# 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. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, it, where t is time and i is the imaginary unit (i$^2$ = –1).
More precisely, in statistical mechanics, the Gibbs measure exp(−H/k$_B$T) describes the relative probability of the system to be in any given state at temperature T, where H is a function describing the energy of each state and k$_B$ is the Boltzmann constant. In quantum mechanics, the transformation exp(−itH/ħ) describes time evolution, where H is an operator describing the energy (the Hamiltonian) and ħ is the reduced Planck constant. The former expression resembles the latter when we replace it/ħ with 1/k$_B$T, and this replacement is called Wick rotation.
Wick rotation is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by the imaginary unit is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude π/2 about the origin.

## Overview

Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (− + + +) convention)
    $ds^{2}=-\left(dt^{2}\right)+dx^{2}+dy^{2}+dz^{2}$
and the four-dimensional Euclidean metric
    $ds^{2}=d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}$
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 = −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.

### Statistical and quantum mechanics

Wick rotation connects statistical mechanics to quantum mechanics by replacing inverse temperature with imaginary time, or more precisely replacing 1/k$_B$T with it/ħ, where T is temperature, k$_B$ is the Boltzmann constant, t is time, and ħ is the reduced Planck constant.
For example, consider a quantum system whose Hamiltonian H has eigenvalues E$_j$. When this system is in thermal equilibrium at temperature T, the probability of finding it in its jth energy eigenstate is proportional to exp(−E$_j$/k$_B$T). Thus, the expected value of any observable Q that commutes with the Hamiltonian is, up to a normalizing constant,
    $\sum _{j}Q_{j}e^{-{\frac {E_{j}}{k_{\text{B}}T}}},$
where j runs over all energy eigenstates and Q$_j$ is the value of Q in the jth eigenstate.
Alternatively, consider this system in a superposition of energy eigenstates, evolving for a time t under the Hamiltonian H. After time t, the relative phase change of the jth eigenstate is exp(−E$_j$it/ħ). Thus, the probability amplitude that a uniform (equally weighted) superposition of states
    $|\psi \rangle =\sum _{j}|j\rangle$
evolves to an arbitrary superposition
    $|Q\rangle =\sum _{j}Q_{j}|j\rangle$
is, up to a normalizing constant,
    $\left\langle Q\left|e^{-{\frac {iHt}{\hbar }}}\right|\psi \right\rangle =\sum _{j}Q_{j}e^{-{\frac {E_{j}it}{\hbar }}}\langle j|j\rangle =\sum _{j}Q_{j}e^{-{\frac {E_{j}it}{\hbar }}}.$
Note that this formula can be obtained from the formula for thermal equilibrium by replacing 1/k$_B$T with it/ħ.

## 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:
    $E=\int _{x}\left[k\left({\frac {dy(x)}{dx}}\right)^{2}+V{\bigy(x){\big )}\right]dx,$
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:
    $S=\int _{t}\left[m\left({\frac {dy(t)}{dt}}\right)^{2}-V{\bigy(t){\big )}\right]dt.$
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:
    $iS=\int _{t}\left[m\left({\frac {dy(it)}{dt}}\right)^{2}+V{\bigy(it){\big )}\right]dt=i\int _{t}\left[m\left({\frac {dy(it)}{dit}}\right)^{2}-V{\bigy(it){\big )}\right]d(it).$

## 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.
Wick rotation also relates a quantum field theory at a finite inverse temperature β to a statistical-mechanical model over the "tube" R$^3$ × S$^1$ with the imaginary time coordinate τ being periodic with period β. However, there is a slight difference. Statistical-mechanical n-point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy reflection positivity.
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.

## Rigorous proof

Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the Osterwalder–Schrader axioms.