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1、TheAlgebraofGrandUnifiedTheoriesJohnBaezandJohnHuertaDepartmentofMathematicsUniversityofCaliforniaRiverside,CA92521USAMay4,2010AbstractTheStandardModelisthebesttestedandmostwidelyacceptedtheoryofelementaryparticleswehavetoday.Itmayseemcomplicatedandarbitrary,but
2、ithashiddenpatternsthatarerevealedbytherelationshipbetweenthree‘grandunifiedtheories’:theoriesthatunifyforcesandparticlesbyextend-ingtheStandardModelsymmetrygroupU(1)×SU(2)×SU(3)toalargergroup.ThesethreeareGeorgiandGlashow’sSU(5)theory,Georgi’stheorybasedonthegr
3、oupSpin(10),andthePati–SalammodelbasedonthegroupSU(2)×SU(2)×SU(4).Inthisexpositoryaccountformathematicians,weex-plainonlytheportionofthesetheoriesthatinvolvesfinite-dimensionalgrouprepresentations.Thisallowsustoreducetheprerequisitestoabareminimumwhilestillgivin
4、gatasteoftheprofoundpuzzlesthatphysicistsarestrugglingtosolve.1IntroductionTheStandardModelofparticlephysicsisoneofthegreatesttriumphsofphysics.Thistheoryisourbestattempttodescribealltheparticlesandalltheforcesofnature...exceptgravity.Itdoesagreatjoboffittingexp
5、erimentswecandointhelab.Butphysicistsaredissatisfiedwithit.Therearethreemainreasons.First,itleavesoutgravity:thatforceisdescribedbyEinstein’stheoryofgeneralrelativity,arXiv:0904.1556v2[hep-th]1May2010whichhasnotyetbeenreconciledwiththeStandardModel.Second,astron
6、omicalobservationssuggestthattheremaybeformsofmatternotcoveredbytheStandardModel—mostnotably,‘darkmatter’.Andthird,theStandardModeliscomplicatedandseeminglyarbitrary.Thisgoesagainstthecherishednotionthatthelawsofnature,whendeeplyunderstood,aresimpleandbeautiful
7、.Forthemoderntheoreticalphysicist,lookingbeyondtheStandardModelhasbeenanendeavorbothexcitingandfrustrating.Mostmodernattemptsarebasedonstringtheory.Therearealsootherinterestingapproaches,suchasloopquantumgravityandtheoriesbasedonnoncommutativegeometry.Butbackin
8、themid1970’s,beforeanyofthecurrentlypopularapproachesrosetoprominence,physicistspursuedaprogramcalled‘grandunification’.ThissoughttounifytheforcesandparticlesoftheStandardMod
