Particle with non-Abelian charge: classical and quantum

TL;DR

This paper explores the classical and quantum dynamics of particles carrying non-Abelian charge using the worldline formalism, deriving wave equations and analyzing the role of isospin constraints.

Contribution

It provides a detailed derivation of the wave equation for non-Abelian charged particles from the path integral, including operator ordering and quantum Hamiltonian construction.

Findings

Derived the wave equation from the path integral formulation.

Identified the operator ordering for the quantum Hamiltonian.

Analyzed the isospin constraints and their classical solutions.

Abstract

We study the action for a non-Abelian charged particle in a non-Abelian background field in the worldline formalism, described by real bosonic variables, leading to the well known equations given by Wong. The isospin parts in the action can be viewed as the Lagrange multiplier term corresponding to a non-holonomic constraint restricting the isospins to be parallel transported. The path integration is performed over the isospin variables and as a result, the worldlines turn out to be constrained by the classical solutions for the isospins. We derive a wave equation from the path integral, constructed as the constrained Hamiltonian operator acting on the wave function. The operator ordering corresponding to the quantum Hamiltonian is found and verified by the inverse Weyl transformation.