Let us look at the 1934 paper by Hans Bethe and Walter Heitler.
The paper is freely readable at the link.
photon k
_______________
/
p_0 e- ------------------------------
| virtual
| photon q
Z+ ------------------------------
When an electron meets a nucleus, Bethe and Heitler consider the above Feynman diagram (though not called a Feynman diagram yet in 1934). They also consider the alternative diagram where the photon with the momentum k is emitted after the virtual photon q line.
Above, after the electron has emitted the photon k, the electron is in an intermediate (virtual) state until it collides with the nucleus and gives up the surplus momentum it had.
Formula (4) in the paper is somehow derived from the density of final states, probably Fermi's golden rule.
The classical limit of the process is a very heavy particle, with the same charge / mass ratio as the electron, meeting the nucleus. If the particle is very heavy, we can consider it a classical charged particle.
Classically, the Larmor formula gives an estimate of the radiated electromagnetic energy in the bypass.
Our rubber plate model qualitatively explains classical radiation.
A. First the nucleus pulls the electron as the electron approaches. The rubber plate lags behind. The released potential energy V is channeled into:
1. kinetic energy of the electron;
2. the stretching and kinetic energy of the rubber plate.
B. Then the electron recedes. Now the rubber plate is moving ahead of the electron and pulls it away from the nucleus.
Kinetic energy of the electron and the rubber plate is channeled into:
3. the potential energy of the electron;
4. the new stretching of the rubber plate.
The end result is that the rubber plate gained oscillation energy (waves), the electron lost kinetic energy, and the extra momentum was absorbed by the nucleus.
We assume that someone has calculated that the Bethe-Heitler formula agrees with Larmor, if a suitable cutoff is made.
What is the significance of the two Feynman diagrams? Do they correspond to phases A and B above?
The Feynman diagrams are an abstract way of describing the end result. In our classical rubber plate model, momentum first flows to the electron, and later flows out. The Feynman diagram just shows the net effect - no details of the process.
What is the virtual electron in the Feynman diagrams?
What is the meaning of the virtual electron in the Feynman diagram? Classically, it is an electron which is interacting both with the Coulomb field of the nucleus, as well as its own Coulomb field.
Bethe and Heitler write in their paper that the electron is simultaneously interacting with the nucleus and the "radiation field" H.
The interaction of the electron with its own field eventually produces the electromagnetic wave. The electromagnetic wave is a "wrinkle" in the electromagnetic field of the electron.
Could we say that there is no electromagnetic field associated with empty space? Electromagnetic waves are just wrinkles in the fields of individual charges? This hypothesis would free us from the hypothetical infinite energy of oscillations of the vacuum electromagnetic field!
Pair production
Bethe and Heitler write that pair production is handled "similarly" to a photon emission.
They write that it is a photoelectric effect. An electron is raised from a -511 keV state to a +511 keV state. The hole that is left behind has the same positive energy 511 keV. But they say that the positron itself has an energy -511 keV.
What does it mean that the positron has a negative energy but the hole has a positive energy? We cannot separate the positron from the hole. This is obscure.
Bethe and Heitler consider the interaction of the created electron and positron. They write that the "matrix element" (18) which denotes a transition from an electron to a positron under their mutual Coulomb interaction, vanishes unless the sum of their 4-momenta is zero. The zero sum would mean that they annihilate but leave no photons behind.
But what if the combined energy of the pair is < 1.022 MeV? Then the pair will annihilate almost certainly. Maybe Bethe and Heitler assume that the combined energy is above 1.022 MeV?
Is there sense in a pair whose combined 4-momentum is zero?
Let us have a schematic Dirac electron:
ψ ~ exp(-i (E t - p x)).
The discussion above brought up the following problem: if we have an electron wave ψ (energy E) in the Dirac equation, we might interpret it also as a positron wave with a negative energy (observed from the outside) -E.
If we create an electron, and a positron whose 4-momentum is opposite to the electron, then the particles have an identical wave function? What is the sense in this? In an earlier blog post we remarked that if a positron has a negative mass-energy, it will in an electric field move like an electron, and the electric field cannot pull it apart from the electron.
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