Next Page
Previous Page
Home
【SEOtext】0. General Formulation, 1. Scalar Representation E0, E0={e|∃E∈E,∀φ∈R(R3);(eφ)(x)=φ(E-1(x))} To note the correspondence explicitly, let's define a mapping E0∈E0(E) as follows. ∀E∈E;[E0(E)φ](x)=φ(E-1(x)) Then the scalar representation of E+ is as follows. E+0={E0(E)|E∈E+} This is parametrized as follows. E0(exp(-iθL-iaP)=exp(-iθL0-iaP0) where: (Lj0φ)(x)=-iεjklLxk∂lφ(x), (Pj0φ)(x)=-i∂jφ(x), {[Lj0,Lk0]=iεjklLl0, [Pj0,Pk0]=0, [Lj0,Pk0]=iεjklPl0, 2. General Case Es, Arbitrary linear representation of E can be constructed as follows. First define generators asd it's domain of definition such that the generators have the same commutation relations as E+0 and they are linear. Second define the representation of E+ as a set of all operators of the form: Es(exp(-iL-iaP))=exp(-iLs-iaPs) Finally find the representation of the reflection -1∈E. Then we get: Es=E+s∪{eEs(-1)|e∈E+s}, 3. Symmetry of Action