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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).

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  • $\begingroup$ In response to votes to close as "opinion-based", I've tried to remove any "opinion-y" language in the question. $\endgroup$ – tcamps Aug 22 '20 at 15:00
  • $\begingroup$ 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? $\endgroup$ – tcamps Feb 1 at 21:41
  • $\begingroup$ Or am I expected to have done a bit more legwork in looking for answers to my question before asking? $\endgroup$ – tcamps Feb 1 at 21:52
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    $\begingroup$ 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). $\endgroup$ – QuantumEyedea Feb 1 at 22:04
  • $\begingroup$ Questions with non-unique answers fit poorly to the SE Q&A format. $\endgroup$ – Qmechanic Feb 3 at 9:55
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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.

[1] G.B.Folland, Quantum field theory. A tourist guide for mathematicians, Math.Surveys & Monographs 149, AMS, 2008.

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  • $\begingroup$ Interesting -- Folland does discuss deducing smoothness via abstract considerations which I think extend beyond the real scalar field. He doesn't mention analyticity explicitly -- that's a nice observation by you to get to the "full support" conclusion! I don't know enough PDEs, but I have to imagine there are arguments for analyticity which are abstract enough to apply to lots of different fields at once (since a drawback of relying on the formula for this is that you have to derive a new formula for each type of field). But as you say, this would require some fancy math! $\endgroup$ – tcamps Aug 24 '20 at 21:43
  • $\begingroup$ @tcamps Surely there should be general tools guaranteeing the analyticity. Explicit analytic expression just simplifies the proof in some particular cases. $\endgroup$ – Mikhail Skopenkov Aug 24 '20 at 21:55
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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.

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