Classical vertex correction and classical bremsstrahlung
If we wave an electric charge in our hand, some energy escapes as radio waves. These waves are classical bremsstrahlung.
The field of the charge temporarily stores energy and momentum which our hand inputs to it. Part of that energy, and almost all momentum is absorbed by our hand later. The emission & absorption process is the classical vertex correction.
Classically, the far field of the charge does not have time to react to the movement of our hand. The charge feels like its inertial mass would be slightly reduced. This is the mechanism which allows energy to escape to infinity as radio waves. If the field would be totally rigid, then all energy would be stored as kinetic energy of the charge (and its field), and no energy would be able to escape to infinity. If we wave a rock in our hand, no energy can escape.
The Larmor formula calculates a good approximation of classical bremsstrahlung.
The significance of the inner field of the charge in the classical setting - classical regularization and renormalization
Classically, the static electric field of a point charge has infinite mass-energy. We could try to remedy this absurd value by letting the point charge have infinite negative mass-energy, but we do not know if this classical "renormalization" procedure yields a sensible physical model.
In the Larmor formula, and presumably in the classical vertex correction, we can assume that the electron is a sphere whose radius is, say, the classical electron radius, and the charge is evenly distributed on the surface of the sphere. The electric field close to the sphere is extremely rigid, so that in classical settings we can assume that almost all mass-energy is contained in the electron itself.
The finite sphere model regularizes the infinite energy of the field of a pointlike electron by setting a cutoff at the classical radius. The classical renormalization is simply that we assume that the combined inertial mass of the sphere and its field is equal to the measured mass of the electron, i.e., 511 keV.
How does the inner field of the electron behave classically?
What is the inner field of the electron, closer than the classical radius, like in classical physics?
We do not know. Studying the inner field would require extremely large accelerations in particle collisions. Those collisions are affected by quantum mechanics. It may be that they have no classical analogue.
Do we have measurements from particle colliders, such that they would reveal the structure of the inner field?
We have found no such experiments. In electron-positron collisions, annihilation prevents us from obtaining data of the inner field.
In electron-electron and other collisions where annihilation does not occur, the inner field is hidden by bremsstrahlung and other processes which happen in an almost head-on collision.
Feynman diagrams may calculate these processes right, but we are not sure if there are accurate enough measurements to make sure that, e.g., Coulomb scattering and bremsstrahlung is accurately described by Feynman diagrams.
The first-order Feynman diagram is essentially the classical Coulomb scattering where we assume that there is no static field, and the electric force is a direct force between point charges.
But the cross section for large bremsstrahlung quanta seems to be so large that it hides the behavior in close encounters.
We need to dig deeper into experimental data. Maybe there is some experiment which can reveal the relation of elastic Coulomb scattering versus bremsstrahlung.
Why do Feynman diagrams calculate bremsstrahlung and the vertex correction correctly?
Our sharp hammer model of the static field of the electron explains this qualitatively to some extent. Let us imagine that the hammer hits the drum skin at finite time intervals Δt. If the electron scatters off another charge within the time
~ Δt,
then part of the wave generated by a single hammer hit escapes as bremsstrahlung, and the rest is absorbed back by the electron. A Feynman diagram describes this process.
In quantum mechanics, energy escapes in packets of size
E = h f,
where h is the Planck constant and f is the frequency of the wave. The Feynman diagram and its associated integral formula takes into account this quantization of bremsstrahlung. Classically, the emitted wave would have very sharp features. Quantum mechanics wipes off sharp features, and quantizes the emitted energy into sizable packets.
In the vertex correction the Feynman integral diverges for high momenta of the emitted virtual photon. Let us compare this to the classical vertex correction. The Feynman formula greatly exaggerates the effect of high frequencies. In the classical process, high frequencies in the sharp hammer hit are quickly absorbed back by the electron. High frequencies do not have time to influence the electron path appreciably. Another way to put this is that the inner field of the electron is very rigid. We can assume that the inner field is rigidly fixed to the point particle electron.
Regularization and renormalization are ad hoc ways to remove the divergence of the Feynman integral. They cut off the high frequencies where the Feynman formula differs from the classical treatment. Therefore, it is not terribly surprising that these ad hoc procedures work. The mystery is: why do they work so well and produce correct estimates?
It is not clear to us if infrared divergences (if any) are a problem for Feynman formulas. In the classical treatment, the Fourier decomposition of any finite feature contains components of arbitrarily low frequency. They should not be a problem.
The real mystery: the structure of the inner field of the electron
We have not introduced a satisfactory classical model of the inner field. So far, we have just assumed that the field closer than the classical electron radius is rigidly fixed to the point particle.
It is not clear to us if one can deduce the structure from particle collider data, since the inner field is hidden behind many different reactions in a high-energy head-on particle collision.
We are not sure if Feynman diagrams and formulas describe these inner field processes correctly.
In upcoming blog posts we will try to analyze the problems in various Feynman loop diagrams, for example, electron self-energy and vacuum polarization.
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