Next Page
Previous Page
Home
【SEOtext】0. General Formulation, the form: [cosθ -sinθ, sinθ cosθ]=e-iθσ2 where: σ2=[0 -i, i 0] 2. Euclidean Group a. Definition, The Euclidean group is defined as a set E. E={E|E∈R3(R3), [E(x)-E(y)]2=(x-y)2}={E|∃R∈O(3), a∈R3; ∀x∈R3; E(x)=Rx+a},E+={E|R∈SO(3), a∈R3; ∀x∈R3; E(x)=Rx+a} b. Parametrization and Generators ∀E∈E+, ∃(θ,a)∈(R3,R3); E=exp(-iθL-iaP) where: Li, Pi∈R3(R3), [Lj(x)]k=-iεjklxl=(Lj)klxl, [Pj(x)]k=iδjk, 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 Γ is defined as a set G of linear operators on some linear space such that each element of Γ corresponds to only one element of G and if γi∈Γ corresponds to gi∈G (i=1,2) then γ1γ2∈Γ corresponds to g1g2∈G and vice versa.