I Attractive force of opposite charges by McIrvin

exponent137

Summary
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:
(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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exponent137

Now I found a paper http://bolvan.ph.utexas.edu/~vadim/Classes/2019f/QED.pdf,
which more precisely explains attraction of charges with opposite signs and repelling of charges with equal signs. This is in pages 15-19. (I hope there are also other papers with this topic?)

1. According to figures 72 and 86 we can understand, why there are two spikes in McIrvin's graph of Photon propagator - because there are 2 x 2 tree graphs in Figs. 72 and 86.

2. But, I do not understand, why there are oppositely equal x-values of spikes in McIrvin's graph. According to Eq. (78), they are not equal. and not at precisely oppositely values of momentum? Probably he made some approximation?

3. McIrvin writes that no-hit wave function (unmodified one) gives the final explanation why repelling is not equalized with by attraction.
Now, by now you might be a little disturbed. We get wave functions by squaring amplitudes. The lump to the right of the origin goes down just as far as the lump to the left goes up. So isn't the probability that the photon knocked the particle's momentum toward the other one just as large as the probability that it knocked it away? No, because there is still some probability amplitude that no photon interaction occurred at all, and since we have no way of unambiguously telling one possibility from the other, we need to add the two wave functions together before squaring them! (There are also amplitudes for larger numbers of interactions, but for short times, we need not worry about those. Also, the "no-hit" wave function is not exactly the unmodified one, because of its own natural time evolution, but for short times all that does to the momentum-space wave function is give it a small imaginary part that we don't care about here.)

How this can be evident in this QED.pdf paper, for instance in Eq. (80)? I suppose that this matrix element is not for unmodified wave function?

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"Attractive force of opposite charges by McIrvin"

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