On the fields and equations of motion of point particles

Abstract

The investigation covers point particles possessing a charge, dipole and higher multipole moments interacting with fields of any spin satisfying the generalized wave equation (8). It is shown that the radiation field defined as the retarded minus the advanced field and all its derivatives is always finite at all points including those on the world line of the point particles. The symmetric field, defined as half the sum of the retarded and advanced fields, is shown to contain a part expressible as an integral along the world line from minus to plus infinity, which is continuous and finite everywhere. This integral vanishes if χ = 0. The modified symmetric field is defined as the symmetric field minus this integral. The actual field is expressed as a sum of the modified symmetric field plus the modified mean field defined as half the sum of the ingoing and outgoing fields plus the integral just mentioned. It is proved that the part of the stress tensor of the field quadratic in the modified symmetric field plays no part in determining the equations of motion of the point particle. Being conserved by itself, it can always be subtracted away, thus defining a new stress tensor which is free from all the highest singularities in the usual stress tensor. The equations of motion of the particle are shown to depend only on the usual 'mixed terms' in the inflow with the modified mean field substituted for the ingoing field. The formulation for several particles is given.