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arXiv:hep-th/9307102 · Journal of Physics A Mathematical and General

Wigner quantum oscillators

T. D. Palev, Nedialka I. Stoilova

Published Feb 7, 1994• 18 citations
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Abstract

We present three groups of non-canonical quantum oscillators. The position and momentum operators of each group generate basic Lie superalgebras, namely sl(1/3), osp(1/6) and osp(3/2). The sl(1/3) oscillators have finite energy spectrum and finite dimensions. The osp(1/6) oscillators are related to the para-Bose statistics. The internal angular momentum s of the osp(3/2) oscillators takes no more than three (half)integer values. In a particular representation s=1/2.

DOI
10.1088/0305-4470/27/3/035

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After a general introduction to Wigner Quantum Systems, we define the three-dimensional n-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra sl(1 vertical bar 3n). In this framework, coordinates, momentum and the angular momentum of the particles are defined and investigated. This investigation is done in state spaces, using representation theory of sl(1 vertical bar 3n). For the case n = 1, the main properties are listed, with an emphasis on the non-commutative coordinates of the particle and its unconventional consequences. For general n, we study energy spectra, angular momentum, and in particular the particle configuration and its interpretation. Throughout, a comparison with the canonical oscillator solution is given.

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The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl (1∣3), are further investigated. Within each state space W ( p ), p = 1, 2, ..., the energy E q , q = 0, 1, 2, 3, takes no more than four different values. If the oscillator is in a stationary state ψ q ∊ W ( p ) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called 'nests'. These lie on a sphere centred on the origin of fixed, finite radius ϱ q . The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases ( p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases ( p = 1, 2), but it is never less than 2. In certain states in W (2) (respectively in W (1)) the oscillator is 'polarized' so that all the nests lie on a plane (respectively on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations.

arXiv:hep-th/0304136•Journal of Physics A Mathematical and General•Apr 2, 2003•25 citations
We present three groups of non-canonical quantum oscillators. The position and momentum operators of each group generate basic Lie superalgebras, namely sl(1/3), osp(1/6) and osp(3/2). The sl(1/3) oscillators have finite energy spectrum and finite dimensions. The osp(1/6) oscillators are related to the para-Bose statistics. The internal angular momentum s of the osp(3/2) oscillators takes no more than three (half)integer values. In a particular representation s=1/2.
The properties of the three-dimensional non-canonical oscillators osp(3/2) (introduced in J. Phys. A: Math. Gen. 27 (1994) 977) are studied further. The angular momentum M of the oscillators can take at most three values M=p-1, p, p+1, which are either all integers or all half-integers. The coordinates anticommute with each other. Depending on the state space the energy spectrum can coincide or can be essentially different from those of the canonical oscillator. The ground state is in general degenerate.
After a general introduction to Wigner Quantum Systems, we define the three-dimensional n-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra sl(1 vertical bar 3n). In this framework, coordinates, momentum and the angular momentum of the particles are defined and investigated. This investigation is done in state spaces, using representation theory of sl(1 vertical bar 3n). For the case n = 1, the main properties are listed, with an emphasis on the non-commutative coordinates of the particle and its unconventional consequences. For general n, we study energy spectra, angular momentum, and in particular the particle configuration and its interpretation. Throughout, a comparison with the canonical oscillator solution is given.
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|3 n ) indexed by an A -superstatistics parameter p . These representations are typical for p ⩾ 3 n but atypical for p < 3 n . The branching rules for the restriction from sl (1|3 n ) 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.
A system of NNN non-canonical dynamically free 3D harmonic oscillators is studied. The position and the momentum operators (PM-operators) of the system do not satisfy the canonical commutation relations (CCRs). Instead they obey the weaker postulates for the oscillator to be a Wigner quantum system. In particular the PM-operators fulfil the main postulate, which is due to Wigner: they satisfy the equations of motion (the Hamiltonian's equations) and the Heisenberg equations. One of the relevant features is that the coordinate (the momentum) operators do not commute, but instead their squares do commute. As a result the space structure of the basis states corresponds to pictures when each oscillating particle is measured to occupy with equal probability only finite number of points, typically the eight vertices of a parallelepiped. The state spaces are finite-dimensional, the spectrum of the energy is finite with equally spaced energy levels. An essentially new feature is that the angular momenta of all particles are aligned. Therefore there exists a strong interaction or correlation between the particles, which is not of dynamical, but of statistical origin. Another relevant feature is that the standard deviations of, say, the kkk-th coordinate and the momenta of ααα-th is $ Δ\hat R_{αk} Δ\hat P_{αk} \le {p \hbar/{|N-3|}}~(~N\ne 3,~ p -$fixed positive integer), namely instead of uncertainty relations one has "certainty" relations. The underlying Lie superalgebraic structure of the oscillator is also relevant and will be explained in the context.
The properties of a non-canonical 3D Wigner quantum oscillator, whose position and momentum operators generate the Lie superalgebra sl (1∣3), are further investigated. Within each state space W ( p ), p = 1, 2, ..., the energy E q , q = 0, 1, 2, 3, takes no more than four different values. If the oscillator is in a stationary state ψ q ∊ W ( p ) then measurements of the non-commuting Cartesian coordinates of the particle are such that their allowed values are consistent with it being found at a finite number of sites, called 'nests'. These lie on a sphere centred on the origin of fixed, finite radius ϱ q . The nests themselves are at the vertices of a rectangular parallelepiped. In the typical cases ( p > 2) the number of nests is 8 for q = 0 and 3, and varies from 8 to 24, depending on the state, for q = 1 and 2. The number of nests is less in the atypical cases ( p = 1, 2), but it is never less than 2. In certain states in W (2) (respectively in W (1)) the oscillator is 'polarized' so that all the nests lie on a plane (respectively on a line). The particle cannot be localized in any one of the available nests alone since the coordinates do not commute. The probabilities of measuring particular values of the coordinates are discussed. The mean trajectories and the standard deviations of the coordinates and momenta are computed, and conclusions are drawn about uncertainty relations.