A Covariant Field Reformulation of Classical Electrodynamics

Classical electrodynamics is here reformulated as a field theory rather than a particle and field theory. Electromagnetic fields are taken to be continuous and differentiable everywhere. Maxwell’s equations are assumed to be valid at all scale lengths, and second order variations on the vector potentials A_k are used to obtain fundamental equations from a covariant action in which the usual term involving particle mass m has been replaced by the covariant mass density µ associated with the fields. These equations lead directly to the Lorentz force plus radiation reaction and to equations from which static charge distributions having finite extent and charge of only one sign are found. These bodies are stabilized by the charge motion in the distribution’s self-field. Their mass is finite and equivalent to the energy of the electromagnetic field integrated over all space. Spin, magnetic moment, and electric dipole and quadrupole moments of these bodies can be calculated. Ordinary classical electromagnetism is thus reproduced without point charges, and the classical structures of fundamental charged bodies may be obtained. Structure of the electron in this theory is briefly discussed.