What happens if I perform a Wick rotation when I work with a signature (+,-,-,-) ? Is a negative euclidean metric a problem? What's the sense for that?
I can perform the wick rotation in two different ways : $t= + i \tau$ or $t= - i \tau$. When my equations doesn't contain poles, which criteria should I use to choose the sign of the rotation?
Is a negative euclidean metric a problem?
Not if you stay awake while performing your calculations, and check the behavior of computer algebra programs you might use, and know that the squared length in Euclidean space is really positive.
If your time evolution is exp (-i H t), with H bounded from below, you should make the continuation t -> -i \tau, with \tau positive. And you should in general consider this as an analytic continuation, not just a replacement.
No problem at all-it's the indefinite sign of the metric that leads to oscillating phase factors. And the sign of the rotation doesn't matter, either. The point is that, at the end, the metric is, up to a global sign, positive definite.
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