Coherent States of the SU(N) groups

TL;DR

This paper explicitly constructs and analyzes coherent states for the SU(N) groups, generalizing the SU(2) case, and explores their classical limits and geometric properties relevant for quantum gauge theories.

Contribution

It provides a detailed construction of SU(N) coherent states, investigates their properties, and connects them to classical phase space and gauge theory analysis.

Findings

Coherent states are parametrized by points in the projective space CP^{N-1}.

They minimize uncertainty and the quadratic Casimir operator's dispersion.

The classical limit relates to Poisson brackets and the Fubini-Study metric.

Abstract

Coherent states $(CS)$ of the $SU(N)$ groups are constructed explicitly and their properties are investigated. They represent a nontrivial generalization of the spining $CS$ of the $SU(2)$ group. The $CS$ are parametrized by the points of the coset space, which is, in that particular case, the projective space $CP^{N-1}$ and plays the role of the phase space of a corresponding classical mechanics. The $CS$ possess of a minimum uncertainty, they minimize an invariant dispersion of the quadratic Casimir operator. The classical limit is ivestigated in terms of symbols of operators. The role of the Planck constant playes $h=P^{-1}$, where $P$ is the signature of the representation. The classical limit of the so called star commutator generates the Poisson bracket in the $CP^{N-1}$ phase space. The logarithm of the modulus of the $CS$ overlapping, being interpreted as a symmetric in the space, gives the Fubini-Study metric in $CP^{N-1}$. The $CS$ constructed are useful for the quasi-classical analysis of the quantum equations of the $SU(N)$ gauge symmetric theories.