Table Of ContentCoherent States and Applications in Mathematical
Physics
Theoretical and Mathematical Physics
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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
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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
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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
