Let us consider waves in water. If we disturb the surface of water and put energy into it, we usually create waves. The energy of a wave moves and spreads out rapidly. Waves have the spin 0. They do not carry spin angular momentum.
A wave consists of hills and valleys in the surface of water.
The vortex model of the electron
If we disturb water in a special way, then we can create a vortex. A vortex is an almost persistent valley in the surface of water, and it may stay at the same place, in contrast to waves.
Now we see an immediate analogy to electromagnetic waves and to the electron. The electron is a persistent source of the electric field, while the field varies rapidly in an electromagnetic wave.
Can a photon have the spin 0? In a classical electromagnetic wave the answer is definitely yes: an oscillating dipole generates a wave which makes a freely floating test charge to move back and forth.
An electron can only have the spin 1/2 or -1/2. The spin cannot be 0. This is like for a vortex: there has to be angular momentum for the vortex to exist.
The analogy might be this:
1. the electric potential corresponds to the height of the surface of water;
2. the electric charge is the water itself;
3. the magnetic field corresponds to the movement of water, just like it in electromagnetism corresponds to the flow of charge.
This analogy explains why the electron must possess a magnetic field: a vortex must contain a rotating motion of water to exist. It also explains why the spin is always ≠ 0.
If we imagine that the electron moves at the speed of light along the zitterbewegung loop, then the magnetic field at the center is ~ 10⁷ tesla.
Vortices in water do not have an analogy for the positron, though. There exists no persistent hill in the surface of water.
Valleys and hills in a water wave can exist because there is inertia in the mass of water. A vortex is, in a vague sense, a wave which formed a loop: the movement and the inertia of water makes the valley persistent.
According to the analogy, there cannot be a truly "static" charge. A static valley in water would immediately get filled. There has to be dynamic motion of charge to maintain the surface of water uneven.
Our analogy works well for a single charge, but for two charges not so well. Suppose that we have two electrons with opposite spins, how are we supposed to maintain their combined valley in water?
Rotation is the standard way to maintain an excited state in classical physics
The photon is a quantum of a classical electromagnetic wave. The electron may be a quantum of a classical Dirac wave.
How do we create a persistent excited state in classical physics? Especially one which can stay at the same place and does not need to move rapidly.
Let us consider a harmonic oscillator. As it oscillates, its energy swings between kinetic and potential energy.
^ rotation
|
● \/\/\/\/\/\/\/\/\ ● mass
fixed spring
point
We may also let a harmonic oscillator to rotate around a fixed point. In a round orbit, the kinetic and the potential energy stay constant.
According to the virial theorem, the ratio of kinetic energy and the potential energy is often 1 : 1 when averaged over a long time.
Rotation is a very common way to maintain an excited state in classical physics. An example is the orbit of Earth around the Sun.
All this suggests that rotation might be the way to make the electron a persistent particle.
The gyromagnetic ratio 2
Suppose that the electron moves at the speed of light around the zitterbewegung loop which is normal to the z axis. We can explain the magnetic moment of the electron with this model.
But the z-spin of the electron in this naive model would be 1 ħ.
The virial theorem may explain why the z-spin is only 1/2 ħ. If half of the mass energy is potential energy which does not take part in the spinning, then the angular momentum is just 1/2 ħ.
The naive spin 1 ħ model is not a good model in classical mechanics. If the entire mass-energy of the electron is orbiting, then what keeps it in the orbit? There has to be a force field to do that. What is the energy of that force field?
These classical considerations suggest that the gyromagnetic ratio of the electron should be larger than 1. The virial theorem gives us the guess 2.
Earlier vortex models of the atom
In the 19th century Hermann Helmholtz and Lord Kelvin tried to explain the nature of atoms as vortices in a hypothetical medium.
"Helmholtz also showed that vortices exert forces on one another, and those forces take a form analogous to the magnetic forces between electrical wires."
These scientists realized the same thing as we did: a vortex is a way to build persistent, localized excited states.
In a sense, the hydrogen atom really is a vortex: the electron orbits the proton.
The semiconductor model of the electron
Suppose that an electron jumps from the valence band of a semiconductor to the conducting band. We then have an excess density of electrons in a certain zone while the hole and the lack of electrons is in another zone.
This would be a fine model for the creation of an electron-positron pair, but in this model no rotation is needed. The particles would have the spin 0.
Our vortex model has a hard time explaining what are positron vortices. Is space filled with two fluids: the electron fluid and the positron fluid? If we create a vortex in the electron fluid, do we always create another vortex to the positron fluid? If yes, how would this happen?
\ / vortex
\ /
+ +
| |
| | ----------------- axis
| |
- -
dipole 1 dipole 2
/ \
/ \ vortex
We can probably create an electron-positron pair from two photons with opposite spins. It is like having two electric dipoles close to each other and rotating to opposite directions. Can they somehow create vortices?
Assume that the dipoles above rotate around an axis which is horizontal in the diagram. They rotate to the opposite directions.
They might create two vortices as drawn above. If we look from up, the vortices rotate to opposite directions.
Now we have a model for the production of an electron-positron pair. The dipoles represent two photons which are circularly polarized with opposite spins. The vortices are the electron and the positron, with opposite spins.
Pair production in a Feynman diagram
How do we calculate the pair production cross section (= probability) from a Feynman diagram?
photon
----------- ------------- e+
|
| virtual
| electron
|
----------- ------------- e-
photon
We assume that the incoming electromagnetic waves disturb the Dirac field according to the QED lagrangian. This disturbance is happening all the time. If we are able to find a history where momentum and energy are conserved, we can calculate its cross section from coupling constants and propagators for virtual particles (= internal lines).
The calculation is perturbative - it is based on the assumption that only very few photons collide. The calculation treats the electron as a wave where the "structure" of the electron is simply the four-component wave function.
It is not easy to visualize how exactly the spin operator in the Dirac formalism processes the four-component wave and outputs its spin.
It is surprising that a perturbative calculation gives probabilities which are very close to measured ones.
How could we visualize the birth of an electron and a positron as two photons collide? We need to think about that.
The production of a photon from a rotating dipole can be visualized quite easily, at least with a computer calculation and computer graphics. The process is not perturbative. Can we do the same for a production of a pair?
Question. Can we find a vortex model which:
1. gives a non-perturbative description of the birth of a pair, and
2. agrees with the perturbative calculation of a Feynman diagram?
A particle vortex model
photon photon
--------- ---------
\ /
e- -----------------------------
In Feynman diagrams, Compton scattering is closely related to pair production. In a Feynman diagram, one is allowed to bend an outgoing line so that it becomes an incoming line and vice versa. One must switch e- and e+ if the line is an electron.
In our vortex model it is hard to understand how a photon, that makes an electron to oscillate, could be related to pair production.
+
|
| ----------- rotation axis
|
-
-----------> velocity
A circularly polarized photon is like an electric dipole rotating and moving fast.
Since the electron and the positron possess a spin, one way to describe them is to make two charges of the same sign to orbit each other.
- ------------ - electron
+ ------------ + positron
|
| rotation axis
Pair production can then be described with this diagram:
photon
+ - ---------------- --------- ++ positron
\ /
/ \
+ - ---------------- --------- - - electron
photon
The sign symbols denote zones of charge density: a photon has both + and - zones which orbit each other. That is, a photon is like a rotating dipole.
An electron contains two (or more) negative charge zones that orbit each other. They cannot be bound by the electric attraction. In the water vortex model they are bound by the pressure which prevents the vortex from coming apart.
One can then describe pair production by two electric dipoles colliding. The ends with the same charge become bound and start to orbit each other.
rotation axis
|
|
<----> separation d
+ +
| |
| | -------- rotation axis
| |
- - D = dipole length
|
|
rotation axis
In the diagram above, the charges + and + break apart from the photons (rotating dipoles) and form a positron.
The negative charges form an electron. If the length of the dipole
D = 2 d,
where d is the separation where the dipoles break, then the spin of the electron is 1/2 ħ the spin of the positron is -1/2 ħ, assuming that the spins of the photons were 1 ħ and -1 ħ.
The end result is:
+ ------- + positron
- ------- - electron
|
| rotation axis
The diagrams resemble the vortex diagram in an earlier section.
In this blog we have earlier speculated that the photon consists of a "virtual" electron orbiting a "virtual" positron. The diagram above clarifies this idea: in a photon we have charge zones + and - orbiting each other. These charge zones are not electrons or positrons, though. The electron is two negative charge zones orbiting each other and a positron consists of two positive charge zones.
We can explain the gyromagnetic ratio 2 in the same way as we did in a previous section. A half of the total mass-energy of the electron is potential energy of the charge zone pair. Only the other half rotates.
Question. Can we explain Compton scattering with the new particle vortex model? Why should the probability amplitude be the same as in pair production?
Further lines of study
Let us continue this study in the next blog post.
How can the spin of the electron be 1/2?
A possible solution:
The electron consists of two or more negative charge zones. The potential, where the negative charge zones orbit, is not -1 / r. If the potential differs from -1 / r, then the orbit will precess. If the orbit precesses, the system may return to its original state after two rotations!
That would be a simple way to construct a system whose spin is 1/2.
Feynman diagrams say that Compton scattering is related to pair production. How is that possible?
In Compton scattering, if the photon is very energetic, say close to 1 MeV, the collision is violent. The charge zones of the photon and the electron, for a short time, form a "plasma" where charge zones move in a random and violent way.
The plasma will soon decay into particles. In the case of Compton scattering, it is the electron and a photon again.
The same plasma model may explain pair production. Now we see that Compton scattering can, indeed, be related to pair production.
Conclusions
The vortex model takes the spin 1/2 of the electron seriously. The spin is not something abstract, but there is real mass-energy rotating.
In the usual -1 / r potential, a system returns to its original state in one rotation. The orbit does not precess. The spin is 1.
The spin 1/2 suggests that the hypothetical potential which keeps an electron together is not a -1 / r potential.
If a small photon hits an electron in Compton scattering, it only gently shakes the electron. However, a large 1 MeV photon collides violently and produces a "plasma" which decays. The plasma model may explain why pair production is related to Compton scattering.
Richard Feynman's parton model may be similar to our plasma model. We will check if it is the same idea.
