# Interpretation of Feynman propagator for massive scalar field in position space

I've always treated propagator in the momentum representation so when it diverges, we are on-shell. But what is the interpretation of light-cone divergences in position space? If it is something we have to renormalize, how can I do it? More precisely, if I have to renormalize light-cone divergences of the following Feynman propagator: $$\begin{equation*} G(s)=\frac{m}{8\pi s} H^{(1)}_1(m s)\,\,,\,\,s^2=(x^0-y^0)^2-|\vec{x}-\vec{y}|^2 \end{equation*}$$ (where $$H^{(1)}_1$$ is a Hankel function of the first type) how should I proceed? Is there a method like dimensional regularization for this problem?

There is no need to renormalize here.

The interpretation is that a quantum field is not an operator-valued function $$\hat{\phi}(x)$$, but an operator-valued distribution, aka generalized function. That means that $$\hat{\phi}$$ is a linear functional from test functions on space-time (e.g. you can choose functions with compact support to get temporal distributions).

Intuitively, $$\hat{\phi}(f) = \int d^4 x f(x) \hat{\phi}(x).$$

But this formula is only formal, because $$\hat{\phi}(x)$$ doesn't mathematically exist.

For any two test functions $$f_{1,2}$$, the quantity

$$\left< \hat{\phi}(f_1) \hat{\phi}(f_2) \right> = \int d^4 x f_1(x) \int d^4 y f_2(y) \cdot G(x - y)$$

is finite and well defined, even when $$G$$ has a divergence on the light cone.