Wick rotation

Summary

Wick rotation replaces the real time variable $t$ by an imaginary time $t = -i\tau$. It turns the Minkowski quadratic form $t^2 - \abs{\mathbf{x}}^2$ into $-(\tau^2 + \abs{\mathbf{x}}^2)$, so problems in Lorentzian spacetime become problems in Euclidean space. In quantum mechanics it turns the oscillating evolution $e^{-itH/\hbar}$ into the decaying semigroup $e^{-\tau H/\hbar}$, and oscillatory path integrals into integrals against genuine measures. The Osterwalder–Schrader reconstruction theorem gives conditions under which a Euclidean theory can be rotated back to a relativistic one.