# Convert propagators from Euclidean to Minkowski spacetime

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 $$$$\langle\phi(x)\phi(y)\rangle_\mathrm{M}= -\frac{i}{(2\pi)^4}\int e^{ip\cdot x}\frac{1}{p^2+m^2-i\varepsilon}\,\mathrm{d}^4p\tag{1}$$$$ and $$$$\langle\phi(x)\phi(y)\rangle_\mathrm{E}= \frac{1}{(2\pi)^4}\int e^{ip\cdot x}\frac{1}{p^2+m^2}\,\mathrm{d}^4p.\tag{2}$$$$ Together with the rule for the Wick rotation $$x^0=-ix^4$$ (implying $$p^0=-ip^4$$ and so on), this would seem to imply that $$$$\langle\phi(x)\phi(y)\rangle_\mathrm{M}= -\langle\phi(x)\phi(y)\rangle_\mathrm{E}\bigr|_{x^4=ix^0, y^4=iy^0}.\tag{3}$$$$

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) $$$$\mathcal{L}_\mathrm{M}= -\overline{\Psi}(\not{\partial}+mI_4)\Psi, \quad \mathcal{L}_\mathrm{E}= \overline{\Psi}(\not{\partial}+mI_4)\Psi.\tag{4}$$$$ Eq. (4) is the Dirac Lagrangian (7.5.34) in Weinberg's QFT vol. 1 book (page 323, 1st ed.). I get $$$$\langle\Psi(x)^a\overline{\Psi(y)}_b\rangle_\mathrm{E}= \frac{1}{(2\pi)^4}\int e^{ip\cdot x}\frac{(-i\!\!\not{p}+mI_4){^a}_b}{p^2+m^2}\,\mathrm{d}^4p\tag{5}$$$$ and $$$$\langle\Psi(x)^a\overline{\Psi(y)}_b\rangle_\mathrm{M}= -\frac{i}{(2\pi)^4}\int e^{ip\cdot x}\frac{(-i\!\!\not{p}+mI_4){^a}_b}{p^2+m^2-i\varepsilon}\,\mathrm{d}^4p.\tag{6}$$$$ 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?

2. All references agree that the Wick rotation in spacetime is $$x^4=ix^0$$, see e.g. this Phys.SE post. It is interesting to develop a consistent continuous Wick-rotation that co-exists in both in $$x^{\mu}$$ and $$p^{\mu}$$ space, cf. my related Phys.SE answer here. However most textbooks in QFT do not work simultaneously in the $$x$$- and the $$p$$-representation, and their Wick rotation in momentum space $$p^0=ip^4$$ is typically the opposite!
3. OP's Minkowskian eqs. (1), (4), (6) agree with eqs. (8.11+15), (36.28), (42.11) in Ref. 2, respectively, if we substitute $$\gamma^{\mu}_{\text{Srednicki}}~\leftrightarrow~i\gamma^{\mu}_{\text{Weinberg}}.\tag{*}$$ Eq. (*) is supported by the fact that Srednicki uses the Clifford algebra $$\{\gamma^{\mu},\gamma^{\nu}\}_+ ~=~ -2g^{\mu\nu}, \tag{47.1}$$ while Weinberg has the opposite sign convention in his Clifford algebra (1.1.21).