which the boundary conditions for the 4-potential in Eq. (\[BC4Pot\]) are recovered. In this way the solution (\[RedGreenFuncA3\]) guarantees that the boundary conditions at the $\theta$-interface are satisfied.

Now we have to solve for the various components $g_{\;\ \sigma }^{\mu }$. To this end we split Eq. (\[RedGreenFuncA3\]) into $\mu = 0$ and $\mu =
j=1,2,3 $ components; $$\begin{aligned}
g ^{0} _{\; \sigma} \left( z , z ^{\prime} \right) &=& \eta ^{0} _{\;
\sigma} \mathfrak{g} \left( z , z ^{\prime} \right) - i \tilde{\theta}
\epsilon ^{3 0 i} _{\;\;\;\ j} p _{i} \mathfrak{g} \left( z , a \right) g
^{j} _{\; \sigma} \left( a , z ^{\prime} \right) ,  \label{g0S} \\
g ^{j} _{\; \sigma} \left( z , z ^{\prime} \right) &=& \eta ^{j} _{\;
\sigma} \mathfrak{g} \left( z , z ^{\prime} \right) - i \tilde{\theta}
\epsilon ^{3 j i} _{\;\;\;\ 0} p _{i} \mathfrak{g} \left( z , a \right) g
^{0} _{\; \sigma} \left( a , z ^{\prime} \right) .  \label{gjS}\end{aligned}$$ Now we set $z = a$ in Eq. (\[gjS\]) and then substitute into Eq. (\[g0S\]) yielding $$g ^{0} _{\; \sigma} \left( z , z ^{\prime} \right) = \eta ^{0} _{\; \sigma} 
\mathfrak{g} \left( z , z ^{\prime} \right) 