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【SEOtext】0. General Formulation, (ii) Real Vector Field and Poincare Group, a. L(x;y)=-(1/4)(yμν-yνμ)(yμν-yνμ)-V(xμxμ), x∈R4, y∈R4×4, I(φ)=∫d4xL(φ*(x);∂*φ*(x)), φ∈R4(R4), b. δI(φ)/δφμ(x)=0⇔∂L/∂φμ(x)-∂/∂xν∂L/∂[∂νφμ(x)]=0⇔∂ν∂νφμ(x)-∂μ∂νφν(x)-V'(φν(x)φν(x))φμ(x)=0, c. Symmetry, 1. Lorentz Group, a. Definition and Fundamental Properties, L={Λ|Λ∈R4×4, gμνΛμρΛνσ=gρσ}, 1. ∀Λ∈R4×4;gρσΛμρΛνσ=gμν⇔(Λ-1)μν=gμρgνσΛσρ⇔Λ∈L, 2. ∀Λ∈R4×4;[∀x∈R4;gμν(Λμρxρ)(Λνσxσ)=gρσxρxσ]⇔Λ∈L, 3. ∀Λ∈L; det Λ=±1, 4. ∀Λ∈L; |Λ00|≧1 and (|Λ00|=1⇔Λj0=0⇔Λ0j=0), 5. L↑+={Λ|detΛ=+1,Λ00≧+1,Λ∈L}, L↓+={Λ|detΛ=+1,Λ00≦-1,Λ∈L}, L↑-={Λ|detΛ=-1,Λ00≧+1,Λ∈L}, L↓-={Λ|detΛ=-1,Λ00≦-1,Λ∈L}, Notice that L↑+,L+=L↑+∪L↓+,L↑=L↑+∪L↑- and L↑+∪L↓- are subgroups of L. b. Parametrization and Generators, ∀Λ∈L↑+, ∃θ∈R[4×4];Λ=exp[-(i/2)θμνMμν] where: (Mμν)ρσ=i(gρμgνσ-gρνgμσ), {[Mμν,Mρσ]=i(gνρMμσ-gμρMνσ-gνσMμρ+gμσMνρ)