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�ySEOtext�z0. General Formulation, SU(2) and SO(3) are representations of each other. If U=exp(-iƒÆƒÑ) and R=exp(-iƒÆL) then: U-1ƒÑkU=RklƒÑl, 2. Symmetry of Action, a. Poincare Invariance, If ƒÓ'ƒ¿(x)=(ƒÎƒÓƒ¿)(x) then: �̓Î�¸P0; I(ƒÓ')=I(ƒÓ), b. SU(n) Invariance, If ƒÓ'ƒ¿(x)=Uƒ¿ƒÀƒÓƒÀ(x) then: �ÍU�¸SU(n); I(ƒÓ')=I(ƒÓ), 3. Application of Noether's Theorem, a. n=1, ƒÓ'(x;ƒÓ,ƒÃ)=e-iƒÃƒÓ(x), x'ƒÊ(x;ƒÃ)=xƒÊ, TƒÊ(x;ƒÓ)=iƒÓ�õ(x)�©�݃Ê�¨ƒÓ(x) Here I abbreviate TƒÊc to TƒÊ. b. n=2, ƒÓ'ƒ¿(x;ƒÓ,ƒÃ)=[exp(-iƒÃƒÆƒÑ)]ƒ¿ƒÀƒÓƒÀ(x), x'ƒÊ(x;ƒÃ)=xƒÊ, TƒÊ(x;ƒÓ)=iƒÆk[ƒÓƒ¿�õ(x)(ƒÑk)ƒÀƒ¿�݃ʃӃÀ(x)-�݃ʃӃ¿�õ(x)�E(ƒÑk)ƒ¿ƒÀƒÓƒÀ(x)], TƒÊ,k(x;ƒÓ)=-i�݃ʃÓ�õ(x)�EƒÑkƒÓ(x)+i[�݃ʃÓ�õ(x)�EƒÑkƒÓ(x)]�õ |
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