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I understand how this is done in perturbative 1-loop calculations, where the pole structure of the Feynman propagators are used to perform the so-called Wick Rotation. This seems to be a perfectly valid procedure in light of complex analysis (in particular, Cauchy's theorem).

However, in path integrals, there are no poles per-se. How is the passage into imaginary time justified? AND does such Euclideanization always yield a Hamiltonian in the exponent?