In QFT, it seems the propagator has a simple closed form in momentum space. The position space representation is more complicated, but could be worse -- basically it's a Bessel function (depending, of course, on what field one is talking about).
Question: What are some applications of knowing the precise analytic form of a Feynman propagator in position space? For instance, if numerical computations of the zeros of Bessel functions (in the context of propagators) have been used to make a physical prediction, I'd be interested to hear about that.
For example, for a 3+1-dimensional massive scalar field, the propagator in momentum space is
$$D(p) = \frac{1}{p^2 - m^2 \pm i \epsilon}$$
whereas in position space, according to the above-linked paper, it's
$$D(x) = \Theta(x^2) \frac{m}{8\pi\sqrt{x^2 - i\epsilon}}H^{(2)}_1(m \sqrt{x^2 - i\epsilon}) + \Theta(-x^2) \frac{m}{4\pi^2\sqrt{-x^2 + i\epsilon}}K_1(m\sqrt{-x^2 + i\epsilon})$$
where $\Theta$ is the Heaviside step function and $H^{(2)}_1$ and $K_1$ are certain Bessel functions.
The only use I happen to have seen made of the position-space representation is to verify that the propagator decays exponentially outside the light cone (for which you can make a crude estimate without knowing anything about Bessel functions).