The electron self-energy Feynman diagram

In our December 19, 2020 blog post we remarked that the Feynman diagram for the electron self-energy seems to break conservation of the speed of the center of mass. If the electron would send a virtual photon to itself, it would change the phase of the electron wave function.

A phase change in the wave function of the Schrödinger equation corresponds to a temporary slowdown or speedup in the movement of the particle. That is, the speed of the center of mass is not constant. We proposed that such phase changes are forbidden for a free particle.

Classical self-energy

Let us assume that the classical electron is static. In the sharp hammer model, it hits a drum skin with a hammer and later absorbs the produced wave. The hammer hit produces a symmetric wave. The electron will not move anywhere. There is no conceptual problem in the classical process.

The Dirac equation versus the self-energy Feynman diagram

                            photon E, q

                               ~~~~~

                             /                \ 

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

         p

In the Feynman diagram, the electron emits a photon (real or virtual) and reabsorbs it.

We cannot measure the photon. There is no wave function collapse. To determine the behavior of the system, we have to sum the probability amplitudes of all possible paths it may take.

Let us prepare a free electron as a wave packet. Conventionally, it is assumed that the packet will spread obeying the free Dirac equation. The Dirac equation describes a relativistic particle.

The Feynman diagram depicts the electron as emitting a photon and then absorbing it back. What does the sum of all possible paths look like? The Feynman integral diverges, which indicates that the sum is not well-defined.

Another question is if the process of emission and reabsorption affects the behavior of the electron wave packet. Does the process make the wave packet to spread faster or slower than what the Dirac equation says?

The success of the Dirac and Schrödinger equations in calculating the hydrogen atom shows that they describe the electron wave function very accurately.

If a wave packet of the electron would spread faster than what the Dirac and Schrödinger equations say, then the uncertainty relation for the electron position and momentum would differ from the conventional one which we know.

The Dirac equation is derived from the principles:

1. the equation has to be Lorentz covariant;

2. the number of particles has to be positive or zero and must be conserved.

It looks natural that whatever the free electron is, it should obey principles 1 and 2.

The Feynman diagram does not make much sense for a free electron which moves, say, 1 meter. How many photons per millimeter is the electron supposed to emit and absorb?

In conventional quantum field theory, the Feynman integral of the self-energy diagram is first regularized and after that, the mass of the electron is renormalized, so that the propagator with the renormalized mass looks like the propagator for the Dirac equation. What is the meaning of this procedure? Why we do not simply define that the electron has the measured mass and obeys the Dirac equation?

The spin and the anomalous magnetic moment

In this blog we believe that the spin and the magnetic moment of the electron are a result of genuine circular physical movement. We have called this movement zitterbewegung because we believe that the zitterbewegung which Erwin Schrödinger in 1930 found in solutions of the Dirac equation somehow reflects this circular movement.

Our March 2, 2021 blog post was able to explain the anomalous magnetic moment of the electron by assuming that the inertia of the far field of the electron does not have time to take part in the circular movement: the electron has a reduced mass.

Thus, the static electric field has a measurable effect on zitterbewegung, even if it would have no effect on linear movement of a free electron.

Are there degrees of freedom in the static field of the electron?

In quantum mechanics, if we attach two particles together through some force, there will be zero-point oscillation even in the lowest energy state. The electron is, in some sense, attached to its static electric field. Are there zero-point oscillations in the field? Could zitterbewegung be zero-point oscillation?

No. The electric field of the electron is 137 times too weak to keep it on a light-speed circular orbit whose length is the Compton wavelength. Moreover, zitterbewegung is a property of the Dirac equation, and the equation does not assume any electric field for the particle.

The static electric field of the electron seems to be mostly a classical thing.

Conclusions

It is not clear to us what the self-energy Feynman diagram for a free electron is supposed to describe. We believe that in this context it is best to treat the static field of the electron as a classical object. For a free electron, the field moves with the electron.

In the classical model the regularization means that we cut off the energy density of the static field somewhere around the classical radius of the electron. The renormalization in the classical model is that we set the system mass at the observed mass of the electron.