Massless representations of the Poincare group

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nrqed
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Never mind, I answered my own question...
 
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By definition

[tex] W_0 = \mathbf{P} \cdot \mathbf{J} [/tex]

Applying this operator to [itex]| p \rangle [/itex] we obtain

[tex] W_0 | p \rangle = P_3 J_3 | p \rangle [/tex]

[itex]J_3 [/itex] is a generator of the "little group" which leaves this vector invariant (up to a constant factor), so (see eq. (2.5.39) in Weinberg's "The quantum theory of fields")

[tex] W_0 | p \rangle = P_3 \lambda| p \rangle = p_0 \lambda| p \rangle [/tex]

where [itex] \lambda[/itex] is helicity.
 
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nrqed
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By definition

[tex] W_0 = \mathbf{P} \cdot \mathbf{J} [/tex]

Applying this operator to [itex]| p \rangle [/itex] we obtain

[tex] W_0 | p \rangle = P_3 J_3 | p \rangle [/tex]

[itex]J_3 [/itex] is a generator of the "little group" which leaves this vector invariant (up to a constant factor), so (see eq. (2.5.39) in Weinberg's "The quantum theory of fields")

[tex] W_0 | p \rangle = P_3 \lambda| p \rangle = p_0 \lambda| p \rangle [/tex]

where [itex] \lambda[/itex] is helicity.
Thank you. My problem was in convincing myself that J_3 leaves the state invariant. I will look at Weinberg when I can

Thanks again
 

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