The electron has a mass and it rotates since it has a nonzero spin. Zitterbewegung suggests that the electron moves around a circular loop at the speed of light.
Maybe the mass-energy of the electron is mainly in its rotation? In our water vortex model that is, indeed, the case.
This has relevance for models of pair annihilation. Maybe the released energy mainly does not come from the attraction between the electron and the positron.
In an earlier blog post we remarked that the formula of the cross section of annihilation suggests at a force which goes as 1 / r³. As if the electric attraction would not be the force which annihilates the pair.
We should develop a model of pair production / annihilation where the spin rotation is the central feature.
The electron and the positron doing the zitterbewegung to opposite directions
e+ e-
| |
| | ------ axis of the orbits
| |
In the diagram, the electron and the positron do a circular orbit which is normal to the screen. They have the opposite spins => they orbit to opposite directions.
| radiation
( ( ( | ) ) )
|
electric dipole
The system is similar to an electric dipole (not rotating) which radiates to its sides.
The new model differs from the vortex model of our previous blog post. The photons are emitted to the directions of the spin axes.
Let us check the literature. If we control the spins of an annihilating pair, to which direction are the photons emitted?
In the link Ali Moh and Tim Sylvester calculate the direction of emitted photons. The spin directions are not controlled, though.
If we set E = 0 and p = 0 in their formula, the cross section is zero. Apparently, a positronium "atom" is formed.
If E is very small, then the cross section goes as
cos²(θ),
where θ is the angle from the direction of the colliding electron and positron.
This is somewhat similar to the power density of a dipole antenna:
sin²(α),
where α is the angle from the dipole axis.
Feynman diagrams destroy information?
p
e- -------------- --------------- photon p'
|
|
e+ -------------- --------------- photon p''
-p
Suppose that the electron arrives from the left and the positron from the right. They have opposite momenta p and -p.
We calculate the value of the formula for two photons of momenta p' and p''.
Since Feynman diagrams only do perturbative calculation, they are expected to lose information in the process.
The phase of the photons depends on where we measure them. The Feynman formula cannot state the phase information. It loses the phase information.
--------------------------- photon
/ --------------- photon
/ /
e- ----------------------------------------
In the clip (18) we have the Feynman formula for the diagram above. The incoming and outgoing electron lines are the 4-component spinors v and u. The internal electron lines are the propagators.
An internal line electron is not described with a spinor, but with a propagator.
Classical Thomson scattering looks very much the same for an electron and a positron. The Feynman diagram should calculate the exact same probability amplitudes for an electron and a positron.
There is no classical annihilation. One might guess that in a classical annihilation process the phase of the two photons is changed by 180 degrees if we switch the electron and the positron. That is because the switch reverses the direction of the electron - positron dipole.
We are looking for a classical wave model for annihilation. In the ideal case the model is non-perturbative and retains all the information which is fed in. It is a unitary model.
Our "spider" model of pair production
A couple of years ago we tried to build a "spider" model of pair production.
O spider
/ | \
-----------------------------------------------
left string right string
A spider stands on tense string and uses its legs to rotate the left part and the right part to opposite directions. Our zitterbewegung model above bears a resemblance to the spider model, if we look at is as a pair production model. The "spider" is the two photons.
But how can we explain Compton scattering with the spider model?
___ O spider
_____/ \_________/ | \_________
wave --->
We can imagine that a wave in the string progresses through the following "spider mechanism": the spider uses the torque in one leg to work against an incoming wave from the left and uses the energy and the torque which it gains from that to create a new wave which proceeds to to the right.
The process resembles pair production. Did we find the relation between Compton scattering and pair production?
The electron as a rotating structured object and the scattering of a 1 MeV photon: a plasma?
^ zitterbewegung loop
|
----------● e-
r = 4 * 10⁻¹³ m
There may be a fundamental difference between low-frequency photons scattering from an electron and high-energy photons.
Let us imagine that the electron is some kind of a rotating structure whose radius is its Compton wavelength divided by two pi:
r = 2.4 * 10⁻¹² m / (2 π)
= 4 * 10⁻¹³ m.
If the wavelength of the photon is much larger, the electron appears as an essentially point charge to the electromagnetic wave. An electron inside a 500 nanometer laser beam oscillates back and forth quite classically. A low-energy photon does Thomson scattering, which can be calculated with a classical model of a point charge.
However, a photon whose wavelength is less than 1/2 of the electron Compton wavelength (> 1 MeV), can see and distort the internal structure of the electron. When it hits the electron, the electron may enter an excited state.
This high-energy process might have a similarity to pair production.
We suggested some kind of a "plasma" model for pair production in our previous blog post. If the internal structure of the electron is severely distorted by the 1 MeV photon, its state might resemble some kind of a plasma.
The emission of the scattered photon happens from this plasma.
No one has ever observed an electron in a stable or metastable excited state. The only stable state is the ground state. However, when a 1 MeV photon hits the electron, it may enter a very short-lived excited state.
How could we fit the spider model to Compton scattering? If there is no photon, the spider harvests the energy and the torque from the incoming wave and "turns the crank" to create the outgoing wave.
If a photon simultaneously is absorbed, we might consider it as some additional energy and angular momentum which the spider puts to "turning the crank".
What is the photon emission then?
The outgoing photon holds a crank which the electron turns?
Maybe we should drop the spider altogether and imagine that the photons are holding the cranks?
The crank model
The name of this model is not a pun :).
Let us model the electron (Dirac) field with a tense string. A photon can interact with it by "turning a crank".
photon
~~~~~~ __
| crank
---------------------------------------- Dirac field
If a photon arrives, it can donate its energy and momentum to a rotating movement of the string. The crank exerts a torque on the string.
The same process backward is the creation of a new photon. The rotating string exerts a torque on the crank.
Now we realize that the model is similar to a Feynman diagram which describes an absorption or an emission of a photon.
But the Feynman diagram is an abstract description, while we aim at a non-perturbative classical description of the process.
Conclusions
A central idea in this blog post is that the spin of the electron is an important classical feature of the particle. It may be that a half of the mass-energy of the electron is kinetic energy of its spinning motion. The other half might be potential energy in the centripetal force which keeps the orbiting parts of the electron in the orbit.
We will next analyze the crank model. Our goal is to construct a classical, non-perturbative model of pair production/annihilation.
I earlier blog posts we have talked about a "generator" which takes strong laser beams as an input and produces a stream of electron-positron pairs. The generator would be analogous to a rotating electric dipole which produces, in a non-perturbative way, a stream of photons. Cranks might be able to function as a generator.
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