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I am currently working through chapter 75 of the book on QFT by Srednicki. There, he considers the example of a single left-handed Weyl field $\psi$ in a $U(1)$ gauge theory. The Lagrangian, written in terms of a full Dirac field $\Psi$, is $$ \mathcal{L} = i\bar\Psi \gamma^\mu (\partial_\mu - igA_\mu) P_L \Psi - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}. $$ Here, $P_L = \frac{1}{2}(1-\gamma_5)$ is the projector on the left-handed component. He then goes on to claim that the Feynman rules are:

  • $i g \gamma^\mu P_L$ for the $\bar\psi\psi A_\mu$ interaction vertex. This makes sense to me.
  • $-P_L \gamma^\mu p_\mu / p^2$ for the fermion propagator. This makes some intuitive sense, you propagate for a while and then afterwards you have to project out only the left-handed component, but I wouldn't know how to derive this propagator from the Lagrangian.

Could someone enlighten me on how to derive this propagator from the Lagrangian?

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  • $\begingroup$ Related: QFT: Propagators are the inverse of the quadratic terms in $\mathcal{L}$? $\endgroup$
    – Qmechanic ♦
    Commented Jun 11 at 11:00
  • $\begingroup$ @Qmechanic I am aware of this relationship, but I have always found it very obscured when dealing with Dirac matrices, as I've been told that $1/(\gamma^\mu p_\mu)$ is not matrix inversion. Moreover, how would this apply here, with $P_L$ not being invertible? $\endgroup$
    – Niels Slotboom
    Commented Jun 11 at 11:05

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