I'm looking for a rule to "convert" the propagators of a quantum field theory formulated in Euclidean spacetime into those of the same theory but in Minkowski spacetime (with the $\operatorname{diag}(-,+,+,+)$ metric). I calculated both for the real scalar field and they turned out to be \begin{equation} \langle\phi(x)\phi(y)\rangle_\mathrm{M}= -\frac{i}{(2\pi)^4}\int e^{ip\cdot x}\frac{1}{p^2+m^2-i\varepsilon}\,\mathrm{d}^4p\tag{1} \end{equation} and \begin{equation} \langle\phi(x)\phi(y)\rangle_\mathrm{E}= \frac{1}{(2\pi)^4}\int e^{ip\cdot x}\frac{1}{p^2+m^2}\,\mathrm{d}^4p.\tag{2} \end{equation} Together with the rule for the Wick rotation $x^0=-ix^4$ (implying $p^0=-ip^4$ and so on), this would seem to imply that \begin{equation} \langle\phi(x)\phi(y)\rangle_\mathrm{M}= -\langle\phi(x)\phi(y)\rangle_\mathrm{E}\bigr|_{x^4=ix^0, y^4=iy^0}.\tag{3} \end{equation}
However, with the Dirac free field, from the Lagrangians ($\gamma^0:=-i\gamma^4$ after the Wick rotation, so as to preserve the slash notation) \begin{equation} \mathcal{L}_\mathrm{M}= -\overline{\Psi}(\not{\partial}+mI_4)\Psi, \quad \mathcal{L}_\mathrm{E}= \overline{\Psi}(\not{\partial}+mI_4)\Psi.\tag{4} \end{equation} Eq. (4) is the Dirac Lagrangian (7.5.34) in Weinberg's QFT vol. 1 book (page 323, 1st ed.). I get \begin{equation} \langle\Psi(x)^a\overline{\Psi(y)}_b\rangle_\mathrm{E}= \frac{1}{(2\pi)^4}\int e^{ip\cdot x}\frac{(-i\!\!\not{p}+mI_4){^a}_b}{p^2+m^2}\,\mathrm{d}^4p\tag{5} \end{equation} and \begin{equation} \langle\Psi(x)^a\overline{\Psi(y)}_b\rangle_\mathrm{M}= -\frac{i}{(2\pi)^4}\int e^{ip\cdot x}\frac{(-i\!\!\not{p}+mI_4){^a}_b}{p^2+m^2-i\varepsilon}\,\mathrm{d}^4p.\tag{6} \end{equation} So far so good, assuming that my calculations are correct, which I'm never sure of even if I did them countless times, but I'm tired of calculating everything twice, so: is this rule correct, and if it is, how can I prove it in a general fashion, so that it is valid for whatever field theory I'm studying?
