The symmetry $g_{\mu \nu }\left( z,z^{\prime }\right) =g_{\nu \mu
}^{\ast }\left( z,z^{\prime }\right) =g_{\mu \nu }^{\dagger }\left(
z,z^{\prime }\right) $ is also manifest.

The various components of the static GF matrix in coordinate representation are obtained by computing the Fourier transform defined in Eq. (\[RedGreenDef\]), with the reduced GF given by Eq. (\[RFSGF\]). The details are presented in Appendix \[IntegralsStaticCase\]. The final results are $$\begin{aligned}
G_{\;0}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=&\frac{1}{|%
\mathbf{x}-\mathbf{x}^{\prime }|}-\frac{\tilde{\theta}^{2}}{4+\tilde{\theta}%
^{2}}\frac{1}{\sqrt{R^{2}+Z^{2}}},  \label{G00} \\G_{\; i}^{0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=& - \frac{2 \tilde{\theta}}{4+\tilde{\theta}^{2}} \frac{\epsilon _{0ij3} R ^{j}}{R ^{2}}\left( 1-\frac{Z}{\sqrt{R^{2}+Z^{2}}}\right) ,  \label{G0i} \\ G _{\; j}^{ i }\left( \mathbf{x},\mathbf{x}^{\prime }\right) &=& \eta ^{i} _{\; j} G _{\; 0}^{ 0}\left( \mathbf{x},\mathbf{x}^{\prime }\right) - \frac{i}{2}\frac{\tilde{\theta}^{2}}{4+ \tilde{\theta}^{2}} \partial ^{i} K _{j} \left( \mathbf{x},\mathbf{x}^{\prime }\right) , \label{Gij}\end{aligned}$$ where $Z=| z - a | + | z^{\prime } - a |$, $R ^{j} = \left( \mathbf{x-x}^{\prime }
\right) _{\parallel } ^{j}  = \left( x - x^{\prime } , y - y^{\prime} \right)$, $R=|\left(  \mathbf{x-x}^{\prime }\right) _{\parallel }|\;$and$$K ^{j} \left( \mathbf{x},\mathbf{x}^{\prime }\right) =2i\frac{\sqrt{%
R^{2}+Z^{2}}-Z}{R^{2}} R ^{j} .$$ Finally, we observe that Eqs. (\[G00\]-\[Gij\]) contain all the required elements of the GF matrix, according to the choices of $z$ and $z^{\prime
}$ in the function $Z$.

Applications
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Point-like charge 