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Massless representations of the Poincare group

  1. Jun 17, 2009 #1

    nrqed

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    Never mind, I answered my own question...
     
    Last edited: Jun 17, 2009
  2. jcsd
  3. Jun 17, 2009 #2
    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.
     
  4. Jun 17, 2009 #3

    nrqed

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    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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