# Applications of position-space formula for a Feynman propagator?

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).

• In response to votes to close as "opinion-based", I've tried to remove any "opinion-y" language in the question. Aug 22, 2020 at 15:00
• Looking back, it still confuses me that this question was closed even after I fixed the superficial issues with phrasing. I think I can understand that a question of the form "Is X ever relevant to physics?" might be seen as problematic. But after rephrasing, I don't understand the issue. Does this site discourage questions of the form "what are some examples of X?" because they are a bit open-ended? Or is the distinction I'm making -- between the use of two mathematically equivalent representations of one object -- not considered to be something one can objectively make? Feb 1, 2021 at 21:41
• Or am I expected to have done a bit more legwork in looking for answers to my question before asking? Feb 1, 2021 at 21:52
• Section 4.5 of paper "Vacuum Noise and Stress Induced by Uniform Acceleration" by Takagi uses the massive position space propagator (the Wightman function) to calculate the rate of excitation of a two-level detector which uniformly accelerated through Minkowski space (this then tells you the Unruh temperature). This calculation requires your formula for the propagator (or more precisely, the Wightman function which is the same as your formula with the $i \epsilon$ in different places). Feb 1, 2021 at 22:04
• Questions with non-unique answers fit poorly to the SE Q&A format. Feb 3, 2021 at 9:55

A mathematical application (see Section 6.5 of [1]): The position-space Feynman propagator is analytic away from the light cone. In particular, there is no domain in $$\mathbb{R}^4$$ where it vanishes identically. This is not so easy to show without an explicit analytic expression.
Section 4.5 of paper "Vacuum Noise and Stress Induced by Uniform Acceleration" by Takagi uses the massive position space propagator (the Wightman function) to calculate the rate of excitation of a two-level detector which uniformly accelerated through Minkowski space (this then tells you the Unruh temperature). This calculation requires your formula for the propagator (or more precisely, the Wightman function which is the same as your formula with the 𝑖𝜖 in different places): $$\langle 0 | \phi(x)\phi(y) | 0 \rangle = \frac{m}{4 \pi^2 \sqrt{ - (\Delta t - i \epsilon)^2 + |\Delta \mathbf{x}|^2 }} K_1\left( m \sqrt{ - (\Delta t - i \epsilon)^2 + |\Delta \mathbf{x}|^2 } \right)$$ where $$\Delta t := x^0 - y^0$$ and $$\Delta \mathbf{x} := \mathbf{x} - \mathbf{y}$$. Note this is not time-ordered, unlike the Feynman propagator.