Summary of our virtual photon models so far

We are stuck with the problem that we cannot find an intuitive quantum mechanical model for the electron flyby of a nucleus.

The trivial classical model is the static electric field of the electron and the nucleus. If we assume a heavy electron a heavy nucleus, they both move at a constant velocity and the finite communication speed of special relativity does not need to come to play.

Rod model

We made a quantum mechanical model of the static field by assuming a massless elastic rod on which the electron tugs on. But the problem with the rod model is that it does not give us an intuitive model of virtual pair production in association with the tugging.

Z ----------------------- e-

In the rod model, the virtual pair should appear like this:

Z ------ e- --- e+ ----- e-

The momentum p which the nucleus Z is sending the original electron is for a short time absorbed by the virtual pair. Since the virtual pair must disappear quickly, it has to give the momentum back to Z or to the original electron. However, the rod model does not give us a natural mathematical formula which should govern the appearance of the virtual pair.

Exchange of two real photons

In another model we made, the nucleus Z and the electron both send real photons whose magnetic field creates the pull. The nucleus and the electron have to move relative to each other so that the magnetic force is present.

This model is not intuitive from the classical point of view. Why should a simple pull be implemented in a complex way through magnetic fields? Why cannot we observe these real photons if they exist? Why should the charges move?

Production of real pairs

One way to study virtual pair production is to study real pair production. A virtual pair is like a real pair which failed to materialize because of lack of energy.

For a real pair to appear, it should tap on the kinetic energy of the electron or the nucleus. If we are working in coordinates where the nucleus Z is static, then the nucleus cannot send out the real photon which would turn into a real pair.

The pair must use the kinetic energy of the electron. The Feynman diagram is something like this:

----- e- -------------->

           |    ----- e+ -->

           | /

             \
                ------ e- -->

                   |

                   |

------- Z ------------->

The electron sends a photon which contains at least 1.022 MeV energy and a lot of momentum p. The momentum is large because the energy had to come from the kinetic energy of the electron. The photon is transformed to a real pair. The pair gives its excessive momentum to Z in a virtual photon.

If the electron would send out a real photon, the diagram would be like this:

              ~~~~~~~
           /
         /
----- e- -------------->
              |
              |
----- Z --------------->

The electron transforms some of its kinetic energy to a real photon. Then the electron gives its excessive momentum to the nucleus Z. This is called bremsstrahlung.

Production of real photons is a dynamic process. The electron cannot be static. It must have kinetic energy which it radiates away. On the other hand, the static electric force is present even if the electron and the nucleus are static. One wonders if virtual pairs can be produced at all if they are static.

Question 1. Is there empirical evidence that the loop correction is needed in the exchange of a virtual photon which has a negligible energy? The Feynman formula does not contain the energy of the virtual photon, just the momentum it carries. Or do the loop corrections cancel each other out if there is only a negligible energy transfer in the flyby?