In our blog post February 16, 2021 we discussed the Feynman diagram of pair annihilation. It looks like scattering where the virtual electron carries (repulsive) momentum.
momentum -p
e+ ---------------- ~~~~~~~~~~ photon -k
|
| virtual
| electron p - k
|
e- --------------- ~~~~~~~~~~ photon k
momentum p
Suppose that the positron and the electron approach from opposite directions at the same speed. The momenta are p and -p.
The momenta of the outgoing photons are k and -k.
The produced photons cannot carry away all the momentum of the particles. The electron gives the excess momentum p - k to the positron in the form of a virtual electron.
The virtual electron as a carrier of a "force"
The annihilation process looks like a scattering event where there is an approximate repulsive 1 / r² potential between the electron and the positron.
The Fourier transform of a 1 / r potential is
~ 1 / q².
The propagator of the photon is of this form.
The Fourier transform of a 1 / r² potential is of the form
~ 1 / |q|,
where q is the "momentum" of the Fourier component.
The propagator of the electron is very crudely of the form 1 / |q|.
Our blog post on February 16, 2021 shows that the potential in annihilation is actually even steeper than 1 / r², according to the cross sections calculated from Feynman diagrams.
We have been wrong in our attempts to explain annihilation by the electric attraction between the pair
The momentum transfer in the Feynman annihilation diagram does not happen through a photon.
Maybe it is not possible to make a (semi)classical model of annihilation using the electric force?
The sharp hammer model again: Huygens
We have explained the static electric field of an electric point charge with a sharp hammer which keeps hitting a "drum skin" at the charge, and makes a depression to the skin.
The Huygens principle is that a wave is absorbed by each point in space, and the point then acts as a new source for the wave.
In the case of the electron and its electric field, the Huygens principle might be something like a sharp hammer hitting a drum skin and creating the electron wave and its electric field.
The hit would produce a "point impulse" both in the Dirac field of the electron and the electromagnetic field.
Now we come to the Feynman diagram principle that all "paths" which conserve energy and momentum are allowed, and their probability amplitudes must be summed.
The idea is that the when the electron wave arrives at a spacetime point x, the wave is "absorbed" and immediately recreated with an impulse which hits both the Dirac field and the electromagnetic field.
Our sharp hammer becomes more versatile: it hits two fields at once. All combinations of responses from the two fields are allowed.
The virtual electron in the annihilation diagram is one component of the impulse response. The photon flying away is another component.
If we hit a drum skin, only sine wave "on-shell" Fourier components can travel over a long distance. Other, "off-shell" components have a short range of the effect. This may explain why the electron and the positron must come close to each other in order to annihilate.
A particle model of electron-positron electric scattering
<------ e+
|
| electric attraction
|
e- ------->
The particle model is very simple and intuitive. Both particles are treated as point charges with an attractive electric force. The paths are calculated with classical relativistic mechanics. This gives results which, according to literature, are (almost?) exactly the same as when calculated using the simplest Feynman diagram.
A wave model of the scattering
<------ e+
| | | | | overlapping
| | | | | waves
e- ------>
Let us then try to form an intuitive wave model from a Feynman diagram. Let us have an average of one electron and one positron in a cubic meter of space.
We do not know the positions of the particles, and represent them with standard plane wave solutions
u(E, p) * exp(-i / ħ * (E t - p • x)),
where u is the spinor.
The Feynman diagram is
e+ ----------------------------------------
|
| virtual photon
|
e- -----------------------------------------
How can we relate this to the wave diagram above?
If we take seriously the wave interpretation, then the scattering of colliding beams of electrons and positrons is a nonlinear effect. If there were just one beam, then scattering would not happen. In a linear system we would be able to sum the solutions of the two beams to obtain a new solution: there would be no scattering.
Suppose that we try to add a "source" to the wave equation of the electron. Something like
D(ψ) = f(φ, ψ),
where ψ is the electron wave function, φ is the positron wave function, and D(ψ) = 0 is the Dirac wave function of the free electron.
The source term f(φ, ψ) depends on both wave functions.
But if the wave functions φ and ψ are essentially constant in the cubic meter (save a phase factor), how can the source generate a wave which is significantly scattered, to an angle, say, 90 degrees?
To simulate the scattering of the point particles we should have scattered waves where the cross section is
~ 1 / α²
where α > 0 is the deflection angle. How to generate such waves without having a localized disturbance of the field?
That looks hard. In the hydrogen atom model, one particle, the proton is treated as a particle while the electron is treated as a wave.
Our view in this blog has been that particles are the "true" nature, and that any wave phenomena are due to path integrals.
An analysis of the electron-positron scattering Feynman diagram
We showed that a pure wave model of the scattering does not work. What does the Feynman diagram then really calculate?
e+ ----------------------------------------
|
| virtual photon
|
e- -----------------------------------------
Let us assume that the underlying process really is the classical Coulomb scattering of point particles. But we do not know the precise position of the particles. We have to calculate the path integral for very many possible paths which the particles can take. The path integral is a collective phenomenon of all the possible paths. The path integral is the "wave" associated with the process.
What is the photon in the Feynman diagram? It summarizes the interaction in these very many paths.
Why is the photon propagator
~ 1 / p²
the "right" way to calculate the effect of the interaction?
It is the Fourier component of the 1 / r potential, but there is no obvious reason why the component would correctly capture the cross section of classical scattering.
\ \ \ scattered flux
________
| |
| | cubic meter
|_______|
\ \ \ scattered flux
Let us again have that cubic meter where electron and positron beams meet. The scattered fluxes come quite uniformly from the entire volume.
We can model the scattering by assuming that for each centimeter of the path of the electron, a small portion of the electron flux gets scattered to various deflection angles, and a corresponding part of the positron flux gets scattered to the opposite direction. It is like both fluxes would travel in a nonuniform medium.
The force which causes the scattering is the electric force, and the propagator for some reason happens to capture it correctly.
Since we cannot interpret the diagram purely with waves, we conclude that the Feynman diagram really does describe the encounter of two particles.
We may use a wave description for one of the particles, though.
What about using a wave packet description for both of the particles? We can make the packets to pass each other at some short distance. That might work reasonably well if the distance is larger than the wavelength.
Classically, we have two point charges, and their electromagnetic fields are kind of "waves".
The interpretation of the annihilation Feynman diagram
Our analysis of the Coulomb scattering diagram concluded that the photon propagator "for some reason" happens to work in that case.
In the annihilation diagram, the "interaction" is not by the photon propagator, but by the electron propagator. How do we analyze this?
Conclusions
A lot of questions but few answers. The key problem in this blog post is what does the electron propagator model in a Feynman diagram. Is it a wave? Is it a path integral of point-like particles?
We will analyze Thomson scattering in the next blog post. There we are able to connect the electron propagator to a classical process.
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