We know that the photon propagator, for an unknown reason, models Coulomb scattering.
The electron attached to its electric field: the "rubber string" model
Classically, the electron possesses a static electric field. We may imagine that the lines of force are made of rubber.
If we shake the electron, waves will propagate along the rubber strings. The waves are electromagnetic wave.
We may interpret that the rubber mesh controls the movement of the pointlike particle electron. The electron is a kind of an oscillating mass attached to the rubber mesh.
Classical Thomson scattering
electric
lines of force
|
| | | --- ● ---
|
laser beam electron e-
The laser beam shakes the electron. New electromagnetic waves propagate in the lines of force of the electron.
We may imagine that the laser beam gives small impulses to the electron at very short time intervals. The response of the electron and the rubber mesh is an impulse response.
Recall that a propagator is a Fourier component of the impulse response of a wave equation. For example, if we hit a drum skin with a sharp hammer, the Fourier decomposition of the wave is the set of propagators.
The impulses to the electron have a cycle determined by the frequency of the laser. Constructive interference strengthens the output at this frequency. Destructive interference wipes out other frequencies.
The electron will act as a radio transmitter and send an electromagnetic wave to many directions. It scatters the incoming laser light.
The propagator in this classical treatment tells us how effective the electron is in outputting energy to the scattered "channel", or scattered wave.
We can, in principle, calculate with a computer how the electromagnetic field behaves, and what is the value of the propagator.
The classical Feynman diagram looks like this:
laser scattered wave
~~~~~~ ~~~~~~~~~~~
\ /
e- ---------------------------------------------------
virtual
electron
It is essentially the same as the quantum electrodynamics (QED) Feynman diagram.
The "virtual electron" is the propagator. We may imagine that it represents the electron in the mesh after an impulse hit the electron.
QED Thomson scattering
As we wrote above, the Feynman diagram is essentially the same as in the classical process. Numerical results from Feynman formulae agree with the classical treatment.
Note the following thing: the propagator in the QED diagram is from the impulse response of the Dirac equation. We were able to connect the Dirac equation to a classical process.
The process in Thomson scattering is non-relativistic. The Dirac equation in that case is equivalent to the Pauli equation, or the Schrödinger equation. The propagator for the Schrödinger equation seems to be complicated.
What is a "virtual" electron and how does an electron "absorb" a photon
In the classical interpretation, a virtual electron means the system electron & its electric field where the system has been disturbed by an impulse.
The absorption of a photon means that the system goes to a disturbed (excited) state.
Classically, the electron is a point particle. It cannot have excited states on its own. An excited state has to be the electron in an interaction with something else, in this case its own electric field.
When an excited electron emits a photon, that means that the oscillation of its electric field moves farther from the electron and starts a life of its own.
A propagator does not make much sense for a point particle. But a propagator for the system the electron & its field makes a lot of sense.
Question. How can we extend the classical propagator to an "electron field"? That is, we would not have a point particle but some kind of a field. This might show the connection between the Dirac equation and the classical model.
It is not clear if we can define an electron field in a reasonable way. If the field simply describes the position and the phase (in a path integral) of a single electron, then there is no obvious interaction between different parts of the field.
If the electron is either in the zone A or the zone B of space, there is no interaction between the zones A and B. This is very different from a drum skin where A and B always interact. In a drum skin, a wave equation is natural, but it is not natural for mutually exclusive histories.
The Dirac equation is used in Feynman diagrams to calculate the electron propagator. That is, a single electron is interacting with something else. Then it makes sense to consider the system the electron & its field.
The Dirac equation does have a conserved probability current and it does predict the magnetic moment of the electron. How can we explain these if the equation only describes an interacting electron?
Maybe the electron is a wave phenomenon, after all? The spin of an electromagnetic wave probably is a wave phenomenon.
But then we face the problem how to attach the electric field to the electron.
The Dirac equation describes the system the electron & its field?
The electron propagator in the Dirac field describes something which is off-shell, or not in its ground state.
If the QED propagator is able to calculate something similar as the classical propagator, then it is natural to assume that the QED propagator describes the combined system the electron and its field. We cannot remove the electric field from the electron. It makes sense that the Dirac equation describes the entire system.
However, it is not clear how the classical system gives rise to the Dirac equation. How do we explain zitterbewegung and the magnetic moment?
Conclusions
The analogue of a photon propagator is a sharp hammer hitting a drum skin (where the skin is actually the three-dimensional space).
The Fourier decomposition of the associated - 1 / r potential has the familiar formula
~ 1 / p²,
where p is the 4-momentum of the Fourier component.
|
_|_
|__| --> ● e-
hammer
The analogue of the electron propagator might be a hammer hitting the electron. The electric field of the electron moves a little bit. That is like adding a dipole where a positron e+ is put to the old position of the electron and a new electron is put to the new position.
The dipole potential is roughly
~ 1 / r²,
and the Fourier decomposition is roughly
~ 1 / |p|,
where p is the 4-momentum of the component.
The formula 1 / |p| is similar to the Feynman electron propagator if we set E = 0 and m = 0, where E is the energy of the electron and m is its mass.
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