Coherent States of $SU(l,1)$ groups

TL;DR

This paper extends the explicit construction of coherent states to all SU(l,1) groups, providing a unified framework for analyzing quantization on complex spaces of constant curvature.

Contribution

It introduces a method to explicitly construct coherent states for SU(l,1) groups, generalizing previous work on SU(N) groups and covering spaces of constant curvature.

Findings

Explicit coherent states for SU(l,1) are constructed.

The parameter space is the open complex ball, a space of constant negative curvature.

Provides tools for quantization analysis on complex spaces of constant curvature.

Abstract

This work can be considered as a continuation of our previous one (J.Phys., 26 (1993) 313), in which an explicit form of coherent states (CS) for all SU(N) groups was constructed by means of representations on polynomials. Here we extend that approach to any SU(l,1) group and construct explicitly corresponding CS. The CS are parametrized by dots of a coset space, which is, in that particular case, the open complex ball $CD^{l}$. This space together with the projective space $CP^{l}$, which parametrizes CS of the SU(l+1) group, exhausts all complex spaces of constant curvature. Thus, both sets of CS provide a possibility for an explicit analysis of the quantization problem on all the spaces of constant curvature.