A polynomial class of $u(2)$ algebras

TL;DR

This paper introduces a new polynomial class of $u(2)$ algebras with multiple parameters, explores their finite unitary representations, and constructs related nonlinear bosonic models and their analytical representations.

Contribution

It defines a new multi-parameter polynomial $u(2)$ algebra, investigates its representations, and develops a related nonlinear bosonic model with a Bargmann representation.

Findings

Finite unitary representations are characterized.

A family of nonlinear bosonic models is constructed.

Analytical Bargmann representation is provided.

Abstract

A $r$-parameter ${u}_{\{\kappa_1, \kappa_2, \cdots, \kappa_r\}}(2)$ algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl-Heisenberg algebra ${\cal A}_{\{\kappa_1, \kappa_2, \cdots, \kappa_r\}}$ (M. Daoud and M. Kibler, J. Phys. A: Math. Theor. {\bf 45} (2012) 244036). A pair of nonlinear (quadratic) bosons of type ${\cal A}_{\kappa}\equiv {\cal A}_{\{\kappa_1=\kappa, \kappa_2=0, \cdots, \kappa_r=0\}}$ are used to construct, \`a la Schwinger, a one parameter family of (cubic) $u_{\kappa}(2)$ algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.