(mass renormalization). The resulting theory should be essentially Schrödinger’s mechanics, but slightly modified by so-called radiative corrections.

Lest the reader think that the self-energy problem is just a mathematical exercise, consideration of the many-body problem will provide a counterexample. Imagine $N$ charged bosons interacting with the radiation field, but neglect any interaction among them such as the Coulomb repulsion. We say that these particles bind if the energy of the combined particles is less than the energy of infinitely separated particles. As we shall show, charged bosons indeed bind and they do it in such a massive way that it will be very likely that this cannot be overcome by the Coulomb repulsion. In particular, the energy of a charged many-boson system is [*not*]{} extensive, and from this perspective it is fortunate that stable, charged bosons do not exist in nature.

The situation is very different for fermions. We are not able to show that they do not bind but we can show — and this is one of the main results of our paper — that the self-energy is extensive, i.e., bounded above and below by a constant times $N$.

We thus have strong evidence that there is [*no*]{} 