Furthermore, we claimed that in the traditional analysis of vacuum polarization, in the integral there are two sign errors which cancel each other out. The Dirac sea is empty. This also solves the problem of the infinite energy density of the vacuum: the energy of the vacuum is zero, not infinite.
Let us analyze this in more detail. Our previous blog posts have taught us something about the ultraviolet divergence in the vertex correction. There, the diagrams with an ultraviolet divergence can be discarded altogether because they have a zero chance of happening. The loop will always send a real photon, which makes the loop integral to converge.
A semiclassical model of vacuum polarization
Suppose that an electron and a positron pass by each other at a very high speed. As the electric field strength grows, it "almost produces" a new electron positron pair. It is a "virtual pair" which electrically polarizes the vacuum.
The pair makes the vacuum to conduct electric lines better: it better "permits" electric lines of force:
e-
• -->
° e+ virtual pair pulls on the
° e- electron and positron
<-- •
e+
In the case an electron meeting an electron, the configuration is like this:
° e- virtual pair pulls on
° e+ the upper electron
e- • -->
<-- • e-
° e+ virtual pair pulls on
° e- the lower electron
In the above diagram, the field strength is the largest at the locations where the virtual pairs form.
If we have a medium where the electric polarization is superlinear in the electric field strength, then charges will behave just like above. We say that the electric susceptibility
χ(E)
is superlinear in the field strength E.
We know that very high energy electrons and positrons will produce real pairs when they meet. It is natural to assume that the electric susceptibility is superlinear in E.
Classically, the extra polarization close to the meeting charges will always produce an electromagnetic wave. The virtual pair is a transient electric dipole which radiates away an electromagnetic wave.
The Feynman diagram above cannot happen. The virtual pair loop always emits a real photon.
The Feynman integral for the virtual pair loop has a logarithmic ultraviolet divergence at large 4-momenta, after using the Ward identity. Adding an emission of a real photon might make the integral to converge. But does that yield a result which matches the traditional renormalization technique? We have to check that.
In our August 27, 2021 post we suggested that destructive interference cancels out virtual pair loops with high 4-momenta. Does the emission of a real photon accomplish the same thing? The real photon makes the diagram asymmetric, which may complicate calculations greatly.
Using the rubber membrane and the sharp hammer model to the virtual pair loop
The rubber membrane model was able to clarify bremsstrahlung and the vertex correction. We hit the membrane twice, but the second hit is a little bit displaced. What escapes from the first Green's function is the bremsstrahlung. That part is not absorbed by the second hit, which, in turn, causes the infrared divergence of the elastic diagram.
#
#====== EM field hits Dirac field
v
__ ___ membrane
\__/
e+ ° ° e- Dirac wave
= created virtual pair
The "hit" to create the pair and the second "hit" to annihilate the pair may be analogous. If there is a lot of energy available, then pair can become real and escape. That is like bremsstrahlung, this time consisting of pairs.
To create a real pair, we need at least 1.022 MeV. We get a strict upper bound on the wavelength of the escaping "pair bremsstrahlung".
*** WORK IN PROGESS ***
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