# Helicity for Zero Rest Mass Field Equations

I'm trying to reconcile the usual definition of the helicity operator, namely

$$h = \hat{p}.S$$

with the definition of a massless helicity $n$ field as a symmetric spinor field $\phi^{A\dots B}$ with $2n$ indices, satisfying the z.r.m. equation

$$\nabla_{CC'}\phi^{A\dots B} = 0$$

I can't find anywhere a proof that these concepts are equivalent, and I can't seem to prove it myself. I feel I need to Fourier transform somewhere and/or use a quantum argument, but I can't see what! Does anyone know how this works, or could point me towards a good reference? Cheers!

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For the Poincare group, the two Casimirs are $m^2$ and $W^2$ where $W^a$ is the Pauli Lubanski vector. For the massless irreps $W_a=sp_a$ where s is the helicity and $p_a$ the momentum. Now the result I think you're wanting is the fact that these spin s irreps are provided by the ZRM equations (with the right number of spinor indices). I think if you look hard enough (the notation is somewhat different!) you may be able to convince yourself that this is what Bargmann and Wigner proved in their paper. – twistor59 Apr 26 '13 at 20:48