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ySEOtextz0. General Formulation, 1. Scalar Representation E0, E0={e|ÎE¸E,ÍƒÓ¸R(R3);(eƒÓ)(x)=ƒÓ(E-1(x))} To note the correspondence explicitly, let's define a mapping E0¸E0(E) as follows. ÍE¸E;[E0(E)ƒÓ](x)=ƒÓ(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(-iƒÆL-iaP)=exp(-iƒÆL0-iaP0) where: (Lj0ƒÓ)(x)=-iƒÃjklLxkÝlƒÓ(x), (Pj0ƒÓ)(x)=-iÝjƒÓ(x), {[Lj0,Lk0]=iƒÃjklLl0, [Pj0,Pk0]=0, [Lj0,Pk0]=iƒÃjklPl0, 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+s¾{eEs(-1)|e¸E+s}, 3. Symmetry of Action |
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