Classical and Quantum Systems Alternative Hamilton

arXiv:math-ph/0504064v121Apr2005CLASSICALANDQUANTUMSYSTEMS:ALTERNATIVEHAMILTONIANDESCRIPTIONSG.Marmo1,3∗,G.Scolarici2†,A.Simoni1,3‡andF.Ventriglia1,3,4§1DipartimentodiScienzeFisiche,Universita’diNapoliFedericoII,ComplessoUniversitariodiMonteSAngelo,Napoli,Italy.2DipartimentodiFisica,Universita’diLecceandINFN,SezionediLecce,Italy3INFN,SezionediNapoli,Italy4INFM,UdRdiNapoli,ItalyFebruary7,2008AbstractIncompleteanalogywiththeclassicalsituation(whichisbrieflyre-viewed)itispossibletodefinebi-HamiltoniandescriptionsforQuantumsystems.WealsoanalyzecompatibleHermitianstructuresinfullanalogywithcompatiblePoissonstructures.KeyWords:Quantum-ClassicalTransition,Quantumbi-Hamiltoniansystems,AlternativeHermitianstructures,Bi-Unitaryoperators.1IntroductionInthepastthirtyyearsalargenumberofnonlinearevolutionequationsweredis-coveredtobeintegrablesystems[1].ItisafactthatalmostinallcasesintegrablesystemsalsoexhibitmorethanoneHamiltoniandescriptions,i.e.theyadmitalternativeHamiltoniandescriptions(theyareoftencalledbi-Hamiltoniansys-tems)[2].Inconnectionwithquantummechanics,therehavebeenproposalsforstudyingcompleteintegrabilityinthequantumsetting.[3],[4]IfwetaketheviewpointofDirac[5]“Classicalmechanicsmustbealimitingcaseofquantummechanics.Weshouldthusexpecttofindthatimportantconceptsinclassicalmechanicscorre-spondtoimportantconceptsinquantummechanicsand,fromanunderstanding∗E-mail:[e-mail]†E-mail:[e-mail]‡E-mail:[e-mail]§E-mail:[e-mail]1ofthegeneralnatureoftheanalogybetweenclassicalandquantummechanics,wemayhopetogetlawsandtheoremsinquantummechanicsappearingassimplegeneralizationsofwellknownresultsinclassicalmechanics”,itseemsquitenaturaltoaskthequestion:whichalternativestructuresinquantummechanics,intheappropriatelimit,willprovideuswithalternativestructuresavailableinclassicalmechanics?Inparticular,isitpossibletoexhibittheanalogofalternativeHamiltoniandescriptionsinthequantumframework?Thisproblemhasbeeninvestigatedinatwo-pagespaperbyWignerincon-nectionwithcommutationrelationsfortheone-dimensionalHarmonicOscillator.[6]SeealsoRefs.[7],[8],[9],[10].Asweareinterestedinthestructuresratherthanonspecificapplications,itisbettertoconsiderthesimplestsettinginordertoavoidtechnicalities.Toclearlyidentifydirectionsweshouldtakeinthequantumsetting,itisappropriatetobrieflyreviewthesearchforalternativeHamiltoniandescriptionsintheclassicalsetting,leavingasidetheproblemofexistenceofcompatiblealternativePoissonbracketswhichwouldgiverisetocompleteintegrabilityoftheconsideredsystems.Thepaperisorganizedinthefollowingway.InSection2wedealwithal-ternativeHamiltoniandescriptionsforclassicalsystems,whileinSection3theparticularcaseofNewtonianequationsofmotionisaddressedandinSection4ameaningfulexampleisdiscussedindetail.TheanalogpictureinQuantumcaseisexposedinSection5usingtheWeylapproachfortheClassical-Quantumtransition.TheSchroedingerpictureistheframeworktostudyalternativede-scriptionsoftheequationsofmotionforQuantumSystemsinSection6,inthefinitedimensionalcase.Thealgebraicresultsobtainedthereinthesearchforin-variantHermitianstructuresareextendedtoinfinitedimensionsinthelastpartofthepaper.Inparticular,inSection7,sometheoremsofNagyarerecalledtoprovideaninvariantHermitianstructureandinSection8,startingwithtwoHermitianstructures,thegroupofbi-unitarytransformationshasbeencharac-terizedandasimpleexampleisusedtoshowhowthetheoryworks.Finally,someconcludingremarksaredrawninSection9.2AlternativeHamiltoniandescriptionsforclas-sicalsystemsAlmostinallcasescompletelyintegrableclassicalsystemsarebi-Hamiltoniansystems.AdynamicalsystemonamanifoldMissaidtobebi-HamiltonianifthereexiststwoPoissonBracketsdenotedby{.,.}1,2andcorrespondingHamil-tonianfunctionsH1,2suchthatdfdt={H1,f}1={H2,f}2,∀f∈F(M).(1)2WithanyPoissonBracketwemayassociateaPoissontensordefinedby{ξj,ξk}=Λjk,Λ=Λjk∂∂ξj∧∂∂ξk.(2)TosearchforalternativeHamiltoniandescriptionsforagivendynamicalsys-temassociatedwithavectorfieldΓonamanifoldM,withassociatedequationsofmotiondfdt=LΓf,(3)wehavetosolvethefollowingequationforthePoissontensorΛ:LΓΛ=0.(4)ThevectorfieldΓwillbecompletelyintegrableifwecanfindtwoPoissontensorsΛ1andΛ2,outofthepossiblealternativesolutionsofequation(4),suchthatanylinearcombinationλ1Λ1+λ2Λ2satisfiestheJacobiidentity.InthiscasethePoissonstructuresaresaidtobecompatible.[11]Inparticular,constantPoissontensorsΛ1andΛ2arecompatible.Summarizing,givenavectorfieldΓΓ=Γj∂∂ξj(5)wesearchforpairs(Λ,H)whichallowtodecomposeΓinthefollowingproductΓj=Λjk∂H∂ξk,(6)alongwiththeadditionalcondition(Jacobiidentity):Λjk∂kΛlm+Λlk∂kΛmj+Λmk∂kΛjl=0.(7)Whenthestartingequationsofmotionaresecondorder,furtherconsidera-tionsarise.3AlternativeHamiltoniandescriptionsforequa-tionsofNewtoniantypeWerecallthat,accordingtoDyson,[12],[13]Feynmanaddressedasimilarprob-lem,withtheadditionalconditionoflocalizability;i.e.writtenintermsofpositionsandmomenta(xj,pj)thelocalizabilityconditionreads{xj,xk}=0.(8)Thus,thesearchofHamiltoniandescriptionsforasecondorderdifferentialequationreads·xj={H,xj},(9)3··xj={H,{H,xj}}=fj(x,·x)(10)No