1. from the Coulomb electric field, using classical physics, or
2. from the simple Feynman diagram, by integrating the "probability amplitude".
It is easy to calculate that the Coulomb model and the Feynman model approximately agree about the scattering probabilities. But why do they agree?
Also, what is the "virtual photon" γ which the electrons exchange?
In the Feynman diagram, the photon is a Fourier component of Green's function which describes the impulse response of the electromagnetic field. What does it mean that the other electron "absorbs" that component?
The rubber foam model
In our previous blog post we introduced the rubber foam model as an analogue of the Coulomb field.
○○○○○○○○○○○
○○●○○○○○●○○○ ○ = rubber foam
○○○○○○○○○○○ ● = electron
Electrons move inside rubber foam. They have to clear enough space for themselves so that they can fit in. They use hammers to hit the foam, so that the foam is squuezed enough and the electron fits in the empty space. The hammers constantly knock the foam around the electron. This creates the required pressure.
If two electrons come close to each other, they have to squeeze the foam more. Let us assume that the foam resists squeezing superlinearly. Then the electrons have problems squeezing the foam between them. The electrons feel a repulsion.
In this model, the "virtual phonon" is easily understood: the first electron hits the foam and the other electron's hammer feels the impulse. The first electron emitted a virtual phonon, and the other one absorbed it.
Virtual phonons are the analogue of virtual photons.
The classical limit
Nothing prevents the particles in Feynman diagrams from having macroscopic masses and charges.
If Feynman diagrams are correct, they must predict the behavior of macroscopic charges approximately right.
That is the case, at least in electron-electron scattering. It is easy to calculate that they correctly predict the scattering from the Coulomb field.
The macroscopic counterpart of the vertex correction is the interaction of a macroscopic charge with its own electric field. We have in this blog introduced the "rubber plate" model for the electric field: it is an elastic object which tries to keep up with the charge if the charge is accelerated.
With the rubber plate model, we may be able to calculate the classical vertex correction. Feynman diagrams should approximately replicate it. If that is the case, why do they calculate the same result? We do not know yet.
The classical limit of the magnetic moment μ is a more difficult case. We do not know what the classical limit is supposed to be. Does the charge move in the zitterbewegung loop at the speed of light? For a macroscopic particle, the loop length is extremely tiny,
h / (m c).
The rubber membrane model once again
In the rubber membrane model, we may imagine that the charge keeps hitting the membrane with a hammer, at short intervals, emitting Green's functions. The hammer also absorbs impulses which come back from previous hits, since the membrane wants to straighten up.
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/ | γ virtual photon
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e-
^ t
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In this model, it is quite a natural assumption that the outgoing electron absorbs impulses which were emitted by the incoming electron. As if the incoming electron and the outgoing electron were different particles.
The Feynman diagram above looks sensible in this interpretation.
The "electric" F term in the vertex correction
In the utexas.edu link, the vertex function has the form above. The term F₁ is the "electric" term, for which the loop correction
F₁(q²) → 0,
when q² → 0. In the rubber plate model of the electric field, we guess that F₁ corresponds to the effects of the elastic electric field "wobbling" when the electron meets another charge and changes course. If the electron passes by a nucleus, for example, at a large distance, then there is very little wobbling of the electric field. The wobbling does not affect much the momentum that the electron gains when it flies past the nucleus.
Question. Does the shape of the electric potential affect the electric vertex correction F₁? The electron can change course abruptly if it passes close to a very small charge. It it passes far from a large charge, then the change in the velocity vector is gradual. In the classical analogy, the wobbling in the first case should be significant, but in the second case insignificant.
In a Feynman diagram, a virtual photon q which contains just momentum, no energy, is a Fourier component wave of a type
exp(i p x / ħ).
It does not care about the shape of the Coulomb potential it was derived from.
But the Fourier decomposition of the time-independent potential does care about the shape of the Coulomb potential. If we have a small charge, then the Fourier decomposition looks different from a large charge?
*** WORK IN PROGRESS ***