In our September 29, 2021 blog post we claimed that the mass-energy of the electrostatic field of the electron has zero mass-energy if the electron is free.
We also claimed that when the electron is in a periodic motion, the far field of the electron stays static and causes a self-force on the electron. The self-force emulates reduction of the inertial mass of the electron.
That is, if the electron is in a circular motion, the self-force cancels a part of the centrifugal "force". If the electron swings back and forth, the self-force helps to return it to the central position.
In the Sommerfeld hydrogen atom model, an electron on an s orbital does a 180 degree turn very close to the proton. We claim that the far electrostatic field lags behind, and the self-force helps to overcome some of the centrifugal "force" in the sharp turn around the proton.
We have claimed that the vertex function (correction), and the major part of the Lamb shift, as well as the anomalous magnetic moment of the electron are all due to reduction in the inertial mass of the electron in sudden movements. We derived the electron anomalous magnetic moment from this assumption (for zitterbewegung) in our March 2, 2021 blog post. We derived the Lamb shift on March 13, 2021.
The anomalous magnetic moment of the electron: reduction in the inertial mass
In the anomalous magnetic moment, the inertial mass of the electron seems to be reduced by a factor
1 - α / (2π),
where α = 1 / 137 is the fine structure constant.
The reduced mass makes the zitterbewegung loop 1 / 861 larger and increases the magnetic moment accordingly. In our model, the magnetism comes from the zitterbewegung loop which the electron circles at the speed of light. The length of the loop is the Compton wavelength of the electron. If the mass is reduced, the wavelength becomes longer.
The classical energy density of an electric field is
1/2 ε₀ E²,
where ε₀ is the vacuum permittivity and E is the electric field strength. Classical physics is unsure if an electrostatic field really contains that energy. An electromagnetic wave does contain, that we know for sure. Let us temporarily assume that there is mass-energy in an electrostatic field, too.
In our blog post March 2021 we assumed that the electrostatic field farther than
1/2 c t
from the electron does not contribute to the inertial mass of the electron. There t is the cycle time of the periodic movement and c is the speed of light.
We motivated our claim by the position of nodes in a standing wave in a string.
With our assumption we were able to explain why the inertial mass of the electron is reduced, the Compton wavelength becomes longer, the zitterbewegung loop is larger, and the magnetic moment 1 / 861 larger.
Let us calculate what is the self-force on the electron. Can it explain the reduction in the inertial mass?
A very crude calculation of the self-force on an electron
Let the electron circle a loop of a radius r at a speed
v = a c,
where c is the speed of light. Let the cycle time be
t = 2 π r / v.
We assume that the field farther than
s = 1/2 c t = π r / a
is static.
| | |
| | | electric field lines
s
|
| \ \ \ oblique
| \ \ \ electric field lines
| \ \ \
|
| | | | electric field lines
| | | |
|
----------- • e-
r (displacement)
We have to connect the (dynamic) field lines going up from the electron to the vertical static field lines up at the distance s. The oblique field lines become slightly longer because of the displacement r. The density of oblique field lines grows a bit larger, which means a larger electric field |E|.
The total energy of the oblique electric field is larger than in the case where the displacement r would be zero.
It makes sense to keep the field closer than s / 2 from the electron undeformed because there the field follows more closely the movement of the electron. The extra energy is taken by the oblique lines higher than s / 2 from the electron.
The "deflection" angle of oblique field lines is
2 r / s
radians. The separation of field lines is
cos(2 r / s) = 1 - 2 r² / s²
relative to the undeformed case. The energy density of the oblique field is
1 + 4 r² / s²
relative to the undeformed field.
The energy of the field between the distances 1/2 s and s from the electron is
[1/2 r_e / (1/2 s) - 1/2 r_e / s] * m_e c²
= 1/2 r_e / s * m_e c²,
where r_e is the classical electron radius, and m_e is the mass of the electron.
The extra energy in the oblique field is
1/2 r_e / s * m_e c² * 4 r² / s².
Let us calculate the derivative of that relative to r, to find the self-force F:
F = 4 r / s² * m_e c² * r_e / s
= 4 a² / (r π²) * m_e c² * r_e / s.
The centrifugal "force" on the electron in the circling motion is
F' = m_e v² / r
= m_e c² a² / r
The ratio F / F' reveals by which proportion the self-force F reduces the effective inertial mass of the electron in the circling motion:
F / F' = 8 / π² * 1/2 r_e / s.
There, 1/2 r_e / s is the ratio of the electrostatic field mass m_s outside the radius s relative to the mass-energy of the electron, m_e. We have
F / F' = 8 / π² * m_s / m_e.
Since 8 / π² = 0.81 is roughly 1, we have the conclusion: the effective inertial mass of the electron is reduced roughly by the mass m_s, which is the mass of the electrostatic field farther than s.
Note that m_s ~ 1 / s. Maybe we should reduce s by a factor π² / 8, so that the self-force would explain the mass reduction exactly?
We do not need to assume any mass-energy in the electrostatic field of a free electron - we get the same effect from the self-force on the electron
We showed that we can explain the reduction in the electron inertial mass either by
1. claiming that the far field has mass-energy m_s, and that mass does not have time to take part in the circling motion, or
2. claiming that the electrostatic field does not have any mass-energy for a free electron, but the deformed field in the circling motion exerts a self-force on the electron.
The very rough calculation supports our claim that the electrostatic field of a free electron has zero energy. An alternative interpretation is that we have renormalized the electrostatic field to have zero energy. Only deformations of the free electron field contain mass-energy.
A precise calculation of the self-force probably requires a computer simulation. There is an open problem, too. We keep the field static farther than s. How does retardation affect the deformation of the oblique field? Does Nature try to minimize the extra energy?
What about the mass-energy in the deformation of the oblique field? It is small compared to m_s. The total mass-energy in the oblique field is m_s. The angle 2 r / s is small in most cases. Thus, the deformation energy is usually very small.
The border conditions at s make the electric field a standing wave at distances < s. Is this wave somehow quantized? Since the electric field is tightly bound to the electron, the wave is not an on-shell photon. The wave is an off-shell photon, or a virtual photon, which contains little energy compared to its wavelength. For an on-shell photon, the energy-momentum relation is
E² = p²,
where E is the photon energy and p is its spatial momentum. If the photon is off-shell, it can contain less energy. In this blog we have stressed that interacting particles are always off-shell. The photon is in a very tight interaction with the electron.
The self-force is purely a classical phenomenon
Nowhere in our analysis did we refer to quantum mechanics. We calculated the self-force from elementary classical electrodynamics.
Many people believe that the vertex function is a quantum mechanical thing which only manifests itself in quantum mechanical calculations with Feynman diagrams. We claim that the vertex function is mostly a classical thing. However, quantum mechanics does affect the process. For example, emission of bremsstrahlung is constrained by the minimum energy of a quantum of light.
The calculation of the anomalous magnetic moment has at least two quantum mechanical features:
1. the length of the zitterbewegung circle is determined by the Planck constant h;
2. there is no bremsstrahlung because the electron is in its lowest energy state.
The calculation of the Lamb shift involves solving the Schrödinger equation for the hydrogen atom. The electron on the 2s orbital is in an almost stationary state, and bremsstrahlung is negligible. These are quantum aspects of the calculation.
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