孔涅 非交换几何与物理

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1、NONCOMMUTATIVEGEOMETRYANDPHYSICSALAINCONNES1.IntroductionThetwoexistingtheorieswhichsuccessfullyencodeourknowledgeofspace-timeare:²GeneralRelativity²TheStandardModelGeneralrelativitydescribesspace-timeasfaraslargescalesareconcerned(cf.Figure1and[2]formanymoresuggest

2、ivethoughtsandpictures)andisbasedonthegeometricparadigmdiscoveredbyRiemann.Itreplacesthe°at(pseudo)metricofPoincar¶e,Einstein,andMinkowski,22222ds=¡dt+dx+dy+dzbyacurvedspace-timemetricwhosecomponentsformthegravitationalpotentialg¹º2¹ºds=g¹ºdxdxFigure1.Suspectedblack

3、holeincenterofgalaxy1NONCOMMUTATIVEGEOMETRYANDPHYSICS2Figure2.CERNcollisionringThebasicEinstein-HilbertactionprinciplegivenbytheactionZ1p4SE[g¹º]=rgdxGMwhichholdsinemptyspacewiththepossibleadditionofacosmologicalterm,isre-placedinthepresenceofmatterbythecombination(

4、1)S=SE+SSMwhereSSMisthestandardmodelactionwhichencapsulatesourknowledgeofspace-timeatsmallscalesasuncoveredbythehighenergyexperimentssuchasthoseper-formedatCERN(Figure2).ThetransitionClassical!QuantumisverysimpletoformulateintermsoftheFeynmanintegralwhicha®ectseachc

5、lassical¯eldcon¯gurationwiththeprobabilityamplitudeiSe~Whilethisprescriptionworksremarkablywellforthequantizationoftheclassical¯eldsinvolvedinthestandardmodelprovidedoneusesthetechniqueofrenormalization,NONCOMMUTATIVEGEOMETRYANDPHYSICS3thislatterperturbativetechniqu

6、efailsdramaticallywhenonetriestodealwiththegravitational¯eldg¹º.Inmanywaysthisresultisnotsurprising.IndeedmanyofthebasicnotionsofthetraditionalformalismofQuantumFieldTheory(QFT),suchasparticles,scatteringmatrices,etc...heavilyrelyonthe°atgeometryofMinkowskispaceandt

7、herelatedPoincar¶esymmetrygroup.Treatingthequantizationoftheg¹ºinthesamewaywould-ifsuccessful-produceaquantum¯eldtheoryoftheg¹ºonMinkowskispace:astrangeresultindeedwhenviewedfromthegeometricstandpoint!Thetechnicalreasonforthenotoriousdi±cultyofquantizingtheg¹ºinthet

8、raditionalperturbativewayistheclashwitheitherrenormalizabilityorunitarity.Insomesensethisclashcontainsaseriouswarning,namelythatoneshouldn