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1、QUANTUMGROUPSANDQUANTUMSPACESBANACHCENTERPUBLICATIONS,VOLUME40INSTITUTEOFMATHEMATICSPOLISHACADEMYOFSCIENCESWARSZAWA1997ONPATHINTEGRATIONONNONCOMMUTATIVEGEOMETRIESACHIMKEMPFDepartmentofAppliedMathematics&TheoreticalPhysicsandCorpusChristiCollegeintheUniversit
2、yofCambridgeSilverStreet,CambridgeCB39EW,U.K.E-mail:a.kempf@amtp.cam.ac.ukAbstract.WediscussarecentapproachtoquantumfieldtheoreticalpathintegrationonnoncommutativegeometrieswhichimplyUV/IRregularisingfiniteminimaluncertaintiesinpositionsand/ormomenta.Oneclasso
3、fsuchnoncommutativegeometriesariseas‘momentumspaces’overcurvedspaces,forwhichwecannowgivethefullsetofcommutationrelationsincoordinatefreeform,basedontheSyngeworldfunction.1.Introduction.Acrucialexampleofnoncommutativegeometry[1]isthequan-tummechanicalphasesp
4、acewithitsnoncommuting`coordinatefunctions'xiandpj.Weinvestigatethepossibilitythatalsothepositionandmomentumspacesacquirenoncom-mutativegeometricfeatures,i.e.weconsiderassociativeHeisenbergalgebrasAgeneratedbyelementsxi;pj,nowallowing[xi;xj]6=0;[pi;pj]6=0(1)
5、andalso:[xi;pj]=ih(ij+ijklxkxl+ijklpkpl+:::)(2)Werestrictourselvestorelationsthatallowtheinvolutionx=x;p=p,i.e.forwhichiiii`'extendstoanantialgebrahomomorphism.TomotivatetheparticularformofrelationEq.2,letthisrelationberepresentedonadensedomainDHinaHil
6、bertspaceH,i.e.boththexiandthepjaretoberepresentedassymmetricoperatorsonD.Assuming,2e.g.inthesimplestcaseofonedimension, ; >0and <1=h,togetherwiththeusualde nitionofuncertainties(x)2:=hj(x hjxji)2ji(3)ji1991MathematicsSubjectClassification:Primary81S05;Sec
7、ondary83C47.Thepaperisinfinalformandnoversionofitwillbepublishedelsewhere.[379]380A.KEMPFyieldsh xp1+(x)2+hxi2+(p)2+hpi2(4)2AsisnotdiculttocheckEq.4impliesthatthereare niteminimaluncertaintiesx0=(1= h2 ) 1=2andp=(1= h2 ) 1=2,sothatthereappearsa`mi
8、nimaluncertainty0gap'(alljinormalised):8ji2D:xjix0andpjip0(5)Physically,sinceandcanbeassumedsmall,wehaveordinaryquantummechanicalbehaviouronmediumscales.Thepresenceofa nitex0,physicallyre
