Why does Feynman use a Green's function to describe the electric field of an electron?

In the electron-electron collision diagram, one electron sends a virtual photon, carrying some 4-momentum. The other electron absorbs this photon and receives a push.

Feynman assumes that the distribution of various 4-momenta in the photon is the Green's function for the massless Klein-Gordon equation. Why?

Let us consider the drum skin analogy of the static electric field of the electron. If I press the drum skin with my finger, it creates a depression into the skin. That depression is analogous to the static electric field of a particle.

We may imagine that instead of pressing with a constant force F, I keep tapping the skin with my finger at a very rapid pace.

The tapping creates a depression. A single tap is equivalent to applying an "impulse source" to the wave equation of the drum skin. The Green's function for the skin wave equation, by definition, is the response of the skin to that impulse.

That is, we may imagine that the static electric field of a particle consists of a very rapid pace of Green's functions emanating from the particle. The electric field does not carry energy away. There has to be a total destructive interference for the "on-shell" waves in the decomposition of the Green's function.

On the other hand, waves carrying just linear momentum p, can progress. Those waves apparently are responsible for the static depression in the drum skin or the static electric field of a particle.

The decomposition for the various p obeys the decomposition of the Green's function.

If there is a planar wave describing another electron nearby, the photon waves for various p disturb the free Dirac equation of that other electron. That is, the equation no longer is equal to zero, but a (small) source term appears.

Each wave p creates a source term. If we perturb the planar wave solution to find a more accurate solution for the source term associated with p, then another wave appears. That wave is interpreted as the wave of an electron which absorbed the photon with a momentum p.

Relationship to the classical scattering from a static Coulomb potential

If we calculate the scattering distribution, assuming that the electrons are charged particles of classical mechanics, the result is the same, or almost the same as when we use the Feynman diagram formula.

Classically, the momentum p which the electrons exchange is roughly proportional to 1 / r, where r is the minimum distance between the electrons. The number of electrons receiving a push > |p| is proportional to

       1 / |p^2|,

which is derived from the fact that the area for passing at a distance < r is proportional to r^2.

There is probably some general mathematical theorem which shows that an 1 / r potential for an incoming flux of particles can be implemented through the absorption of quanta of the Green's function for the massless Klein-Gordon wave equation.