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ySEOtextz0. General Formulation, (iii) Complex Scalar Fields and SU(N), a. L(x;y)=yʃyʃ-V(xx), xCn, yC4~n, I()=d4xL(*(x),**(x)) Here, I abbreviate Lc(x,x;y,y) to L(x;y). b. I()/ƒӃ(x)=0(L/݃Ӄ(x)-/xʁL/[݃ʃӃ(x)]=0,L/݃Ӄ(x)-/xʁL/[݃ʃӃ(x)]=0́݃ʁ݃ʃӃ(x)+V'(Ӄ(x)Ӄ(x))Ӄ(x)=0, c. Symmetry, 1. SU(N), a. Definition, SU(N)={U|UCN~N, UU=1, det U=+1}, b. Parametrization and Generator, USU(N),΃ƁRN2-1;U=exp(-ikk) where: iCN~N,i=i (i=1,2,EEE,N2-1) fC(N2-1)~(N2-1)~(N2-1);[j,k]=fjkll, c. Example, 1. SU(1)={e-i|ƁR}, SU(1) and SO(2) are representations of each other. SU(1)e-iƁ[cos -sin, sin cos]SO(2), 2. SU(2), i=i (i=1,2,3);[j,k]=ijkll, exp(-ikk)=0coskk-il(lkk)sinjj, 0=[1001],1=[0110],2=[0 -i,i 0],3=[1 0,0 -1]

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