It seems to me that in a path integral, since you are integrating over all field configurations, that going into Euclidean space is not valid because some field configurations will give poles in the integrand of your action, and when the integrand has poles you can't make the rotations required...
Continuing to Euclidean Space Justified in Path Integral?
Continuing to Euclidean Space Justified in Path Integral?
Continuing to Euclidean Space Justified in Path Integral?
Continuing to Euclidean Space Justified in Path Integral?
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