In the Schwartz book "Quantum field theory and the standard model" on page 301, it mentions that "the Coulomb potential is given by the exchange of a single photon", and later "[Coulomb potential] is just the Fourier transform of the propagator".
It's not very clear to me where that comes from physically and mathematically
Moreover, on page 310 he gets the $V(r)$ from $\tilde V(p)$ merely by mentioning the Fourier transform
For both I guess mathematically it comes from the Fourier transform (of the Photon propagator and of $\tilde V(p)$), but I don't get how, since in the Photon propagator and in $\tilde V(p)$ we have the 4-vector momentum, while in the Coulomb potential we have the 3-vector position. So I don't get how to use a 4D Fourier transform to get a function in the 3D domain.
