; TeX output 2001.11.05:1602yi7)"DtqGcmr17DeformationBQuanuNtizationandQuantummecuNhanics#􍍍)%XQ ff cmr12Akira/Y4oshiok7xa= !", cmsy10.25/09/2001,/W4aseda99<"V 3 cmbx10AbstractMƍL̻9K`y 3 cmr10ResultsCAofexistence,Wclassi cationinformaldeformationquan!tiza- <_tion=arereview!ed.A=ystarproMductisintroMducedwitharealnumbMeras<_a*deformationparameter.Aneigen!vdDalueproblemisdiscussedforcon-<_v!ergentostarproMduct,withanexpampleofonedimensionalharmonic<_oscillator.(VBNG cmbx12B1<(Inutro=ductionb#XQ cmr12Inawrord,deformationQuantizationistointroSduceanassociativrenoncom-mrutativeHproSductinrtofunctionspacesonmanifolds.TheproductisgivrenbrydeformingthetrivialcommutativepSointwizemultiplicationoffunctions.0In thistalk, rstIwillsurvreymainresultsofCN cmbx12Cformaldeformationquan-tizationsconcerningtheexistenceandtheequivXalenceproblems.0ThenFIFwillmensionanobservXationonwhatcanbSedonewhentheformaldeformationparameterFg cmmi12FaisreplacedwitharealnrumbSerR msbm10R~.XIwillshorwbyusingastarproSductacertainmecrhanicalsystemonaphasespacecanbeconsideredequivXalenrttosomequantumsystem.B2<(FaGormalzdeformationquanutizationb#Letmestartwithshorwingthetypicalexample,theX@ cmti12XMoyal35prffoduct.#@ffd g^ O!cmsy7K`y cmr10e-mail:ayoshiokqa@rs.kagu.sut.ac.jp.ٺThisQworkissuppGortedbyGrant-in-AidforSci- enti cUUResearch(C)(#13640088.),JapanSoGcietyforthePromotionofScience.1*yiCPoissonbracket OnRthe2-dimensionaleuclideanspaceCR2D|{Ycmr8D2 zI!", cmsy10I3(Fx;yn9)wreharveabiderivXationcalledtheXPoisson35brffacketP-Z}IfFf;gn9IgUR=ōF@f[z ΍2@xō@gܒ[z ߟ ΍@y%LIōF@f۟[z ΍t@yōI@g1[z R ΍@x"m; fG(Fx;y)F;g(Fx;y)URI2FCܞ JK cmsy8J1 ܦ(CR D2)|(1)„Norwusingtheover-leftandover-rightarrows,wewritethebracketasō\nIfFf;gn9IgUR=FfG Lu cmex10L rI 7 F@ G2cmmi8GxI !7F@ GyI  7F@ GyI !7F@ GxH L}~FgË=Ff GI 7.F@^GxI^ !7F@ GyFg|(2)ƍHere^wreremarktheleftarrowoverpartialderivXativemeansthepartialderivXa-tivre#oftheleftfunctionFfG,'Bandso-on.QWiththisnotationwede nebidi er-enrtialopSeraotorby퍍J L Q2I 7TcF@_Gxf"I^ !7F@ Gy L"]D28;=UR L I 7 /F@_GxI !7F@ GyI  7F@ GyI !7F@ GxH L{ ]D2ύ8;=UR L I 7 /F@_GxI !7F@ Gy L?F]D2FI2 L *I 7 gF@ GxsI !7F@ Gy L@ZI L TI 7 F@*GyjI !7F@ GxH L+ L TI 7 F@*GyjI !7F@ GxH L>\]D2kӍ8;= URI 7F@ D2ڍGxBI !7F@  D2ڍ GyI2 I 7=1F@ UaGx HI 7yF@Gy'^I !7F@ Gy IRu!7:GF@"RwGx*F+ I 7F@  D2ڍ GyI !7F@  D2ڍ Gx|(3)\SimilarlyV,bryusingthebinomialtheoremweobtainbidi erentialopSerators]i L dCI 7gFF@qvGxyI^ !7F@ Gy Lȟ]Gnj= kGnURLX'؍?Gk6D=0qLōFnYyk&qL/X(I1) Gk #I 7`F@ xGnJGkڍxx #aMI 7&~F@ 0Gkڍ0y8I !7F@  Gkڍ x HIO!7!F@ "QGnJGkڍ"Qy|(4)!7(FnUR=3F;4F;I).ICMoyalz>pro`ductID0Norw,weintroSduceaformalparameterF.74FVorsmoothfunctionsFfG(Fx;yn9)F;g(Fx;y)URI2FCܞ2J1 ܦ(CR2D2),wreconsideraformalpSowerseriesqS._FfID0jFgs==URFfGexpōnFn[z ΍X2"1ݟ L )\I 7,F@6Gx>yI^ !7F@ Gy Ld:Fg|(5)ԟ L KiI 7NF@XGx`pI^ !7F@ Gy L3]GnFgI2FCܞ J1 ܦ(CR D2)[[F]]!_DThisopSerationcanbetakrenforelementsofFCܞ2J1 ܦ(CR2D2)[[F]]intheobviouswayV.QR0TheproSductID0onFCܞ2J1 ܦ(CR2D2)[[F]]hasthefollorwingpropertryV.?(D1).HFVorfunctionsFfG(Fx;yn9)F;g(Fx;y)URI2FCܞ2J1 ܦ(CR2D2)theproSductisexpandedas%QXFfID0jFgË=URFfGg+ōF۟[z ΍X2 IfFf;gn9Ig+Fǟ D2FD2(Ff;gn9)+IUN+Fǟ GnjFGnP(Ff;gn9)+Im0<_ThismeanstheproSductisadeformationofthetrivialmrultiplication<_offunctionsandthe rstordertermofFoistheProissonbrakcket.2yi?(D2).HTheproSductisassociativre.?(D3).HEacrhFGn isabidi erentialopSeratorforFnUR=2F;3F;I.RemarkdthatonCR2D2Gn =I3UR(FxD1F;IF;xGnPF;yD1F;IF;yGnP)dtheproSductID0$canbede nedsimilarlybrythePoissonbracket ԍ rI 7uF@GxI^ !7F@ GyRh= kGnURLX ㇍GjvD=1 L TI 7!F@+*Gx8:H;cmmi6Hj7 I !7F@ Gy8:HjII  7F@ Gy8:HjI !7F@ Gx8:Hj3˟ L#ɍFVurther,thisconceptisalsoextendedonthegeneralProissonmanifolds.CDeformationquantization A.]skrew.nbiderivXationonamanifoldFMoRisex-pressedloScallyastRIfFf;gn9IgUR=LX ㇍lRGi;jFaGijJ(Fx)F@Gx8:HiuYFfG@Gx8:Hj 3Fg$uforTFf;gwI2 FCܞ2J1 ܦ(FM@).vATskrewderivXationIfF;Igsatis yingtheJacobiidenrtityTiscalledaXPoisson35brffacket.CDe nition1jXA2nCR[[F]]X-bilineffar(orCC(FM@)[F]]X-bilineffar)productIXonthefunc-tionHspffaceFCܞ2J1 ܦ(FM@)[[F]]XiscalledCstarĘpro`ductXwhenitsatis estheproperties(D1){(D3).EThe`algebrffa(CC(FM@)[F]]F;I)XiscalledaCDeformationquantiza-tionX.CThebackground OneXseesthecommrutatoroftheMoyalproSductID0\sat-is es}f,[Fx;yn9]UR=FxID0jFyIFyID0jFxUR=Fxy+ōF۟[z ΍X2KI(FxyIōF۟[z ΍X2 )=FtheHeisenrbSerg'scanonicalcommutationrelation.7DThenthealgebraofpSoly-nomilasuL(IP[Fx;yn9;]F;ID0)canbSeregardedasthealgebraofoperatorsgeneratedbrypthederivXationF@GxWandthemultiplicationFx,Handthealgebra(FCܞ2J1 ܦ(CR2D2)[[F]]F;I)is acompletionundercertaintopSologyV. MotivXatedbrytheconsiderationabSorve,deformation7quantizationwasinitiatedinordertogeneralizethequan-tizationfromoneuclideanspacetoonmanifolds([BFFLS%]).CMainresults1.ċExistence OnOhanryPoissonmanifold(FM;IfF;Ig),ntthereOhexistsastarproSd-uct.wTheexistenceproblemwrassolvedforasymplecticmanifoldby[DLP],[F],[OMY]andforaProissonmanifoldby[K "w].3 8yiC2.Classi cation  min10!!BTwro]starproSductsI,9I2J0arecalledXeffquivalenti thereexistsanCR[[F]]-linear(orCC[[F]]-linear)isomorphismɍ*FT:URFCܞ J1 ܦ(FM@)[[F]]I!FCܞ J1(FM@)[[F]]F;$΍,*+T=URFI++FTD1j+F D2FTD2j+IUN+F GkYFTGk:+IF; TGk #(di erenrtialopSeratorsj)sucrhthatq FfI J0xFgË=URFTƟ JD1 B(FTfIFTgn9)F; f;gËI2URFCܞ J1 ܦ(FM@)[[F]]F:'0TheequivXalenceclassesofstarproSductsIfgF=RIareknorwn; forasym-plecticFM@,wrehave#IfgF=URI=FHV D2Z(FM@)[[F]]([BCG],[F],[OMMY'z]),andteforaProissonFM@,theequivXalenceclassesareisomorphictothespaceofformalProissonbrackets[K "w].C3.EQuantizedDarb`ouxtheorem FVoriasymplecticmanifoldFM@,astarproSductwGIislocallyequivXalenrttotheMoyalproSduct,oi.e,forwGeverypSointFpURI2FM@,therer{existsacanonicalcoSordinateneighrbourhoodFU_whereproductsIandID0areequivXalenrt([G 8(],[LWs]).C4.-#Geometric picture WhenFM8issymplectic,astarproSductisgivreninthegeometricwrayasfollorws([OMY]).:-\h1.<_Thereexista bSerbundlewithWVeylalgebra ber(FWr;TLbjI)ɍ9FË:URFWGM BI!FMx-\h2.<_There 2isansubalgebraIF1(FWGM )ofsmoSothsectionsofthebundleFWGM,<_(algebrastructureofsectionsaregivrenbythepSointwisemultiplication<_ofFWGM ),andalsoanalgebraisomorphism!㍍#FË:UR(FCܞ J1 ܦ(FM@)[[F]]F;I)I!(IF1(FWGM )F;TLbjI)|(6)ThecalgebraIF1(FWGM )isgivrenbyanullspaceofacertain atconnectionofFWGM ,4so-called%aFVedosorvconnection([F],[X]). FVurther,thereexistsaconrtactalgebraKbundleICJM^conrtainingFWGM asasubbundleandthereexistsanon- atconnectionwhoserestrictiontoFWGMisthepreviousconnection([Y1],Z[Y2],[Y3]).4-yiB3<(Starzpro=ductswitharealparameterdG msbm10d~b#IfwresubstituteanumbSerintotheparameterF,BtheMoyalproSductFf|=ID0BFgdivrerges*forfunctionsFf;gËI2URFCܞ2J1 ܦ(CR2D2)ingeneral.ThemainresultsuptonowareobasedonthefactthatF6isaformalparameter,HandthemosttecrhniquesarenotusefulforanrumericaldeformationparematerR~.$WVehavenogeneraltheory&ofconrvergent&starproSductsatpresenrt,ubut,on&theotherhandweharvesomeinrterestingobServXationasfollows.JCVacuum LetǿusputF=URFiR~, R~F>0ǿandletusconsidertheMoryalǿproSductID0onCR2D2.8InthesequalwredeontesimplybyID0V=URI.0Norwwecanconsiderafunction,calledaXvacffcum Í5xFvD0V=URexpHҟqLō#Q2!Q][z } ΍R~Fi-txyn9qL čsinceR~isanrumbSer.8AneasycalculationgivresӍ@(eFyIFvD0V=URFyn7expōF[z ΍X2!X L (I 7+F@5$Gx=˳I^ !7F@ Gy LavFvD0=URFyn9vD0j+ōFiR~۟[z } ΍N2Fy(I 7=1F@ UaGyRn)( IU_!7=1F@ UaGxH)FvD0=UR0F:|(7)+aFxIFvD0V=UR2FxvD0F:|(8)ObrviousidentitiessFxIFxUR=Fx D2F; xIUNFxp L|p5{zp5}Ս$.pGnC=Fx Gn%yieldthatapSolynomialFp(Fx)satis esPvFyIFp(Fx)IFvD0A=URFp(Fx)IFyIFvD0jI[Fp(Fx)F;yn9]IFvD0V=[Fyn9;p(Fx)]IFvD0A=FuSppmsbm8S~zvꍑGi X.F@GxHFp(Fx)IFvD0&FFxIFp(Fx)IFvD0V=URFxp(Fx)IFvD0Thrus,5onthesetIPQ=URIfFp(Fx),IFvD0IjFp(Fx)pSolynomialsAlIgthefollowingstarproSductopSerationsarerepresenrtedasderivXationandmultiplicationopSerators|Fyn9IURUTq lasy10U;FuS~zvꍑGi X.F@GxHF; xIU;FxI|(9)59PyiCThe onedimensionalharmonicoscillator Usingcbthedeformationpa-rameter%jR~,4wrecanhandleasystemwiththeMoyalproSductID0nwhichseemsequivXalettotheonedimensionalquanrtumharmonicoscillator.0WVesetasfollorws.-\h1.<_WVeidenrtitfyCR2D2V=URCCviaFz5=Fy+Fix.-\h2.<_WVeconsiderthehamiltonianfuncitonFHB=FuD1z@2 (Fx2D2j+Fyn92D2.=)UR=FuD1z@2Fz?Fz.-\h3.<_WVesetanelemenrtFuD0 "=exp GLIFualD133zvS~(Fx2D2j+Fyn92D2.=)GLgl=exp GLIFualD133zvS~Fz?FzGLER)calledthe<_Xvacuum.-\h4.<_WVeMinrtroSduceaninnerproductbry(Ff;gn9)=FuD10z ǀGI{S~+UQLR+UQR +*Y2@cmbx8YREAacmr6E2.(Ff6IXFg dxdyandwe<_considerafunctionspacebryRIHr=URIfFfG(Fz)IFuD0IjFf(Fz)UR=LX ҍ kGnFaGnPFz Gn13F;LX ҍ ~GnUPIjFaGnIj D2Fn9(2R~) GnFn!URF.Then,"FuD0bSecomes0whenregardedasaformalparwerseiresofR~.(Thrus,StheargumentabSorvecannotbesetintheformaldeformationquanrtizationtheoryV.CConclusion IfkwreintroSducearealnumbSerR~asadeformationparame-ter,Swre candealwithaneigenvXalueprobleminthedeformationquantiza-tiontheorysimilarlyasintheopSeratorquanrtummechanics.60However,5mosttecrhniqueseofformaldeformationquantizationthoSeryarenotusefulforcon-vrergentstarproSducts,hencewrehavetoseeknewtechniquesindeformationquanrtizationwithanumbSerparameter.'OBReferencesb#[BCG]EmM.Y.MaedaandA.YVoshiokXa,XWeylmanifoldsandDefor-7' mation35quantization,Adv.Math.C85(1991),224-255.[OMMY]RH.Omori,BY.Maeda,N.MiyrazakiandA.YVoshiokXa: oXPoincffar3Le-7' CartansAclassanddeformationquantizationofKalermanifolds,AXComm.7' Math.35Phys.C194(1998),207{230.7Uyi[X]7' X.Ping:XFeffdosovIX-productsandquantummomentummaps.XComm.7' Math.35Phys.C197(1998),167{197.[Y1]:A. YVoshiokXa,7XContact(structurffesonWeylmanifoldsandDeformation7' quantization,