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A non-commutative n-particle 3D Wigner quantum oscillator

A non-commutative n-particle 3D Wigner quantum oscillator
A non-commutative n-particle 3D Wigner quantum oscillator
An n-particle three-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p >= 3n but atypical for p < 3n. The branching rules for the restriction from sl(1|3n) to gl(1) + so(3) + sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n 2. It is shown that measurements of the coordinates of the individual particles give rise to a set of discrete values defining nests in the three-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realization of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator.
0305-4470
11999-12019
King, R.C.
76ae9fb3-6b19-449d-8583-dbf1d7ed2706
Palev, T.D.
d76c10dd-851a-4422-b649-fd7b98776e68
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
Palev, T.D.
d76c10dd-851a-4422-b649-fd7b98776e68
Stoilova, N.I.
c5aabcae-ce39-4841-8183-4d9b531e6546
Van der Jeugt, J.
dcba948f-0e7c-41f6-bc33-42c21b84ad8a

King, R.C., Palev, T.D., Stoilova, N.I. and Van der Jeugt, J. (2003) A non-commutative n-particle 3D Wigner quantum oscillator. Journal of Physics A: Mathematical and General, 36, 11999-12019. (doi:10.1088/0305-4470/36/48/007).

Record type: Article

Abstract

An n-particle three-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p >= 3n but atypical for p < 3n. The branching rules for the restriction from sl(1|3n) to gl(1) + so(3) + sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n 2. It is shown that measurements of the coordinates of the individual particles give rise to a set of discrete values defining nests in the three-dimensional configuration space. The fact that the underlying geometry is non-commutative is shown to have a significant impact on measurements of particle separation. In the atypical case, exclusion phenomena are identified that are entirely due to the effect of A-superstatistics. The energy spectrum and associated degeneracies are calculated for an infinite-dimensional realization of the Wigner quantum oscillator model obtained by summing over all p. The results are compared with those applying to the analogous canonical quantum oscillator.

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Published date: 2003
Organisations: Applied Mathematics

Identifiers

Local EPrints ID: 29531
URI: http://eprints.soton.ac.uk/id/eprint/29531
ISSN: 0305-4470
PURE UUID: 5d8921e3-72b8-42c5-8adc-7c6b9d0b1cad

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Date deposited: 12 May 2006
Last modified: 08 Jan 2022 06:55

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Contributors

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

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