Classical Maxwell Theory with Finite-Particle Sources

A Lorentz-invariant finite-particle model is introduced into the Maxwell theory by extending the space from space-time to all (time-like) space-time spheres. The properties of the model are examined in the classical theory as a preliminary to the quantized case. The space-time sphere radius $\lambda$ is the parameter of finiteness; it has the effect of smearing point particles into bell-shaped bounded distributions which go over into the $\delta$-function point-particle distributions in the limit $\lambda =0\mathrm{}$. The smeared particles give rise to fields in which the Coulomb infinity no longer exists. It is shown that the finite-particle 4-current has various indispensable formal properties: that charge is conserved; and that, in interaction with its field, momentum and energy are conserved, the integrals representing the electromagnetic self-energy and self-force being convergent for $\lambda \ne 0$. This replacement of point by finite particles results in corrections to calculations which are probably negligible where the classical theory is valid, but which might be appreciable in the quantum domain at distances comparable to $\lambda$.