I consider Yukawa non-equilibrium theory with interaction $g\bar{\psi}\phi\psi$ with massive fermionic field $\psi$ with mass $m$ and massive scalar field $\phi$ with mass $M$. I would like to understand how one can derive Feynman rules for this lagrangian. My attempt is to start with introducing $+$ and $-$ components of field and than use Keldysh rotation, i.e. introduce $\phi_{cl}$, $\phi_{q}$ and $\psi_1$, $\psi_2$ (and their conjugated). But I misunderstand this moment. For instance, for free fermionic term I have obtained something like that $$\begin{pmatrix}\bar{\psi_1} & \bar{\psi_2}\end{pmatrix}\begin{pmatrix}i\gamma\partial-m & 0 \\ 0 & i\gamma\partial-m\end{pmatrix}\begin{pmatrix}\psi_1 \\ \psi_2\end{pmatrix}.$$ Then, for interaction I have found $$g\begin{pmatrix}\bar{\psi_1} & \bar{\psi_2}\end{pmatrix}\begin{pmatrix}\phi_{cl} & \phi_{q} \\ \phi_{q} & \phi_{cl}\end{pmatrix}\begin{pmatrix}\psi_1 \\ \psi_2\end{pmatrix}.$$ I do not understand how using rewritten lagrangian (with fields $\phi_{cl,q}$ and $\psi_{1,2}$) one can calculated, for instance, 1-loop correction for fermionic propagator. I see that in many textbooks Keldysh-Schwinger technique is derived from cananonical quantization but I would like to use functional methods specifically.


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