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 (i2 = –1). More precisely, in statistical mechanics, the Gibbs measure exp(−H/kBT) 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 kB 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/kBT, 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. Instantons are Wick-rotated time solutions to certain potentials that allow for the calculation of eigenenergies and decay rates.
Overview Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (− + + +) convention)
d s 2 = − ( d t 2 ) + d x 2 + d y 2 + d z 2 {\displaystyle ds^{2}=-\left(dt^{2}\right)+dx^{2}+dy^{2}+dz^{2}}
and the four-dimensional Euclidean metric
d s 2 = d τ 2 + d x 2 + d y 2 + d z 2 {\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 = −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/kBT with it/ħ, where T is temperature, kB 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 Ej. When this system is in thermal equilibrium at temperature T, the probability of finding it in its jth energy eigenstate is proportional to exp(−Ej/kBT). Thus, the expected value of any observable Q that commutes with the Hamiltonian is, up to a normalizing constant,
∑ j Q j e − E j k B T , {\displaystyle \sum _{j}Q_{j}e^{-{\frac {E_{j}}{k_{\text{B}}T}}},}
where j runs over all energy eigenstates and Qj 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(−Ejit/ħ). Thus, the probability amplitude that a uniform (equally weighted) superposition of states
| ψ ⟩ = ∑ j | j ⟩ {\displaystyle |\psi \rangle =\sum _{j}|j\rangle }
evolves to an arbitrary superposition
| Q ⟩ = ∑ j Q j | j ⟩ {\displaystyle |Q\rangle =\sum _{j}Q_{j}|j\rangle }
is, up to a normalizing constant,
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