# 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. This transformation is also used to find solutions to problems in quantum mechanics and other areas.

## Overview

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (−1, +1, +1, +1) convention)

${\displaystyle ds^{2}=-\left(dt^{2}\right)+dx^{2}+dy^{2}+dz^{2}}$

and the four-dimensional Euclidean metric

${\displaystyle ds^{2}=d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}}$

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 = − 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.

## Statistical and quantum mechanics

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

${\displaystyle \sum _{j}Q_{j}e^{-{\frac {E_{j}}{k_{\text{B}}T}}},}$

where the j runs over all states, ${\displaystyle Q_{j}}$  is the value of Q in the jth state, and ${\displaystyle E_{j}}$  is the energy of the jth state. Now consider a single quantum harmonic oscillator in a superposition of basis states, evolving for a time t under a Hamiltonian H. The relative phase change of the basis state with energy E is ${\displaystyle \exp(-Eit/\hbar ),}$  where ${\displaystyle \hbar }$  is reduced Planck's constant. The probability amplitude that a uniform (equally weighted) superposition of states

${\displaystyle |\psi \rangle =\sum _{j}|j\rangle }$

evolves to an arbitrary superposition

${\displaystyle |Q\rangle =\sum _{j}Q_{j}|j\rangle }$

is, up to a normalizing constant,

${\displaystyle \left\langle Q\left|e^{-{\frac {iHt}{\hbar }}}\right|\psi \right\rangle =\sum _{j}Q_{j}e^{-{\frac {E_{j}it}{\hbar }}}\langle j|j\rangle =\sum _{j}Q_{j}e^{-{\frac {E_{j}it}{\hbar }}}.}$

## Statics and dynamics

Wick rotation relates statics problems in n dimensions to dynamics problems in n − 1 dimensions, trading one dimension of space for one dimension of time. A simple example where n = 2 is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve y(x). The spring is in equilibrium when the energy associated with this curve is at a critical point (an extremum); this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy we integrate the energy spatial density over space,

${\displaystyle E=\int _{x}\left[k\left({\frac {dy(x)}{dx}}\right)^{2}+V(y(x))\right]dx,}$

where k is the spring constant and V(y(x)) is the gravitational potential.

The corresponding dynamics problem is that of a rock thrown upwards. The path the rock follows is that which extremalizes the action; as before, this extremum is typically a minimum, so this is called the "principle of least action". Action is the time integral of the Lagrangian,

${\displaystyle S=\int _{t}\left[m\left({\frac {dy(t)}{dt}}\right)^{2}-V(y(t))\right]dt}$

We get the solution to the dynamics problem (up to a factor of i) from the statics problem by Wick rotation, replacing y(x) by y(it) and the spring constant k by the mass of the rock m:

${\displaystyle iS=\int _{t}\left[m\left({\frac {dy(it)}{dt}}\right)^{2}+V(y(it))\right]dt=i\int _{t}\left[m\left({\frac {dy(it)}{dit}}\right)^{2}-V(y(it))\right]d(it)}$

## Both thermal/quantum and static/dynamic

Taken together, the previous two examples show how the path integral formulation of quantum mechanics is related to statistical mechanics. From statistical mechanics, the shape of each spring in a collection at temperature T will deviate from the least-energy shape due to thermal fluctuations; the probability of finding a spring with a given shape decreases exponentially with the energy difference from the least-energy shape. Similarly, a quantum particle moving in a potential can be described by a superposition of paths, each with a phase exp(iS): the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle.

## Further details

The Schrödinger equation and the heat equation are also related by Wick rotation. However, there is a slight difference. Statistical mechanics n-point functions satisfy positivity whereas Wick-rotated quantum field theories satisfy reflection positivity.[further explanation needed]

Wick rotation 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 about the origin.

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.

Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no effect.