1 + r_e / λ_e = 1 + α / (2π),
where
r_e = 2.82 * 10^-15 m
is the classical radius of the electron,
λ_e = 2.43 * 10^-12 m
is the Compton wavelength of the electron, and
α = 1 / 137
is the fine structure constant.
A classical model where the electron moves at the speed of light along a circular orbit of the length λ_e would explain the Dirac equation magnetic moment of the electron.
λ_e Compton wavelength
O --------
e- orbits
along the letter O
In the rubber plate model of the electron electric field, the field which is far from the electron "lags behind".
Let us guess that the electric field farther than λ_e / 2 does not contribute to the effective mass of the electron in its circulating motion. Why is the guess reasonable? In a standing wave, the first node is at λ / 4 from the central oscillation. Since the mass at the node does not move, it does not contribute to the effective mass. It is a reasonable guess that the total reducing effect might be equivalent to the entire mass farther than λ / 2 away. (In the future, we need to calculate an estimate somehow.)
The effective mass of the electron is reduced by the factor
1 - r_e / λ_e.
The effective Compton wavelength grows by the factor
1 + r_e / λ_e.
The electron must then circulate a path which is a little bit longer. The magnetic moment of the path grows by the factor
1 + r_e / λ_e,
which is the right value.
The photograph model may explain the spin 1 / 2
The electron angular momentum in the above model is
L = h-bar,
where h-bar is the Planck constant divided by 2 π. That is, the spin of the electron would be 1.
But the measured spin of the electron is 1/2.
We suggest the following model, which is inspired by our photograph model of quantum field theory.
The classical electron deep down does circulate at the speed of light along a path whose length is only
λ_e / 2.
The spin of that path is 1/2.
But the path integral determines what size of features in the motion we can "see". There would be total destructive interference if we would try to put the electron into a path which is that short.
The smallest circular orbit, which the path integral allows, has the length of one Compton wavelength. That may be the reason why the electric field of the electron appears to circulate at the speed of light along a path of that length when we measure the magnetic moment of the electron. That is the shortest path we can "see".
In zitterbewegung, the electron moves at the speed of light along a path whose length is one Compton wavelength.
If we see the electric field in that way, why we do not see the angular momentum of the electron in the same way? That is a problem of our model. Maybe we can only measure the angular momentum through direct interaction with the classical electron deep down?
The model may explain the strange nature of the spinor: the spinor has to be "rotated" twice through 360 degrees to get it to the original configuration. It is like walking around a Möbius strip.
In our model, the classical electron makes two rounds when we outside observers only see one round.
____
|O O|
|O O|
------
Imagine four rotating discs placed side by side like in the diagram above. Let us put a tight rubber band loop around the discs. The lines in the diagram depict the rubber band.
If we rotate the discs two full rounds, then the rubber band makes approximately one full round.
The path integral may behave like the rubber band. Two rounds of classical rotation may appear as one rotation in the path integral.
How does the vertex function know how to calculate the rubber plate effect?
virtual photon p
~~~~~~~
/ \
e- --------------------------------------------
/
/
~~~~~~
virtual photon q
from the magnetic field B
In the vertex correction diagram, the spatial momentum from the external magnetic field B, q, can be arbitrarily small.
The virtual photon which the electron sends to itself may contain an arbitrary 4-momentum p. Its magnitude does not depend on q at all. It is kind of an internal process of the electron.
In our classical model of the electron, the reduction in the effective mass of the electron is an "internal process". It depends only on the frequency of the circulation of the electron in the zitterbewegung orbit. It does not depend on q at all.
How does the vertex correction "know" about the zitterbewegung orbit, so that it can calculate the effect correctly?
The answer has to lie in the way how the magnetic moment in the Dirac equation is coupled to the external magnetic field B. The coupling might be somewhat similar to a classical electron in the zitterbewegung orbit. The coupling then "knows" something about the orbit.
Let us imagine an unknown force which makes a "scalar" pseudoelectron to move in the zitterbewegung orbit. The effective mass of the electron would be reduced just like in our above model.
Actually, when an electron orbits a proton in the hydrogen atom, the electron has a slightly reduced mass because its far electric field lags behind. The orbital frequency is
f = 6.6 * 10^15 / s.
Light moves
s = c / f = 4.5 * 10^-8 m
in that time.
The effective mass reduction factor is
1 - r_e / s = 1 - 6 * 10^-8,
or 0.03 eV.
Let us check if this effective mass reduction is known in the literature.
Regularization and renormalization of the anomalous magnetic moment vertex correction
In quantum field theory, the infrared and the ultraviolet divergences in the vertex correction are cut off in an ad hoc fashion.
"Too large" and "too small" 4-momenta p are banned in the virtual photon which the electron sends to itself. The cutoff procedure has been chosen in a "natural" way, for example, adding a little mass for the photon, and imagined heavy particles in Pauli-Villars regularization. The cutoff, for an unknown reason, yields exactly the measured results.
How the rubber plate model removes ultraviolet and infrared divergences
In our rubber plate model, we do not need any ad hoc cutoffs of infinite values.
We do cut off the far electric field which lags behind and does not follow the movement of the electron, but that is a finite cutoff and has an intuitive physical explanation.
Suppose that there is an impulse on 1 square meter of the rubber plate, and the impulse lasts for 1 second. We use Green's functions to calculate the response. A Green's function calculates the response to a Dirac delta impulse. We have to sum such "spike" impulses over the 1 square meter and the 1 second.
Very high frequencies are removed by destructive interference. What removes very slow frequencies? Apparently, it is the small extent of the impulse which removes very slow frequencies. Waves of length, say, 1 kilometer would require either a long lasting impulse or an impulse which is 1 kilometer wide.
The impulse itself is kind of a wave packet. The packet contains a negligible spectrum of very high and very low frequencies. When we use Green's functions to calculate the response, very high and very low frequencies cannot contribute much to the response.
The problem of divergences arises from new degrees of freedom which are introduced by loops in Feynman diagrams. One can put an arbitrary 4-momentum p to circulate in the loop, and one gets a valid Feynman diagram.
The way out of too much freedom may be to take constraints from the deep down classical level of processes. In an earlier posting, we already removed the divergence from photon-photon scattering by appealing to classical virtual pairs.
The rubber plate model may explain the cancelation of bremsstrahlung against infrared divergence in vertex correction
According to Amita Kuttner (2016), the infrared divergence in the vertex correction cancels against a similar bremsstrahlung divergence.
In the rubber plate model, bremsstrahlung is born when the far electric field of the electron lags behind and pumps energy out from the vibration of the near field.
In the rubber plate model, vertex correction is the result of the far field lagging behind, and reducing the effective mass of the electron.
These processes may be related, which might explain why their infrared divergences cancel each other.
Bremsstrahlung is banned in an elastic collision. How to ban it? One may introduce an imagined wave which makes the far field to stand still. That might be equivalent to reducing the effective mass of the electron by cutting off the far field.
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