In the previous blog posts we have suggested that the classical analogue of vacuum polarization is superlinear electric polarization of a material.
https://en.wikipedia.org/wiki/Nonlinear_optics
Nonlinear optics studies phenomena in such materials. The Wikipedia article says that above the Schwinger limit
1.3 * 10¹⁸ V / m
"the vacuum itself is expected to become nonlinear."
Let us calculate the electric field strength of a proton at the distance of the electron reduced Compton wavelength
4 * 10⁻¹³ m.
It is
k e / r² = 10¹⁶ V / m,
where k is the Coulomb constant and e is the electron charge. The Uehling potential shows that there is slight nonlinearity already at that distance scale. If r is 1/10 of the reduced Compton wavelength, then the Schwinger limit is reached.
What material is analogous to vacuum polarization?
Our material should be a gas, because we believe that virtual pairs can move around in the vacuum just like molecules or atoms in a gas. Though the movement may not be relevant because the collision of, e.g., the electron and the proton happens quickly. Let us try to visualize the polarization process.
e- ------->
+
-
● proton
One effect is that superlinearity causes an extra electric field to appear between the proton and the electron. Superlinear polarization, in a sense, brings some of the charge of the electron closer to the proton, which results in larger attraction.
There is also an opposite effect. The electric field strength is ~ 1 / r². But the area of a 3D sphere is ~ r². Therefore, polarization which depends linearly on the field strength, hides some fixed amount of the proton charge when viewed by the electron at any distance. If polarization is superlinear, then the closer the electron gets to the proton, the more of the proton charge is hidden.
Which of the two effects wins? Does the Coulomb force appear stronger when we go closer?
It seems to depend on the formula of superlinearity. If there is 1% extra polarization after the first doubling of the electric field, and more than 2% after the second doubling, then the attractive force might win. That is, if superlinearity grows faster than
ln(E),
where E is the electric field strength. In practice, polarization grows very steeply once we approach the ionization threshold.
Visible light inside glass
Let us consider visible light inside glass. The speed of light is slower there. Let the refractive index be, for instance, n = 1.1. If v = 0.9 c, then the familiar γ of special relativity is
γ = 1 / sqrt(1 - v² / c²)
= 2.3.
That is, the effective "rest mass" of a photon is roughly half of its total energy.
The photon probably satisfies the massive Klein-Gordon equation.
What happens if we raise photon energy so much that it can ionize the material? What is the effective mass of the photon then? We could not find any empirical data. Apparently, the material becomes opaque when photon energy is high enough.
For low photon energies the refractive index of various materials is fairly constant.
Photon propagator in a polarizable material
Maybe the on-shell photon propagator in a polarizable material should be something like
1 / ( p² - m² ),
where m is the effective photon "mass" at the "scale" of the momentum p? The scale of the momentum should be taken in the euclidean norm.
The Coulomb force in a linearly polarizable material is weaker than in the vacuum. The propagator for a pure momentum photon is
1 / (p² n),
where n >= 1 is the refractive index.
The propagators are very different. It looks like we cannot define a single photon propagator in a polarizable medium.
Empirical values for the Coulomb force strength
Empirical measurements of the Uehling potential (through the Lamb) shift show that the Coulomb force is ~ 1 / 30,000 stronger at the distance of the electron reduced Compton wavelength
4 * 10⁻¹³ m
from the proton.
In the collision of a 100 GeV electron and a positron, significant deflection requires that they come within 1 / 100,000 of the electron reduced Compton wavelength, that is,
4 * 10⁻¹⁸ m.
At that distance, the Coulomb force appears roughly 7% stronger.
Vacuum polarization always strengthens the Coulomb force?
In our previous blog post we claimed that a virtual pair cannot absorb any angular momentum because it could not give it to the proton. That dictates that the momentum q transfers to the virtual pair have to be simultaneous.
The proton, the electron, and the virtual pair apparently have to be on the same line. Can the virtual pair exert a repulsive force on the electron?
• e+ virtual positron
e- ● real electron
• e- virtual electron
● proton
There is something strange in the diagram. How can the virtual positron end up "behind" the real electron, even though the summed electric field there pulls it toward the real electron?
If the virtual pair is between the proton and the electron, it makes attraction stronger.
Let us assume that the electron alone produces some polarization on the line of the diagram. The polarization pulls it equally up and down. There is no net force.
When we add the proton, it makes polarization stronger below the electron, and weaker up from the electron. In both cases the new polarization causes a downward force on the electron. This suggests that vacuum polarization always strengthens the Coulomb force. It cannot make the force weaker.
What about a solid? We know that polarization of a solid makes the Coulomb force weaker.
• e+ virtual positron
● e-
• e- virtual electron
● proton
In the diagram above, the virtual pair denotes polarization which is caused by the proton. It is hard to make a diagram where the torque on the virtual pair is zero, unless all the particles are on the same line. In a solid the torque is no problem. The solid can absorb the torque.
Hypothesis. Vacuum polarization always makes the Coulomb force stronger, while polarization of a solid in most cases makes the force weaker.
Our hypothesis would explain why the Feynman diagram of vacuum polarization makes the Coulomb force stronger. In the Feynman diagram we have corrected the sign error in the effect of polarization - an error which has lingered in literature ever since the first paper by Dirac in 1934.
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