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

Representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis and Wigner quantum oscillators
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 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.
0305-4470
5763-5785
King, R.C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Stoilova, N.I.
c5aabcae-ce39-4841-8183-4d9b531e6546
Van der Jeugt, J.
dcba948f-0e7c-41f6-bc33-42c21b84ad8a
King, R.C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Stoilova, N.I.
c5aabcae-ce39-4841-8183-4d9b531e6546
Van der Jeugt, J.
dcba948f-0e7c-41f6-bc33-42c21b84ad8a

King, R.C., Stoilova, N.I. and Van der Jeugt, J. (2006) Representations of the Lie superalgebra gl(1|n) in a Gel'fand-Zetlin basis and Wigner quantum oscillators. Journal of Physics A: Mathematical and General, 39, 5763-5785. (doi:10.1088/0305-4470/39/20/010).

Record type: Article

Abstract

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.

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Published date: 2006
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 29537
URI: http://eprints.soton.ac.uk/id/eprint/29537
ISSN: 0305-4470
PURE UUID: 682b1f49-5ed3-474a-813d-170697bdd45a

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Date deposited: 11 May 2006
Last modified: 08 Jan 2022 18:56

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Contributors

Author: R.C. King
Author: N.I. Stoilova
Author: J. Van der Jeugt

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