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I have been provided that Spinor $\psi(x)$ satisfying Dirac's equation with $M\neq 0$, how one should start evaluating Pauli-Lubanski scalar on it, I mean how to evaluate $W^{\mu}W_{\mu}\psi(x)?$ Where, we have given $$(i\gamma^{\mu}\partial_{\mu}\pm M)\psi(x)=0$$ $W_{\mu}=\frac{1}{2}\epsilon_{\mu\alpha\beta\sigma}\sum^{\alpha\beta}\frac{\partial}{\partial x^{\sigma}}$ and $\sum^{\alpha\beta}=\frac{i}{4}[\gamma^{\alpha},\gamma^{\beta}]$

Any helpful suggestions are welcome and thanks for that.

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  • $\begingroup$ What have you tried? $\endgroup$ Jan 24, 2021 at 20:54
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    $\begingroup$ Since $W^\mu W_\mu$ is a Lorentz scalar, you can use any Lorentz frame you like to evaluate its effect. For a massive particle, there is a Lorentz frame in which things are particularly simple. $\endgroup$
    – G. Smith
    Jan 24, 2021 at 21:01
  • $\begingroup$ Another approach, when all else fails, is brute force. (Sometimes it helps to get extremely explicit.) Write out $W_0$,$W_1$,$W_2$, and $W_3$ as $4\times 4$ matrices of partial derivatives. Then compute the matrix $W_\mu W^\mu$. It will be proportional to the identity matrix times a d'Alembertian operator. $\endgroup$
    – G. Smith
    Jan 25, 2021 at 1:54
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    $\begingroup$ Finally, one can use identities for the product of two Levi-Civita symbols, and identities for contractions of products of gamma matrices, to simplify $W^2$ using index notation. $\endgroup$
    – G. Smith
    Jan 25, 2021 at 5:13
  • $\begingroup$ Thank you G.Smith, your suggestions works but calculation using Levi-Civata symbol is very messy. $\endgroup$
    – user286848
    Jan 25, 2021 at 18:25

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Fields, I think that the last one was a really useful advice. I can add that you should recognize the derivative as the momentum operator. $W_{\mu}=\frac{1}{2}\epsilon_{\mu\alpha\beta\sigma}\sum^{\alpha\beta}P^{\sigma}$. Then you can go to the rest frame (since it's a massive particle) and evaluate the scalar. You have $P^\mu=(m,0)$ so the index $\sigma$ should be $0$ otherwise you get no contribution. Finally you should notice/define the spin $\Sigma_i=\frac{1}{2}\epsilon_{ijk}\Sigma^{jk}$ and you should find the value $-m^2\Sigma_i\Sigma_i$ for the scalar, that it's actually $-m^2s(s+1)$ with $s=\frac{1}{2}$. Hope to be helpful.

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