# Plane Wave Relations for Dirac Spinors

I am trying to show the following relationships: $$\bar{u}_{\pm p\sigma}\gamma^\mu u_{\pm p\sigma'} = 2p^\mu \delta_{\sigma\sigma'}$$, $$\bar{u}_{\pm p\sigma} u_{\pm p\sigma'} = \pm 2m\delta_{\sigma\sigma'}$$, and $$\bar{u}_{+ p\sigma}u_{-p\sigma'} = 0$$.

Now, for the plane wave solution of the Dirac equation, we have $$$$\psi_{\pm p \sigma} = \frac{1}{\sqrt{2\epsilon_\mathbf{p}V}}u_{\pm p\sigma}e^{\mp ipx},$$$$ where $$px = p^\mu x_\mu$$, and has a normalization of $$\psi^\dagger_{\pm p\sigma}\psi_{\pm p\sigma} = 1/V$$, then $$\frac{1}{2\epsilon_\mathbf{p}V}u^\dagger_{\pm p\sigma}u_{\pm p\sigma} = 1/V\implies u^\dagger_{\pm p\sigma}u_{\pm p\sigma} = \bar{u}_{\pm p\sigma}\gamma^0u_{\pm p\sigma} = 2\epsilon_\mathbf{p}$$, but I am not sure how to get the full covariant form with $$\gamma^\mu$$, or is it just something I argue such as, "due to Lorentz invariance..." and similar for the mass relationship, but I know how to get out the $$2m$$ by using the Dirac equation, plug in the plane wave solution and simplify, but I don't know how to get out the Delta function above. But for the third relationship, is that doing exactly what I did above, but specify the momentum (I am not sure)?

EDIT: The link provided in the first answer helped with finding the mass relationship, but for the rest I am at a hard wall, without going the route of the general Gordon Identity, which is something I don't want to do (i.e when $$p = p'$$).

For the mass part, I used the vector representation of of the spinors, aka equation 38.6 and equation 38.10, but I believe this only gets me the $$+p$$ solution, not the minus, unless I am interpreting something wrong, or wrong in my calculation. So I am very lost admittedly.

• Do you mean $\delta_{\sigma \sigma'}$ rather than $\delta(\sigma-\sigma')$? Commented Oct 30, 2022 at 17:49
• @mikestone Yes, it wasn't clear was the case, so I assumed the dirac delta. Commented Oct 30, 2022 at 18:33

• I think the source of your confusion is the notation you are given. First notice that you can identify your $u_{+}$ with $u$ in the book and $u_{-}$ with $v$ respectively. Then take all possible product combinations of (38.6) and (38.10) and this will yield the last two equations that you want to prove.
• As for the Gordon identities, I am not sure I understand the issue. Setting $p'=p$ immediately makes the term that is analogous to $S^{\mu\nu}$ vanish and you are left with just the momentum term as you wanted. Then you can use the second identity to get rid of the masses and that's it. From a broader perspective it would be very instructive to also prove the Gordon identities as they are useful in calculations of tree level amplitudes.