Representations of the Lie Superalgebra gl(1|n) in a Gel'fand-Zetlin Basis and Wigner Quantum Oscillators

TL;DR

This paper explicitly constructs all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis and explores their applications to Wigner Quantum Oscillators, revealing new physical properties and spectra.

Contribution

It provides a comprehensive construction of all finite-dimensional irreducible representations of gl(1|n) and applies these to analyze Wigner Quantum Oscillators beyond Fock spaces.

Findings

New physical properties for WQOs using different unitary representations

Explicit spectra of Hamiltonian, position, momentum, and angular momentum operators

Classical limit consistent with the correspondence principle

Abstract

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis is given. Particular attention is paid to the so-called star type I representations (``unitary representations''), and to a simple class of representations V(p), with p any positive integer. Then, the notion of Wigner Quantum Oscillators (WQOs) is recalled. In these quantum oscillator models, the unitary representations of gl(1|DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1|DN) WQOs were described only in the so-called Fock spaces W(p), leading to interesting concepts such as non-commutative coordinates and a discrete spatial structure. Here, we describe physical properties of WQOs for other unitary representations, including certain representations V(p) of gl(1|DN). These new solutions again have remarkable properties following from the spectrum of the Hamiltonian and of the position, momentum, and angular momentum operators. Formulae are obtained that give the angular momentum content of all the representations V(p) of gl(1|3N), associated with the N-particle 3-dimensional WQO. For these representations V(p) we also consider in more detail the spectrum of the position operators and their squares, leading to interesting consequences. In particular, a classical limit of these solutions is obtained, that is in agreement with the correspondence principle.