Wick rotation

"Wick rotation is a very basic procedure for inter-playing Lorentzian and Riemannian geometry. The simplest example applies to \mathbb R^{n+1} endowed with both the standard Minkowski metric {-dx_0}^2+\cdots+dx_{n-1}^2+dx_n^2\, and the Euclidean metric dx_0^2+\cdots+dx_n^2\,. By definition ... these are related via a Wick rotation directed by the vector field \frac{\partial}{\partial x_0}. Sometimes one refers to it as "passing to the imaginary time"."

"The celebrated Schrödinger equation is mathematically close to the ordinary diffusion equation. What is the main difference is that time is imaginary, there is the Wick rotation. This means that classical and quantum are related partly by a rotation of 90 degrees in the complex plane (multiplying by the imaginary unit)."

"The Wick rotation idea is now so commonly used in quantum field theory that it is often taken as an automatic procedure in numerous different kinds of situation, with barely a mention, and its validity is hardly ever questioned. It does, in fact, have a broad applicability, but it is not a universally valid procedure. Most particularly. it is highly questionable in the context of the curved space-times arising in general relativity, when in normal circumstances the procedure cannot even be applied, because there is no natural time coordinate. In string theory, this is a problem both in the 10-dimensional space-time, in general curved-space situations, and also in the string world-sheet ..."