Wick rotation

In physics, Wick rotation, named after Italian physicist Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. Quotes Wick rotation is a very basic procedure for inter-playing Lorentzian and Riemannian geometry. The simplest example applies to endowed with both the standard Minkowski metric and the Euclidean metric . By definition ... these are related via a Wick rotation directed by the vector field . Sometimes one refers to it as "passing to the imaginary time". R. Benedetti and Francesco Bonsante: Canonical Wick Rotations in 3-Dimensional Gravity https://books.google.com/books?id=Mhq9AwAAQBAJ&pg=PA2 6 March 2009 American Mathematical Soc. 978-0-8218-4281-2 2 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). Jussi Lindgren and Jukka Liukkonen: Quantum Mechanics can be understood through stochastic optimization on spacetimes Scientific Reports 9 1 2019 2045-2322 10.1038/s41598-019-56357-3 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 ... Roger Penrose: Faith, Fashion, and Fantasy in the New Physics of the Universe 2016 Princeton University Press 65–66 https://books.google.com/books?id=T09kCwAAQBAJ&pg=PA65 External links Category:Quantum mechanics