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ySEOtextz0. General Formulation, the form: [cos -sin, sin cos]=e-iƃ2 where: 2=[0 -i, i 0] 2. Euclidean Group a. Definition, The Euclidean group is defined as a set E. E={E|ER3(R3), [E(x)-E(y)]2=(x-y)2}={E|RO(3), aR3; xR3; E(x)=Rx+a},E+={E|RSO(3), aR3; xR3; E(x)=Rx+a} b. Parametrization and Generators EE+, (,a)(R3,R3); E=exp(-iL-iaP) where: Li, PiR3(R3), [Lj(x)]k=-ijklxl=(Lj)klxl, [Pj(x)]k=ijk, Notice that Pi is nonlinear and the commutation relations are different from those of linear representation. Especially: [Pj,Pk]0. c.Linear Representation of E, A linear representation of a group is defined as a set G of linear operators on some linear space such that each element of corresponds to only one element of G and if i corresponds to giG (i=1,2) then 12 corresponds to g1g2G and vice versa. |
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