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Andrii Neronov (2017)
We were not able to find from the Internet figures for the total cross section for pair production, but let us guess it is of the same order of magnitude as for 5 MeV gamma rays, if the electron is in the MeV range. That is, around 0.1 barn = 10^-29 m^2.
That suggests that the electron has to come within 1.5 * 10^-15 m from the center of the proton, to produce pair(s).
We could satisfy energy and momentum conservation even if the electron would pass the proton at a very large distance, say 2 * 10^-12 m, or the Compton wavelength of the electron. Any output from a collision satisfies conservation principles - there is no need for the particles to be born close to each other.
Tunneling, with 1 MeV of energy borrowed, is easy through a barrier of 0.1 Compton wavelengths, or 2 * 10^-11 m. The tunneling barrier cannot explain why the electron and the proton must come very close.
Let us imagine that the process creates a positron close to the electron, and an electron close to the proton. Both particles have to get their energy from the kinetic energy of the electron. To give up that kinetic energy, the electron must lose a lot of momentum. To lose that momentum, it must come very close to the proton. The Compton wavelength is way too far away. Here we have a semiclassical explanation why the cross section is ~ 0.1 barn.
The process is described using a particle model of the electron, and almost classical physics.
What are virtual electrons?
To analyze vacuum polarization we should understand what exactly are the virtual electrons in various processes. There is a virtual electron line in the Feynman diagram for pair annihilation or the converse process. There is a virtual electron in Thomson/Compton scattering.
The virtual electron in Thomson/Compton scattering is a "very real" electron that moves in interaction with the photon field(s). There is nothing unreal or mysterious in that electron.
The virtual electron in pair production
The virtual electron in pair annihilation, or the converse process, is mysterious. When we analyzed pair production from colliding photons using a the drum skin wave model, we suggested that the virtual electron is some kind of a malformed wave which transfers momentum in the Dirac field.
But later we used a particle model, and tunneling to potential wells, to explain pair production. How can we reconcile a particle model with a malformed wave?
As we have noted, the wave equation source, which creates an electron or a positron, is quite small, ~ 10^-15 m (in the MeV range). The source is as close as we get to a pointlike "particle" if we work in a wave model.
A source so small will produce also electrons in the GeV range, unless we use a cutoff, or sum over many different source locations, to eliminate waves which have too high a 4-momentum.
A possible model:
1. The virtual electron in pair production is just a symbol for the (complex) process which happens, in a particle model, in a volume of a size 10^-15 m.
This is analogous to the fact that the virtual photon which transfers momentum in Coulomb scattering is just a symbol for Coulomb interaction.
2. We external observers cannot discern details which are much smaller than the Compton wavelength 2 * 10^-12 m. For us, the electron and the positron just pop up in a volume ~ 2 * 10^-12 m.
An external observer may interpret the 2 * 10^-14 m scale process in the Feynman way: a positron coming from the future was scattered by a photon into a virtual electron, and that virtual electron was scattered by a second photon to a real electron.
This model reconciles the almost pointlike character of the photons and the pair, with the larger scale view of waves scattering from each other.
Collapse from a wave to a particle: the teleportation model
The "collapse" of a wave to a particle can be explained by the teleportation model which we introduced in an earlier posting. The original particle system which produced the wave is magically teleported close to the system which observes the particle. The original system is (in the MeV range) ~ 10^-15 m in size. This explains why the particle appears quite pointlike.
The Wheeler-Feynman absorber theory had some similarity to the above idea. In the absorber theory, a mysterious mechanism simultaneously decides the emitter and the absorber of a photon. It is like the teleportation process.
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