2006/02/17 by R. C. King, R C King, N. I. Stoilova +3
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Quantum Mechanics and Non-Hermitian Physics #hep-th #math-ph #math.MP #math.RT #quant-ph
paper · pdf · doi:10.1088/0305-4470/39/20/010
published as J.Phys.A39:5763-5786,2006
arxiv created 2006/02/17 · openalex publication_date 2006/05/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/30
An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra 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 are physical state spaces of the N -particle D -dimensional oscillator. So far, physical properties of 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 , associated with the N -particle three-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.