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【SEOtext】0. General Formulation, (iii) Complex Scalar Fields and SU(N), a. L(x;y)=yμα†yμα-V(xα†xα), x∈Cn, y∈C4×n, I(φ)=∫d4xL(φ*(x),∂*φ*(x)) Here, I abbreviate Lc(x,x†;y,y†) to L(x;y). b. δI(φ)/δφα(x)=0⇔(∂L/∂φα(x)-∂/∂xμ∂L/∂[∂μφα(x)]=0,∂L/∂φα†(x)-∂/∂xμ∂L/∂[∂μφα†(x)]=0⇔∂μ∂μφα(x)+V'(φβ†(x)φβ(x))φα(x)=0, c. Symmetry, 1. SU(N), a. Definition, SU(N)={U|U∈CN×N, U†U=1, det U=+1}, b. Parametrization and Generator, ∀U∈SU(N),∃θ∈RN2-1;U=exp(-iθkτk) where: τi∈CN×N,τi†=τi (i=1,2,・・・,N2-1) ∃f∈C(N2-1)×(N2-1)×(N2-1);[τj,τk]=fjklτl, c. Example, 1. SU(1)={e-iθ|θ∈R}, SU(1) and SO(2) are representations of each other. SU(1)∋e-iθ←→[cosθ -sinθ, sinθ cosθ]∈SO(2), 2. SU(2), τi=σi (i=1,2,3);[τj,τk]=iεjklτl, exp(-iθkτk)=σ0cos√θkθk-iσl(θl√θkθk)sin√θjθj, σ0=[1001],σ1=[0110],σ2=[0 -i,i 0],σ3=[1 0,0 -1]