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 can be seen as a useful trick that works because of the similarity between the equations of two seemingly distinct fields of physics. This can be seen by the similarity between two central objects in quantum mechanics and statistical mechanics, where H is the Hamiltonian relating to conserved energy: The transformation $exp(−iHt/&hbar;)$ derived from the Schrödinger equation and the Gibbs measure $exp(H/k_{B}T)$ arising when considering systems in an environment (where t is time, ℏ is the Planck constant, T is temperature and kB is the Boltzmann constant).

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 $i=$ 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 $(−1, +1, +1, +1)$ 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 $$1/(k_\text{B} T)$$ 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 $$\exp(-E/k_\text{B} T)$$, where $k_{B}$ is the Boltzmann constant. The average value of an observable $Q$ is, up to a normalizing constant,
 * $$\sum_j Q_j e^{-\frac{E_j}{k_\text{B} T}},$$

where the $j$ runs over all states, $$Q_j$$ is the value of $Q$ in the $j$-th state, and $$E_j$$ is the energy of the $j$-th 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 $$\exp(-E it/ \hbar),$$ where $$\hbar$$ is the reduced Planck constant. 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}}. $$

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:
 * $$E = \int_x \left[ k \left(\frac{dy(x)}{dx}\right)^2 + V\big(y(x)\big) \right] dx,$$

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:
 * $$S = \int_t \left[ m \left(\frac{dy(t)}{dt}\right)^2 - V\big(y(t)\big) \right] dt.$$

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 $m$:
 * $$iS = \int_t \left[ m \left(\frac{dy(it)}{dt}\right)^2 + V\big(y(it)\big) \right] dt = i \int_t \left[ m \left(\frac{dy(it)}{dit}\right)^2 - V\big(y(it)\big) \right] d(it).$$

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. However, there is a slight difference. Statistical-mechanical $n$-point functions satisfy positivity, whereas Wick-rotated quantum field theories satisfy reflection positivity.

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 $β$.

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 theorem.