UPDATE August 29, 2021: In our August 27, 2021 post we found a classical analogue for vacuum polarization. If we have a solid, for which electric polarization is superlinear on the field strength, then that material "conducts" strong electric fields better than weak fields.
----
In our blog we have claimed that the vertex function (correction) is mostly a classical phenomenon.
The vertex function
virtual photon
~~~~~~~~
/ \
e- -----------------------------------------
|
| virtual
| photon
Z+ -----------------------------------------
The static electric field of the electron lags behind in sudden movements. Some of the inertia in the electric field is temporarily shaken off the electron. The effective mass of the electron appears then smaller than 511 keV.
The reduced mass of the accelerating electron explains how a radio transmitter can function. Some of the work to accelerate the electron goes to bending the electric field lines rather than giving kinetic energy to the electron. The deformation energy can escape as vibration: electromagnetic waves or bremsstrahlung.
If the field of the electron would be absolutely rigid, then its mass would always appear as 511 keV and no electromagnetic waves could escape. There would be no bremsstrahlung. There would be no vertex correction either.
Electron self-energy
The Feynman diagram for electron self-energy is like for the vertex function, but this time there is no external disturbance caused by the nucleus.
virtual photon
~~~~~~~~
/ \
e- ----------------------------------------
The process does not appear at all for a classical electron. If there is no external disturbance, the static electric field flies along the electron and nothing happens.
Does self-energy have any classical analogue? If we interpret that the static electric field of the electron consists of virtual photons, then we could say that the field of the electron is the self-energy.
The inner field of the electron at distances
< 1.4 * 10⁻¹⁵ m = r₀ / 2,
where r₀ is the classical radius of the electron, poses a problem in classical physics. Why does the inner field appear to be massless even though its energy density
1/2 ε₀ E²
integrated down to the radius zero yields an infinite result? Above, ε₀ is vacuum permittivity and E is the strength of the electric field.
We have not found a solution for this problem of classical physics. One can claim that it constitutes a classical regularization and renormalization problem.
Can quantum mechanics solve the classical problem? Imagine that the Planck constant h is much smaller (but it cannot be > 137 times smaller since then α > 1 and many QED formulas diverge) than the current value. Then the electron and its inner field will be mostly classical objects. It looks like quantum mechanics will not help us.
Another question is if we have to "correct" Feynman diagrams somehow for the self-energy. Classically, if the electron is not under an external disturbance, its electric field follows it without any effect on the electron. Thus, there is no correction needed.
In the Feynman diagram above, there is a problem in conservation of the speed of the center of mass. Suppose that the electron is initially static. It then emits a virtual photon carrying momentum k. The electron starts to move. The electron subsequently absorbs the photon. The electron moved from its initial position even though no external force was present.
In our "sharp hammer" model of the electron electric field this problem does not exist because the hammer sends a shock wave symmetrically to all directions. The sharp hammer model is classical. In quantum mechanics we are used to treating individual quanta. An individual virtual photon would breach the conservation rule.
Vacuum polarization
e- ----------------------------------
| virtual photon
|
/ \
e- | | e+ virtual pair
\ /
|
| virtual photon
Z+ ---------------------------------
We have not found a classical analogue for vacuum polarization and, consequently, suspect that the phenomenon does not exist at all in quantum electrodynamics.
In the Feynman diagram, a virtual photon is "reflected" from a virtual pair. The phase of the reflected photon changes 180 degrees and the photon has the opposite effect to a photon which would move unhindered.
The process is suspicious. A particle is reflected from itself, or from "nothing". By nothing we mean that the virtual pair without input from the photon would have zero energy and zero momentum.
In classical physics, such a Baron Munchausen trick cannot happen. If an object has zero energy and zero momentum, it cannot affect anything. And a particle cannot be reflected by itself.
In the Feynman diagram above, the virtual photons contain no energy, just spatial momentum. It is an elastic process - the nucleus does not give any energy to the electron. Could it be that the virtual pair somehow temporarily loosens the attraction between the nucleus and the electron, but the whole process eventually ends up being elastic?
Classical electric polarization in a solid
In a solid polarizable material, classical electric polarization does exist. Let us analyze the basic mechanics.
Let the nucleus Z+ be immersed in the material, as well as the electron at some distance away.
Polarization of the material at the distance of the electron puts some negative charge between the nucleus and the electron.
polarization polarization
+ - + -
<----- f -f ----->
e- ● ● Z+
F -------> <------ -F
There is an attractive force F which pulls the electron toward the nucleus. Some of that force is offset by a polarization force f on the electron.
The electron, in turn, exerts a force -f on the polarized material close to it. Where does the momentum from this force go? It is balanced by an opposite force which the nucleus exerts on polarized material close to the nucleus.
There has to be pressure, or rigidity, in the material for the balancing of the forces to occur.
If there is no pressure, then the polarized material close to the electron will obtain momentum.
If we replace the solid with a vacuum, then either there has to be rigidity in the vacuum, or the part of the vacuum close to the electron will obtain momentum. Both of these hypotheses sound strange. This suggests that vacuum polarization does not exist.
Vacuum polarization as failed pair production
If a forming pair receives less than 1.022 MeV of energy, it will annihilate. Could this process be vacuum polarization?
Classically, the pair has to receive at least some energy to start the process of separation. But in the vacuum polarization diagram above, no energy is contained in the virtual photons.
It might be that the electron reabsorbs the energy which it initially gave to the forming pair. What would then be the net effect? The electron eventually does not lose any energy, and receives some momentum. It is like a rubber band between the nucleus and the electron, which almost - but not quite - breaks. The net effect would be that the Coulomb force appears slightly weaker if the electron passes the nucleus very close.
We need to think about this. The classical process is quite complicated. Can we somehow map it to a quantum process?
Our analysis thus far suggests that in Feynman diagrams, loops which only have two lines out do not make sense in classical physics. There has to be an external disturbance to make something to happen.
The vertex function has the virtual photon line to the nucleus, and is very sensible in classical physics.
Electron self-energy is a loop with no external influence and does not make much sense classically.
The vacuum polarization loop occurs in the middle of a virtual photo line. We have failed to make sense of it classically.
e- ------------------------------------------------------
| / virtual
| e+ / photon
|----------- /
| \___/
| /
|-----------
| e-
|
Z+ ------------------------------------------------------
Above we have a Feynman diagram of failed pair production. The electron and the nucleus kick components of a zero-energy pair. The pair fails to make it to the outside world, annihilates, and the electron absorbs the result of the annihilation.
In the rubber band analogy, the band almost breaks, but eventually the electron absorbs whatever energy and momentum there were in the band.
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