Using Wick Rotation to calculate Generating Function in Minkowski Space 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?
 A: 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.
A: 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.
