I’m sorry to be yet another person confused about Wick rotation, but it’s been all day and I’m still not sure I’ve got it right.
Let’s start by considering the field expansion of a scalar field:
$$ \phi(x) = \int \frac{d^4k}{(2\pi)^4} \, \tilde{\phi}(k) \, e^{-ikx} \tag{1} $$
Now, I would like to perform a Wick rotation. In order to leave the argument of the exponent invariant, I map $$x^0 \rightarrow -i x^0\qquad\text{and}\qquad k^0 \rightarrow i k^0.\tag{1b}$$
This way, I also achieve that $k^2 = (k^0)^2 + |\vec{k}|^2 = k_E^2$, so the norm of the 4-momentum vector becomes Euclidean (hence the subscript $E$).
This rotation is consistent with removing the $i\epsilon$ prescription from the denominator of the propagator, since now the term $k^2 + m^2$ is positive definite and never zero.
Although everything seems to work out nicely, I still encounter a problem I can't resolve.
Consider the Fourier transform of a 4D Euclidean field:
$$ \phi_E(x) = \int \frac{d^4k}{(2\pi)^4} \, \tilde{\phi}_{E}(k) \, e^{-ik_E x_E} \tag{2} $$
This seems different from the Wick-rotated field expansion. After the Wick rotation, the argument of the exponent of equation (1) becomes:
$$ -ikx = -i(-k^0 x^0 + \vec{k} \cdot \vec{x}) $$
On the other hand the one of equation (2) is :
$$ -ik_E x_E = -i(k^0 x^0 + \vec{k} \cdot \vec{x}) $$
So my first question is: is there a way to reconcile the two?
More importantly, once I perform the Wick rotation, the Lagrangian also changes as $\mathcal{L} \rightarrow -\mathcal{L}_E$, where the subscript $E$ stands for Euclidean.
This confuses me: given the difference between the Wick-rotated field expansion and the standard Euclidean Fourier transform, I’m unsure how to correctly write down the explicit partition function of a theory.
Take, for example, $\phi^3$ theory. I thought the Euclidean partition function would look something like:
$$ Z_E[J] = \sum_{V=0}^\infty \left( -g \int d^4x_E \, \frac{\delta}{\delta J(x)} \right)^3 \sum_{P=0}^\infty \left( \frac{1}{2P!} \int d^4y_E \, d^4z_E \, J(y) \, \Delta(y-z) \, J(z) \right)^P $$
This last equation would be valid if I can use the canonical Euclidean Fourier transform — which, at the moment, I’m not entirely sure is justified.
EDIT:
I think I was overthinking it. To reconcilliate 1 with 2 I just need to equate the two and map $k^0$ into $-k^0$ in the first then the identity follows by imposing
$$ \tilde{\phi_{E}}=i\tilde{\phi(-ik^0,...)} $$
