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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.

## Contents

## Overview

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.

## Statistical and quantum mechanics

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

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 where is reduced Planck's constant. The probability amplitude that a uniform (equally weighted) superposition of states:

evolves to an arbitrary superposition

is up to a normalizing constant,

## 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,

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,

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:

## 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" **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.

## See also

## References

- Wick, G. C. (1954). "Properties of Bethe-Salpeter Wave Functions".
*Physical Review*.**96**(4): 1124–1134. Bibcode:1954PhRv...96.1124W. doi:10.1103/PhysRev.96.1124.

## External links

- A Spring in Imaginary Time — a worksheet in Lagrangian mechanics illustrating how replacing length by imaginary time turns the parabola of a hanging spring into the inverted parabola of a thrown particle
- Euclidean Gravity — a short note by Ray Streater on the "Euclidean Gravity" programme.