A Continuous Field Formulation of Classical Electrodynamics

Classical electrodynamics is here formulated as a field theory rather than as a particle and field theory. Electromagnetic fields are taken to be continuous and differentiable everywhere. Maxwell’s equations are assumed to be valid always, and second order variations are used to obtain fundamental equations from a covariant action in which the usual term involving particle mass m has been replaced by a term involving an integral over the covariant mass density µ associated with the fields. These equations lead directly to an equation of motion for a body’s center of inertia: the Lorentz force plus a radiation reaction term without the anomalies present in the usual particle and field formulation. The solutions to these fundamental equations are also solutions for the time-independent Dirac and Schrodinger equations but here give arrangements of mass densities in a body interacting with its environment, not probabilities of finding an electron or other body. Stable bodies comprising charge densities of one sign have been found. The mass density µ of a body interacting with its environment is equal to a term involving the Hamiltonian, which describes environmental effects, plus an invariant term proportional to (E² -H²) for the mass density in the charged region of the body itself. This purely classical formulation of continuous electromagnetic fields thus reproduces and improves ordinary classical electromagnetism, yields spatial charge distributions and energy levels familiar from quantum mechanics, and suggests a new approach to finding the masses of fundamental bodies.