The Gauged O(3) Sigma Model: Schr\"odinger Representation and Hamilton-Jacobi Formulation

TL;DR

This paper develops a Schr"odinger and Hamilton-Jacobi framework for a free particle on a sphere and extends it to the O(3) nonlinear sigma model, deriving energy spectra and BRST Lagrangian.

Contribution

It introduces a novel Hamilton-Jacobi formulation for the gauged O(3) sigma model and provides explicit energy spectra and BRST structure.

Findings

Derived the energy spectrum of a free particle on a sphere.

Extended the formulation to the O(3) nonlinear sigma model.

Obtained the BRST Lagrangian for the model.

Abstract

We first study a free particle on an $(n-1)$-sphere in an extended phase space, where the originally second-class Hamiltonian and constraints are now in strong involution. This allows for a Schr\"odinger representation and a Hamilton-Jacobi formulation of the model. We thereby obtain the free particle energy spectrum corresponding to that of a rigid rotator. We extend these considerations to a modified version of the field theoretical O(3) nonlinear sigma model, and obtain the corresponding energy spectrum as well as BRST Lagrangian.