Mechanics of a Particle in a Gauge Field

TL;DR

This paper explores the classical and quantum mechanics of particles in gauge fields, extending the action principle to include non-Abelian fields, revealing wave-like properties and deriving a generalized Schrödinger equation.

Contribution

It introduces a novel extension of the action principle for gauge fields, leading to a new path-integral formalism and a generalized Schrödinger equation.

Findings

Wave-like properties of particles in gauge fields identified.

Generalized Schrödinger equation derived from the extended framework.

Reduction to Klein-Gordon equation demonstrated.

Abstract

The action principle is frequently used to derive the classical equations of motion. The action may also be used to associate group elements with curves in the space-time manifold, similar to the gauge transformations. The action principle is shown here to be an equivalence relation between the infinitesimal elements so defined for a collection of closed curves and the identity element. The action principle is then extended by requiring the equivalence of global elements with the identity and by considering all curves. The resulting equation is generalized further to include the non-Abelian gauge fields. The extended equation has an infinite number, but not all, trajectories as solutions. The properties of these paths are shown to impart wave-like properties to the particles in motion. These results provide an insight into the wave-particle duality and lead to a modified path-integral formalism. The motion of a particle is formulated within the resulting framework which yields a generalized Schrodinger equation. This equation is shown to reduce to a set of equations, one of them being the Klein-Gordon equation.