# Does Wick rotation work for time-dependent Hamiltonian?

Consider a quantum system that is governed by a Hamiltonian with explicit time dependence $$H(t)$$.

Is it always legitimate to perform a Wick rotation $$t \rightarrow -i\tau$$, and calculate the time-dependent ground state with imaginary-time Schrodinger equation?

If not, what are the sufficient and necessary conditions to apply Wick rotation?

• What problem are you trying to solve? Do you want the ground state, or time-evolution of a particular state, or the full time-evolution operator? Commented Sep 2, 2020 at 22:43
• I just want to calculate the time-evolution of the ground-state wave function. Commented Sep 2, 2020 at 22:47
• Why do you need to perform a Wick rotation? Are you just asking how to compute the time evolution operator?
– Alex
Commented Sep 2, 2020 at 22:54
• The imaginary-time evolution operator is much easier to calculate with numerical methods than the real-time operator. It is also conceptually important to understand when one can use the imaginary-time formulation. Commented Sep 3, 2020 at 2:58

A Wick rotation is a method of finding a solution to a mathematical problem in Minkowski spacetime from a solution to a related problem in four dimensional Euclidean space by relating the two spaces via the substitution of the real time $$t$$ variable with an imaginary time $$i \tau$$ variable.
The integral along the real line, $$t$$ real variable from $$-\infty \to + \infty$$, is extended to a contour integral in the complex plane ($$\pm$$ half plane).
The integral along the $$\pm$$ half circle, closing the loop with an infinite radius, vanishes.
The poles, if any, of the function to be integrated are included in the original contour and remain included in the $$\pm \pi/2$$ rotated contour.
The rotated contour corresponds to the substitution $$t \to \pm i \tau$$.