# Spinor satisfying Dirac's equation with $M\neq 0$, how one should start evaluating Pauli-Lubanski scalar on it

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.

• What have you tried? Jan 24, 2021 at 20:54
• 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. Jan 24, 2021 at 21:01
• 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. Jan 25, 2021 at 1:54
• 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. Jan 25, 2021 at 5:13
• Thank you G.Smith, your suggestions works but calculation using Levi-Civata symbol is very messy.
– user286848
Jan 25, 2021 at 18:25

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.