Classical and Quantum Mechanics of Non-Abelian Chern-Simons Particles

TL;DR

This paper explores the classical and quantum dynamics of non-Abelian Chern-Simons particles, analyzing their phase space, constraints, and quantization, and providing a coherent state framework for their quantum properties.

Contribution

It offers a detailed symplectic and constraint analysis of $SU(N)$ NACS particles and introduces a coherent state quantization approach with applications to the Knizhnik-Zamolodchikov equation.

Findings

Derived the symplectic structure of $SU(N)$ coadjoint orbits.

Performed Dirac constraint analysis and computed isospin brackets.

Presented a coherent state representation of quantum NACS particles.

Abstract

We investigate the classical and quantum properties of a system of $SU(N)$ non-Abelian Chern-Simons (NACS) particles. After a brief introduction to the subject of NACS particles, we first discuss about the symplectic structure of various $SU(N)$ coadjoint orbits which are the reduced phase space of $SU(N)$ internal degrees of freedom or isospins. A complete Dirac's constraint analysis is carried out on each orbit and the Dirac bracket relations among the isospin variables are calculated. Then, the spatial degrees of freedom and interaction with external gauge field are introduced by considering the total reduced phase space which is given by an associated bundle whose fiber is one of the coadjoint orbits. Finally, the theory is quantized by using the coherent state method and various quantum mechanical properties are discussed in this approach. In particular, a coherent state representation of the Knizhnik-Zamolodchikov equation is given and possible solutions in this representation are discussed.