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  • $\begingroup$ You should probably look at the equivalent treatment in statistical thermodynamics, where it is Euclidean from the start. Quite a lot of the transition between Minkowski and Euclidean versions are done by comparing the equations from both sides and then seeing what minimal steps can be done to do this conversion. The proper thing to do is analytical continuation and shifting of contours, but we actually are not doing that; we are doing a shortcut that is justified by the fact that it must be possible to do that analytical continuation. $\endgroup$ Commented Jun 4 at 16:06
  • $\begingroup$ Yes, disregarding mathematical details, is the procedure correct then? $\endgroup$ Commented Jun 4 at 16:09
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    $\begingroup$ Related: physics.stackexchange.com/q/106292/2451 and links therein. $\endgroup$ Commented Jun 4 at 16:50
  • $\begingroup$ This question is similar to: How to Perform Wick Rotation in the Lagrangian of a Gauge Theory (like QCD)?. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. $\endgroup$ Commented Jun 4 at 16:54
  • $\begingroup$ Oh, Qmechanic's answer is way more comprehensive. I was also already rather wary of your Wick rotation, because your middle equation expression still had $i$ in there. Your last expression should have a $\mathrm d^4x_E$ too. The details, just check Qmechanic's answer. Sooo good. $\endgroup$ Commented Jun 4 at 16:56