I think I have a pretty good handle on how scalar field scattering works in QFT, so now I'm trying to wrap my head around spin 1/2 particles, and I'm having a bit of trouble with it. For instance, in [itex]N + \phi \rightarrow N + \phi[/itex] scattering, an application of the Feynman rules leads to an expression like(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

(2\pi)^4\delta(p + q - (p' + q'))\bar{u_{p'}}\frac{i}{-\not{p} -\not{q} - m + i\epsilon}u_{p} + crossing[/tex]

So far so good--I see how the Wick expansion plus the definition of the fermion propagator lead to that. What I want to know (and I have a feeling this is going to turn out to be kind of a stupid question) is: how exactly do I go about evaluating this? I know that [itex]\bar{u_p}u_p = 2m[/itex], so that's a start on how to multiply the spinors, but a) that only works if the initial and final momenta are the same, and b) I still have to deal with the slashed momenta in the denominator.

For part a), my first thought is that when the momenta are different, the result might be proportional to the dot product of the momenta--something like [itex]\bar{u_{p'}}u_p \propto p'\cdot p [/itex], but I don't know how to show that. Is that right?

For part b), I know that the u's have been constructed such that [itex]\not{p}u_p = mu_p[/itex], so that would tell me how to evaluate the fraction if the slashed momentum were in the same direction as the spinor. But I'm not sure I understand how to handle the case where they're different, for instance [itex]\not{a}u_p[/itex]. Going along the same lines as above, my guess is that it'd work out to something like [itex]\not{a}u_p = (a\cdot p)u_p[/itex] but again I'm not really sure how to prove that. Am I on the right track here, or completely off-base?

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# Feynman diagrams for spin 1/2 particles

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