Wick rotation justification in path integrals without poles

[TriTertButoxy]

I can't find any good references on Euclideanizing path integrals (from Minkowski to Euclidean metric).

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?

 

[DarMM]

This is a complex topic.

Basically you can show that a field theory whose fields obey a reasonable set of axioms (Wightman-Garding axioms) give a set of correlation distrbutions obeying a separate reasonable set of axioms (Wightman axioms).

These axioms imply the correlation functions have an analytic continuation called Schwinger functions and that these functions obey another set of axioms, the Osterwalder-Schrader axioms. A slight additional assumption on the analytic behaviour of the Schwinger functions (provably satisfied in all realistic field theories) then shows they are derivatives of a functional obeying another set of axioms (Frohlich's axioms). Via Minlos theorem this then implies the existence of a probability theory of which this Functional is the Fourier transform. This probability theory is the path integral.