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?


1 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.

  • $\begingroup$ About your first argument, if I understand correctly, you say we almost always do QFT computations in Euclidean spacetime and this has worked so far. So, is the idea that the Wick rotation is valid based on the fact that we use it all the time and it works? $\endgroup$
    – adithya
    May 21, 2020 at 10:06
  • $\begingroup$ @adithya I'd say that Wick rotation is part of the definition of most QFTs. It's built into the basic computations. $\endgroup$
    – user1504
    May 21, 2020 at 10:50

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