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ySEOtextz0. General Formulation, -1Mʃ˃=ʃσ˃Mσ (L), Mjk=Mjk, M0k=-M0k, 2. Poincare Group, a. Definition, P={|R4(R4), [(x)-(y)]2=(x-y)2}={|΃L, aR4; xR4; (x)=x+a}, Psr={|΃Lsr, aR4; xR4; (x)=x+a}, (r=+,-; s=,), b. Parametrization and Generators, ̓P+, (,a)(R[4~4],R4); =exp[-(i/2)ƃʃMʃ+iaP] where: [Mʃ(x)]=(Mʃ)σx, [P(x)]=-igʃ, c. Linear Representation of P, 1. Scalar Representation P0, P0={|΃P, ̓ӁC(R4); (΃)(x)=(-1(x))}, [0()](x)=(-1(x)), 0(exp[-(i/2)ƃʃMʃ+iaP])=exp[-(i/2)ƃʃMʃ0+iaP0]P+0, {(Mʃ0)(x)=i(xʁ݃-xˁ݃)(x), (P0)(x)=i݃ʃ(x), {[Mʃ0,Mσ0]=i(g˃Mʃ0-gʃM˃0-g˃Mʃ0+gʃM˃0), [P0,P0]=0, [Mʃ0,P0]=i(g˃P0-gʃP0), 2.Vector Representation P1, P1={|΃P, ̓ӁR4(R4); (΃)(x)=ʃ˃Ӄ(-1(x))} where: [(x)]=ʃx+[(0)], {(Mʃ1)(x)=[Mʃ˃*(x)]+(Mʃ0Ӄ)(x), (P1)(x)=(P0Ӄ)(x)

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