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 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.
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).
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: Taken together, the previous two examples show how the path integral formulation of quantum mechanics is related to statistical mechanics.
Wick rotation also relates a quantum field theory 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.
Dirk Schlingemann proved that a more rigorous link between Euclidean and quantum field theory can be constructed using the Osterwalder–Schrader axioms.