Are the Feynman virtual photon waves longitudinal?

Richard Feynman uses as the photon propagator a Green's function of the massless Klein-Gordon equation.

https://en.wikipedia.org/wiki/Propagator

An electromagnetic wave can be described by the electric field E. A real photon is a transverse wave where, for example, the electric field alternately points up and down relative to the direction to which the wave moves.

If a virtual photon carries just spatial momentum, no energy, then it does not look natural to draw it as a transverse wave. Since it pushes or pulls on another charge, a longitudinal wave looks more natural.

     Z+ ●

               ^

               |

               |

               •  e-

           

When the electron in the diagram passes the nucleus, the electron first feels a weak pull to the left, then a strong pull, and then again a weak pull. It is like the electron would have absorbed a half of a longitudinal wave.

Let the electron pass at the distance of the classical electron radius 3 * 10^-15 m.

The transferred momentum is of the order the electron mass m divided by c. The associated wavelength of the transferred momentum is the Compton wavelength 2 * 10^-12 m of the electron.

The Compton wavelength is ~ 1000X larger than the physical dimensions of the encounter. The Compton wavelength does not describe the actual, physical, length of the strong pull.

Anyway, it may be more intuitive to imagine virtual photons as longitudinal waves. That conveys the pushing or pulling aspect of such a photon.

Real photons are described as transverse waves.