It is well known that we can switch from a statistical system to a quantum mechanical system by a Wick rotation. Has this rotation some implication on the way the time flow? namely, this is an accident or has a deep meaning?
In perturbation theory, the Wick rotated Euclidian Green functions are exactly the time ordered vacuum expectations. The reason behind that is that the Wick rotation in the complex plane is only possible given the analyticity properties of the causal Green functions. The situation in nonperturbative situations is more subtle, because not always the action functional becomes negative after the Wick rotation, please see the following article by t'Hooft.