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

Abstract

An n-particle 3-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\geq 3n but atypical for p<3n. The branching rules for the restriction from sl(1|3n) to gl(1) \oplus so(3) \oplus sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n\leq 2. It is shown that measurements of the coordinates of the individual particles gives rise to a set of discrete values defining nests in the 3-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 realisation 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.

26 pages, to be published in J.Phys. A

References in corpus (4)

• Many-Body Wigner Quantum Systems
• The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator
• Jacobson generators, Fock representations and statistics of sl(n+1)
• Microscopic and macroscopic properties of A-superstatistics

Cited by in corpus (8)

• Harmonic oscillator chains as Wigner Quantum Systems: periodic and fixed wall boundary conditions in gl(1|n) solutions
• Harmonic oscillators coupled by springs: discrete solutions as a Wigner Quantum System
• Representations of the Lie Superalgebra gl(1|n) in a Gel'fand-Zetlin Basis and Wigner Quantum Oscillators
• Superconformal Quantum Mechanics via Wigner-Heisenberg Algebra
• Analytically solvable Hamiltonians for quantum systems with a nearest neighbour interaction
• Solutions of the compatibility conditions for a Wigner quantum oscillator
• Wigner quantization of some one-dimensional Hamiltonians
• A classification of generalized quantum statistics associated with basic classical Lie superalgebras