# Modified Bessel Function vs. Yukawa Potential in the propagator for spacelike separations

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)

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.
• @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 $K_{1/2}$). – Sean E. Lake Oct 28 '19 at 15:35