I have recently learned a bit about the Osterwalder-Schrader theorem. From my understanding, this tells you when a Euclidean path integral can be analytically continued to a valid relativistic Hermitian quantum field theory (one needs reflection positivity etc.). I am curious about corresponding results for non-relativistic theories. Let's say I have a Euclidean field theory where there isn't full rotational symmetry; if a definite example is needed, let's say I want it to Wick-rotate to a Galilean-invariant theory. Is there a corresponding theorem for what properties are needed in the Euclidean theory for the Wick rotation (which is used all the time) to guarantees a valid Hermitian quantum field theory? I don't have an extremely strong background in the mathematics behind AQFT. I did find this statement of the OS theorem by Urs Schreiber (who I know posts here), which gives rotational invariance as an assumption. So my question boils down to whether this assumption can be relaxed.