Using Wick Rotation to calculate Generating Function in Minkowski Space

Question

The question arises when I'm reading over the section "3.3.1 Minkowski Space" in page 16-17 in the following link: https://www-thphys.physics.ox.ac.uk/people/JohnCardy/qft/qftcomplete.pdf

It is discussing the technique of using Wick Rotation to calculate the generating function in Minkowski space.

It mentioned that simply inserting $τ=it$ into the results of the generating function in Euclidean space (i.e. imaginary time) provides the generating function in Minkowski space.

However, on top of page 17, it mentioned that I also have to let $p_0 \to ip_0$ as well. Why do I have to do this as well? How is that related to defining $τ=it$ and making a Wick Rotation?

Answer 1

A Wick-rotation in spacetime $x^{\mu}$ implies via Fourier transformation a Wick rotation in Energy-momentum space $p_{\mu}$. Perhaps the easiest way to convince oneself that this must be so is to consider the Fourier-integral representation $$\delta^4(x)~=~\int_{\mathbb{R}^4} \frac{d^4p}{(2\pi\hbar)^4}~\exp\left(\frac{ip\cdot x}{\hbar} \right)\tag{A}$$ of the Dirac delta distribution. It cannot be analytically continued to the ambient complexified spacetime. The real integration region can at most be deformed, i.e. the $x^0$ and $p_0$ Wick-rotations must be balanced. See also e.g. this and this related Phys.SE posts.

Answer 2

Cardy discusses how to pass from euclidean space to Minkowski space.

The Wick rotation can be thought of as a coordinate transformation $x' \rightarrow x$, where the $x' \equiv (\tau, \vec{x}')$ are the euclidean ones and the $x \equiv (t,\vec{x})$ are the ones for Minkowski space (see below for a caveat). As stated in the question $\tau = \mathrm{i} t$.

A covector transforms according to

$$ \omega_\mu = \frac{ \partial x'^\nu }{ \partial x^\mu } \omega'_\nu. $$ Using this transformation law for the vector $p_\mu$ we get for $p_0$ $$ p_0 = \frac{ \partial x'^\mu }{ \partial x^0 } p'_\mu = \frac{ \partial x'^0 }{ \partial x^0 } p'_0 = \frac{ \partial \tau }{ \partial t } p'_0 = \mathrm{i} p'_0 . $$

Now for the caveat. Viewing the Wick transformation as a coordinate transormation gives the following metric

$$ g_{\mu\nu} = \frac{ \partial x'^\alpha }{ \partial x^\mu } \frac{ \partial x'^\beta }{ \partial x^\nu } g'_{\alpha\beta} = \mathrm{diag}(-1,1,1,1)_{\mu\nu} $$ which in Cardy's conventions is the negative of the Minkowski metric.