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are used to obtain fundamental equations from a covariant action in which the usual term involving particle mass m has been replaced by the covariant mass density µ associated with the fields. Point, line, and surface charges are not allowed. Stable 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. The Lorentz force law plus radiation reaction is shown to govern the motion of these charged objects in an imposed electromagnetic field. Ordinary classical electromagnetism is thus reproduced, and the classical structures of fundamental charged bodies are obtained. Structures of the electron and proton in this theory are briefly discussed.