quantum mechanics – Why is imaginary time evolution non …

If $H$ is hermitian then $U=e^{-itH}$ is unitary if and only if $t$ is real. Making a change of variables $t=itau$ won't change that. The point is that when you do a Wick rotation to imaginary time you are not making a simple change of variables - a change of variables after all can't actually affect the physics.

The basic place where an imaginary time quantity arises is the thermal density matrix$$ rho = frac{e^{-beta H}}{Z}$$with $beta=1/(k_BT)$ the inverse temperature, which to have physical meaning must be real. This is the same thing as $U$ for an imaginary time $t=-ibeta$. This should be enough to convince you that in the vast majority of cases when talking about imaginary time one really does consider the time to be imaginary, and not purely real as needed for $U$ to be unitary.

In a QFT context, the Wick rotation is less physical and more of a mathematical trick - there you decide that the observables $O(t)$ asked for are hard to compute along the real line and instead compute them along the imaginary axis $O(it)$ and hope that the resulting formulas are analytically continuable to the entire complex plane.

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quantum mechanics - Why is imaginary time evolution non ...