A classical particle model of QED vacuum polarization loop divergence

                <---- e-     e+ ---->

                    -p             p

Let us put a classical particle electron and a positron very close to each other, less than 10^-15 m apart. If the pair is static, its negative potential energy is < -1.022 MeV. The pair has negative total energy.

We can raise the energy by giving the pair large opposite momenta. Then the total energy of the pair may be zero or positive.

This may be a classical analogue of the vacuum polarization loop divergence in a Feynman diagram. In principle, we can give the particles arbitrarily high momenta, if we place them close enough.

Can we give the particles arbitrary energy? Let us work in a frame where the sum of momenta is zero. Then the system is symmetric. The natural way to assign energies is that both particles have the same energy. But if we allow ourselves to divide the large negative potential energy arbitrarily between the particles, then we can also give them arbitrary energies.

This has a lot of similarity to the strange Feynman diagram vacuum polarization loop, where e- and e+ can have huge opposite momenta and "opposite" energies.

The magnetic field of fast moving particles complicates the calculation of their potential energy, but that probably does not spoil the setup.

Electromagnetic radiation from the fast pair

Our pair can be understood as a (time-reversed) final stage of a classical pair crashing into each other. The negative potential energy as well as the kinetic energies of the particles grow without limit.

But our setup is not realistic. The pair would radiate away electromagnetic radiation at a fast rate. It would quickly lose, for example, 1.022 MeV in radiation.

What about Delbrück scattering? How much energy in radiation does the virtual pair lose, if we calculate the radiation with the Larmor formula?

Actually, annihilation of a pair might be that the pair radiates away all its energy in electromagnetic radiation which is produced by the acceleration.

In the Larmor formula, the power of radiation is proportional to the acceleration squared. The acceleration in the static case is 1 / r^2, where r is the distance. The power is 1 / r^4 if the speed is not relativistic.

The negative potential is 1 / r. In the nonrelativistic case, putting the particles at half a distance (double the negative potential) makes the power of radiation out 16-fold. This classical model suggests that the lifetime of a virtual pair with high opposite momenta is extremely short.