That suggests that the "bare mass" of the electron is minus infinite, and it only becomes 511 keV once we add the infinite mass-energy of the field.
Having an object with negative mass-energy is problematic in classical physics. We would like all objects to have mass-energy which is > 0.
Furthermore, there is a "naturalness problem": if the bare mass is minus infinite, why does the field happen to raise the observed mass to a tiny value 511 keV? Why not 10¹⁰⁰ times larger? There has to exist incredible fine-tuning.
In a periodic movement, the effective inertial mass can appear negative
(The drawing is by KiraLakira. See https://commons.wikimedia.org/wiki/File:A_mechanical_model_giving_rise_to_the_negative_effective_mass_effect..jpg#mw-jump-to-license about licensing.)
Wikipedia has a drawing of a mechanical spring device which appears to have negative inertial mass if subjected to a periodic force of a certain frequency.
The idea is that a spring overcompensates the centrifugal "force" (the centrifugal "force" is a real force in a rotating, co-moving, coordinate system). We move a body with our hand along a circle, and feel then a centrifugal "force" from the inertia of the body. Suppose then that there is a spring which helps to keep the body in the orbit. We feel that the centrifugal "force" is weaker, as if the body would have less inertia. If the spring is very strong, we have to pull the body outward to keep it in its orbit. The inertial mass of the body appears to be negative.
The static electric field of the electron can be viewed as a spring system around the electron. If the electron does not radiate real photons, then the far field has to be static and helps to keep the electron in a periodic orbit, effectively reducing the inertia of the electron for that movement. In many quantum mechanical processes, no real photons are radiated.
The field only reduces the effective inertia for a periodic movement. It does not do that for a linear movement.
In this blog we have been claiming that the far field of the electron cannot keep up with the electron in sudden movements, and the effective inertial mass of the electron is thus reduced in such movements. We were thinking that the mass-energy of the far field is cut off from the mass of the electron. But another interpretation is that in a periodic movement, the far field of the electron exerts a force on the electron and helps to keep the electron in the orbit, reducing the effective inertia of the electron.
In this new interpretation, the mass-energy of the static electric field is zero. This solves the classical renormalization problem of the infinite energy of the electron electric field. The energy is not infinite - it is zero. The 511 keV mass of the electron is all in the particle, not in the field.
The new interpretation also solves the famous "4/3" problem of the Poynting vector. Richard Feynman has written a lengthy discussion of that problem.
The self-energy of the electron in quantum electrodynamics is zero
virtual photon k
~~~~~~~~
/ \
e- -------------------------------------------
Quantum electrodynamics calculates the "self-energy" of the electron from the Feynman diagram above. We integrate over all 4-momenta k. The integral is divergent. The mass of the electron is renormalized to straighten things up.
This bears a resemblance to the classical renormalization problem of the static field of the electron.
In this blog we have been claiming that the diagram above breaks conservation of the speed of the center of mass, and the diagram is thus prohibited. Imagine that the electron is initially static. The electron emits a photon with spatial momentum k to the left. The electron starts moving to the right. Then the electron absorbs the photon. The end result is that the electron moved to the right. The center of mass moved.
We have also been claiming that destructive interference wipes out the whole integral over all 4-momenta k. Imagine a static electron. There is no reason why it should emit the photon k at a time t rather than at a slightly later time t'. Summing over all the times produces total destructive interference.
Question. Should we claim that photons with pure spatial momentum k do exist? They do not "live in time" and carry no energy.
t
^
| | / crests of the waves of
| | / photons k with
| | / / energy != 0
| | /
| | /
| |
| e-
-----------------------------------------> x
In the diagram we have a static (= free) electron which emits photons. If the photon contains energy, the crests of the wave of the photon are oblique lines. There is no reason why the photon would be emitted at a time t rather than at a slightly later time t'. We conclude that destructive interference wipes out such waves.
However, a photon with pure spatial momentum k in the diagram has vertical crests of the waves. Why there should be destructive interference of such waves? Maybe they do exist.
Anyway, photons carrying energy:
1. seem to break conservation of the speed of the center of mass;
2. are totally wiped out by destructive interference for a free electron.
We have good reasons to claim that the field of a free electron contains no energy. The "self-energy" of the electron is zero. There is no need to renormalize the mass of the electron. Its "bare mass" is the same as the observed mass.
If we allow photons with pure spatial momentum k to exists, the Feynman self-energy integral is still zero because the set of such photons has a zero measure in the 4-momentum space ℝ⁴.
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