Variational Approach to Gaussian Approximate Coher

arXiv:gr-qc/9612054v118Dec1996VARIATIONALAPPROACHTOGAUSSIANAPPROXIMATECOHERENTSTATES:QUANTUMMECHANICSANDMINISUPERSPACEFIELDTHEORYA.A.MinzoniFENOMEC/DepartmentofMathematicsandMechanics,IIMASUniversidadNacionalAut´onomadeM´exicoA.Postal20-726M´exico01000D.F.,MEXICOMarcosRosenbaumandMichaelP.Ryan,Jr.InstitutodeCienciasNuclearesUniversidadNacionalAut´onomadeM´exicoA.Postal70-546M´exico04510D.F.,MEXICOPACSnumbers:04.60.Kz98.80.Hw03.70.+k03.65.-w03.65.GeABSTRACTThispaperhasadualpurpose.Oneaimistostudytheevolutionofcoherentstatesinordinaryquantummechanics.ThisisdonebymeansofaHamiltonianapproachtotheevolutionoftheparametersthatdefinethestate.Thestabilityofthesolutionsisstudied.Thesecondaimistoapplythesetechniquestothestudyofthestabilityofminisuperspacesolutionsinfieldtheory.Foraλϕ4theoryweshow,bothbymeansofperturbationtheoryandrigorously,bymeansoftheoremsoftheK.A.M.type,thatthehomogeneousminisuperspacesectorisindeedstableforpositivevaluesoftheparametersthatdefinethefieldtheory.I.INTRODUCTIONThispaperhasadualpurpose.Ithasitsorigininthestudyofsuperspacesinfieldtheory,thatis,thefunctionspacesofsolutionsofanyfieldtheory(eventhoughthenamesuperspaceoriginatedinthestudyofthegravitationalfield).However,wewouldliketoemphasizethatthetechniqueswewilluseheremaybeappliedtothestudyofcoherentstatesinordinaryquantummechanics,andtheexampleswewillgiveare,infact,equivalenttoone-andtwo-dimensionalnonrelativisticquantummechanics.Thesimplestexampleofasuperspaceisthatofaone-dimensionalrealscalarfieldϕ(z,t).IfweexpandϕinarealFourierseries(assumingthedomainofϕtobeconfinedto−L/2zL/2withtheendpointsidentified)ϕ(z,t)=ϕ0(t)+∞Xn=1ϕn(t)cos2nπzL+ϕ−n(t)sin2nπzL,(1.1)theevolutionofϕ(independentoftheactionthatgeneratestheevolution)isnothingmorethanacurveinthespaceofcountablyinfinitedimensiondefinedbythe“coordinates”ϕn,−∞≤n≤+∞.Ofcourse,classicallyanynonlinearactionforϕgivesanextremelycomplicatedinfinitesetofcoupledODE’sfortheϕn,andthisapproachtofieldtheoryisrarelyusedindirectcalculations.Nevertheless,suchFourierexpansionswerethebasisforfieldquantizationformanyyears,andarestillusedinmanycontexts.ItisnotdifficulttoshowthatforarealFourierseriessuchas(1.1)thequantumevolutionofϕisjustthequantummechanicsofaparticlemovinginϕn-spaceundertheinfluenceofwhatmaybeaverycomplicatedpotential(eveninthesimplestcasesofnonlinearactionsforϕ).Therehavebeenanumberofstudiesofnonlinearfieldtheorieswhichhaveattemptedtogleaninformationaboutthebehavioroffieldssuchasϕ(z,t)bystudyingmodeltheorieswheretheconfigurationspaceofthesystemisreducedbyputtingallbutafinitenumberoftheϕnequaltozero[1,2,3,4].Thesearecalled“minisuperspace”fieldtheories.Noticethatsuchtheoriesareequivalenttoone-particlequantummechanicsinaspaceofdimensionofthesurvivingϕn.Onemay1askwhethersuchtheoriesarejustmodelsorwhethertheycouldbeapproximatesolutionstothefullfieldtheory.Onemethodforansweringthisquestionistostudyapproximate“coherentstates”whosecenterissupposedtomoveonatrackinϕn-spacethatiscenteredonsomeclassicalpathwhichisrestrictedtothereducedconfigurationspace.Thisproblemleadstothestudyofthesimplerquantummechanicalproblemoffindingconsistentapproximationstothetime-dependentevolutionoflocalizedsolutionstotheSchr¨odingerequationindimensionscorrespondingtothereducedconfigurationspace.Ofcourse,aproblemofthissortiscompletelyindependentofanyfieldtheory,andwefeelthatourresultsareusefulinthegeneralstudyofcoherentstates.Aninterestingmethodforthestudyofthesestatesisbasedontheideasofaveragevariationalprinciples.Thisapproachiswidelyusedinthecontextofnonlinearwavesandoscillations[5]andmorerecentlyintheproblemofsolitonpropagationundertheinfluenceofvariousperturbingeffects[6].Theemphasisherewillbeonobtainingapproximateandrigorousstabilityresultsratherthanondetailedcalculationsoftheevolutionofthesolutionwhichisthepurposeofmanyworksonwavepropagation.Thesestabilityargumentsareofmuchmoreinterestforfieldtheory,sincewhathasbeencalledthe“quantumstability”ofminisuperspacesolutionsisdirectlyrelatedtotheproblemoftheuseofquantummin-isuperspacesolutionsasapproximations,butwewillfindevolutionequationsforcoherentstatesolutionswhichcouldbeofuseinthestudyofcoherentstatesinordinaryquantummechanics.Thetechniquewewilluseisbasedontheconsiderationoftime-dependenttrialfunc-tionsintheLagrangianoftheSchr¨odingerequation.TheLagrangianisthenaveragedoverthespacevariablestoobtainaneffectiveactionwhichinvolvesonlythetime-dependentparametersofthetrialfunction.TheEulerequationsofthisnewLagrangiangiveustheevolutionoftheparameters.ThisproceduregivesaconsistentwaytoapproximatetheinfinitelymanydegreesoffreedomofthewavefunctionbymeansofafinitenumberofparameterswhilepreservingtheLagrangianstructure.Ontheotherhand,pointwiseap-2proximationscould,inprinciple(andofteninpractice),produceinthetruncationspuriousnon-conservativeterms.SinceweareinterestedinaHamiltonianformulationoftheprob-lem,itsstabilityisnotdeterminedbydampingtermsbutratherbythenatureofoftheHamiltonian.BecauseofthistheapproximationmustbeconsistentwiththeunderlyingHamiltonianstructure.Wewouldliketoemphasizeoncemorethatthisideagi