Classical mappings of the symplectic model and the

arXiv:nucl-th/9307021v121Jul1993Classicalmappingsofthesymplecticmodelandtheirapplicationtothetheoryoflarge-amplitudecollectivemotionAbrahamKleinandNielsR.Walet∗DepartmentofPhysics,UniversityofPennsylvania,Philadelphia,Pennsylvania19104-6396,USAAbstractWestudythealgebraSp(n,R)ofthesymplecticmodel,inparticularforthecasesn=1,2,3,inanewway.StartingfromthePoisson-bracketrealizationwederiveasetofpartialdifferentialequationsforthegeneratorsasfunctionsofclassicalcanonicalvariables.Weobtainasolutiontotheseequationsthatrepresentstheclassicallimitofabosonmappingofthealgebra.Weshowfurtherthatthismappingplaysafundamentalroleinthecollectivedescrip-tionofmany-fermionsystemswhoseHamiltonianmaybeapproximatedbypolynomialsintheassociatedalgebra,asisdoneinthesimplestversionsofthesymplecticmodel.Therelationshiptothecollectivedynamicsisformu-latedasatheoremthatassociatesthemappingwithanexactsolutionofthetime-dependentHartreeapproximation.Thissolutiondeterminesade-coupledclassicalsymplecticmanifold,thussatisfyingthecriteriathatdefineanexactlysolvablemodelinthetheoryoflargeamplitudecollectivemotion.Themodelsthusobtainedalsoprovideatestofmethodsforconstructinganapproximatelydecoupledmanifoldinfullyrealisticcases.Weshowthatanalgorithmdevelopedinoneofourearlierworksreproducesthemainresultsofthetheorem.TypesetusingREVTEX∗addressafterSept.1,1993:Instit¨utf¨urtheoretischePhysikIII,Universit¨atErlangen-N¨urnberg,D-91058Erlangen,Germany1I.INTRODUCTIONWehavebeenengagedforadecadeinanefforttoformulateatheoryoflargeamplitudecollectivemotionwiththespecialaimofapplyingittonuclearphysics.Thetheoryhasbothaclassicalandaquantumdimension.Theclassicalaspecthasbeenmostfullydevelopedanddescribedinareview[1].Thequantumaspectispresentlyinastageofvigorousdevelopment[2,3],supplantingtheearlyworkonthispartofthetheory[4,5].Atthesametimeaprogramofapplicationstoproblemsofnuclearstructurehasbeenundertaken[6–9].Earlyinthelatterwork,webecameawareofapaucityofsolvablemodelswithsomephysicalcontent.Theusefulnessofsuchmodelsisthattheyprovideatestinggroundforthealgorithmsthatwouldlaterbeappliedtomorerealisticmodels.Forourinitialinvestigationweselectedawell-knownmodelofmonopolevibrations[10],exactlysolvablebecausetheHamiltonianisapolynomialinthegeneratorsofthealgebraSp(1,R)(orSU(1,1)).Westud-iedthismodelintwoways.First,byutilizingtheclassicallimitofthealgebra,wewereabletoproduceanexactsolutionofthetime-dependentHartreeequations(derivedpreviouslybyratherlesstransparenttechniques[11])andbymeansofthissolutionadecoupledcollectiveHamiltonianforthemonopolevibration.Second,andmoreimportant,wecouldcheckifthesameHamiltonianemergedfromtheapplicationofthetheoryoflargeamplitudecollectivemotion.Themonopolemodelprovideduswithanapparentlyidealtestofthesoundnessofouralgorithms.Thistestfailedinitially,forcinguseventuallytorecognizeandcorrectanincompletenessinourprevioustheory.ThefirstgoalofthispaperistoshowthatthemethoddevelopedforthealgebraofSp(1,R)canbeextendedtothealgebraofSp(n,R).Inparticular,weworkoutfullythecasesn=2andn=3.TheformerleadstoaHamiltonianwiththreecollectivecoordinates,describingtheinteractionofamonopoledegreeoffreedomwithaquadrupoletensorintwospatialcoordinatesandisthusonlyofinterestasatoymodel.Ontheotherhandn=3leadstoaHamiltonianwithsixdegreesoffreedomdescribingbothamonopoleandathree-dimensionalquadrupole.Thisdefinesthemodelasnotonlyoneofphysicalinterestperse,butalsobecauseofitsconnectionwiththesymplectic[12]andpseudo-symplectic[13]models.Thelatter,inparticular,providesapossiblyusefultruncationschemeforshell-modelcalculationsforotherthanthelightestdeformednuclei.Thoughnotoneoftheaimsofthepresentpaper,thisidentificationwillallowusonafutureoccasiontocomparethequantumconsequencesofthecollectiveHamiltoniantobederivedinthispaperwiththeresultsofanexactdiagonalizationcarriedoutfortheoriginalmanybodyHamiltonian.InfactsuchacomparisonhasbroaderimplicationsthantheaccuracyachievedforthespecialHamiltonianconsidered,sinceithasbeendemonstatedthatHamiltoniansconsistingofsuitablychosenpolynomialsinthegeneratorscangivearatherprecisefittothelow-energyspectraandotherpropertiesofevendeformednuclei[13].Thesecondaimofthispaperistodemonstratethattheextendedalgorithmformulatedinconnectionwiththemonopolemodelalsoprovidescorrectresultsforthegeneralizedmodels.Thepresentationisorganizedasfollows:InSec.2wegiveabriefsummaryoftheprop-ertiesofthealgebraofSp(n,R)neededintheensueingdevelopmentaswellasadiscussionofthemodelHamiltonianstobestudied.Inthefollowingthreesectionswethenstudy2separatelythecasesn=1,2,3.Wedescribeineachinstancethemappingofthealgebraontoaclassicalsymplecticmanifold,whichistiedtotheexistenceofamanifoldofsolutionsofthetime-dependentHartreeequationandanassociateddecoupledcollectiveHamiltonian.WethenshowhowthesamecollectiveHamiltoniancanbederivedfromourtheoryoflarge-amplitudecollectivemotion.Thematerialforthemonopolecaseisarearrangementwithdifferentemphasisofresultspresentedpreviously[6].Allotherresultsarenew.InSec.6,wemakesuggestionsforfurtherwork,involvingbothapplicationsandextensionsofthere