# Wick rotation on Ward identities

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) $$$$0=-k_{\mu}G^{\alpha,\mu\nu}_E(k)-iF_{\mu}^{\ \nu}G^{\alpha,\mu}_E(k)+k^{\nu}\langle J^{\alpha}\rangle -k_{\mu}\delta^{\alpha\nu}\langle J^{\mu}\rangle \tag{53}$$$$ where $$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 $$$$0=-k_{\mu}G^{\alpha,\mu\nu}_R(k)\color{red}{+}iF_{\mu}^{\ \nu}G^{\alpha,\mu}_R(k)+k^{\nu}\langle J^{\alpha}\rangle -k_{\mu}\eta^{\alpha\nu}\langle J^{\mu}\rangle. \tag{55}$$$$

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) $$$$k_{\mu}^EG^{\mu}_E=k_0^EG^0_E+k_i^EG^i_E=(-ik_0^M)(iG^0_R)+k_i^MG^i_M=k_0^MG^0_R+k_i^MG^i_M$$$$ but then?

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

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