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I would like a general method to derive the Feynman rules for any scalar Lagrangian from the generating functional $$Z[J]=\int \mathcal{D}\phi \exp \left[iS[\phi]+i\int d^4xJ(x)\phi(x)\right].$$ There are many examples in the common QFT books (Schwartz, Peskin & Schroeder, Srednicki) but I failed to find a general and comprehensive method.

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    $\begingroup$ If you have read the standard sources and still desire more detail, the only next step would be Weinberg. You may also enjoy Nair's QFT book. $\endgroup$
    – Richard Myers
    Commented Dec 8, 2020 at 20:51
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    $\begingroup$ You write that you want the Feynman rules for "any Lagrangian," but the partition function you have written down has the action for one scalar field, with arbitrary self interactions. Do you want Feynman rules for any scalar self-interaction around a vacuum state like $\phi=0$? Do you want to allow for a non-trivial classical background? Multiple scalar fields? Fields with spin greater than zero? Interactions that violate Lorentz invariance? The best way to answer the "any Lagrangian" question is to work many, many problems and gain experience. A restricted version could be manageable here. $\endgroup$
    – Andrew
    Commented Dec 8, 2020 at 21:03
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    $\begingroup$ Quantum Fields: From the Hubble to the Planck Scale by Michael Kachelriess has the derivation from page 56 to 58. It is my favorite QFT book. $\endgroup$
    – Kian Maleki
    Commented Dec 8, 2020 at 22:00

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