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The analytic continuation from Euclidean space to Minkowski spacetime is perturbatively well (uniquely) defined to all orders for the Feynman propagator for Dirac fields with the so called "$i\epsilon$ prescription" that guarantees positive energy (unitarity) under time reflection transformations. Mathematically this is well understood from the point of view of complex analysis and it is the most natural way to uniquely choose the right poles in the contour integrals, in this sense it appears to me just to be the opposite to a "trick" or a "hack" as it is often named and there is probably no other way to ascertain convergence in the integral for the propagator.

However, in trying to understand how could this lead to being able to define a nonperturbative theory this perturbative method would be approximating I stumble into this (probably silly) kind of catch-22 situation perspective where if one was to guarantee that this analytical continuation held not just perturbatively it would lead to the propagator being analytic (holomorphic) which is obviously impossible in Minkowski spacetime, so I have to conclude that whatever the final nonperturbative 4-dimensional QFT will be like in the end it will not be based on this kind of $i\epsilon$ perturbative prescription but it will be something completely different altogether. Is this reasoning correct?

EDIT: Maybe the key point in my question got a bit hidden. Let's suppose the analytic continuation $i\epsilon$ prescription for the Feynman propagator of massive chiral fermions was no longer just perturbative, so it would be no longer an $\epsilon$ (arbitrarily small quantity) prescription. It seems obvious to me that this is not compatible with a non-conformal QFT. But I'm not sure this implies that analytic continuation in Euclidean space is discarded for a non-perturbative QFT of massive chiral fermions in 4 dimensions or my naive reasoning is off, can anyone clarify this point?

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