For example, the density of negative energy electrons is larger close to a proton. The density of negative energy electrons is imagined to be lower somewhere far away, perhaps at the infinity.
We do not like the idea of the vacuum being full of particles like negative energy electrons. We would like the vacuum to be totally empty. How could we simulate Dirac's idea without assuming a Dirac sea?
Dirac has to put a cutoff on the momenta |p| of negative energy electrons in, say, one cubic meter of empty space. Then the number of negative electrons in a stationary state in that cubic meter is finite, and we can calculate a finite amount of polarization caused, for example, by a proton.
The cutoff is an ugly feature. The second ugly problem is that negative energy electrons have to be able to store momentum to be able to interact with real particles. What guarantees that empty space does not soak up a macroscopic amount of momentum?
Feynman diagrams solve the second ugly problem by requiring that energy and momentum are always conserved and exit the process in real particles.
Though, as we remarked in an earlier blog post, Feynman diagrams do not conserve the speed of the center of mass. As if empty space could contain mass-energy which our process can displace. Clearly, we must add conservation of the speed of the center of mass to the rules which a Feynman diagram must obey.
The cutoff problem exists in Feynman diagrams, just as in Dirac sea vacuum polarization.
Uehling and Serber in 1935 calculated vacuum polarization from the Dirac model, and got the correct result. The Lamb shift has been measured very precisely, and it must include the effect from the Uehling potential.
The Feynman diagram
In modern literature, the Uehling potential is calculated from the vacuum polarization Feynman diagram. The result is the same as Uehling got from Dirac's model. Clearly, the Feynman diagram has to calculate the same thing as Dirac's model, though it is not yet clear to us how it does that trick.
e- -------------------------------------
|
|
O vacuum polarization loop
|
| virtual photon p
Z+ -------------------------------------
The Feynman diagram at the first sight does not assume that the vacuum is full of particles. No such particles are drawn into the diagram. But a vacuum full of particles creeps in when we set a cutoff on the virtual pair 4-momenta in the vacuum polarization loop.
Maybe the solution is to set the cutoff at the nucleus energy?
Dirac in the papers which can be found in the book in the link, suggested that the cutoff should be something like 1 GeV. That is the mass of the proton.
We have in this blog suggested that destructive interference wipes off virtual pairs whose 4-momentum is large. In wave phenomena, long waves typically cannot create shorter waves. The nucleus in the diagram has lots of energy. Its wavelength in the time direction is very short. It could create wave phenomena whose 4-momentum is large.
virtual pair loop
large virtual photon
~~~ O ~~~
/ \
Z+ ------------------------------------------
In the Feynman diagram above, the nucleus creates a large virtual photon, which in turn can create a virtual pair with large 4-momentum. The electron might interact with the pair. A problem is that the diagram will have more vertices than the simpler vacuum polarization diagram, and consequently, its amplitude is smaller in the Feynman calculus.
This idea does not work, though. If the cutoff would depend on the mass of the nucleus, then the charge visible to a far-away observer would depend on that mass.
A possible solution: vacuum polarization is temporary pairs which "conduct" electric field lines better?
The Uehling potential shows that the proton attracts the electron stronger than we would expect if the electron comes within the reduced Compton wavelength distance 4 * 10⁻¹³ m from the proton.
How to make electric attraction stronger? Create a temporary electric dipole between the electron and the proton. The dipole will make the attraction stronger. It "conducts" electric field lines. The energy for the creation of the dipole would come from the approaching electron.
In this blog we have thus far used electric polarization of a solid as the model of polarization. If polarization grows superlinearly with the field strength, then the attraction between the electron and the proton will appear stronger.
Since a very strong (changing?) electric field can create a real pair, we suspect that polarization really is superlinear.
But how to reconcile this very different model with the idea of the Dirac sea?
The Uehling "force" is mediated by an electron-positron pair?
The Uehling potential becomes important at the distance of the electron reduced Compton wavelength, and it drops off exponentially at larger distances.
This immediately brings into mind that the Uehling potential is an attractive force caused by a dipole virtual electron-positron pair. The mediator "particle" of the Uehling force is a pair.
proton
● - + • e-
dipole
A dipole is born between the proton and the electron which is passing it. The pair makes the attraction between the proton and the electron stronger. A dipole "conducts" the electric line of force and makes attraction stronger.
The energy-time uncertainty principle tells us that the pair, whose energy as a real system would be 1.022 MeV, can at most reach to the distance 1/2 times the reduced Compton wavelength of the electron.
How do we calculate the probability of a dipole forming?
When the electron passes the proton, it produces a sharp pulse in the electric field of the electron. The pulse contains frequencies which are high enough to create a 1.022 MeV pair, but the total energy of the pulse is too small. A real pair cannot be created, but maybe we can create a very short-lived virtual pair?
There is a sign error in the Feynman diagram and it calculates the "complement" of the correct effect?
The following could explain things.
The Feynman vacuum polarization diagram erroneously reduces the calculated cross section. It should increase it.
We wrote on December 25, 2020 that a particle cannot change its phase on its own. The photon (momentum p) in the diagram should keep its phase, not change it by 180 degrees.
The Feynman formula, when we increase the exchanged spatial momentum |p| between the proton and the electron (make the electron go closer), leaves out the effect of virtual pairs with low 4-momentum. Think of Dirac's model. Negative energy electrons with low momentum have the polarization spread over a large spatial area. Decreasing the distance between the electron and the proton leaves out their effect.
But in the "Uehling force" model above, low 4-momentum is necessary for the pair, because otherwise destructive interference wipes it out. In that model, it is the low-4-momentum pairs which create the attractive force.
Thus, when we increase |p|, the Feynman integral (with its negative sign, or 180 degree phase change) leaves out the probability amplitude contribution, say, P, of low-4-momentum pairs.
e- ---------------------------------------------
| virtual
| photon
| p
Z+ ---------------------------------------------
Above we have the diagram of plain Coulomb scattering.
Since P in the Feynman calculus has a 180 degree phase shift relative to the plain Coulomb scattering probability amplitude, adding P to it would reduce the cross section. Leaving P out increases the cross section: the Coulomb force looks a little bit stronger, which is the right end result.
We suggest that in the correct calculation, P should not have a 180 degree phase shift, and it should be added to the plain Coulomb scattering amplitude. We need to study this in detail.
P is kind of a "complement" of the Feynman integral.
If this is true, empty space is truly empty. It is the proton and the electron themselves who create virtual pairs, which make the electric line of force stronger.
Open problems: why does Dirac's model calculate the Uehling potential right, even though in his model it is the bending paths of negative energy electrons which cause the polarization? It is not virtual pair dipoles like in our model.
Why does the Feynman integral formula calculate the same thing as Dirac's model? In Dirac's model the negative energy electron enters the process as an independent particle. In the Feynman diagram, it is one half of a virtual pair.
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