Phase Space Structure of Non-Abelian Chern-Simons Particles

TL;DR

This paper analyzes the classical phase space of non-Abelian Chern-Simons particles, revealing how external gauge fields modify symplectic structure and connecting to quantum models.

Contribution

It constructs the phase space using associated bundles and demonstrates the equivalence of minimal substitution to symplectic modification, linking classical and quantum descriptions.

Findings

Derived the symplectic structure modification by gauge fields.

Connected classical phase space to quantum NACS particle models.

Clarified the role of gauge connections in phase space geometry.

Abstract

We investigate the classical phase space structure of $N$ $SU(n+1)$ non-Abelian Chern-Simons (NACS) particles by first constructing the product space of associated $SU(n+1)$ bundle with ${\bf CP}^n$ as the fiber. We calculate the Poisson bracket using the symplectic structure on the associated bundle and find that the minimal substitution in the presence of external gauge fields is equivalent to the modification of symplectic structure by the addition of field strength two form. Then, we take a direct product of the associated bundle by the space of all connections and choose a specific connection by the condition of vanishing momentum map corresponding to the gauge transformation, thus recovering the quantum mechanical model of NACS particles in Ref.\cite{lo1}.