wave of an
emitted photon
electron wave \ \
--------- \ \
^ time ---------
|
| ____________
\ ____________ positron wave
\ \ ____________
\ \ \
\ \ \
photon wave
1. Let us imagine that there is a positron around. The positron is a solution of the Dirac equation with no electromagnetic field.
2. A (virtual) photon causes a disturbance in the positron field. The disturbance is a source term in the Dirac equation.
3. We try to remedy the solution of the Dirac equation by using Green's functions of the Dirac equation to cancel the source term.
4. Green's functions produce an electron wave. We may interpret that the positron traveling backward in time absorbed the photon and turned into an electron.
5. Next we imagine that there is an electromagnetic wave which corresponds to the electromagnetic wave which would be produced by the electron emitting the photon which it absorbed earlier.
6. The imagined wave disturbs the wave of the electron. The disturbance produces a positron wave which matches the original positron solution. An emitted photon wave is also produced.
The loop is complete! The positron, which we first just imagined, was "produced" by the scattering of the electron backwards in time, and the scattering also produced the emitted photon wave, which we originally only imagined to exist.
It is like trying to find solutions for the perturbed Dirac equation by assembling Lego blocks. We can use a block where an incoming photon produces an electron-positron pair.
If we turn that block around, we have a block where an incoming electron and a positron produce a photon.
As long as we can assemble a diagram which obeys certain rules, we are free to "imagine" the existence of whatever particle.
Note that in the diagram, all the waves really span the entire diagram area, and are overlapped. There is a large spatial uncertainty about the location of each particle.
What if the waves were classical waves?
Classically, we cannot just imagine the existence of any non-zero wave. In the diagram, there would be no positron wave present. The photon wave would proceed undisturbed.
What about the magnitudes of each wave? Let us use classical mechanics. Let us assume that the imagined waves do exist.
The electron flux is typically very small compared to the positron flux. It cannot "produce" the entire positron flux which exists in the diagram.
https://en.wikipedia.org/wiki/Münchhausen_trilemma
https://en.wikipedia.org/wiki/Münchhausen_trilemma
Baron Münchhausen told the story where he pulls himself out of a swamp by his own pigtail.
The Baron Münchhausen type trick of creating an electron-positron loop from (almost) nothing cannot work in classical mechanics if the disturbance is small. The "feedback" of the loop should be strictly equal to one, to allow a Münchhausen type of a process.
We know that pairs are produced in high-energy collisions of electrons. In quantum mechanics, a disturbance seems to have the ability to "concentrate" its effect on a very small spatial area, such that the feedback of a loop becomes strictly 1.
The diverging of the Feynman integral over a loop
The diverging of the Feynman integral indicates that something is wrong with the assumption that quantum mechanics can conjure up Baron Münchhausen type loops without any restriction. Feynman's rules allow the loop to carry any 4-momentum around, without any restriction.
In previous blog posts we developed the particle model of a photon as a rotating electric dipole.
If we assume that all the particles, including photons, obey certain restrictions of classical mechanics, then it is impossible for a loop to carry an arbitrarily large 4-momentum. No diverging of integrals is possible.
But does that restrict Feynman diagrams too much, so that they would no longer agree with empirical data?
But does that restrict Feynman diagrams too much, so that they would no longer agree with empirical data?