Representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis and Wigner quantum oscillators

An explicit construction of all finite-dimensional irreducible representations of the Lie superalgebra gl(1 vertical bar 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 vertical bar DN) are physical state spaces of the N-particle D-dimensional oscillator. So far, physical properties of gl(1 vertical bar 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 vertical bar 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 vertical bar 3N), 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.