Renormalization and regularization of electron self-energy

Our blog post on December 19, 2020 was about this problem. Let us look at it again.

The Feynman diagram for the electron self-energy is shown below.

                           virtual photon E, q

                               ~~~~

                            /               \

           e- -------------------------------------

           momentum p

The electron emits a virtual photon and subsequently absorbs it.

Our sharp hammer model explains what this is about. The electron is a source in the massless Klein-Gordon equation which describes the electric field. It hammers the field at a very high frequency to make a pit into the field. The pit is the static Coulomb field of the electron.

https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/current/qed/renorm-new.pdf

In the link there is a short exposition of renormalization in QED. It is assumed that the electron has a "bare mass". We calculate the propagator of the electron, including the additional effect of the self-energy diagram. The effect turns out to be infinite. We remove the infinity with regularization. The new propagator looks similar to the original electron propagator if we redefine the mass of the electron.

Classical regularization and renormalization of the mass of a point charge

The infinite energy in the field of a point charge is a classical regularization problem. A way to remedy the classical problem is to assume an infinite negative energy for the bare electron. When we add the infinite energy of the field, we get the measured electron mass 511 keV - this is classical renormalization. Classical renormalization is ugly. The combined mass-energy of the electron and the field located at a radius < half of the classical electron radius becomes negative. Negative energies can have surprising consequences.

In the diagram, the electron has constant momentum p. If there is no external interaction, then it does not matter where the mass-energy of the electron is located. Renormalization is only relevant when the electron is under interaction - that is, it is the vertex correction where mass regularization and renormalization might have an effect on phenomena.

How does a negative mass move classically?

Let us investigate further the effect of negative mass-energies in classical physics.

                     spring                     ● charge P

            ● \/\/\/\/\/\/\/\/\/\ ●          

       mass M                  mass -m charge Q

Suppose that we have a smaller negative mass -m attached with a spring to a larger positive mass M. How does the system move?

Let us assume that the negative mass -m is charged. Let us assume that the charge P repels the charge Q.

Since -m is negative, Q starts to move to the wrong direction, toward the charge P. This is because of momentum conservation. Then -m pulls on M. The pulling moves both -m and M closer to P. This does not make sense.

Above we assumed a direct force between P and Q. Maybe we should assume that P and Q only interact through their electric fields?

Another option is to return to the December 19, 2020 hypothesis that a static electric field contains no mass-energy. All the mass-energy of the electron would be in the point particle itself.

Feynman diagrams do not restrict the mass and the charge of a particle. Therefore, they should be able to handle the classical renormalization problem, too. All the problems of the classical case should pop up in the Feynman diagrams, too.

A tentative solution to the classical renormalization problem

If the acceleration of the electron is not extremely high, we can assume that it is not a point particle. We can then assign a positive mass-energy both to the electron and its field. Calculations can be done using classical Maxwell equations.

If the acceleration is huge, then we have to assume that the electron is a point particle. To avoid negative mass-energies, we have to assume that the mass-energy of the static electric field of the electron is zero.

Question. Does it conflict with classical electrodynamics if we set the energy of a static field zero?

Especially, we are interested in how our model of reduced electron mass works in the anomalous magnetic moment of the electron, if we assume the static field to have zero mass-energy.

The fact that the electron in high-energy collisions (e.g. LEP 100 GeV) behaves like a point particle, suggests that the 511 keV mass-energy of the electron really resides in the particle itself, not in the field.