Proving that a Wick rotation is valid for a quantum field theory

Question

While trying to find out if there is a rigorous justification for Wick rotating a QFT, I came across this other question (link below [1]) that mentions the Osterwalder-Schrader Theorem that gives a set of conditions under which Wick rotation is valid.

Now, my question is the following: Are the theories in which we normally use Wick rotation, such as QED or QCD, known to satisfy these conditions? In non-abelian gauge theories, we calculate instanton contributions to the path integral in Euclidean spacetime. I don't understand how this is valid in case the Euclidean fields don't satisfy the conditions in the Osterwalder-Schrader theorem.

[1] Wick rotation in field theory - rigorous justification?

Answer 1

Just about every particle physics computation is done via analytic continuation from Euclidean signature, either via Euclidean lattice simulations or via the $+i\epsilon$ prescription in perturbation theory. So in that sense, yes, Wick rotation is always valid for QFTs. (There are a handful of cases like Chern-Simons theory where the analytic continuation gets rather subtle.)

This observation -- that QFT computations are really Euclidean computations -- predates the Osterwalder-Schrader Theorem. The Osterwalder-Schrader theorem is one attempt for formalize it within a particular setting. The theorem is a tool; it tells you that analytic continuation works if your Euclidean model satisfies the axioms. But it's not necessary for a theory to satisfy the OS axioms in order to exhibit a link between its Euclidean and Lorentzian versions.

That said, most of the OS axioms are pretty uncontroversial: Analyticity, Euclidean invariance, reflection positivity, and ergodicity are very reasonable demands. Any Lorentzian QFT we use in the real world should satisfy them, at least in spirit. (QED, for example, has a Landau pole, and probably doesn't exist non-perturbatively, but one can still work with it as a formal perturbation series.)

The other axiom -- regularity -- is "technical", meaning that it's not really clear if it's physically necessary or just a convenient mathematical tool. I wouldn't want to bet my life on QCD satisfying an appropriate analogue.