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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: http://arxiv.org/pdf/hep-th/9403084v2.pdf 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.

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Are you considering flat or curved spacetime? –  Valter Moretti Feb 25 at 18:21
    
I think it is just an analytic continuation. It connect propagator and partition function of one system. For $\tau=it$, it gives the partition function directly, but $\tau=-it$ does not. –  qfzklm Feb 25 at 18:57
    
I was thinking of flat space. –  symanzik138 Feb 25 at 21:53

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