When studying perturbation theory and Feynman rules, the standard derivation seems to start from the S-Matrix element in the interaction picture, and expands it into some series, after which Wick's theorem is applied. Another common approach starts from the path integral formalism. Whereas the latter is often done both in Euclidean and Minkowski spacetime, I have not found a lot on the former approach in Euclidean spacetime.
Is it thus correct that the former approach cannot be easily formulated in Euclidean space? Or am I missing something?