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I'm trying to understand why the equal time propagator for a scalar field can sometimes be written in terms of the modified Bessel function $K_1(mr)$ and sometimes in terms of a Yukawa-like potential $\sim \frac{e^{-mr }}{4\pi r} $.

On the one hand, it is regularly argued that the integral that shows up in the Klein-Gordon equal time propagator can be written in terms of the modified Bessel function $K_1$: $$ D(\vec x,\vec x') \equiv \langle \vec x' | \vec x\rangle = \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{- i\vec k \cdot (\vec x ' -\vec x)} =\frac{m}{4\pi^2 r} K_1(mr),$$ where $r \equiv |\vec x - \vec x'|$.

On the other hand, the Green's function of the Klein-Gordon equation for a static field configuration $\phi$ reads $$G(\vec x,\vec x') = \int \frac{dk^3}{(2\pi)^3} \; \frac{e^{-i \vec k \cdot (\vec x - \vec x')}}{-k^2 + m^2}= \frac{e^{-mr }}{4\pi r} $$ where again $r \equiv | \vec x - \vec x'| $. (The integral is solved explicitly in Zee's book on page 29.)

Shouldn't the equal time propagator be equal to the Green's function for a static field configuration? If yes, why do we get such as simple solution (a Yukawa potential) in the second case and a much more complicated solution (a modified Bessel function) in the former case?


A useful hint is probably that the propagator for spacelike separations decays approximately like $e^{-mr }$: $$ D(\vec x,\vec x') \equiv \langle \vec x' | \vec x\rangle = \int \frac{ \mathrm{d }k^3 }{(2\pi)^3 2\omega_{k} } {\mathrm{e }}^{- i\vec k \cdot (\vec x ' -\vec x)} \sim e^{-mr }$$ (This is demonstrated at page 18 here, for example, or on page 27 in Peskin & Schröder's book)

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Shouldn't the equal time propagator be equal to the Green's function for a static field configuration?

Nope. The equal time propagator lives in a $4$-d Minkowski space-time, and the Yukawa potential lives in a $3$-d Euclidean space. They are, in fact, related. You can see the relationship in the propagator to the wave equation versus the Coulomb potential.

Short version: $$G_3(\mathbf{x};\mathbf{x}') = \int_{-\infty}^\infty \mathrm{d}t\ G_4(x;x'),$$ where $G_4$ is one of (perhaps any of? maybe only the causal ones?) the position space propagators for the Klein-Gordon equation, and the Green's function for the D'Alembertion for the wave equation. $G_3$ is the Yukawa potential for the KG equation, and the Coulomb potential for the Poisson's equation.

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  • $\begingroup$ Thanks a lot! This seems to imply there is indeed a close relationship between the modified Bessel function and the form of the Yukawa potential. I will try to figure it out in explicit terms and then get back to you. $\endgroup$ – jak Oct 28 '19 at 14:57
  • $\begingroup$ @jak Closer than you think. The Yukawa potential is a modified Bessel function of the second kind. In the Wikipedia Green's function article linked above, plug $n=3$ and $k=m$ into the $\Delta - k^2$ row of the table and you get the Yukawa potential (poke around in the "Asymptotic forms" section of the Wikipedia Bessel function article to find $K_{1/2}$). $\endgroup$ – Sean E. Lake Oct 28 '19 at 15:35
  • $\begingroup$ Ah perfect! A problem I encountered when I thought about your answer is that the time dependence drops out from the greens function for equal times. Thus I don’t see how we can integrate over t to find the yukawa potential. $\endgroup$ – jak Oct 28 '19 at 15:43
  • $\begingroup$ @jak Yeah, you can't integrate over time after evaluating, you need to do the integral with the full propagator. $\endgroup$ – Sean E. Lake Oct 28 '19 at 15:46
  • $\begingroup$ "the Yukawa potential lives in a 3-d Euclidean space." Can you please elaborate on that? Does it not experience time? Is that the same for nuclear force? $\endgroup$ – Árpád Szendrei Oct 29 '19 at 2:38

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