1st comment:

It's worth thinking for a second about where Wick rotation comes from. You can do this in the context of the quantum mechanics of a free particle. In QFT, all of the details are more complicated, but the basic idea is the same.

In free particle, QM, we get the path integral by inserting sums over intermediate states at various times. The need for Wick rotation arises as soon as you do this just once.

$\langle q' | e^{- \frac{ iP^2 t}{2m\hbar}}|q\rangle = \int_{-\infty}^\infty \langle q'| p \rangle \langle p |e^{- \frac{ iP^2 t}{2m\hbar}}|q\rangle dp = \frac{1}{2\pi\hbar} \int_{-\infty}^\infty e^{\frac{-i t }{2m\hbar} p^2 + i\frac{q' - q}{\hbar} p} dp$.

This is an oscillatory integral. The integrand has norm 1 because the argument of the exponential is purely imaginary. Such integrals don't converge absolutely, so the right hand side of this equation is not a well-defined expression as it stands. To make it well defined we need to supply some additional information.

Wick rotation provides a way of doing this. You observe that the left hand side is analytic in $t$, and that the right hand side is well-defined if $Im(t) < 0$. Then you can define the integral for real $t$ by saying that it's analytic continued from complex $t$ with negative imaginary part.

2nd comment:

As V. Moretti pointed out, in QFT, it's in some sense backwards to think of analytically continuing from Minkowski signature to Euclidean signature. Rather, one finds something in Euclidean signature which has nice properties and then analytically continues from Euclidean to Minkowski. However, one can often begin this process by taking a Minkowski action and finding its Euclidean version, and then trying to build up a QFT from there. There's no guarantee that this will work though. Spinor fields may have reality conditions that depend on the signature of spacetime. Or the Euclidean action you derive may be badly behaved. This is famously the case for Einstein's gravity; the Euclidean action is not bounded below, so one does not get a sensible Euclidean theory.

This post imported from StackExchange Physics at 2014-05-04 11:30 (UCT), posted by SE-user user1504