Quantum stochastic differential equations for boso

arXiv:math-ph/0304023v114Apr2003Quantumstochasticdifferentialequationsforbosonandfermionsystems—MethodofNon-EquilibriumThermoFieldDynamicsA.E.Kobryn1,T.HayashiandT.Arimitsu∗InstituteofPhysics,UniversityofTsukuba,Ibaraki305-8571,JapanAbstractAunifiedcanonicaloperatorformalismforquantumstochasticdifferentialequa-tions,includingthequantumstochasticLiouvilleequationandthequantumLangevinequationbothoftheItˆoandtheStratonovichtypes,ispresentedwithintheframe-workofNon-EquilibriumThermoFieldDynamics(NETFD).ItisperformedbyintroducinganappropriatemartingaleoperatorintheSchr¨odingerandtheHeisen-bergrepresentationswithfermionicandbosonicBrownianmotions.Inordertode-cidethedoubletildeconjugationruleandthethermalstateconditionsforfermions,ageneralizationofthesystemconsistingofavectorfieldandFaddeev-PopovghoststodissipativeopensituationsiscarriedoutwithinNETFD.Keywords:Non-EquilibriumThermoFieldDynamics,stochasticdifferentialequations,martingaleoperator,fermionicBrownianmotion,bosonicBrownianmotionPACS:05.30.-d,02.50.Ey1IntroductionInthispaperwestudytime-dependentbehaviorofnon-equilibriumquantumsystemsinvolvingstochasticforceswhichcanbebosonorfermiontypeandarecalledquantumBrownianmotion.Presentconsiderationisanextensionofpreviousanalysisreportedcomprehensivelybyoneoftheauthors[1]andisgivenintermsofNon-EquilibriumThermoFieldDynamics(NETFD)[2,3,4].∗CorrespondingauthorEmailaddress:[e-mail](T.Arimitsu).1Presentaddress:InstituteforMolecularScience,Myodaiji,Okazaki444-8585,JapanPreprintsubmittedtoElsevierPreprint7February2008NETFDisaunifiedformalism,whichenablesustotreatdissipativequan-tumsystemsbythemethodsimilartousualquantummechanicsorquantumfieldtheory,whichaccommodatestheconceptofthedualstructureintheinterpretationofnature,i.e.intermsoftheoperatoralgebraandtherepre-sentationspace.TherepresentationspaceinNETFDiscomposedofadirectproductoftwoHilbertspaces:oneisfornon-tildefields,andtheotherfortildefields.Withinthestatisticaloperator(densityoperator)formalismthereisentanglementbetweenoperatorsandstatisticaloperatorduetotheirnon-commutativity.Introductionoftwokindsofoperators,withouttildeandwithtilde,madeitpossibletoresolvetheentanglementbetweenrelevantoperatorsandthestatisticaloperator.Wearederivingaunifiedsystemofquantumstochasticdifferentialequations(QSDEs)undertheinfluenceofquantumBrownianmotion,includingthequantumstochasticLiouvilleequationandthequantumLangevinequation.ThequantumFokker-PlanckequationisderivedbytakingtherandomaverageofthecorrespondingstochasticLiouvilleequation.TherelationbetweentheLangevinequationandthestochasticLiouvilleequation,aswellasbetweentheHeisenbergequationforoperatorsofgrossvariablesandthequantumFokker-Planckequationobtainedhere,issimilartotheonebetweentheHeisenbergequationandtheSchr¨odingerequationinquantummechanicsandfieldtheory.Ourextensionofanalysis[1]consistsofessentiallythreeitems.TwoofthemincludedefinitionoffermionicBrownianmotionandtreatmentoffermionsinNETFD,i.e.thetildeconjugationruleandthethermalstateconditionsinthecaseoffermionsystems.Thirditemisthesimultaneousconsiderationofhermitianandnon-hermitianinteractionHamiltonians.Tobeginwith,wefirstremindbrieflysomestandardstepsthatpeopleusuallytakeinordertoobtaintheirreversibleevolutionofmacroscopicsystemsstart-ingfromthemicroscopiclevel.Atpresent,therearemanyviewpointsgivingustoolshowtodescribeN-bodysystemsoutofequilibrium.Atthesametime,oneusuallyfollowsoneofseveralbasicapproaches:(i)thebehaviorofthesystemsisexpressedintermsofnotthetotal(N-particle)distributionfunctionbuts-particleones(withsbeingusually1and/or2),(ii)thedy-namicsofthesystemsischaracterizedbytheevolutionofa“coarsegrained”phase-spacedistributionfunctionorstatisticaloperator,and(iii)theevolu-tionofthesystemsisdescribedbytheequationsofmotionforthedynamicalgrossvariables.Theapproach(i)isintimatelyrelatedtotheBogoliubovmethodofareduceddescriptionofmany-particlesystems[5],whichiswidelyusedforconstruc-tionofkineticequationsbasedontheLiouvilleortheLiouville-von-Neumannequation.Bogoliubov’shypothesisthatthetimedependenceofhigher-particledistributionfunctionsenterthroughtheone-particledistributionprovidesafundamentalimportanceinvariousschemesoftruncationoftheBBGKYhi-2erarchy.Intheapproach(ii),themostfrequentlyusedtoolsareprojectionoperatorsintroducedbyNakajima[6]andZwanzig[7,8].Thebasicideaunderlyingtheapplicationoftheirtechniquestocomplexsystemsistoregardtheoperationoftracingovertheenvironmentasaformalprojectioninthespaceofthetotalsystem.Itbecameespeciallypopularinquantumopticswheretheso-calledquantummasterequationforreducedstatisticaloperatorofarelevantsystemnowbearstheirnamesandiscalledtheNakajima-Zwanzigequation[9].Thegeneralframework,calledsub-dynamics,attheBrusselsschoolisalsorelatedtotheapproach(ii)buttheunderlyingconceptisdifferentfromtheonebyNakajimaandZwanzig.Themainpointhereisthenotionoftheincreaseofthenumberofcorrelationswithinasystemintime.IthasbeenexpoundedindetailbyPrigogineandcoauthors,seee.g.[10,11].Regardingtotheapproach(iii),weshouldmentionprojectionoperatorbyMori[