We think that we have solved the mystery of Feynman diagrams and regularization / renormalization.
| hits at a very high frequency
v
__
| |====== hammer
\ /
------- -------- drum skin
\ /
pit
Recall the drum skin & sharp hammer model where a sharp hammer at a very high frequency hits the drum skin at a specific location. Hitting the skin makes a pit in the skin. That pit is analogous to the static electric field of the electron.
e- -------->
constant velocity v
If the electron moves at a constant velocity, the setup stays static. Destructive interference completely removes any dynamic waves made by the hitting hammer.
e- ---->
v
● Z+
But if the electron is scattered by a nucleus, then dynamic waves are created by the hitting hammer. The waves may be absorbed by the electron itself, or they may escape to infinity as bremsstrahlung.
If the waves are absorbed by the electron itself, then the phenomenon is the vertex correction.
In a Feynman diagram, an electron creates a photon from a Dirac delta impulse, as a Green's function. It is a single hit by the hammer. We need to figure out how this is related to our model with high-frequency hammer hits.
High-frequency components of the Green's function are wiped out by destructive interference in the fly-by of the electron. Only "medium-frequency" and low-frequency components remain.
Regularization is a method where we "subtract" the waves created in the constant velocity case (that is, the electron self-energy diagram) from the waves created in the scattering case. It is not a miracle that the method recovers a sensible and correct result in the scattering case. Regularization is a way to implement total destructive interference of high-frequency waves.
The drum skin model is classical. Consequently, the vertex correction is fundamentally a classical thing which is we look at through the microscope of quantum mechanics.
In our rubber plate model of the static electric field of the electron we never specified how the electron is attached to its rubber plate. Now we see that the attachment can be explained by the sharp hammer model. The static electric field is the response of the electromagnetic field to constant hammering by the electron charge.
------ ------ drum skin
\●/ charge presses the skin
pit
Alternatively, we may imagine that the charge is a small weight which makes a pit to the electromagnetic field. That is classically equivalent to the sharp hammer model.
A classical renormalization problem: the energy of the static field of a point charge is infinite
We will write a more detailed analysis of this new sharp hammer model.
A renormalization problem remains: a point charge makes an infinitely deep pit into the electric field. The energy of its field is infinite. We have to assume that the potential energy of the charge in the pit is infinitely negative, so that the total energy of the system is 511 keV. This is ugly.
Conclusions
The Feynman way of creating photons is explained by the sharp hammer model. The Dirac delta impulse of a Green's function is a single hit by the hammer.
Regularization works because it essentially implements the destructive interference in the classical model. The divergences in Feynman formulas are created because the formulas fail to consider destructive interference. The ad hoc method of Pauli-Villars regularization probably works because it somehow imitates the case where the electron moves at a constant velocity, and therefore implements the destructive interference.
If we are right, we have solved the 70-year-old mystery of regularization: why does it work?
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