Let us write yet another post about the renormalization problem of a classical point charge.
Suppose that an electron collides with a positron. If the positron could survive very close to the electron, both would "see" each other having negative mass-energy. That is bad because negative mass-energies lead to strange phenomena.
Frank Rieger gives the cross sections for pair annihilation. For low-energy pairs, it is
π r₀² * 1 / β,
where r₀ is the classical electron radius and the velocity of the particles is β c. For high-energy pairs, the cross section is
π r₀² * 1 / γ * (ln (2 γ) - 1),
where γ is the Lorentz factor.
For a low-energy electron, the field farther than r₀ / 2 has mass-energy 511 keV. Annihilation prevents the particles from coming that close to each other. We ignored here the focusing effect of Coulomb attraction, though. Anyway, annihilation is a way to prevent the particles from seeing each other with negative energy.
What about high-energy pairs? If the pair would come so close to each other that they would almost see each other with negative energy, the particles would scatter to large deflection angles, say, ~ 90 degrees. The particles roughly obey classical Coulomb scattering. If the pair would come within
r₀ / γ
of each other, they would scatter to a large angle. Again, annihilation prevents the particles from seeing each other with negative energy.
The far field of an electron must have inertia of its own
In the previous blog post we speculated that the static field of the electron may have zero mass-energy. It might be so, but the far field must have inertia. Let us analyze the following thought experiment.
We push an electron with our hand for a time t, with some force. The pushing accelerates the electron and creates an electromagnetic wave of energy E. Besides giving the electron some kinetic energy, our hand does work against an additional force F(t) which produces the extra energy E. Let the velocity of our hand be v(t). Our hand does the work
E = ∫ F(t) v(t) dt
and gives an impulse (momentum)
p = ∫ F(t) dt.
A photon of energy E can only carry away a momentum
p' = E / c.
The velocity v(t) is much less than c. Thus, the static far field of the charge must absorb almost all of the momentum p. That is only possible if the far field has inertia.
The total inertial mass of the electron is 511 keV / c². Some of that inertia must reside in the far field.
What happens if opposite charges cannot annihilate each other?
Suppose that an electron collides with an up quark whose charge is +2/3 e. What prevents the quark from seeing the electron with negative mass-energy?
Suppose that the electron is ~ 1 MeV. Then no pair production is possible. Classically, the electron may pass very close to the quark. The electron sends some classical bremsstrahlung as it passes the quark. The amount of classical bremsstrahlung would exceed 1 MeV if the electron would come much closer than r₀ from the quark. We conclude that we cannot model very close encounters with the classical model. Feynman diagrams calculate the amount of bremsstrahlung correctly, as far as we know.
If the electron has high energy, say, 100 MeV, then pair production may obfuscate the behavior in close encounters.
We also need to check if there is some electroweak mechanism which blocks negative energies in this case.
What prevents an electron having negative energy in the hydrogen atom? It is the uncertainty principle which forbids an orbit which is very close to the proton. Uncertainty blocks negative energies.
Conclusions
The far field of the electron must have inertia. That is the only way to explain how a radio transmitter works, and ensure momentum conservation.
In QED, annihilation hides the structure of the very near field from us. It hides a possible negative energy of the electron plus the inner field.
In the standard model, we cannot model a very close encounter of a quark and an electron classically.
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