Wick rotation

In physics, Wick rotation, named after 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. This transformation is also used to find solutions to problems in quantum mechanics and other areas.

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with $(-1, +1, +1, +1)$ convention for the metric tensor)

and the four-dimensional Euclidean metric

are equivalent if one permits the coordinate $t$ to take on imaginary 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 $t = - iτ$ , sometimes yields a problem in real Euclidean coordinates $x, y, z, τ$ which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.

Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature $1/(k_{B}T)$ with imaginary time $it/\hbar$ . Consider a large collection of harmonic oscillators at temperature $T$ . The relative probability of finding any given oscillator with energy $E$ is $\exp(-E/k_{B}T)$ , where $k B$ is Boltzmann's constant. The average value of an observable $Q$ is, up to a normalizing constant,

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