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What is called Wick rotation (after Gian-Carlo Wick) is a method in physics for finding a solution to a problem in Minkowski spacetime from the solution to a related problem in Euclidean space. It is motivated by the observation that the Minkowski metric (with the $-1,1,1,1$ convention) and the four-dimensional Euclidean metric are equivalent if the time components of either are allowed to have imaginary values.
Specifically, in quantum field theory Wick rotation is a method for identifying QFT partition functions with expectation values of a corresponding system of statistical mechanics.
In some special cases Wick rotation has been rigorously understood and takes the form of a theorem. Notably the Osterwalder-Schrader theorem gives a precise meaning to Wick rotation for quantum field theory on Minkowski spacetime formalized by the axioms of AQFT.
However, Wick rotation is sometimes also appealed to in situations where the assumptions of theorems like this are evidently violated. For instance, it has been appealed to a lot in an approach to quantum gravity often known as “Euclidean quantum gravity”, where, however, by definition the assumption of global spacetime translation invariance is manifestly violated. In such a context the exact meaning of Wick rotation remains mysterious, and yet on this basis some subtle relations between quantum mechanics and thermodynamics, such as the Bekenstein-Hawking entropy, find elegant explanations, at least at the level of the manipulation of formulas.
Consider the Minkowski metric with the $-1,1,1,1$ convention for the tensor:
$d s^{2}= -(d t)^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$
and the four-dimensional Euclidean metric:
$d s^{2}= d \tau^{2} + (d x)^{2} + (d y)^{2} + (d z)^{2}$.
Notice that if $d t = i\cdot d \tau$, the two are equivalent.
A typical method for employing Wick rotation would be to make the substitution $t=i\tau$ in a problem in Minkowski space. The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem.
Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals.
See also at Osterwalder-Schrader theorem.
Dirk Schlingemann, From euclidean field theory to quantum field theory (arXiv:hep-th/9802035)
Graeme Segal, Wick rotation and the positivity of energy in quantum field theory (video)
Edward Witten, The Feynman $i \epsilon$ in String Theory (arXiv:1307.5124)