Holstein-Primakoff Realizations on Coadjoint Orbits

TL;DR

This paper derives Holstein-Primakoff oscillator realizations on coadjoint orbits of SU(N+1) and SU(1,N) groups using symplectic reduction and action-angle variables, leading to canonical quantization and Dyson realizations.

Contribution

It introduces a novel method to obtain Holstein-Primakoff realizations on coadjoint orbits via symplectic reduction and Darboux variables, expanding the theoretical framework.

Findings

Holstein-Primakoff realizations derived for SU(N+1) and SU(1,N)

Canonical quantization with normal ordering achieved

Dyson realizations also obtained and discussed

Abstract

We derive the Holstein-Primakoff oscillator realization on the coadjoint orbits of the $SU(N+1)$ and $SU(1,N)$ group by treating the coadjoint orbits as a constrained system and performing the symplectic reduction. By using the action-angle variables transformations, we transform the original variables into Darboux variables. The Holstein-Primakoff expressions emerge after quantization in a canonical manner with a suitable normal ordering. The corresponding Dyson realizations are also obtained and some related issues are discussed.