Some Arguments for the Wave Equation in Quantum Theory 3

TL;DR

This paper demonstrates the existence of charge and current solutions to the 1D wave equation that satisfy the continuity equation and produce no radiation at infinity when extended to Maxwell's equations, challenging traditional views.

Contribution

It provides a rigorous proof of charge-current solutions satisfying the continuity equation with zero radiation, linking wave equations to electromagnetic theory.

Findings

Existence of charge solutions to the 1D wave equation

Construction of smooth solutions satisfying the continuity equation

Demonstration of zero radiated power at infinity

Abstract

We prove there exists a charge solution to the 1-dimensional wave equation, and a corresponding current, such that the pair satisfy the continuity equation. We show that when they are extended to a smooth solution of the continuity equation on a vanishing annulus containing the unit circle, with a corresponding causal solution to Maxwell's equations, obtained from Jefimenko's equations, the power radiated at infinity in a time cycle is zero.