Schroedingers equation with gauge coupling derived from a continuity equation

TL;DR

This paper derives Schrödinger's equation from a continuity equation for a particle ensemble, introduces a generalized gauge coupling via an extended Ansatz, and discusses the emergence of electromagnetic potentials through gauge transformations.

Contribution

It presents a novel derivation of Schrödinger's equation from statistical and continuity principles, and introduces a generalized gauging procedure leading to electromagnetic coupling.

Findings

Derivation of Schrödinger's equation from a continuity equation.

Introduction of a generalized gauge coupling with a time-dependent Planck constant.

Identification of gauge transformations that eliminate certain modifications of the Schrödinger equation.

Abstract

We consider a statistical ensemble of particles of mass m, which can be described by a probability density \rho and a probability current \vec{j} of the form \rho \nabla S/m. The continuity equation for \rho and \vec{j} implies a first differential equation for the basic variables \rho and S. We further assume that this system may be described by a linear differential equation for a complex state variable \chi. Using this assumptions and the simplest possible Ansatz \chi(\rho,S) Schroedingers equation for a particle of mass m in an external potential V(q,t) is deduced. All calculations are performed for a single spatial dimension (variable q) Using a second Ansatz \chi(\rho,S,q,t) which allows for an explict q,t-dependence of \chi, one obtains a generalized Schroedinger equation with an unusual external influence described by a time-dependent Planck constant. All other modifications of Schroeodingers equation obtained within this Ansatz may be eliminated by means of a gauge transformation. Thus, this second Ansatz may be considered as a generalized gauging procedure. Finally, making a third Ansatz, which allows for an non-unique external q,t-dependence of \chi, one obtains Schroedingers equation with electromagnetic potentials \vec{A}, \phi in the familiar gauge coupling form. A possible source of the non-uniqueness is pointed out.