Some mathematical problems in a neoclassical theory of electric charges

TL;DR

This paper develops a neoclassical model of electromagnetic phenomena where charges are wave functions, proving convergence to classical trajectories, finding localized solutions, and connecting quantum energy levels to classical limits.

Contribution

It introduces a neoclassical theory with wave-function charges, proves convergence to Newtonian dynamics, and links quantum energy levels to classical results.

Findings

Wave centers converge to Newtonian trajectories with Lorentz forces.

Localized accelerating solitons are explicitly constructed.

Quantum energy levels approach classical levels as charge size increases.

Abstract

We study here a number of mathematical problems related to our recently introduced neoclassical theory for the electromagnetic phenomena in which charges are represented by complex valued wave functions as in the Schrodinger wave mechanics. Dynamics of charges in the non-relativistic case is governed by a system of nonlinear Schrodinger equations coupled with the electromagnetic fields, and we prove for it that the centers of wave functions converge in macroscopic regimes to trajectories of points governed by the Newton's equations with the Lorentz forces provided the wave functions remain localized. Exact solutions in the form of localized accelerating solitons are found. Our studies of a class of time multiharmonic solutions of the same field equations show that they satisfy Planck-Einstein relation and that the energy levels of the nonlinear eigenvalue problem for the hydrogen atom converge to the well-known energy levels of the linear Schrodinger operator when the free charge size is much larger than the Bohr radius.