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【SEOtext】0. General Formulation, Λ-1MμνΛ=ΛμρΛνσMρσ (Λ∈L), Mjk†=Mjk, M0k†=-M0k, 2. Poincare Group, a. Definition, P={Π|Π∈R4(R4), [Π(x)-Π(y)]2=(x-y)2}={Π|∃Λ∈L, a∈R4; ∀x∈R4; Π(x)=Λx+a}, Psr={Π|∃ΛLsr, a∈R4; ∀x∈R4; Π(x)=Λx+a}, (r=+,-; s=↑,↓), b. Parametrization and Generators, ∀Π∈P↑+, ∃(θ,a)∈(R[4×4],R4); Π=exp[-(i/2)θμνMμν+iaμPμ] where: [Mμν(x)]ρ=(Mμν)ρσxσ, [Pμ(x)]ρ=-igμρ, c. Linear Representation of P, 1. Scalar Representation P0, P0={π|∃Π∈P, ∀φ∈C(R4); (πφ)(x)=φ(Π-1(x))}, [Π0(Π)φ](x)=φ(Π-1(x)), Π0(exp[-(i/2)θμνMμν+iaμPμ])=exp[-(i/2)θμνMμν0+iaμPμ0]∈P↑+0, {(Mμν0φ)(x)=i(xμ∂ν-xν∂μ)φ(x), (Pμ0φ)(x)=i∂μφ(x), {[Mμν0,Mρσ0]=i(gνρMμσ0-gμρMνσ0-gνσMμρ0+gμσMνρ0), [Pμ0,Pν0]=0, [Mμν0,Pρ0]=i(gνρPμ0-gμρPν0), 2.Vector Representation P1, P1={π|∃Π∈P, ∀φ∈R4(R4); (πφ)μ(x)=Λμνφν(Π-1(x))} where: [Π(x)]μ=Λμνxν+[Π(0)]μ, {(Mμν1φ)ρ(x)=[Mμνφ*(x)]ρ+(Mμν0φρ)(x), (Pμ1φ)ρ(x)=(Pμ0φρ)(x)