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1、1
2、ClassicalSymmetriesTheconceptofsymmetrywillplayacrucialroleinnearlyallaspectsofourdiscussionofweakinteractions.Atthelevelofthedynamics,thefundamentalinteractions(oratleastthatsubsetofthefundamentalinteractionsthatweunderstand)areassociatedwithgaugesymmetries".Butmorethanthat,theunderlyingmathem
3、aticallanguageofrelativisticquantummechanics
4、quan-tumeldtheory
5、ismucheasiertounderstandifyoumakeuseofallthesymmetryinformationthatisavailable.Inthiscourse,wewilmakeextensiveuseofsymmetryasamathematicaltooltohelpusunderstandthephysics.Inparticular,wemakeuseofthelanguageofrepresentationsofLiealgebr
6、as.1.1Noether'sTheorem{ClassicalAttheclassicallevel,symmetriesofanactionwhichisanintegralofalocalLagrangiandensityareassociatedwithconservedcurrents.Considerasetofelds,(x)wherej=1toN,andanactionjZ4S[]=dxL((x)@(x))(1.1.1)whereListhelocalLagrangiandensity.Theindex,j,iswhatparticlephysicistsca
7、lla
avor"index.Dierentvaluesofjlabeldierenttypes,or
avors",oftheeld.Thinkoftheeld,,withoutanyexplicitindex,asacolumnvectorin
avorspace.Assume,forsimplicity,thattheLagrangiandependsonlyontheelds,,andtheirrstderivatives,@.TheequationsofmotionareLL@=:(1.1.2)(@)Notethat(1.1.2)isav
8、ectorequationin
avorspace.Eachsideisarowvector,carryingthe
avorindex,j.Asymmetryoftheactionissomeinnitesimalchangeintheelds,,suchthatS[+]=S[](1.1.3)orL(+@+@)=L(@)+@V(@)(1.1.4)whereVissomevectorfunctionoftheorderoftheinnitesimal,.Weassumeherethatwecan4throwaways
9、urfacetermsinthedxintegralsothattheVtermsmakesnocontributiontotheaction.ButLLL(+@+@);L(@)=+@(1.1.5)(@)1WeakInteractions
10、HowardGeorgi
11、draft-February10,1998
12、2because@=@.Notethat(1.1.5)isasingleequationwithnojindex.Thetermsontherighthandsideinvolveamatrixmultiplication
13、in
avorspaceofarowvectorontheleftwithacolumnvectorontheright.From(1.1.2),(1.1.4)and(1.1.5),wehave@N=0(1.1.6)whereLN=;V:(1.1.7)(@)Often,wewilbeinterestedinsymmetriesthataresymmetriesoftheLagrangian,notj