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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?

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Comments:

  1. I have some bad news: In practice, everybody makes mistakes, it is very hard to detect your own errors, and there is never such time where you never should check your calculation again. The situation is made worse by the fact that different authors often have different conventions.

  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).

References:

  1. S. Weinberg, Quantum Theory of Fields, Vol. 1, 1995; Section 11.2 p. 476.

  2. M. Srednicki, QFT, 2007; p. 55 + p. 99. A prepublication draft PDF file is available here.

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