Renormalisation and hierarchies

Abstract

We describe a new method of calculating the renormalised energy of a field obeying the Maxwell and Dirac equations. The method does not involve evaluating integrals but relies instead on summing a geometric series. We show that the new solution is equivalent to summing over the levels in the gravitational hierarchies explored previously by Bell and Diaz. We augment our gravitational model by the introduction of a shell composed of a smooth mixture of negative and positive energy. This separates the charge from the mass of a body and leads to a more general hierarchy. Summing over the levels in this then produces the renormalisation series found above for the electromagnetic field, as well as the renormalisation series required by gravity.