Coherent states and applications in mathematical physics PDF
Preview Coherent states and applications in mathematical physics
Coherent States and Applications in Mathematical Physics Theoretical and Mathematical Physics The series founded in 1975 and formerly (until 2005) entitled Texts and Monographs in Physics (TMP) publishes high-level monographs in theoretical and mathematical physics. The change of title to Theoretical and Mathematical Physics (TMP) signals thattheseriesisasuitablepublicationplatformforboththemathematicalandthethe- oretical physicist. The wider scope of the series is reflected by the composition of the editorialboard,comprisingbothphysicistsandmathematicians. Thebooks,writteninadidacticstyleandcontainingacertainamountofelementary background material, bridge the gap between advanced textbooks and research mono- graphs. They can thus serve as basis for advanced studies, not only for lectures and seminarsatgraduatelevel,butalsoforscientistsenteringafieldofresearch. EditorialBoard W.Beiglböck,InstituteofAppliedMathematics,UniversityofHeidelberg,Heidelberg, Germany P.Chrusciel,GravitationalPhysics,UniversityofVienna,Vienna,Austria J.-P. Eckmann, Département de Physique Théorique, Université de Genève, Geneva, Switzerland H.Grosse,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria A.Kupiainen,DepartmentofMathematics,UniversityofHelsinki,Helsinki,Finland H.Löwen,InstituteofTheoreticalPhysics,Heinrich-Heine-UniversityofDuesseldorf, Duesseldorf,Germany M.Loss,SchoolofMathematics,GeorgiaInstituteofTechnology,Atlanta,USA N.A.Nekrasov,IHÉS,Bures-sur-Yvette,France M.Ohya,TokyoUniversityofScience,Noda,Japan M. Salmhofer, Institute of Theoretical Physics, University of Heidelberg, Heidelberg, Germany S.Smirnov,MathematicsSection,UniversityofGeneva,Geneva,Switzerland L.Takhtajan,DepartmentofMathematics,StonyBrookUniversity,StonyBrook,USA J.Yngvason,InstituteofTheoreticalPhysics,UniversityofVienna,Vienna,Austria Forfurthervolumes: www.springer.com/series/720 Monique Combescure (cid:2) Didier Robert Coherent States and Applications in Mathematical Physics MoniqueCombescure DidierRobert BatimentPaulDirac LaboratoireJean-Leray IPNL DepartementdeMathematiques Villeurbanne NantesUniversity France NantesCedex03 France ISSN1864-5879 e-ISSN1864-5887 TheoreticalandMathematicalPhysics ISBN978-94-007-0195-3 e-ISBN978-94-007-0196-0 DOI10.1007/978-94-007-0196-0 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2012931507 ©SpringerScience+BusinessMediaB.V.2012 Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurpose ofbeingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface The main goal of this book is to give a presentation of various types of coherent states introduced and studied in the physics and mathematics literature during al- mostacentury.Wedescribetheirmathematicalpropertiestogetherwithapplication toquantumphysicsproblems.Itisintendedtoserveasacompendiumoncoherent statesandtheirapplicationsforphysicistsandmathematicians,stretchingfromthe basicmathematicalstructuresofgeneralizedcoherentstatesinthesenseofGilmore andPerelomov1viathesemiclassicalevolutionofcoherentstatestovariousspecific examples of coherent states (hydrogen atom, torus quantization, quantum oscilla- tor). We have tried to show that the field of applications of coherent states is wide, diversifiedandstillalive.Becauseofourownabilitylimitationswehavenotcovered thewholefield.Besidesthiswouldbeimpossibleinonebook.Wehavechosensome partsofthesubjectwhicharesignificantforus.Othercolleaguesmayhavedifferent opinions. Thereexistseveraldefinitionsofcoherentstateswhicharenotequivalent.Nowa- daysthemostwellknownistheGilmore–Perelomov[84,85,155]definition:aco- herent state system is an orbit for an irreducible group action in an Hilbert space. Fromamathematicalpointofviewcoherentstatesappearlikeapartofgrouprep- resentationtheory. In particular canonical coherent states are obtained with the Weyl–Heisenberg groupactioninL2(R)andthestandardGaussianϕ (x)=π1/4e−x2/2.Modulomul- 0 tiplication by a complex number, the orbit of ϕ is described by two parameters 0 (q,p)∈R2 andtheL2-normalizedcanonicalcoherentstatesare ϕ (x)=π−1/4e−(x−q)2/2ei((x−q)p+qp/2). q,p Wavelets are included in the group definition of coherent states: they are obtained from the action of the affine group of R (x (cid:3)→ax +b) on a “mother function” ψ∈L2(R).Thewaveletsystemhastwoparameters:ψ (x)= √1 ψ(x−b). a,b a a 1Theyhavediscoveredindependentlytherelationshipwithgrouptheoryin1972. v vi Preface Oneofthemostusefulpropertyofcoherentsystemψ isthattheyarean“over- z complete”systemintheHilbertspaceinthesensethatwecananalyzeany η∈H withitscoefficient(cid:6)ψ ,η(cid:7)andwehaveareconstructionformulaofηlike z (cid:2) η= dzη˜(z)ψ , z whereη˜ isacomplexvaluedfunctiondependingon(cid:6)ψ ,η(cid:7). z Coherent states (being given no name) were discovered by Schrödinger (1926) when he searched solutions of the quantum harmonic oscillator being the closest possibletotheclassicalstateorminimizingtheuncertaintyprinciple.Hefoundthat thesolutionsareexactlythecanonicalcoherentstatesϕ . z Glauber (1963) has extended the Schrödinger approach to quantum electro- dynamicandhecalledthesestatescoherentstatesbecausehesucceededtoexplain coherencephenomenainlightpropagationusingthem.AftertheworksofGlauber, coherentstatesbecameaverypopularsubjectofresearchinphysicsandinmathe- matics. There exist several books discussing coherent states. Perelomov’s book [156] playedanimportantroleinthedevelopmentofthegroupaspectofthesubjectandin itsapplicationsinmathematicalphysics.Severalotherbooksbroughtcontributions to the theory of coherent states and worked out their applications in several fields of physics; among them we have [3,80,126] but many others could be quoted as well. There is a huge number of original papers and review papers on the subject; we have quoted some of them in the bibliography. We apologize the authors for forgottenreferences. Inthisbookweputemphasisonapplicationsofcoherentstatestosemi-classical analysisofSchrödingertypeequation(timedependentortimeindependent).Semi- classicalanalysismeansthatwetrytounderstandhowsolutionsoftheSchrödinger equationbehaveasthePlanckconstant(cid:2)isnegligibleandhowclassicalmechanics is a limit of quantum mechanics. It is not surprising that semi-classical analysis andcoherentstatesarecloselyrelatedbecausecoherentstates(whichareparticular quantum states) will be chosen localized close to classical states. Nevertheless we think that in this book we have given more mathematical details concerning these connectionsthanintheothermonographsonthatsubjects. Letusgivenowaquickoverviewofthecontentofthebook. The first half of the book (Chap. 1 to Chap. 5) is concerned with the canonical (standard)GaussianCoherentStatesandtheirapplicationsinsemi-classicalanalysis ofthetimedependentandthetimeindependentSchrödingerequation. ThebasicingredienthereistheWeyl–Heisenbergalgebraanditsirreduciblerep- resentations.TherelationshipbetweencoherentstatesandWeylquantizationisex- plained in Chaps. 2 and 3. In Chap. 4 we compute the quantum time evolution of coherentstatesinthesemi-classicalrégime:theresultisasqueezedcoherentstates whoseshapeisdeformed,dependingontheclassicalevolutionofthesystem.The mainoutcomeisaproofoftheGutzwillertraceformulagiveninChap.5. Thesecondhalfofthebook(Chap.6toChap.12)isconcernedwithextensions ofcoherentstatessystemstoothergeometrysettings.InChap.6weconsiderquan- tizationofthe2-toruswithapplicationtothecatmapandanexampleof“quantum chaos”. Preface vii Chapters 7 and 8 explain the first examples of non canonical coherent states where the Weyl–Heisenberg group is replaced successively by the compact group SU(2)andthenon-compactgroupSU(1,1).Weshallseethatsomerepresentations of SU(1,1) arerelatedwithsqueezedcanonicalcoherentstates,withquantumdy- namicsforsingularpotentialsandwithwavelets. WeshowinChap.9howitispossibletostudythehydrogenatomwithcoherent statesrelatedwiththegroupSO(4). In Chap. 10 we consider infinite systems of bosons for which it is possible to extendthedefinitionofcanonicalcoherentstates.Thisisusedtoprovemean-field limitresultfortwo-bodyinteractions:thelinearfieldequationcanbeapproximated byanonlinearSchrödingerequationinR3inthesemi-classicallimit(largenumber ofparticlesorsmallPlanckconstantaremathematicallyequivalentproblems). Chapters11and12areconcernedwithextensionofcoherentstatesforfermions withapplicationstosupersymmetricsystems. Finally in the appendices we have a technical section A around the stationary phasetheorem,andinsectionBwerecallsomebasicfactsconcerningLiealgebras, Liegroupsandtheirrepresentations.Weexplainhowthisisusedtobuildgeneral- izedcoherentsystemsinthesenseofGilmore–Perelomov. Thematerialcoveredinthesebookisdesignedforanadvancedgraduatestudent, or researcher, who wishes to acquaint himself with applications of coherent states in mathematics or in theoretical physics. We have assumed that the reader has a good founding in linear algebra and classical analysis and some familiarity with functionalanalysis,grouptheory,linearpartialdifferentialequationsandquantum mechanics. WewouldliketothankourcolleaguesofLyon,Nantesandelsewhere,fordiscus- sionsconcerningcoherentstates.InparticularwethankourcollaboratorJimRalston withwhomwehavegivenanewproofofthetraceformula,StephanDebièvre,Alain JoyeandAndréMartinezforstimulatingmeetings. M.C.alsothanksSylvieFloresforofferingvaluablesupportinthebibliography. To conclude we wish to express our gratitude to our spouses Alain and Marie- Francewhoseunderstandingandsupporthavepermittedtoustospendmanyhours forthewritingofthisbook. LyonandNantes,France MoniqueCombescure DidierRobert Contents 1 IntroductiontoCoherentStates . . . . . . . . . . . . . . . . . . . . . 1 1.1 TheWeyl–HeisenbergGroupandtheCanonicalCoherentStates. . 2 1.1.1 TheWeyl–HeisenbergTranslationOperator . . . . . . . . 2 1.1.2 TheCoherentStatesofArbitraryProfile . . . . . . . . . . 6 1.2 TheCoherentStatesoftheHarmonicOscillator . . . . . . . . . . 7 1.2.1 DefinitionandProperties . . . . . . . . . . . . . . . . . . 7 1.2.2 The Time Evolution of the Coherent State fortheHarmonicOscillatorHamiltonian . . . . . . . . . . 11 1.2.3 AnOver-completeSystem . . . . . . . . . . . . . . . . . . 12 1.3 FromSchrödingertoBargmann–FockRepresentation . . . . . . . 16 2 WeylQuantizationandCoherentStates . . . . . . . . . . . . . . . . 23 2.1 ClassicalandQuantumObservables . . . . . . . . . . . . . . . . . 23 2.1.1 GroupInvarianceofWeylQuantization . . . . . . . . . . . 27 2.2 WignerFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3 CoherentStatesandOperatorNormsEstimates . . . . . . . . . . . 35 2.4 ProductRuleandApplications . . . . . . . . . . . . . . . . . . . 40 2.4.1 TheMoyalProduct . . . . . . . . . . . . . . . . . . . . . 40 2.4.2 FunctionalCalculus . . . . . . . . . . . . . . . . . . . . . 43 2.4.3 PropagationofObservables . . . . . . . . . . . . . . . . . 45 2.4.4 ReturntoSymplecticInvarianceofWeylQuantization . . . 47 2.5 HusimiFunctions,FrequencySetsandPropagation . . . . . . . . 49 2.5.1 FrequencySets . . . . . . . . . . . . . . . . . . . . . . . . 49 2.5.2 AboutFrequencySetofEigenstates . . . . . . . . . . . . . 52 2.6 WickQuantization . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6.1 GeneralProperties . . . . . . . . . . . . . . . . . . . . . . 52 2.6.2 ApplicationtoSemi-classicalMeasures . . . . . . . . . . . 55 3 TheQuadraticHamiltonians . . . . . . . . . . . . . . . . . . . . . . 59 3.1 ThePropagatorofQuadraticQuantumHamiltonians . . . . . . . . 59 3.2 ThePropagationofCoherentStates . . . . . . . . . . . . . . . . . 61 3.3 TheMetaplecticTransformations . . . . . . . . . . . . . . . . . . 69 ix x Contents 3.4 Representation of the Quantum Propagator in Terms oftheGeneratorofSqueezedStates . . . . . . . . . . . . . . . . . 71 3.5 RepresentationoftheWeylSymboloftheMetaplecticOperators . 78 3.6 Traps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6.1 TheClassicalMotion . . . . . . . . . . . . . . . . . . . . 81 3.7 TheQuantumEvolution . . . . . . . . . . . . . . . . . . . . . . . 82 4 TheSemiclassicalEvolutionofGaussianCoherentStates . . . . . . 87 4.1 GeneralResultsandAssumptions . . . . . . . . . . . . . . . . . . 87 4.1.1 AssumptionsandNotations . . . . . . . . . . . . . . . . . 88 4.1.2 TheSemiclassicalEvolutionofGeneralizedCoherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.1.3 RelatedWorksandOtherResults . . . . . . . . . . . . . . 100 4.2 ApplicationtotheSpreadingofQuantumWavePackets . . . . . . 100 4.3 EvolutionofCoherentStatesandBargmannTransform . . . . . . 103 4.3.1 FormalComputations . . . . . . . . . . . . . . . . . . . . 103 4.3.2 WeightedEstimatesandFourier–BargmannTransform . . . 105 4.3.3 LargeTimeEstimatesandFourier–BargmannAnalysis . . 107 4.3.4 ExponentiallySmallEstimates . . . . . . . . . . . . . . . 110 4.4 ApplicationtotheScatteringTheory . . . . . . . . . . . . . . . . 114 5 TraceFormulasandCoherentStates . . . . . . . . . . . . . . . . . . 123 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2 TheSemi-classicalGutzwillerTraceFormula. . . . . . . . . . . . 127 5.3 PreparationsfortheProof . . . . . . . . . . . . . . . . . . . . . . 131 5.4 TheStationaryPhaseComputation . . . . . . . . . . . . . . . . . 135 5.5 APointwiseTraceFormulaandQuasi-modes . . . . . . . . . . . 143 5.5.1 APointwiseTraceFormula . . . . . . . . . . . . . . . . . 144 5.5.2 Quasi-modesandBohr–SommerfeldQuantizationRules . . 145 6 QuantizationandCoherentStatesonthe2-Torus . . . . . . . . . . . 151 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.2 TheAutomorphismsofthe2-Torus . . . . . . . . . . . . . . . . . 151 6.3 TheKinematicsFrameworkandQuantization . . . . . . . . . . . 155 6.4 TheCoherentStatesoftheTorus . . . . . . . . . . . . . . . . . . 162 6.5 TheWeylandAnti-WickQuantizationsonthe2-Torus . . . . . . . 166 6.5.1 TheWeylQuantizationonthe2-Torus . . . . . . . . . . . 166 6.5.2 TheAnti-WickQuantizationonthe2-Torus . . . . . . . . 168 6.6 QuantumDynamicsandExactEgorov’sTheorem . . . . . . . . . 170 6.6.1 QuantizationofSL(2,Z) . . . . . . . . . . . . . . . . . . 170 6.6.2 TheEgorovTheoremIsExact . . . . . . . . . . . . . . . . 173 6.6.3 PropagationofCoherentStates . . . . . . . . . . . . . . . 174 6.7 EquipartitionoftheEigenfunctionsofQuantizedErgodicMaps onthe2-Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.8 SpectralAnalysisofHamiltonianPerturbations . . . . . . . . . . . 177
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