NOTE January 3, 2021: the heavy string B is not like the Dirac field, but just another model of the massless Klein-Gordon equation. A better model for a Dirac field is the massive Klein-Gordon equation, where the term m^2 ψ can be interpreted as springs attached to the string, to make it kind of a harmonic oscillator.
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Feynman diagrams are based on the QED lagrangian, which is an equation between fields. The fields are the electromagnetic field and the Dirac field.
The precise meaning and interpretation of the QED lagrangian is fuzzy at some points, as we have remarked in this blog earlier.
Let us study a classical field equation of two fields A and B, to make the setup concrete.
massive string
=========================== B
| | | | rubber membrane
----------------------------------------------------- A
lightweight, very tense string
Let us have a heavy, moderately tense string, and a lightweight, very tense string. They are coupled with a thin, easily stretchable rubber membrane.
The heavy string B is like the Dirac field, with the non-zero mass of the electron. The lightweight string A is like the electromagnetic field. Waves travel at a very great speed in the lightweight string.
The rubber membrane tries to transmit waves between the strings, which is not easy as the strings behave in quite different ways. The rubber membrane is the interaction.
Let there be a wave in the field A. The interaction causes disturbance in the field B. We estimate the impact on field B by using a Green's function of B.
Each spacetime point acts as a Dirac delta source in the B field equation. The intensity of the source depends on A at that point.
Let a wave in B be of the form
sin(E t - k x),
where we call E the energy and k the momentum.
Intuitively, the Green's function method should produce some kind of an estimate for the disturbance in B. But applying the Green's function at just a single spacetime point of B does not make sense. The Green's function contains arbitrarily high momenta k, which corresponds to an arbitrary small wavelength in B.
We must consider the sum of Green's functions at various points, to get the estimate of the disturbance in B. Destructive interference cancels then high momenta k.
The size of "features" or "detail" in the disturbance in B is roughly the same as in A. A typical interaction does not magically create fine detail in a disturbance.
An exception is turbulence, where fine detail spontaneously appears in a field.
We may further use Green's functions to calculate what effect the disturbance in B has on A. Since B behaves in quite a different way from A, there is little "resonance", and B generally returns the disturbance completely back to A.
We may interpret that a "real particle" in A is converted to a "virtual particle in B", and is returned to A as a real particle.
What this means for vacuum polarization?
In a Feynman diagram, a virtual pair whose components have arbitrarily high momenta k appears. If we try to integrate over all k, there is divergence.
There might well be divergence in our string example if we would not take into account destructive interference in B.
This suggests that one should take into account destructive interference in vacuum polarization. It should cancel high momenta whose wavelength is less than the smallest "feature size" of the electromagnetic field in question.
If there is no such destructive interference in vacuum polarization, then the physics of the phenomenon is quite strange: the physics cannot be modeled with a typical field equation.
Is there need for a high-momentum cutoff?
In our string example, we may imagine that the strings consist of atoms. If we would have problems with high momenta k in our analysis of the strings, maybe we should use the atom hypothesis to cut off high momenta k?
Maybe that would work, but first we have to sort out destructive interference. If we still have divergences, then we might resort to atoms.
Regularization and renormalization in QED is like resorting to the atom hypothesis without first checking what is the effect of destructive interference on, e.g., a vacuum polarization loop.
An analogue of virtual pair production
In out string example, let us classify waves using the circular polarization framework. A wave can rotate clockwise or counter-clockwise, as seen from the left. A sine wave is a sum of these circular polarization states.
If A has a sine wave, it will tend to create transient vertical oscillation in B. It always creates a clockwise and counter-clockwise transient wave in B at the same time. This reminds us the production of an electron and a positron. They are Dirac waves that rotate to opposite directions.
If there is an external electric field present, then the electron and the positron accelerate to opposite directions. Simulating that in a string is hard. A clockwise wave should repel other clockwise waves?
--> <--
A _________/\_______________/\_________
wave wave
What about a collision of waves in A? Could it somehow create a pair in B?
Waves in B tend to move slowly, and a collision creates a transient standing wave in A whose frequency is some f. If there is some nonlinearity in the system A & B & interaction, the (strong) standing wave does create a wave in B of the same frequency f. This works nicely.
However, we cannot model conservation of charge in our string model. Any oscillation in the massive string B quickly leaks into oscillation in A. That would mean that an electron spontaneously decays into photons.
An electron in B, viewed from the left, consists of clockwise rotation of the string. A positron is counter-clockwise rotation. The interaction (i.e., the rubber membrane) should be such that a pure circularly polarized wave cannot leak from B to A. Then we could ensure charge conservation.
Our string model illustrates the following rules:
1. A single massless particle cannot decay into a massive particle, but two colliding massless particles can do that.
2. A massive particle can decay into massless particles.
A massless particle is a wave in the lightweight string A, and a massive particle is a wave in the massive string B.
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