Wick rotation
From Exampleproblems
In physics, Wick rotation is a method of finding a solution to a problem in Minkowski space from a solution to a related problem in Euclidean space, by analytic continuation.
It is motivated by the observation that the Minkowski metric
- ds2 = − (dt2) + dx2 + dy2 + dz2
and the four-dimensional Euclidean metric
- ds2 = dt2 + dx2 + dy2 + dz2
are not distinct if one permits the coordinate t to take on complex values. The Minkowski metric becomes Euclidean when t is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates x,y,z,t, and substituting w = it, sometimes yields a problem in real Euclidean coordinates x,y,z,w which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.
Wick rotation connects quantum mechanics to statistical mechanics in a surprising way. The Schrödinger equation and the heat equation are related by Wick rotation, for example. The reasons for this are not understood.
Wick rotation is named after Gian-Carlo Wick. It is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of π / 2.
When Stephen Hawking wrote about "imaginary time" in his famous book A Brief History of Time, he was referring to Wick rotation.
Wick rotation also relates a QFT at a finite inverse temperature β to a statistical mechanical model over the "tube" R3×S1 with the imaginary time coordinate τ being periodic with period β.
