A virtual photon and a virtual phonon

Our elastic rod hypothesis makes the interaction between a nucleus and a passing electron more tangible, literally.

The Feynman integral which describes a virtual pair creation from a virtual photon diverges. But apparently, numerically right results can be acquired by putting a momentum cutoff to the virtual pair.

Our goal is to understand pair creation better. We believe that if we can model the creation in an intuitive way, then we can cure the divergence in the integral.

As we wrote in an earlier blog post, an obvious way to try pair creation is to make use of the acceleration of the electron. The electron can pull on the virtual positron and expel the virtual electron, and then accelerate away, donating them energy. If the pair stays virtual, the energy soon returns back to the electron.

In our previous blog post we modeled the flyby of an electron with classical point particles and a classical rod. What is the classical analogue of a pair creation?

e- -------- e+

A classical analogue of a virtual pair might be an electron and a positron which are tightly locked together with a very short rod. It is a positronium atom whose all mass-energy has been lost when the pair came very close together.

Alternatively, we can consider a real positronium atom and what kind of interaction might free the electron and the positron to go their own ways.

A real photon whose electric field pulls the pair apart can try pair creation. The photon tries to pull the pair apart. If the pulling does not succeed, then in the Feynman diagram we mark the real photon as temporarily decayed to a virtual pair.

A real photon can certainly pull a positronium atom apart.

What about an electron flyby of a nucleus? Can the longitudinal phonon in the rod somehow cause the virtual pair (or positronium) to separate? Is there a rapidly changing electric field that could give an energy boost to the virtual pair?

Let us make a thought experiment where the nucleus Z and the electron e- are very heavy. In that case, the flyby happens essentially at a constant speed. There is essentially no acceleration and the dynamics is low close to the electron.

Thus, the virtual pair creation by an accelerating electron cannot be responsible for the virtual pair loop in the Feynman diagram.

There is some dynamics, though. The dipole electric field of the heavy electron and nucleus does change direction and its dipole moment during the flyby. It is like a dipole antenna which sends out one half of a radio wave.

A positronium atom could harvest some energy from the changing electric field. Thus, virtual pairs will play some role. Maybe the Feynman diagram captures this dynamics and calculates the effect of virtual pairs correctly if we put a momentum cutoff?

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An aside: Matt Strassler's take on virtual photons:
https://profmattstrassler.com/articles-and-posts/particle-physics-basics/virtual-particles-what-are-they/
According to him, they are temporary "ripples" in the electromagnetic field.
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Above we were able to reduce the virtual photon conundrum into half of a cycle of a dipole radio transmitter. The virtual photon in some way describes what happens inside the transmitter.

The outside world will see half a cycle of radio waves. What does that look like? A dipole field turns 180 degrees.

If the electron is very heavy, then our elastic rod model has the nucleus tugging on the rod. The tugs do not alter the path of the electron significantly. It is like tugging on a rope which is attached to a heavy train passing by.

A single tug on the rod is half a cycle of a pressure wave in the rod. It is not a real phonon, because a real phonon must exist "free" and must contain one or more cycles.

A single tug is an "impulse", that is, a transfer of momentum.

Here we have a new characterization for a virtual photon or phonon: it contains only half a cycle and implements an impulse operation.

What is the lagrangian like for a tug of a rod? The lagrangian determines the speed at which the phase of the wave function rotates. The lagrangian is typically total energy minus potential energy, that is, the kinetic energy. The more kinetic energy, the faster the wave function of a particle spins.