I'm having trouble performing a Wick rotation back to Minkowski spacetime ($\eta_{\mu\nu}=(-1,1,1,\dots)$), following page 19 in the lecture notes here by C.P. Herzog.

I have this expression (equation 53 from the notes linked) \begin{equation} 0=-k_{\mu}G^{\alpha,\mu\nu}_E(k)-iF_{\mu}^{\ \nu}G^{\alpha,\mu}_E(k)+k^{\nu}\langle J^{\alpha}\rangle -k_{\mu}\delta^{\alpha\nu}\langle J^{\mu}\rangle \tag{53} \end{equation} where $k$ is the momentum 4-vector, $G_E$ are (Fourier transformed) 2-points functions and $\langle J^{\mu}\rangle$ is a 1-point function.

By Wick rotation from Euclidean to Minkowski the Euclidean Green functions become retarded ones $G_R$. Furthermore it states that

Each upper index zero carries a factor of $i$ while each lower index zero carries a factor of $-i$. Thus we have $J^0_E\rightarrow iJ^0_M$, $T^{0i}_E\rightarrow iT^{0i}_M$, $T^{00}_E\rightarrow -T^{00}_M$, $k^E_0\rightarrow -ik_0^M=i\omega$, and a similar rule for $F^{\mu}_{\nu}$.

With this prescription the paper claims that the above equation (53) becomes \begin{equation} 0=-k_{\mu}G^{\alpha,\mu\nu}_R(k)\color{red}{+}iF_{\mu}^{\ \nu}G^{\alpha,\mu}_R(k)+k^{\nu}\langle J^{\alpha}\rangle -k_{\mu}\eta^{\alpha\nu}\langle J^{\mu}\rangle. \tag{55} \end{equation}

However I have no idea how to obtain this result (55) from the first equation (53). I'd say that the free indices ($\alpha$ and $\nu$) don't matter, whether they are temporal or spatial index they are the same for all the terms and always carry the same number of $i$ factors for each term, so they cannot be the reason for the extra minus sign that the $F_{\mu}^{\ \nu}$ term gains going from Euclidean to Minkowski spacetime.

Let's study the first term of the first expression and use the prescription to Wick rotate (I'm suppressing the free indices) \begin{equation} k_{\mu}^EG^{\mu}_E=k_0^EG^0_E+k_i^EG^i_E=(-ik_0^M)(iG^0_R)+k_i^MG^i_M=k_0^MG^0_R+k_i^MG^i_M \end{equation} but then?

In the same spirit I get for the second term \begin{equation} F_{\mu}^{\ \nu}G^{\mu}_E=F_0^{\ \nu}G^0_E+F_i^{\ \nu}G^i_E=(-iF_0^{\ \nu})(iG^0_R)+F_i^{\ \nu}G^i_R=F_0^{\ \nu}G^0_R+F_i^{\ \nu}G^i_R. \end{equation}

I'm quite sure there's something really simple that I'm forgetting. Can anyone help?


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