Extended Quantum Mechanics

arXiv:math-ph/9909022v615May2003ExtendedQuantumMechanics∗PavelB´ona,e-mail:[e-mail]lPhysics,ComeniusUniversitySK-84248Bratislava,SlovakiaFebruary6,2008AbstractTheworkcanbeconsideredasanessayonmathematicalandconceptualstructureofnon-relativisticquantummechanics(QM)whichisrelatedheretosomeother(moregeneral,butalsotomorespecial–“approximative”)theories.QMishereprimarilyequivalentlyreformu-latedintheformofaPoissonsystemonthephasespaceconsistingofdensitymatrices,wherethe“observables”,aswellas“symmetrygenerators”arerepresentedbyaspecifictypeofrealvalued(denselydefined)functions,namelytheusualquantumexpectationsofcorrespond-ingselfadjointoperators.Itisshowninthisworkthatinclusionofadditional(“nonlinear”)symmetrygenerators(i.e.“Hamiltonians”)intothisreformulationof(linear)QMleadstoaconsiderableextensionofthetheory:twokindsofquantum“mixedstates”shouldbedis-tinguished,andoperator–valuedfunctionsofdensitymatricesshouldbeusedintherˆoleof“nonlinearobservables”.Ageneralframeworkforphysicaltheoriesisobtainedinthisway:Bydifferentchoicesofthesetsof“nonlinearobservables”weobtain,asspecialcases,e.g.classicalmechanicsonhomogeneousspacesofkinematicalsymmetrygroups,standard(linear)QM,ornonlinearextensionsofQM;alsovarious“quasiclassicalapproximations”toQMareallsubtheoriesofthepresentedextensionofQM-theextendedquantummechanics(EQM).AgeneralinterpretationschemeofEQMextendingtheusualstatisticalinterpretationofQMisalsoproposed.Eventually,EQMisshowntobe(includedinto)aC∗-algebraic(hencelinear)quantumtheory.Mathematicalformulationofthesetheoriesispresented.Thepresentationincludesananalysisofproblemsconnectedwithdifferentiationoninfinite–dimensionalmanifolds,aswellasaso-lutionofsomeproblemsconnectedwithworkononlydenselydefinedunboundedreal–valuedfunctionsonthe(infinitedimensional)“phasespace”correspondingtounboundedoperators(generators)andtotheirnonlineargeneralizations.Also“nonlineardeformations”ofunitaryrepresentationsofkinematicalsymmetryLiegroupsareintroduced.Possibleapplicationsarebrieflydiscussed,andsomespecificexamplesarepresented.ThetextcontainsalsobriefreviewsofHamiltonianclassicalmechanics,aswellasofQM.Mathematicalappendicesmaketheworknearlyselfcontained.PACSnumbers:03.65.-w;03.70.+k;11.10.Lm.;02.20.+b.∗Publishedin:actaphys.slov.50(2000)1–1981CONTENTS2Contents1Introduction41.1NotesonMotivation,BackgroundIdeas,andHistory.................51.1-aOninitialideasandconstructions.......................61.1-bRelationtoinfinitesystems...........................91.1-cQuestionable“subsystems”...........................111.1-dSomebasicbuildingblocksofEQM......................121.2ABriefDescriptionoftheContents...........................141.3RemarksontheText...................................191.4AGeneralSchemeofHamiltonianClassicalMechanics................201.4-aClassicalphasespaceanddynamics......................211.4-bObservablesandstatesinclassicalmechanics.................241.4-cSymplecticstructureoncoadjointorbits....................251.5BasicConceptsofQuantumMechanics.........................271.5-aPurestatesanddynamicsinQM........................271.5-bStatesandobservables..............................301.5-cSymmetriesandprojectiverepresentationsinQM..............351.5-dOnthecausalityprobleminQM........................382ExtendedQuantumMechanics402.1ElementaryQuantumPhaseSpace...........................412.1-aBasicmathematicalconceptsandnotation...................412.1-bThemanifoldstructureofS∗..........................432.1-cPoissonstructureonquantumstate-space...................472.1-dHamiltonianvectorfieldsandflows.......................502.1-eOninterpretation:Subsystemsandtwotypesofmixedstates........552.2UnboundedGenerators..................................592.2-aSomeprobabilisticaspectsofselfadjointoperators..............602.2-bUnbounded“linear”generators.........................612.2-cOnunboundednonlineargenerators......................652.2-dNonlineargeneratorsfromgrouprepresentations...............702.3Symmetries,DynamicsandObservables........................813SpecificationsandApplications973.1AReviewofConsideredSpecifications.........................973.2StructureofProjectiveHilbertSpace..........................993.3SymplecticFormofQMandNLQM;RestrictionsofQM...............1043.3-aGeneralizedquantummechanicsonP(H)...................1053.3-bTheWeyl–HeisenberggroupandCCR.....................1093.3-cRestrictedflowswithlineargeneratorsonO̺(GWH).............1153.3-dTimedependentHartree–Focktheory.....................1183.3-eNonlinearSchr¨odingerequationandmixedstates...............1233.4“Macroscopic”ReinterpretationofEQM........................1273.5SolutionofSomeNonlinearSchr¨odingerEquations..................1303.6OnanAlternativeFormulationofNLQM.......................132CONTENTS3APPENDICES135ASelectedTopicsofDifferentialGeometry135A.1Introductiontotopology.................................135A.2ElementsofdifferentiationonBanachspaces.....................138A.3Basicstructuresonmanifolds...............