GAUGE THEORIES IN PARTICLE PHYSICS ed .3rd vol2 solutions外文书

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1、SolutionstoproblemsinGaugeTheoriesinParticlePhysics",ThirdEdition,volume2:QCDandtheElectroweakTheory"Chapter1212.1Theelementsoftheset(whichwewanttoshowformagroup)areyyall22matricesVwhichareunitary(i.e.VV=VV=I)and2havedeterminantequalto+1.Weneedtoshowthatconditions(i)-(iv)listedinsectionM.1ofAp

2、pendixMaresatised,wherethelawofcombinationisordinarymatrixmultiplication.(i)LetVandVbeanytwomembersoftheset.Theirproductis12VV.Clearly(bytherulesofmatrixmultiplication)thisisa2212ymatrix.Toseeifitisunitary,weform(VV)(VV).Usingthe1212yyygeneralmatrixresult(AB)=BAthisbecomesyyy(VV)(VV)=VV(V)(V)12

3、121221y=VI(V)=I1212usingrsttheunitarityofVandthenthatofV.Sotheproductof21anytwosuchmatricesisaunitary22matrix.Similarly,det(VV)=12(detV)(detV)=1usingthe`detV=1'conditionforeachmatrix.So12theproductofanytwosuchmatricesisaunitary22matrixwithdeterminant+1,andthereforetheproductalsobelongstothesam

4、eset,verifyingproperty(i).(ii)Matrixmultiplicationisassociative.(iii)TheidentityelementisI.2y(iv)TheinverseofanyelementVisV,bytheunitarityproperty.12.2Visgivenbyequation(12.17)whichwemaywriteasin !1+ii1112V=:in i1+i2122HencedetV=(1+i)(1+i)+in 112212211+i(+);1122torstorderin,sothatthe

5、conditiondetV=1reducesto+in 11=0,whichisjustTr=0.2212.3Thecommutationrelationscanbecheckedbystraightforwardmatrixmultiplicationwiththe22matrices.12.4Wehave111111[T;T]=++:::;++:::ij(1)i(2)i(A)i(1)j(2)j(A)j222222111111=;+;+:::;(1)i(1)j(2)i(2)j(A)i(A)j222222111=i++:::i

6、jk(1)k(2)k(A)k222=iT;(1)ijkkwhereinthesecondlineweusedthefactthatthe'sassociatedwithdierentnucleonscommute,andinthethirdlineweusedequation(12.28)foreachoftheindependent's.(1)12.5The(j;k)element(j=rowlabel,k=columnlabel)ofTisi.The1jk1(1)onlynon-zeroentriesareforj=2;k=3,giving(T)=i,andfor231

7、(1)j=3;k=2,giving(T)=i.Hence32101000(1)BCT=00i:@A10i0Similarly,0100i(1)BCT=000;@A2i00and010i0(1)BCi00T=:@A3000Itisthenamatterofstraightforwardmatrixmultiplicationtocheck(12.47)forthecasei;j;k=1;2;3.12.6Weexpandequat