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1、NoncommutativegeometryandrealityAlainConneslnstitutdesHautesEtudesScientijques,CollegedeFrance,35RoutedeChartres,F-91440Bures-sur-Yvette,France(Received4April1995;acceptedforpublication7June1995)Weintroducethenotionofrealstructureinourspectralgeometry.Thisno
2、tionismotivatedbyAtiyah’sKR-theoryandbyTomita’sinvolutionJ.Itallowsustoremovetwounpleasantfeaturesofthe“Connes-Lott”descriptionofthestandardmodel,namely,theuseofbivectorpotentialsandtheasymmetryinthePoincaredualityandintheunimodularitycondition.01995American
3、InstituteofPhysics.1.ONTHENOTIONOFGEOMETRICSPACEThegeometricconceptshavefirstbeenformulatedandexploitedintheFrameworkofEu-clideangeometry.ThisframeworkisbestdescribedusingEuclid’saxioms(intheirmodernformbyHilbert’).TheseaxiomsinvolvethesetXofpointspEXofthege
4、ometricspaceaswellasfamiliesofsubsets:thelinesandtheplanesfor3-dimensionalgeometry.Besidesincidenceandorderaxiomsoneassumesthatanequivalencerelation(calledcongruence)isgivenbetweensegments,i.e.,pairsofpoints@,q),p,qEXandalsobetweenangles,i.e.,triplesofpoints
5、(a,O,b);a,O,bEX.TheserelationseventuallyallowustodefinethelengthI(p.q)[ofasegmentandthesizeK(a,O,b)ofanangle.Thegeometryisuniquelyspecifiedoncethesetwocongruencerelationsaregiven.Theyofcoursehavetosatisfyacompatibilityaxiom:uptocongruenceatrianglewithvertice
6、sa,O,bEXisuniquelyspecifiedbytheangleQ(a,O,b)andthelengthsof(a,O)and(0,b)(Fig.1).Besidesthecompletenessorcontinuityaxiom,thecrucialoneistheaxiomofuniqueparallel.Theeffortsofmanymathematicianstryingtodeducethislastaxiomfromtheothersledtothediscoveryofnon-Eucl
7、ideangeometry.Onecandescribenon-EuclideangeometryusingtheKleinmodelorthePoincaremodel.IntheKleinmodel,sayfor2-dimensionalgeometry,thesetXofpointsofthegeometryistheinteriorofanellipse(Fig.2).Thelines/aretheintersectionsofEuclideanlineswithX(Fig.2)andthemeasur
8、ementsoflengthandanglesaregivenbyI(p,q)1=log(crossratio(p,q;r,s)),(1.1)wherer,sarethepointsofintersectionoftheEuclideanlinep,qwiththeellipse,asshowninFig.2Q(a,O,b)=&log(crossratio(cr,P;&y)),(1.2
