrasaLiehomomorphismisalinearmapχ:g→hsuchthat[χX,χY]=[X,Y].Asub-Liealgebrahingisasubspaceofgsuchthat[h,h]⊆hhisanidealiffurthermorewehave[h,g]⊆h.IfχisaLiehomomorphismthenkerχisanideal.InthefollowingweassumeforsimplicitythattheLiealgebrasgconsideredarefinite-dimensional.M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,391TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0,©SpringerScience+BusinessMediaB.V.2012 392BLieGroupsandCoherentStatesgisabelianif[g,g]=0.gissimpleifitisnon-abelianandcontainsonlytheideals{0}andg.ThecenterZ(g)isdefinedasZ(g)=X∈g,[X,Y]=0,∀Y∈g.Z(g)isanabelianideal.Ifghasnoabelianidealexcept{0}thengissaidsemi-simple.InparticularZ(g)={0}.TheKillingformongisthesymmetricbilinearformB(X,Y)definedasB(X,Y)=Tr(adX)(adX).gissemi-simpleifandonlyifitsKillingformisnon-degenerate(Cartan’scriterion).B.2.2LieGroupsALiegroupGisagroupequippedwithamultiplication(x,y)→x·yandequippedwiththestructureofasmoothconnectedmanifold(wedonotrecallheredefinitionsandpropertiesconcerningmanifolds,see[105]fordetails)suchthatthegroupop-eration(x,y)→x·yandx→x−1aresmoothmaps.WealwaysassumethatLiegroupsconsideredhereareanalytic.AusefulmappingistheconjugationC(x)(y)=xyx−1,x,y∈G.Itstangentmappingaty=eisdenotedAd(x).Ad(x)∈GL(g)andx→Ad(x)isahomomor-phismfromGintoGL(g).ItiscalledtheadjointrepresentationofGandAd(G)istheadjointgroupofG.Wedenoteadthetangentmapofx→Ad(x)atx=e.TheLiealgebragassociatedwiththeLiegroupGisthetangentspaceTe(G)attheuniteofG.TheLiebracketongisdefinedasfollows:LetX,Y∈g=Te(G)anddefine[X,Y]=(adX)(Y).gistheLiealgebraas-sociatedwiththeLiegroupG.AfirstexampleofLiegroupisGL(V)thelineargroupofafinite-dimensionallinearspaceV.Herewehaveg=L(V,V)andtheLiebracketisthecommutator[X,Y]=XY−YX.MuchinformationonLiegroupscanbeobtainedfromtheirLiealgebrasthroughtheexponentialmapexp.Theorem54LetGbeaLiegroupwithLiealgebrag.Thenthereexistsauniquefunctionexp:→Gsuchthat(i)exp(0)=e.(ii)dexp(tX)|t=0=X.dt(iii)exp((t+s)X)=exp(tX)exp(sX),forallt,s∈R.(iv)Ad(expX)=eadX. B.2OnLieGroupsandLieAlgebras393ThereisanopenneighborhoodUof0ingandanopenneighborhoodVofeinGsuchthattheexponentialmappingisadiffeomorphismformUontoV.ThislocaldiffeomorphismcanbeextendedinaglobaloneifmoreoverthegroupGisconnectedandsimplyconnected.Otherpropertiesoftheexponentialmappingaregivenin[72].AusefultoolonLiegroupisintegration.Definition42LetμbeaRadonmeasureontheLiegroupG.μisleft-invariant(leftHaarmeasure)iff(x)dμ(x)=f(yx)dμ(x)foreveryy∈Gandμisright-invariant(rightHaarmeasure)iff(x)dμ(x)=f(xy)dμ(x)foreveryy∈G.Ifμisleftandrightinvariantwesaythatμisabi-invariantHaarmeasure.IfthereexistsonGabi-invariantHaarmeasure,Gissaidunimodular.Theorem55OneveryconnectedLiegroupGthereexitsaleftHaarmeasureμ.Thismeasureisuniqueuptoamultiplicativeconstant.IfGisacompactandconnectedLiegroupthenaleftHaarmeasureisarightHaarmeasureandthereexistsauniquebi-invariantHaarprobabilitymeasurei.e.everycompactLiegroupisunimodular.Remark73TheaffinegroupGaff={x→ax+b,a,b∈R}isnotunimodularbuttheHeisenberggroupHnandSU(1,1)are.Thisremarkisimportanttoun-derstandthedifferencesbetweenthecorrespondingcoherentstatesassociatedwiththesegroups.Coherentstatesassociatedwiththeaffinegrouparecalledwavelets.InmanyexamplesaLiegroupGisaclosedconnectedsubgroupofthelineargroupGL(n,R)(orGL(n,C)).ThenwecancomputealeftHaarmeasureasfollows(see[172]fordetails).Letusconsiderasmoothsystem(x1,...,xn)onanopensetUofGandthematrixofone-forms=A−1∂Adxj.Thenwehavethefollowing[172]:1≤j≤n∂xjProposition143isamatrixofleft-invariantone-formsinU.Moreoverthelinearspacespannedbytheelementsofhasdimensionn.Thereexistnindependentleft-invariantone-formsω1,...,ωnandω1∧ω2∧···∧ωndefinesaleftHaarmeasureonG.ExplicitexamplesofHaarmeasures:(i)Onthecircle§dx1≡R/(2πZ)theHaarprobabilitymeasureisdμ(x)=.2π(ii)HaarprobabilitymeasureonSU(2).ConsidertheparametrizationofSU(2)bytheEulerangles(seeChap.7).cos(θ/2)e−i/2(ϕ+ψ)−sin(θ/2)ei/2(ψ−ϕ)g(θ,ϕ,ψ)=sin(θ/2)e−i/2(ψ−ϕ)cos(θ/2)ei/2(ϕ+ψ).(B.1) 394BLieGroupsandCoherentStatesAstraightforwardcomputationgives−1−i(cosθdϕ+dψ)eiψ(dθ−isinθdϕ)2gdg=−iψ.e(dθ+isinθdϕ)i(cosθdϕ+dψ)Soweget,afternormalizationtheHaarprobabilityonSU(2):1dμ(θ,ϕ,ψ)=sinθdθdϕdψ.16π2(iii)HaarmeasureforSU(1,1).ThesamemethodasforSU(2)usingtheparametrizationcoshtei(ϕ+ψ)/2sinhtei(ϕ−ψ)/2g(ϕ,t,ψ)=22.sinhtei(ψ−ϕ)/2coshte−i(ϕ+ψ)/222Aftercomputationsweget−1i(coshtdφ+dψ)e−iψ(dt+icoshtdφ)2gdg=iψe(dt−icoshtdφ)i(coshtdφ−dψ)andaleftHaarmeasure:dμ(t,φ,ψ)=coshtdtdφdψ.(B.2)(iv)LetusconsidertheHeisenberggroupHn(seeChap.1).Thisgroupcanalsoberealizedasalineargroupasfollows.Let⎛⎞1x1x2···xns⎜010···0y1⎟⎜⎟⎜001···0y2⎟⎜⎟g(x,y,s)=⎜......⎟,⎜⎜............⎟⎟⎝000···1yn⎠000···01wherex=(x1,...,xn)∈Rn,y=(y1,...,yn)∈Rn,s∈R.Wecaneasilycheckthat{g(x,y,s),x,y∈Rn,s∈R}isaclosedsubgroupH˜nofthelineargroupGL(2n+1,R).H˜nisisomorphictotheWeyl–HeisenberggroupHnbytheisomorphismx·yx−iyg(x,y,s)→s−,√.22TheLebesguemeasuredxdydsisabi-invariantmeasureonH˜ni.e.theWeyl–Heisenberggroupisunimodular. B.3RepresentationsofLieGroups395WeshallseethatwhenconsideringcoherentstatesonageneralLiegroupGitisusefultoconsideraleft-invariantmeasureonaquotientspaceG/HwhereHisaclosedsubgroupofG;G/Histhesetofleftcoset,itisasmoothanalyticmanifoldwithananalyticactionofG:foreveryx∈G,τ(x)(gH)=xgH.Thefollowingresultisprovedin[105].Theorem56ThereexistsaGinvariantmeasuredμG/HonG/HifandonlyifwehavedetAd=detAd,∀h∈H,G(h)H(h)whereAdGistheadjointrepresentationforthegroupG.Moreoverthismeasureisuniqueuptoamultiplicativeconstantandwehaveforanycontinuousfunctionf,withcompactsupportinG,f(g)dμG(g)=f(gh)dμH(h)dμG/H(gH),G/HHwheredμGanddμHareleftHaarmeasuresuitablynormalized.InthisbookwehaveconsideredthethreegroupsHn,SU(2)andSU(1,1)andtheirrelatedcoherentstates.IneachcasetheisotropysubgroupHisisomorphictotheunitcircleU(1)andwehavefoundthequotientspaces:Hn/U(1)≡R2n,SU(2)/U(1)≡S2andSU(1,1)/U(1)≡PS2withtheircanonicalmeasure.Eachofthesespacesisasymplecticspaceandcanbeseenasthephasespaceofclassicalsystems.B.3RepresentationsofLieGroupsThegoalofthissectionistorecallsomebasicfacts.B.3.1GeneralPropertiesofRepresentationsGdenotesanarbitraryconnectedLiegroup,V1,V2,VarecomplexHilbertspaces,L(V1,V2)thespaceoflinearcontinuousmappingfromV1intoV2,L(V)=L(V,V),GL(V)thegroupofinvertiblemappingsinL(V),U(V)thesubgroupofGL(V)ofunitarymappingsi.e.A∈U(V)ifandonlyifA−1=A∗.Definition43ArepresentationofGinVisagrouphomomorphismRˆfromGinGL(V)suchthat(g,v)→R(g)vˆiscontinuousfromG×VintoV.IfR(g)ˆ∈U(V)foreveryg∈Gtherepresentationissaidtobeunitary. 396BLieGroupsandCoherentStatesDefinition44ThesubspaceE⊆VisinvariantbytherepresentationRˆifR(g)Eˆ⊆Eforeveryg∈G.TherepresentationRˆisirreducibleinViftheonlyinvariantclosedsubspacesofVare{V,{0}}.Definition45Tworepresentations(Rˆ1,V1)and(Rˆ2,V2)areequivalentifthereexistsaninvertiblecontinuouslinearmapA:V1→V2suchthatR2(g)A=AR1(g),∀g∈G.Irreduciblerepresentationsareimportantinphysics:theyareassociatedtoele-mentaryparticles(see[194]).LetdμbealeftHaarmeasureonG.ConsidertheHilbertspaceL2(G,dμ)anddefineL(g)f(x)=f(g−1x)whereg,x∈G,f∈L2(G,dμ).LisaunitaryrepresentationofGcalledtheleftregularrepresentation.TheSchurlemmaisanefficienttooltostudyirreducibledimensionalrepresen-tations.Lemma80(Schur)SupposeRˆ1andRˆ2arefinite-dimensionalirreduciblerepre-sentationsofGinV1andV2,respectively.SupposethatwehavealinearmappingA:V1→V2suchthatARˆ1(g)=Rˆ2(g)Aforeveryg∈G.ThenorAisbijectiveorA=0.InparticularifV1=V2=VandAR(g)ˆ=R(g)Aˆforallg∈GthenA=λ1forsomeλ∈C.Supposethat(R,V)ˆisaunitaryrepresentationintheHilbertspaceHthenitisirreducibleifandonlyiftheonlyboundedlinearoperatorsAinVcommutingwithRˆ(AR(g)ˆ=R(g)Aˆforeveryg∈G)areA=λ1,λ∈C.AusefulpropertyofarepresentationRˆisitssquareintegrability(see[93]fordetails).Definition46Avectorv∈Vissaidtobeadmissibleifwehave2R(g)v,vˆdμ(g)<+∞.(B.3)GTherepresentationRˆissaidsquareintegrableifRˆisirreducibleandthereexistsatleastoneadmissiblevectorv=0.IfGiscompacteveryirreduciblerepresentationissquareintegrable.ThediscreteseriesofSU(1,1)aresquareintegrable(provethat1isadmissibleusingtheformula(B.2)fortheHaarmeasureonSU(1,1)).WehavethefollowingresultduetoDuflo–MooreandCarey(see[93]foraproof).Theorem57LetRˆbeasquareintegrablerepresentationinV.Thenthereexistsauniqueself-adjointpositiveoperatorCinVwithadensedomaininVsuchthat: B.3RepresentationsofLieGroups397(i)ThesetofadmissiblevectorsisequaltothedomainD(C).(ii)Ifv1,v2aretwoadmissiblevectorsandw1,w2∈VthenwehaveR(g)vˆ2,w2R(g)vˆ1,w1dg=Cv1,Cv2w1,w2.(B.4)G(iii)IfGisunimodularthenC=λ1,λ∈R.Remark74IfGisunimodularcoherentstatescanbedefinedasfollows.Westartfromanadmissiblevectorv0∈V, v0 =1andanirreduciblerepresentationRˆinV.Definethecoherentstate(ortheanalyzingwavelet)ϕg=R(g)vˆ0.Thenthefamily{λ−1/2ϕg|g∈G}isovercompleteinV:−1ϕλg,ψ1ϕg,ψ2dμ(g)=ψ1,ψ2,ψ1,ψ2∈V,Gwhereλ=|R(g)vˆ0,v0|2dμ(g).GB.3.2TheCompactCaseRepresentationtheoryforcompactgroupiswellknown(foraconcisepresentationsee[129]orformoredetails[130]).TypicalexamplesareSU(2)andSO(3)consid-eredinChap.7.HereGisacompactLiegroup.Themainfactsarethefollowing:1.Everyfinite-dimensionalrepresentationisequivalenttoaunitaryrepresentation.2.EveryirreducibleunitaryrepresentationofGisfinite-dimensionalandeveryuni-taryrepresentationofGisadirectsumofirreduciblerepresentations.3.IfRˆ1,Rˆ2arenonequivalentirreduciblefiniterepresentationsofGonV1andV2thenRˆ1(g)v1,w1Rˆ2(g)v2,w2=0,forallv1,w1∈V1,v2,w2∈V2.G4.IfRˆisanirreducibleunitaryrepresentationofG,thenwehave(dimV)R(g)vˆ1,w1R(g)vˆ2,w2=v1,v2w1,w2,Gforallv1,w1∈V1,v2,w2∈V2.(B.5)5.(Peter–WeylTheorem)Ifwedenoteby(Rˆλ,Vλ),λ∈Λ,thesetofallirre-duciblerepresentationsofGandMλ,v,w(g)=Rˆλ(g)v,w,thenthelinearspacespannedby{Mλ,v,w(g)|g∈G,v,w∈Vλ}isdenseinL2(G,dμ). 398BLieGroupsandCoherentStatesB.3.3TheNon-compactCaseThiscaseismuchmoredifficultthanthecompactcaseandtherearenotyetageneraltheoryofirreducibleunitaryrepresentations.ThetypicalexampleisSU(1,1)orequivalentlySL(2,R)consideredinChap.8.Thesegroupshavethefollowingproperties.Definition47(i)ALiegroupGissaidreductiveifGisaclosedconnectedsub-groupofGL(n,R)orGL(n,C)stableunderinverseconjugatetranspose.(ii)ALieGissaidlinearconnectedsemi-simpleifGisreductivewithfinitecenter.Proposition144IfGisalinearconnectedsemi-simplegroupitsLiealgebragissemi-simple.ItisknownthatacompactconnectedLiegroupcanberealizedasalinearcon-nectedreductiveLiegroup([127],Theorem1.15).LetusconsidertheLiealgebragofG.ThedifferentialofthemappingΘ(A)=A−1,∗ate=1isdenotedθ.Wehaveθ2=1soθhastwoeigenvalues±1.Sowehavethedecompositiong=l⊕pwherel=ker(θ−1)andp=ker(θ+1).LetK={g∈G|θg=g}.Thefollowingresultisageneralizationofthepolarde-compositionformatricesoroperatorsinHilbertspaces.Proposition145(PolarCartandecomposition)IfGisalinearconnectedreductivegroupthenKisacompactconnectedgroupandisamaximalcompactsubgroupofG.ItsLiealgebraislandthemap:(k,X)→kexpXisadiffeomorphismfromK×pontoG.B.4CoherentStatesAccordingGilmorePerelomovHerewedescribeageneralsettingforatheoryofcoherentstatesinaarbitraryLiegroupfromthepointofviewofPerelomov(formoredetailssee[155,156]).WestartfromanirreducibleunitaryrepresentationRˆoftheLiegroupGintheHilbertspaceH.Letψ0∈Hbeafixedunitvector( ψ0 =1)anddenoteψg=R(g)ψˆ0foranyg∈G.InquantummechanicsstatesintheHilbertspaceHaredeterminedmoduloaphasefactorsowearemainlyinterestedintheactionofGintheprojectivespaceP(H)(spaceofcomplexlinesinH).WedenotebyHtheisotropygroupofψ0intheprojectivespace:H={h∈G|R(h)ψˆ0=eiθψ0}.Sothecoherentstatessystem{ψg|g∈G}isparametrizedbythespaceG/HofleftcosetinGmoduloH:ifπisthenaturalprojectionmap:G→G/H.Choos-ingforeachx∈G/Hsomeg(x)∈Gwehave,withx=π(g),ψg=eiθ(g)ψg(x).Moreoverψgandψgdefinethesamestatesifandonlyifπ(g1)=ψg:=x;hence122wehaveψg=eiθ1ψg(x)andψg=eiθ2ψg(x).12 B.4CoherentStatesAccordingGilmore–Perelomov399Soforeveryx∈G/Hwehavedefinedthestate|x={eiαψg}wherex=π(g).Itisconvenienttodenote|x=ψg(x)andx(g)=ψ(x).ThisparametrizationofcoherentstatesbythequotientspaceG/Hhasthefollowingniceproperties.Wehaveψg=eiθ(g)|x(g)andθ(gh)=θ(g)+θ(h)ifg∈Gandh∈H.TheactionofGonthecoherentstate|xsatisfiesR(gˆiβ(g1,x)|g1.x,(B.6)1)|x=ewhereg1.xdenotesthenaturalactionofGonG/Handβ(g1,x)=θ(g1g)−θ(g)whereπ(g)=x(βdependsonlyonx,notong).Computationofthescalarproductoftwocoherentstatesgivesxi(θ(g1)−θ(g2))−11|x2=e0|Rˆg1g2|0,(B.7)wherex1=x(g1)andx2=x(g2).Moreoverifx1=x2wehave|x1|x2|<1andi(β(g,x1)−β(g,x2))xg.x1|g.x2=e1|x2.(B.8)ConcerningcompletenesswehaveProposition146AssumethattheHaarmeasureonGinducesaleft-invariantmea-suredμ(x)onG/H(seeTheorem56)andthatthefollowingsquareintegrabilityconditionissatisfied:20|xdx<+∞.(B.9)MThenwehavetheresolutionofidentity:1x|ψψxdμ(x)=ψ,∀ψ∈H,(B.10)dMwhered=|0|x|2dx.MoreoverwehavethePlancherelidentityM12ψ|ψ=x|ψdμ(x).(B.11)dMRemark75(i)AswehaveseeninChap.2,usingformula(B.10)and(B.11),wecanconsiderWickquantizationforsymbolsdefinedonM=G/H.(ii)Whenthesquareintegrabilitycondition(B.9)isnotfulfilled(thePoincarégroupforexample)thereexistsanextendeddefinitionofcoherentstates.Thisisexplainedin[3].Remark76WhenGisacompactsemi-simpleLiegroupandRˆisaunitaryirre-duciblerepresentationofGinafinite-dimensionalHilbertspaceHthenitispossi-bletochooseastateψ0inHsuchthatifHistheisotropygroupofψ0thenG/HisaKählermanifold(see[148]).Someresultsconcerningcoherentstatesandquantizationhavealsobeenobtainedfornon-compactsemi-simpleLiegroupsextendingresultsalreadyseeninChap.8forSU(1,1). AppendixCBerezinQuantizationandCoherentStatesWehaveseeninChap.2thatcanonicalcoherentstatesarerelatedwithWickandWeylquantization.Berezin[20]hasgivenageneralsettingtoquantize“classicalsystems”.LetusexplainhereverybrieflytheBerezinconstruction.LetMbeaclassicalphasespace,i.e.asymplecticmanifoldwithaPoissonbracketdenoted{·,·},andanHilbertspaceH.Assumethatforasetofpositivenumbers,with0aslimitpoint,wehavealinearmappingA→AˆwhereAisasmoothfunctiononMandAˆisanoperatoronH.TheinversemappingisdenotedS(Aˆ).Ingeneralitisdifficulttodescribeindetailthedefinitiondomainandtherangeofthisquantizationmapping.Someexampleareconsideredin[187,188].Neverthelessforaquantizationmapping,thetwofollowingconditionsarere-quired,topreserveBohr’scorrespondencecondition(semi-classicallimit):limSAˆBˆ(m)=A(m)B(m),∀m∈M,(C1)→01limSAˆ,Bˆ(m)={A,B}(m),∀m∈M.(C2)→0iWehaveseeninChap.2thattheseconditionsarefulfilledfortheWeylquantiza-tionofR2n.In[20]theauthorshaveconsideredthetwodimensionalsphereandthepseudosphere(Lobachevskiiplane).InthesetwoexamplesthePlanckconstantisreplacedby1wherenisanintegerparameterdependingontheconsiderednrepresentation.Thesemi-classicallimitisthelimitn→+∞.Forthepseudospheren=2k,wherekistheBargmannindex.InthissectionweshallexplainsomeofBerezin’sideasconcerningquantizationonthepseudosphereandweshallprovethattheBohrcorrespondenceprincipleissatisfiedusingresultstakenfromChap.8.Thesameresultscouldbeprovedforquantizationofthesphere[20],usingresultsofChap.7.Nowadaysthequantizationproblemhasbeensolvedinmuchmoregeneralset-tings,inparticularforKählermanifolds(thePoincarédiscDortheRiemannsphere§2areexamplesofKählermanifolds),wheregeneralizedcoherentstatesarestillM.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,401TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0,©SpringerScience+BusinessMediaB.V.2012 402CBerezinQuantizationandCoherentStatespresent.ThisdomainisstillveryactiveandisnamedGeometricQuantization;itsstudyisoutsidethescopeofthisbook(seethebook[201]andtherecentre-view[173].Wehavedefinedbeforethecoherentstatesfamilyψζ,ζ∈D,fortherepresenta-tionsDn+ofthegroupSU(1,1)whichisasymmetrygroupforthePoincarédiscD.Recallthat{ψζ}ζ∈DisanovercompletesysteminHn(D);hencethemapϕ→ϕ,whereϕ(z)=ψz,ϕ,isanisometryfromHn(D)intoL2(D).LetAˆbeaboundedoperatorinHn(D).ItscovariantsymbolAc(z,w)¯isdefinedasψz,AψˆwAc(z,w)¯=.ψz,ψwItisaholomorphicextensionin(z,w)¯oftheusualcovariantsymbolAc(z,z)¯.More-over,theoperatorAˆisuniquelydeterminedbyitscovariantsymbolandwehaveAϕ(z)ˆ=Ac(z,w)ϕ(w)¯ψz,ψwdνn(w).(C.1)DFrom(C.1)wegetaformulaforthecovariantsymbolproductoftheproductoftwooperatorsA,ˆBˆ.If(AB)cdenotesthecovariantsymbolofAˆBˆthenwehavetheformula2(AB)c(z,z)¯=Ac(z,w)B¯c(w,z)¯ψz,ψwdνn(w).(C.2)DFromourpreviouscomputations(Chap.8)wehave22n2(1−|z|)(1−|w|)ψw,ψz=2.|1−¯zw|Wehavetoconsiderthefollowingoperator:22nn−1(1−|z|)(1−|w|)TnF(z,z)¯=F(w,w)¯dμ(w)4πD|1−¯zw|2forFboundedinDandC2-smooth.Proposition147Wehavethefollowingasymptoticexpansion,forn→+∞:n12∂2FT2nF(z,z)¯=F(z,z)¯1−+1−|z|(z,z).¯(C.3)(n−2)2n∂z∂z¯ProofUsinginvariancebyisometriesofDweshowthatitisenoughtoprovefor-mula(C.3)forz=0.ζ−zLetusconsiderthechangeofvariablew=.DenoteG(ζ,ζ)¯=F(w,w)¯;1−¯zζthenwegetTnF(z,z)¯=TnG(0,0).Adirectcomputationgives CBerezinQuantizationandCoherentStates403∂2G2∂2F(0,0)=1−|zz¯|(z,z)¯=F(z;¯z),∂ζ∂ζ¯∂z∂z¯whereistheLaplace–BeltramioperatoronD.Sowehaveproved(C.3)foranyzifitisprovedforz=0.Toprove(C.3)forz=0weusetherealLaplacemethodforasymptoticexpan-sionofintegrals(seeSect.A.4).Wewrite2π1n−1iθ−iθ−(n−2)φ(r)TnF(0,0)=dθdrFre,rere,(C.4)π00whereφ(r)=log((1−r2)−1).Notethatφ(r)>0forr∈]0,1]andφ(0)=1.SowecanapplytheLaplacemethod(seeSect.A.4foraprecisestatement)toestimate(C.4)withthelargeparameterλ=n−2.Moduloanexponentiallysmalltermitisenoughtointe-grateoverrin[0,1/2].Usingthechangeofvariableφ(r)=s2,s>0,wehaver(s)=1−e−s2and2πcn−1−λs−s2−∞TnF(0,0)=dθdseKr(s)se+Oλ,π00whereK(r)=F(reiθ,re−iθ).Nowtogettheresultwehavetocomputetheasymp-toticexpansionats=0forL(s)=K(r(s))se−s2.NotethatL(s)isperiodicinθandwehavetoconsideronlythepartoftheexpansionindependentinθ.IfL0(s)isthispart,wegetaftercomputationn−1n−1∂2F1L0(s)=F(0,0)+(0,0)−F(0,0)+O2π(n−2)2π(n−2)2∂z∂z¯n2andformula(C.4)follows.Itisnotdifficult,usingProposition147,tocheckthecorrespondenceprinciple(C1)and(C2).Weget(C1)byapplyingthePropositiontoFAB(w,w)¯=Ac(z,w)B¯c(w,z)¯.Sowehave(AB)c(z,z)¯=Tn(z,z)¯−→Ac(z,z)B¯c(z,z).¯n→+∞For(C2)wewritenFAB(z,z)¯=Ac(z,z)B¯c(z,z)¯1−(n−1)2122∂Ac∂Bc1+1−|z|(z,z)¯(z,z)¯+O.n∂w¯∂wn2 404CBerezinQuantizationandCoherentStatesSoweget22∂Ac∂BcnFAB(z,z)¯−FBA(z,z)¯=1−|z|(z,z)¯(z,z)¯∂z¯∂z∂Ac∂Bc1−(z,z)¯(z,z)¯+O.∂z∂z¯nButweknowthatFAB−FBAisthecovariantsymbolofthecommutator[A,ˆBˆ]andthePoissonbracket{A,B}is22∂Ac∂Bc∂Ac∂Bc{A,B}(z,z)¯=i1−|z|(z,z)¯(z,z)¯−(z,z)¯(z,z)¯.∂z¯∂z∂z∂z¯Soweget(C2).WehaveseenthatthelinearsymplecticmapsandthemetaplectictransformationsareconnectedwithquantizationoftheEuclideanspaceR2n.Heresymplectictrans-formationsarereplacedbytransformationsinthegroupSU(1,1)andmetaplectictransformationsbytherepresentationsg→R(g)ˆ=Dn−(g).ThenwehaveProposition148(i)ForanyboundedoperatorAˆinHn(D)thecovariantsymbolAgofR(g)ˆAˆ×R(g)ˆ−1isA−1−1g(ζ)=Acgζ,gζ.(C.5)(ii)ThecovariantsymbolR(g)cofR(g)ˆisgivenbytheformula1−|ζ|2R(g)inarg(α+βζ)(C.6)c(ζ)=eα¯+β¯ζ¯−βζ−α|ζ|2αβwhereg=β¯α¯.Proof(i)isadirectconsequenceofdefinitionofDn−(g)andcomputationsofChap.8.For(ii)usingformula(8.74)ofChap.8incomplexvariablesweget,aftercom-putation,n2α+βz1−|ζ|ψζ,ψg−1ζ=2|α+βz|α¯+β¯ζ¯−βζ−α|ζ|whichgives(C.6). References1.Abe,S.,Suzuki,N.:Wignerdistributionfunctionofasimpleopticalmodel:anextended-phase-spaceapproach.Phys.Rev.A45,520–523(2001)2.Aharonov,Y.,Anandan,J.:Phasechangeduringacyclicquantumevolution.Phys.Rev.Lett.58,1593–1596(1987)3.Ali,S.-T.,Antoine,J.-P.,Gazeau,J.-P.:CoherentStatesWaveletsandTheirGeneralizations.Springer,Berlin(2000)4.Aravind,P.K.:Spincoherentstatesasanticipatorsofthegeometricphase.Am.J.Phys.10(October),899–904(1999)5.Arecchi,F.T.,Courtens,E.,Gilmore,R.,Thomas,H.:Atomiccoherentstatesinquantumoptics.Phys.Rev.A6,2211–2237(1972)6.Arnaud,J.A.:Hamiltoniantheoryofbeammodepropagation.In:Wolf,E.(ed.)ProgressinOpticsXI.North-Holland,Amsterdam(1973)7.Arnold,V.:MathematicalMethodsofClassicalMechanics.Springer,NewYork(1999)8.Babich,V.:Eigenfunctionsconcentratedinaneighborhoodofaclosedgeodesic.In:Babich,V.M.(ed.)Math.ProblemsinWavePropagationTheory.Sem.Math.,V.A.SteklovMath.Inst.,vol.9.SteklovMathematicalInstitute,Leningrad(1970).TranslatedbyConsultantsBureau,NewYork(1970)9.Babich,V.M.,Buldyrev,V.S.:AsymptoticMethodsinShortWaveDiffractionProblems.Nauka,Moscow(1972)10.Balazs,N.,Voros,A.:Thequantizedbaker’stransformation.Ann.Phys.190,1–31(1989)11.Balazs,N.L.,Voros,A.:Chaosonthepseudosphere.Phys.Rep.143(3),109–240(1986)12.Balian,R.,Bloch,C.:Distributionofeigenfrequenciesforthewaveequationinafinitedo-main.Ann.Phys.69,76–160(1972)13.Balian,R.,Bloch,C.:Distributionofeigenfrequenciesforthewaveequationinafinitedo-main.Ann.Phys.85,514–545(1974)14.Bander,M.,Itzykson,C.:Grouptheoryandthehydrogenatom.Rev.Mod.Phys.38,330–345(1966)15.Bargmann,V.:IrreducibleunitaryrepresentationsoftheLorentzgroup.Ann.Math.(2)48,568–640(1947)16.Bargmann,V.:Onunitaryrayrepresentationsofcontinuousgroups.Ann.Math.(2)59,1–46(1954)17.Bargmann,V.:OnaHilbertspaceofanalyticfunctions.Commun.PureAppl.Math.14,187–214(1961)18.Beals,R.:Propagationdessingularitéspourdesopérateurs.In:JournéesE.D.PSaintJeandeMonts,conf.,No.19(1980)19.Berezin,F.A.:TheMethodofSecondQuantization.AcademicPress,NewYork(1966)20.Berezin,F.A.:Generalconceptofquantization.Commun.Math.Phys.40,153–174(1975)M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,405TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0,©SpringerScience+BusinessMediaB.V.2012 406References21.Berezin,F.A.:ModelsofGross–Neveutypearequantizationofaclassicalmechanicswithnonlinearphasespace.Commun.Math.Phys.63,131–153(1978)22.Berezin,F.A.:IntroductiontoSuperanalysis.Reidel,Dordrecht(1987)23.Berezin,F.A.,Shubin,M.A.:TheSchrödingerEquation.KluwerAcademic,Dordrecht(1991)24.Berry,M.:Quantalphasefactorsaccompanyingadiabaticchanges.Proc.R.Soc.Lond.Ser.A,Math.Phys.Sci.292,47–57(1984)25.Bertrand,J.,Irac-Astaud,M.:Characterizationofsu(1,1)coherentstatesintermsofaffinewavelets.J.Phys.A,Math.Gen.35,7347–7357(2002)26.Bily,J.M.:Propagationd’étatscohérentsetapplications.Ph.D.thesis,UniversitédeNantes(2001)27.Bonechi,F.,DeBièvre,S.:Controllingstrongscarringforquantizedergodictoralautomor-phisms.DukeMath.J.117,571–587(2003)28.Bouclet,J.M.,DeBièvre,S.:Longtimepropagationandcontrolonscarringforperturbedquantizedhyperbolictoralautomorphism.Ann.HenriPoincaré6,885–913(2005)29.Bouzouina,A.:Comportementsemi-classiquedesymplectomorphismesdutorequantifiés.Ph.D.thesis,UniversitédeParis-Dauphine(1997)30.Bouzouina,A.,DeBièvre,S.:Equipartitionoftheeigenfunctionsofquantizedergodicmapsonthetorus.Commun.Math.Phys.178,83–105(1996)31.Bouzouina,A.,Robert,D.:Uniformsemiclassicalestimatesforthepropagationofquantumobservables.DukeMath.J.111,223–252(2002)32.Boya,L.J.,Perelomov,A.M.,Santander,M.:BerryphaseinhomogeneousKählermanifoldswithlinearHamiltonians.J.Math.Phys.42(11),5130–5142(2001)33.Bratteli,O.,Robinson,D.W.:OperatorAlgebraandQuantumStatisticalMechanicsII.Springer,NewYork(1981)34.Brocker,T.,tomDieck,T.:RepresentationsofcompactLiegroups.In:GraduateTextsinMathematicsvol.17,pp.7–87.Springer,NewYork(2007)35.Burq,N.:Mesuressemi-clasiquesetmesuresdeDéfaut.In:SéminaireN.Bourbaki(1996–1997)36.Buser,P.:GeometryandspectraofcompactRiemannsurfaces.In:GeometryandSpectraofCompactRiemannSurfaces.ProgressinMathematics,vol.106.Birkhäuser,Boston(1992)37.Cahill,K.E.,Glauber,R.Y.:Densityoperatorsforfermions.Phys.Rev.A59(2),1538–1555(1999)38.Cassanas,R.:Hamiltoniensquantiquesetsymétries.Ph.D.thesis,UniversitédeNantes(2005)39.Cesari,L.:AsymptoticBehaviourandStabilityProblemsinOrdinaryDifferentialEquations.Springer,Berlin(1963)40.Charbonnel,A.M.,Popov,G.:Semiclassicalasymptoticsforseveralcommutingoperators.Commun.PartialDiffer.Equ.24,283–323(1998)41.Chazarain,J.:FormuledePoissonpourlesvariétésriemanniennes.Invent.Math.24,65–82(1974)42.Chazarain,J.:Spectred’unhamiltonienquantiqueetmécaniqueclassique.Commun.PartialDiffer.Equ.6,595–644(1980)43.Chen,L.,OonLee,J.:RateofconvergenceinnonlinearHartreedynamicswithfactorizedinitialdata.J.Math.Phys.52,1–25(2011)44.ColindeVerdière,Y.:SpectreduLaplacienetlongueursdesgéodésiquespériodiquesI.Com-ment.Math.Helv.27,83–106(1973)45.ColindeVerdière,Y.:Equipartitionoftheeigenfunctionsofquantizedergodicmapsonthetorus.Commun.Math.Phys.102,497–502(1985)46.ColindeVerdière,Y.:Bohr–Sommerfeldrulestoallorders.Ann.HenriPoincaré6(5),925–936(2005)47.Combescure,M.:Aquantumparticleinaquadrupoleradio-frequencytrap.Ann.Inst.HenriPoincaré.Phys.Théor.44,293–314(1986) References40748.Combescure,M.:Ageneralizedcoherentstateapproachofthequantumdynamicsforsuit-abletime-dependentHamiltonians.In:JournéesEquationsauxDérivéesPartielles,Saint-Jean-de-Monts(1992)49.Combescure,M.:Thesqueezedstatesapproachofasemi-classicallimitofthetimedepen-dentSchrödingerequation.J.Math.Phys.33,3870–3880(1992)50.Combescure,M.:Circulantmatrices,Gausssumsandmutuallyunbiasedbases,I.Theprimenumbercase.CUBO11,73–86(2009)51.Combescure,M.,Robert,D.:Semiclassicalspreadingofquantumwavepacketsanapplica-tionsnearunstablefixedpointsoftheclassicalflow.Asymptot.Anal.14,377–404(1997)52.Combescure,M.,Robert,D.:Quadraticquantumhamiltoniansrevisited.CUBO8(1),61–86(2006)53.Combescure,M.,Ralston,J.,Robert,D.:AproofoftheGutzwillersemi-classicalformulausingcoherentstatesdecomposition.Commun.Math.Phys.202,463–480(1999)54.Cordoba,A.,Fefferman,C.:WavepacketsandFourierintegraloperators.Commun.PartialDiffer.Equ.3(11),979–1005(1978)55.Cornfeld,I.P.,Fomin,S.V.,Sinai,Y.G.:ErgodicTheory.Springer,Berlin(1982)56.Courant,R.,Hilbert,D.:MethodsofMathematicalPhysics.Wiley,NewYork(1966)57.Davis,P.:CirculantMatrices.Wiley,NewYork(1979)58.DeBièvre,S.,Faure,F.,Nonnenmacher,S.:Scarredeigenstatesforquantumcatmapsofminimalperiods.Commun.Math.Phys.239,449–492(2003)59.DeBièvre,S.,DegliEsposti,M.,Giachetti,R.:Quantizationofaclassofpiecewiseaffinetransformationsofthetorus.Commun.Math.Phys.176,73–94(1995)60.DeGosson,M.:MaslovClasses,MetaplecticRepresentationandLagrangianQuantization.AkademieVerlag,Berlin(1997)61.DeGosson,M.:TheWeylrepresentationofmetaplecticoperators.Lett.Math.Phys.72,129–142(2005)62.DeWitt,B.:Supermanifolds.CambridgeUniversityPress,Cambridge(1984)63.DegliEsposti,M.:Quantizationoftheorientationpreservingautomorphismofthetorus.Ann.Inst.HenriPoincaré.Phys.Théor.58,323–341(1993)64.DegliEsposti,M.,Graffi,S.,Isola,S.:StochasticpropertiesofthequantumArnold’scatintheclassicallimit.Commun.Math.Phys.176,73–94(1995)65.Deligne,P.,Etingof,P.,Freed,D.,Kazhdan,D.,Morgan,J.,Morrison,D.,Witten,E.:Quan-tumFieldsandStrings:aCourseforMathematicians,vol.1.AMS,Providence(1999)66.Derezinski,J.,Gérard,C.:ScatteringTheoryofClassicalandQuantumn-ParticleSystems.Springer,Berlin(1997)67.Derezinski,J.,Napiorkowski,M.,Solovej,J.P.:OntheminimizationofHamiltoniansoverpureGaussianstates.arXiv:1102.2931[math-ph](2011)68.Dimassi,M.,Sjöstrand,J.:SpectralAsymptoticsintheSemi-classicalLimit.LectureNoteSeries,vol.268.Lond.Math.Soc.,London(1999)69.Dirac,P.A.:LecturesonQuantumMechanics.Dover,Princeton(2001)70.Dozias,S.:Opérateursh-pseudodifférentielsàflotpériodique.Ph.D.thesis,UniversitédeParis-Nord(1994)71.Duistermaat,J.J.,Guillemin,V.:Thespectrumofpositiveellipticoperatorsandperiodicbicharacteristics.Invent.Math.29,39–79(1975)72.Duistermaat,J.J.,Kolk,J.A.C.:LieGroups.Springer,NewYork(1999)73.Egorov,Y.V.:Oncanonicaltransformationsofpseudodifferentialoperators.Usp.Mat.Nauk25,235–236(1969)74.Ehrenfest,P.:Bemerkungüberdieangenahertegultigkeitderklassischenmechanikinnerhalbderquantenmechanik.Z.Phys.45,455–456(1927)(Remarksontheapproximatevalidityofclassicalmechanicsinquantummechanics)75.Erikson,F.:Onthemeasureofsolidangles.Math.Mag.63,184–185(1999)76.Fedosov,B.:DeformationQuantizationandIndexTheory.AkademieVerlag,Berlin(1996)77.Folland,G.B.:HarmonicAnalysisinthePhaseSpace.PrincetonUniversityPress,Princeton(1989) 408References78.Folland,G.B.:QuantumFieldTheory.MathematicalSurveysandMonographs.Am.Math.Soc.,Providence(2008)79.Galaviz,I.,Garcia-Compean,H.,Przanowski,M.,Turrubiates,F.J.:Weyl–Wigner–MoyalformalismforFermiclassicalsystems.Ann.Phys.323,267–290(2008)80.Gazeau,J.-P.:CoherentStatesinQuantumPhysics.Wiley-VCH,Berlin(2009)81.Gelfand,I.,Neumark,M.:UnitaryrepresentationsoftheLorentzgroup.Acad.Sci.USSRJ.Phys.X(3),93–94(1946)82.Gérard,P.:Microlocaldefectmeasures.Commun.PartialDiffer.Equ.16,1761–1794(1991)83.Gérard,P.,Leichtnam,E.:ErgodicpropertiesoftheeigenfunctionsfortheDirichletproblem.Commun.Math.Phys.71,559–607(1993)84.Gilmore,R.:Geometryofsymmetrizedstates.Ann.Phys.(NY)74,391–463(1972)85.Gilmore,R.:Onpropertiesofcoherentstates.Rev.Mex.Fis.23,143–187(1974)86.Ginibre,J.,Velo,G.:Theclassicalfieldlimitofscatteringtheoryofnonrelativisticmanybosonssystems.II.Commun.Math.Phys.68,45–68(1979)87.Ginibre,J.,Velo,G.:Theclassicalfieldlimitofnonrelativisticbosons.I.Borelsummabilityforboundedpotentials.Ann.Phys.128(2),243–285(1980)88.Ginibre,J.,Velo,G.:Theclassicalfieldlimitofnonrelativisticbosonssystems,II.Asymp-toticexpansionsforgeneralpotentials.Ann.Inst.HenriPoincaré.Phys.Théor.33,363–394(1980)89.Ginibre,J.,Velo,G.:OnaclassofnonlinearSchrödingerequationwithnonlocalinterac-tions.Math.Z.170,109–136(1980)90.Glauber,R.:Coherentandincoherentstatesoftheradiationfield.Phys.Rev.131,2766–2788(1963)91.Glauber,R.:Thequantumtheoryofopticalcoherence.Phys.Rev.130,2529–2539(1963)92.Goldstein,H.:ClassicalMechanics.Addison-Wesley,Reading(1980)93.Grossmann,A.,Morlet,J.,Paul,T.:Transformsassociatedtosquareintegrablegrouprepre-sentations.I.Generalresults.J.Math.Phys.26,2473–2479(1985)94.Guichardet,A.:Histoired’unvecteurtricentenaire.Gaz.Math.117,23–33(2008)95.Guillemin,V.,Sternberg,S.:GeometricAsymptotics.MathematicalSurveys,vol.14.AMS,Providence(1977)96.Guillemin,V.,Uribe,A.:Circularsymmetryandthetraceformula.Invent.Math.96,385–423(1989)97.Gutzwiller,M.:Periodicorbitsandclassicalquantizationconditions.J.Math.Phys.12,343–358(1971)98.Gutzwiller,M.:ChaosinClassicalandQuantumMechanics.Springer,Berlin(1991)99.Hagedorn,G.:SemiclassicalquantummechanicsIII.Ann.Phys.135,58–70(1981)100.Hagedorn,G.:SemiclassicalquantummechanicsIV.Ann.Inst.HenriPoincaré.Phys.Théor.42,363–374(1985)101.Hagedorn,G.,Joye,A.:Semiclassicaldynamicswithexponentiallysmallestimates.Com-mun.Math.Phys.207,439–465(1999)102.Hagedorn,G.,Joye,A.:Exponentiallyaccuratesemi-classicaldynamics:propagation,lo-calization,Ehrenfesttimes,scatteringandmoregeneralstates.Ann.HenriPoincaré15,837–883(2000)103.Hall,K.R.,Meyer,G.R.:IntroductiontoHamiltonianDynamicalSystemsandtheN-bodyproblem.AppliedMathematicalSciences,vol.90.Springer,Berlin(1991)104.Hannay,J.,Berry,M.:ErgodicpropertiesoftheeigenfunctionsfortheDirichletproblem.PhysicaD1,267–291(1993)105.Helein,F.:Arepresentationformulaformapsonsupermanifolds.J.Math.Phys.49,1–19(2008)106.Helffer,B.,Martinez,A.,Robert,D.:Ergodicitéetlimitesemi-classique.Commun.Math.Phys.109,313–326(1987)107.Helffer,B.,Robert,D.:CalculfonctionnelparlatransformationdeMellinetopérateursadmissibles.J.Funct.Anal.53(3),246–268(1983)108.Helffer,B.,Robert,D.:Puitsdepotentielsgénéralisésetasymptotiquesemi-classique.Ann.Inst.HenriPoincaré.Phys.Théor.41,291–331(1984) References409109.Helffer,B.,Sjöstrand,J.:Multiplewellsinthesemi-classicallimitI.Commun.PartialDiffer.Equ.9,337–408(1984)110.Heller,E.J.:Timedependentapproachtosemiclassicaldynamics.J.Chem.Phys.62,1544–1555(1975)111.Heller,E.J.:Quantumlocalizationandtherateofexplorationofphasespace.Phys.Rev.A35,1360–1370(1987)112.Henneaux,M.,Teitelboim,C.:QuantizationofGaugeSystems.PrincetonUniversityPress,Princeton(1994)113.Hepp,K.:Theclassicallimitofquantummechanicalcorrelationfunctions.Commun.Math.Phys.35,265–277(1974)114.Hirshfeld,A.,Henselder,P.:Deformationquantizationforsystemswithfermions.Ann.Phys.302,59–77(2002)115.Hislop,P.,Sigal,M.:IntroductiontoSpectralTheory.AppliedMathematicalSciences,vol.113.Springer,Berlin(1996)116.Hörmander,L.:Thespectralfunctionofanellipticoperator.ActaMath.121,173–218(1968)117.Hörmander,L.:TheAnalysisofLinearPartialDifferentialOperators.GrundlehrenderMathematischenWissenschaften,vol.256.Springer,Berlin(1983)118.Hörmander,L.:SymplecticclassificationofquadraticformsandgeneralMelherformulas.Math.Z.219,413–449(1995)119.Howland,J.:TwoProblemswithTime-DependentHamiltonians.LectureNotesinPhysics,vol.130(1980)120.Hudson,R.L.:WhenistheWignerquasi-probabilitynonnegative?Rep.Math.Phys.6,249–252(1974)121.Inönü,E.,Wigner,E.:Onthecontractionofgroupsandtheirrepresentations.Proc.Natl.Acad.Sci.USA39,510–524(1953)122.Ivrii,V.:MicrolocalAnalysisandPreciseSpectralAsymptotics.Springer,Berlin(1998)123.Katok,A.,Hasselblatt,B.:IntroductiontotheModernTheoryofDynamicalSystems.Cam-bridgeUniversityPress,Cambridge(1995)124.Keller,J.B.,Streifer,W.:ComplexrayswithanapplicationtoGaussianbeams.J.Opt.Soc.Am.61,40–43(1971)125.Khuat-Duy,D.:Formuledestracessemi-classiquepourunénergiecritiqueetconstructiondequasi-modesàl’aided’étatscohérents.Ph.D.thesis,UniversitédeParis-Dauphine(1996)126.Klauder,J.R.,Skagerstam,B.S.:CoherentStates.WorldScientific,Singapore(1985)127.Knapp,A.W.:RepresentationTheoryofSemisimpleGroups.PrincetonMathematicalSeries,vol.36.PrincetonUniversityPress,Princeton(1986).Anoverviewbasedonexamples128.Knapp,A.W.:LieGroupsBeyondanIntroduction.PrincetonMathematicalSeries,vol.36.Birkhäuser,Boston(1996).Anoverviewbasedonexamples129.Knapp,A.W.,Trapa,P.E.:RepresentationsofsemisimpleLiegroups.In:RepresentationThe-oryofLiegroups,ParkCity,UT,1998.IAS/ParkCityMath.Ser.,vol.8,pp.7–87.Am.Math.Soc.,Providence(2000)130.Kosmann-Schwarzbach,Y.:GroupsandSymmetries.Universitext.Springer,NewYork(2010).FromfinitegroupstoLiegroups,Translatedfromthe2006French2ndeditionbyStephanieFrankSinger131.Leboeuf,P.,Voros,A.:Chaos-revealingmultiplicativerepresentationofquantumeigenstates.J.Phys.A23,1165–1174(1990)132.Leites,D.A.:Introductiontothetheoryofsupermanifolds.Russ.Math.Surv.35,3–57(1980)133.Leray,J.:LagrangianAnalysisandQuantumMechanics.MIT,Cambridge(1981)134.Leray,J.:LagrangianAnalysisandQuantumMechanics.AMathematicalStructureRelatedtoAsymptoticExpansionsandtheMaslovIndex.MITPress,Boston(1981)(trans.fromFrench)135.Lerner,N.:TheWickcalculusofpseudo-differentialoperatorsandsomeofitsapplications.CUBO5(1),213–236(2003) 410References136.Lieb,E.H.:Theclassicallimitofquantumspinsystems.Commun.Math.Phys.31,327–340(1973)137.Lions,P.L.,Paul,T.:SurlesmesuresdeWigner.Rev.Mat.Iberoam.9,553–618(1993)138.Littlejohn,R.:Thesemiclassicalevolutionofwavepackets.Phys.Rep.138,193–291(1986)139.Maas,H.:Siegel’sModularFormsandDirichletSeries.LectureNotesinMathematics,vol.216.Springer,Berlin(1971)140.Manè,R.:ErgodicTheoryandDifferentiableDynamics.Springer,Berlin(1987)141.Martinez,A.:AnIntroductiontoSemiclassicalandMicrolocalAnalysis.Universitext.Springer,NewYork(2002)142.McLachland,N.W.:TheoryandApplicationofMathieuFunctions.OxfordUniversityPress,NewYork(1947)143.Mehlig,B.,Wilkinson,M.:Semiclassicaltraceformulausingcoherentstates.Ann.Phys.(Leipz.)10,501(2001)144.Meinrenken,E.:Semi-classicalprincipalsymbolsandGutzwiller’straceformula.Rep.Math.Phys.31,279–295(1992)145.Moyal,J.E.:Quantummechanicsasastatisticaltheory.Proc.Camb.Philos.Soc.45,99–124(1959)146.Mukada,N.,Aravind,P.,Simon,R.:Wignerrotations,Bargmanninvariantsandgeometricphases.J.Phys.A,Math.Gen.36,2347–2370(2003)147.Muller,C.:Sphericalharmonics.In:LectureNotesinMathematics,vol.17,pp.7–87.Springer,NewYork(1966)148.Onofri,E.:AnoteoncoherentstaterepresentationsofLiegroups.J.Math.Phys.16,1087–1089(1975)149.Pancharatnam,S.:Generalizedtheoryofinterferenceanditsapplications.Proc.IndianAcad.Sci.,Sect.A,Phys.Sci.44,247–262(1956)150.Paul,T.,Uribe,A.:Surlaformulesemi-classiquedestraces.C.R.Math.Acad.Sci.Ser.I,Math.313(1),217–222(1991)151.Paul,T.,Uribe,A.:Aconstructionofquasi-modesusingcoherentstates.Ann.Inst.H.Poincaré.Phys.Théor.59(4),357–381(1993)152.Paul,T.,Uribe,A.:Thesemiclassicaltraceformulaandpropagationofwavepackets.Com-mun.Math.Phys.132,192–242(1995)153.Paul,T.,Uribe,A.:Onthepointwisebehaviorofsemiclassicalmeasures.Commun.Math.Phys.175,229–256(1996)154.Pauli,W.:GeneralPrinciplesofQuantumMechanics.Springer,Berlin(1980).TranslatedfromtheGermanbyP.AchuthanandK.Venkatesan,withanintroductionbyCharlesP.Enz155.Perelomov,A.M.:CoherentstatesforarbitraryLiegroup.Commun.Math.Phys.26,222–236(1972)156.Perelomov,A.M.:GeneralizedCoherentStatesandTheirApplications.TextsandMono-graphsinPhysics.Springer,Berlin(1986)157.Petkov,V.,Popov,G.:SemiclassicaltraceformulaandclusteringofeigenvaluesforSchrödingeroperators.Ann.Inst.HenriPoincaré.Phys.Théor.68,17–83(1998)158.Petkov,V.,Robert,D.:Asymptotiquesemiclassiqueduspectred’hamiltoniensquantiquesettrajectoiresclassiquespériodiques.Commun.PartialDiffer.Equ.10,365–390(1985)159.Ralston,J.:Ontheconstructionofquasimodesassociatedwithstableperiodicorbits.Com-mun.Math.Phys.51,219–242(1976)160.Ralston,J.:Gaussianbeamsandpropagationofsingularities.In:StudiesinPartialDifferen-tialEquations.MAAStud.Math.,vol.23,pp.246–248(1982)161.Ramis,J.P.:Lessériesk-sommablesetleursapproximations.In:Analysis,MicrolocalCal-culusandRelativisticQuantumTheory,Proceedings,LesHouches,1979.LectureNotesinPhysics,vol.126,pp.178–199.Springer,Berlin(1980)162.Reed,M.,Simon,B.:Fourier-Analysis.Self-Adjointness.AcademicPress,NewYork(1979)163.Robert,D.:AutourdeL’approximationSemi-classique.ProgressinMathematics,vol.68.Birkhäuser,Basel(1987) References411164.Robert,D.:Propagationofcoherentstatesinquantummechanicsandapplications.In:Wang,X.P.,etal.(ed.)PartialDifferentialEquationsandApplications.ProceedingsoftheCIMPASchool,Lanzhou,China,July2004.SociétéMathématiquedeFrance,Paris(2004).Sémin.Congr.15,181–252(2007)165.Robert,D.,Tamura,H.:Asymptoticbehaviorofscatteringamplitudesonsemi-classicalandlowenergylimits.Ann.Inst.Fourier39,155–192(1989)166.Robin,J.,Salamon,D.:TheMaslovindexforpath.Topology32,827–844(1993)167.Robinson,S.L.:ThesemiclassicallimitofthequantumdynamicsII:Scatteringtheory.Ann.Inst.HenriPoincaré48,281–296(1988)168.Rodnianski,I.,Schlein,B.:Quantumfluctuationsandrateofconvergencetowardsmeanfielddynamics.Commun.Math.Phys.291,31–61(2009)169.Rogers,A.:Supermanifolds.TheoryandApplications.WorldScientific,Singapore(2007)170.Rudin,W.:RealandComplexAnalysis.McGraw-Hill,NewYork(1966)171.Ruelle,D.:StatisticalMechanics.Benjamin,NewYork(1969)172.Sattinger,D.H.:LieGroupsandAlgebraswithApplicationstoPhysics,GeometryandMe-chanics.Springer,NewYork(1986)173.Schlichenmaier,M.:Berezin–ToeplitzquantizationforcompactKählermanifolds.Areviewofresults.Adv.Math.Phys.2010,1–38(2010)174.Schnirelman,A.:Ergodicpropertiesofeigenfunctions.Usp.Mat.Nauk29,181–182(1974)175.Schrödinger,E.:DerstetigeübergangvonderMikro-zurMakromechanik.Naturwis-senschaften14,664–666(1926)176.Schrödinger,E.:Furtherstudiesonsolvingeigenvalueproblemsbyfactorization.Proc.R.Ir.Acad.,AMath.Phys.Sci.46,183–206(1940)177.Simon,B.:Theclassicallimitofquantumpartitionfunctions.Commun.Math.Phys.71,247–276(1980)178.Sjöstrand,J.:PropagationofanalyticsingularitiesforsecondorderDirichletproblems.Com-mun.PartialDiffer.Equ.5(1),41–95(1980)179.Sjöstrand,J.:Singularitésanalytiquesmicrolocales.Astérisque95(1982)180.Sokal,A.:AnimprovementofWatson’stheoremonBorelsummability.J.Math.Phys.21,261–263(1980)181.Soto,F.,Claverie,P.:WhenaretheWignerfunctionofmultidimensionalsystemnonnega-tive?J.Math.Phys.24,97–100(1983)182.Takhtajan,L.A.:Quantummechanicsformathematicians.In:QuantumMechanicsforMath-ematicians.GraduateStudiesinMathematics,vol.95.AMS,Providence(2008)183.Tataru,D.:Phasespacetransformandmicrolocalanalysis.Publ.Cent.Ri.Mat.EnnioGiorgi,pp.505–524(2004)184.Thaller,B.:TheDiracOperator.Springer,Berlin(1990)185.Thomas,L.E.,Villegas-Blas,C.:AsymptoticsofRydbergstatesforthehydrogenatom.Com-mun.Math.Phys.187,623–645(1997)186.Unterberger,A.:Oscillateurharmoniqueetopérateurspseudo-différentiels.Ann.Inst.FourierXXXIX(3),201–221(1979)187.Unterberger,A.:Quantization:someproblems,tools,andapplications.In:Contemp.Math.,vol.212,pp.285–298.AMS,Providence(1998)188.Unterberger,A.,Unterberger,J.:Representationsofsl(2,r)andsymboliccalculi.IntegralEqu.Oper.Theory18,303–334(1994)189.Uribe,A.:AsymbolcalculusforaclassofpseudodifferentialoperatorsonSnandbandasymptotics.J.Funct.Anal.59,535–556(1984)190.Varadarajan,V.S.:SupersymmetryforMathematicians:AnIntroduction.Am.Math.Soc.,Providence(2004)191.Villegas-Blas,C.:TheLaplacianonthen-sphere,thehydrogenatomandtheBargmannspacerepresentation.Ph.D.thesis,MathematicsDepartment,UniversityofVirginia(1996)192.Wang,X.P.:Time-delayoperatorsinsemiclassicallimitII.Shortrangepotentials.Trans.Am.Math.Soc.322,1509–1563(1990)193.Weil,A.:Surcertainsgroupesd’opérateursunitaires.ActaMath.111,143–211(1964) 412References194.Weinberg,S.:TheQuantumTheoryofFields,Vols.I,II,III.CambridgeUniversityPress,Cambridge(1995)195.Weinstein,A.:AsymptoticsoftheeigenvaluesclustersfortheLaplacianplusapotential.DukeMath.J.44,883–892(1977)196.Weyl,H.:TheClassicalGroups.PrincetonLandmarksinMathematics.PrincetonUniversityPress,Princeton(1997).Theirinvariantsandrepresentations,Fifteenthprinting,PrincetonPaperbacks197.Wigner,E.:OnunitaryrepresentationsoftheinhomogeneousLorentzgroup.Ann.Math.(2)40(1),149–204(1939)198.Wigner,E.P.:GroupTheoryandItsApplicationstoQuantumMechanicsofAtomicSpectra.AcademicPress,NewYork(1959)199.Wilkinson,M.:Asemi-classicalsumruleformatrixelementsofclassicallychaoticsystems.J.Phys.A,Math.Gen.20,2415–2423(1987)200.Witten,E.:SupersymmetryandMorsetheory.J.Differ.Geom.17,661–692(1982)201.Woodhouse,N.M.J.:GeometricQuantization.Clarendon,Oxford(1991)202.Yajima,K.:ScatteringtheoryforSchrödingerequationswithpotentialsperiodicintime.J.Math.Soc.Jpn.29,729–743(1977)203.Yajima,K.:Thequasiclassicallimitofquantumscatteringtheory.Commun.Math.Phys.69,101–129(1979)204.Yajima,K.:ExistenceofsolutionsforSchrödingerevolutionequations.Commun.Math.Phys.110,415–426(1987)205.Yajima,K.:Schrödingerevolutionequationswithmagneticfields.J.Anal.Math.56,29–76(1991)206.Zelditch,S.:Uniformdistributionoftheeigenfunctionsoncompacthyperbolicsurfaces.DukeMath.J.55,919–941(1987)207.Zinn-Justin,J.:IntégraledeCheminenMécaniqueQuantique:Introduction.CNRSEdi-tions/EDPSciences,Paris(2003) IndexACleanintersectionhypothesis,129Adjointrepresentation,395Cliffordalgebra,315Affinegroup,258,259Cliffordalgebrarelations,356Anosovsystem,154Completeness,326Anti-Wicksymbol,53Continuousseries,234Anticommutationrelations,288ContractionsofLiegroups,221Automorphismofthe2-torus,152Contravariantsymbol,25Covariantsymbol,25BCreationandannihilationoperators,7,290Baker–Campbell–Hausdorffformula,2Criticalset,383Bargmann–Fockrepresentation,16Berezian,329DBerezin–Liebinequality,217Densityofstates,124Berryphase,207Densityoperator,336Birkhoffaverage,152Dickemodel,379Blochcoherentstates,201Dickestates,189Blochdecomposition,159Diracbracket,375Bogoliubovtransformation,253,345DiracquantumHamiltonian,375Bohrcorrespondenceprinciple,23DiscreteFouriertransform,158Bohr–Sommerfeldformula,129Discreteseries,234Bohr–Sommerfeldquantizationrule,149Dispersion,199Borelsummability,113,304Duhamelformula,302Boreltransform,304Duhamel’sprinciple,91Bosons,285Dysonseries,306CECalderon–Vaillancourt,35Egorovtheorem,173CanonicalAnticommutationRelation,312Ehrenfesttime,97CanonicalCommutationRelations,288,297Eigenvalue,123Casimiroperator,190,197,231Energyshell,127Cayleytransform,350Equipartitionofeigenfunctions,175Circulantmatrices,161Equivalentrepresentations,396Classicalaction,94,127Ergodic,152Classicalscatteringmatrix,115Eulerangles,186Classicalwaveoperators,114Eulertriangle,206,207Cleanintersectioncondition,138Euler–Lagrangeequations,366M.Combescure,D.Robert,CoherentStatesandApplicationsinMathematicalPhysics,413TheoreticalandMathematicalPhysics,DOI10.1007/978-94-007-0196-0,©SpringerScience+BusinessMediaB.V.2012 414IndexFLFermions,285Laplaceintegral,387Fieldoperator,292Laplace–Runge–Lenzoperator,272FirstreturnPoincarémap,141Laplace–Runge–Lenzvector,271Floquetoperator,81Linearizedflow,128Fockspaceofbosons,287Liouvillemeasure,130Fockspaceoffermions,287Lyapunovexponent,102Fokker–Planckequation,380Fouriertransform,6,323MFourier–Bargmanntransform,13Maslovindex,126,130Fourier–Laplacetransform,260Mathieuequations,82Frequencyset,49Mehlerformula,21,81Mehlig–Wilkinsonformula,348Functionalcalculus,123Metaplecticgroup,28Metaplectictransformations,69GMicrolocal,126Gaussdecomposition,235Minkowskimetric,225Gaussianbeams,100,126Mixing,153Gaussiandecomposition,201,202Möbiustransformation,214GeneralizedPaulimatrices,161Monodromymatrix,128Generatingfunction,314Moyalformula,341Generatorofsqueezedstates,71,73Moyalproduct,40,339Geodesictriangle,204Geometricphase,209NGevreyseries,112,304Nelsoncriterium,292Gilmore–Perelomovcoherentstates,398Noethertheorem,366Grandcanonicalensemble,342Nontrapping,115Grassmannalgebra,313Nondegenerate,129Grassmannanalyticcontinuationprinciple,Nondegenerateorbit,128363Numberoperator,287Gutzwillertraceformula,124PHPaulimatrices,185Haarmeasure,393Pfaffian,320Hadamardmatrix,161Poincarédisc,230Hamilton’sequations,89Poincarémap,128Hardypotential,301Poissonbracket,24Harmonicoscillator,8Poissonsummationformula,124Hartreeequation,301Pseudosphere,225Heisenbergcommutationrelation,2QHeisenbergpicture,298Quasi-modeofenergyE,148Hilbert–Schmidtclass,15Quasi-modes,143Husimifunctions,49Quaternionfield,264Hydrogenatom,271Quaternions,192Hyperbolicgeometry,225Hyperbolictriangle,248RResolutionoftheidentity,14,249IRiemannsphere,213Irreduciblerepresentations,396SKScarring,177Kählerform,388Scatteringoperator,115,116KeplerHamiltonian,263Scatteringtheory,114 Index415Schrödingerequation,60Superlinearspace,357Schurlemma,6,13,396Supermanifold,361Schwartzspace,24Supermechanics,369Secondquantization,287Supersymmetricharmonicoscillator,355,376Semi-classical,124Supersymmetry,353Semi-classicalmeasures,55Supersymmetrygenerators,376SemiclassicalGardinginequality,55Symplecticmatrix,60SemiclassicalGutzwillertraceformula,127Symplecticproduct,3Semisimplegroups,398Siegelspace,62,116TSL(2,R),229,259Temperatedistributions,24Spaceofsymbols,39Temperateweight,39Spectrum,123Thermodynamiclimit,220Sphericalharmonics,196Trace,328Spincoherentstates,201Traceclass,15Spinobservables,190Spinrepresentation,345USpinstates,190Uniquelyergodic,153Squareintegrablerepresentations,396Squeezedstate,73VSqueezedstates,62,252Stationaryphase,384Vacuumstate,288Stationaryphaseexpansion,134Virialtheorem,274Stereographicprojection,201,213Superalgebra,358WSuperLiealgebra,359,363Waveoperators,115SuperLie-group,362Wavelets,258Super-coherentstates,377Weylasymptotic,45Super-derivation,360Weylasymptoticformula,44,138Super-Poisson-bracket,374Weyltranslationoperators,293Super-spaces,353Weyl–Heisenbergalgebra,4Super-trace,328Wignerfunctions,28Supercoherentstates,353,376WittensupersymmetricHamiltonian,375
