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- Questions, how to understand better the paper of McIrvin

I am reading the paper of McIrvin:

In fourth picture, two opposite spikes of imaginary magnitude are drawn in momentum space. This is a photon propagator.

Photon propagator in p space can be written as:

##\frac{-ig^{\mu\nu}}{(k-k')^2}##

Below the picture, McIrvin writes:

Can this connection with the common QED calculation be explained more?

How does this picture change, if we are more and more precise with QED calculations?

Why there are two spikes?

Is this calculation not possible If ##k=k'=0##, because then one photon approximation is not possible? This is example where both particles are resting.

In fourth picture, two opposite spikes of imaginary magnitude are drawn in momentum space. This is a photon propagator.

Photon propagator in p space can be written as:

##\frac{-ig^{\mu\nu}}{(k-k')^2}##

### Bhabha scattering - Wikipedia

en.wikipedia.org

Below the picture, McIrvin writes:

*(A note for experts only: The somewhat QED-savvy may be puzzled by the total nonresemblance of this to any well-known photon propagator. That's because I'm not going into momentum space in every direction, just in the x-direction. The more QED-savvy will notice that I am making some pretty monstrous oversimplifications here. Actually, they are not so bad; what I'm doing is the equivalent of assuming that the potential can be locally approximated by a sinusoid! If the wave packet is small enough in position space, a Coulomb potential and a sinusoidal one are both effectively a constant-force potential, so I can do this. Neglecting all magnetic effects and taking the nonrelativistic limit, the amplitude for transfer of a given momentum by a single virtual photon—which is essentially what I am colorfully, and without much prevarication, labeling the "photon's momentum-space wave function"— has to have an imaginary part odd in p_x because the potential is real, so in any case the qualitative effect will be the same as what I describe below, and for essentially the same reasons. It's just so much easier to convolute spikes. As for the single-particle "wave functions" of the charged particles, I can speak of them with fair correctness because the particles are far apart and slowly moving.*Can this connection with the common QED calculation be explained more?

How does this picture change, if we are more and more precise with QED calculations?

Why there are two spikes?

Is this calculation not possible If ##k=k'=0##, because then one photon approximation is not possible? This is example where both particles are resting.

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