The Green's function for the d'Alembertian operator with Feynman boundary conditions is given in many QFT textbooks as (in four-dimensions and $(+,-,-,-)$ signature)
$$ \Box G(x)=\delta^4(x)\\ G(x)=\frac{1}{x^2-i\epsilon}. $$
However, I could not find any textbook that discusses the correct way to interpret the $-i\epsilon$ here. Do both lightcone singularities occur in the upper half of the complex $x^0$ plane i.e. $x^0=\pm|\vec{x}|+i\epsilon$, or is it like the momentum space propagator where we interpret $k^2-i\epsilon$ to mean that the poles occur on opposite sides of the real $k^0$ axis i.e. $k^0=\pm|\vec{k}|\mp i\epsilon$?
