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Lahiri's A First Book on Quantum Field Theory states on problem 4.24 that from the Pauli-Lubansky vector $$W_\mu=-\frac{1}{2}\epsilon_{\mu\nu\lambda\rho}P^\nu J^{\lambda\rho}$$ one can prove that for eigenstates of momentum and angular momentum $W^\mu W_\mu=-m^2s(s+1)$ where $m$ is the mass of the particle and $s$ is the spin of the particle. I don't understand however how it is that one actually constructs those states from a given field theory.

On the other hand, I am asked to show by using $$J_{\mu\nu}=i(x_\mu\partial_\nu-x_\nu\partial_\mu)+\frac{1}{2}\sigma_{\mu\nu}$$ where $\sigma_{\mu\nu}=\frac{i}{2}[\gamma_\mu,\gamma_\nu]$ that the spin of a particle satisfying Dirac's equation is $1/2$. I don't even know how to begin since I don't know how to construct an eigenstate of momentum and angular momentum from a Dirac field. I don't think they want us to naively apply the Pauli-Lubansky vector to the field by assuming $P_\nu=-i\partial_\nu$. I've tried this and the math becomes quite unbearable.

As you may observe, I am quite confused with the relationship particle-field. In my mind the Fourier decomposition of a field yields the creation and anhilation operators to construct the momentum eigenstates. On the other hand, through the Lagrangian one can construct from the field a unitary representation of the Poincaré group by calculating its generators from the conserved quantities. So, do I have to go through the whole quantization procedure and construct the correct operators for $P^\nu$ and $J^{\mu\nu}$ to apply to a state created from the creation operators of the field?

Any help in the right direction is much appreciated!

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    $\begingroup$ I've added the homework-and-exercises tag. In the future, please use this tag on this type of question. $\endgroup$ – Ben Crowell Apr 22 '18 at 19:40

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