Types of quantum field thories
Wick rotation is a method for finding a solution to a problem in Minkowski space from the solution to a related problem in Euclidean space. It is motivated by the observation that the Minkowski metric (with the convention) and the four-dimensional Euclidean metric are equivalent if the time components of either are allowed to have imaginary values.
Consider the Minkowski metric with the convention for the tensor:
and the four-dimensional Euclidean metric:
Notice that if , the two are equivalent.
A typical method for employing Wick rotation would be to make the substitution 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.