The classical electron self-energy divergence problem

Our previous blog post mentioned the classical renormalization problem of a point charge.

Let us have a hamiltonian

        H = ∫ ε₀ E² dv + V + m c²

which contains the energy of the static electric field of a charge, the potential energy V of the charge in its own field, and the mass-energy of the charge. The hamiltonian is like for a point mass sitting on an elastic rubber sheet.

Let us start from a configuration where the static field is zero. We can release energy by making a pit in the field and letting V become negative. The charge will fall into ever lower potential, and keeps releasing more energy.

Classically, the energy of the field of a point charge is infinite. To recover a sensible value for the total energy of the combined system of the charge and the field, we have to assume that the charge itself has negative infinite energy.  That is awkward.

A way to save us from infinities and negative energies is to make the electron into a ball whose radius is the classical radius of the electron. But particle collider experiments show that the electron radius cannot be that big. The radius of the charge has to be 10^-18 m or smaller.

Can zitterbewegung save the day?

If the electron moves at a speed of light in a small circle, that affects the geometry of its static field. Could it be that the infinite energy in the field of a point charge does not appear because of zitterbewegung?

The Darwin term of the hydrogen spectrum can be derived by assuming "smearing" of the charge in the zitterbewegung area, or more formally, using the Foldy-Wouthuysen transformation. It is probably not a coincidence that both approaches produce the correct result. If the charge is, in some sense, smeared, then there is no infinite energy in its field.

If a classical point electron moves at (almost) the speed of light, can we extract infinite energy by dropping an opposite charge into its field?

The classical renormalization problem is not really fundamental because the electron is a very small particle and it has to be treated quantum mechanically anyway.