I often see that action is written with a Euclidean metric instead of the original Minkowski metric. My question is basically this : Under what conditions is okay to make a wick rotation? I am thinking of the Wick-rotation in a manner analagous to a change of contour in a contour integral. In the case of contour integral, before making a similar change in contour, I would consider issues like location of poles of the integrand, behavior of the integrand at infinity of the complex plane etc. But in case of Wick rotation, I don't see such things discussed.
I would appreciate if some of you could answer or give some inputs to answer the following questions I have:
Is the rotation just as trivial as a change of variable : $\tau=\imath t$ ?
How do I know that the rotation should be $\tau=\imath t$ and not $\tau=-\imath t$
Can I always make a wick rotation for any Lagrangian?
Thanks.
EDIT/PARTIAL-ANSWER/GENERAL-UPDATE: https://arxiv.org/abs/hep-th/9403084 sheds some light on the issue. The short section on lack of an analytic continuation tells me that in a general case wick rotation needs to be justified.
