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arXiv:hep-th/9405125 · Journal of Physics A Mathematical and General

Wigner quantum oscillators. osp(3/2) oscillators

T. D. Palev, Nedialka I. Stoilova

Published Nov 21, 1994• 20 citations
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Abstract

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.

DOI
10.1088/0305-4470/27/22/014

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Journal of Physics Conference Series•Dec 6, 2011•4 citations
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.
Abstract. After introducing Wigner Quantum Systems, we give a short review of the one-dimensional Wigner Quantum Oscillator. Then we define the three-dimensional N-particle Wigner Quantum Oscillator, and its relation to the Lie su-peralgebra sl(1|3N). In this framework (and first for N = 1), energy, coordinates, momentum and the angular momentum of the particles are investigated. Wigner Quantum Systems (WQSs) are quantum systems in which the canonical com-mutation relations (CCRs) are replaced by a compatibility condition between the Heisenberg equations and Hamilton’s equations. By dropping the CCRs, WQSs offer a natural framework for quantum mechanics with non-commutative coordinates. Here we consider in particular the three-dimensional N-particle Wigner Quantum Oscillator (WQO). As an introductory ex-ample, some properties of the one-dimensional WQO are reviewed, with an emphasis on the different particle probability distributions (as compared to the ordinary one-dimensional os-cillator). For the three-dimensional N-particle WQO, a solution related to the Lie superalge-bra sl(1|3N) is considered. For N = 1 we investigate the operators corresponding to coordi-nates, momentum and angular momentum of the particle in a class of representation spaces.
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.
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.
We study the quantum mechanical harmonic oscillator in two and three dimensions, with particular attention to the solutions as basis states for representing their respective symmetry groups — O(2), O(1,1), O(3), and O(2,1). The goal of this study is to establish a correspondence between Hilbert space descriptions found by solving the Schrodinger equation in polar coordinates, and Fock space descriptions constructed by expressing the symmetry operators in terms of creation/annihilation operators. We obtain wavefunctions characterized by a principal quantum number, the group Casimir eigenvalue, and one group generator whose eigenvalue is m + s , for integer m and real constant parameter s . For the three groups that contain O(2), the solutions split into two inequivalent representations, one associated with s = 0, from which we recover the familiar description of the oscillator as a product of one-dimensional solutions, and the other with s > 0 (in three dimensions, solutions are found for s = 0 and s = 1/2) whose solutions are non-separable in Cartesian coordinates, and are hence overlooked by the standard Fock space approach. The O(1,1) solutions are singlet states, restricted to zero eigenvalue of the symmetry operator, which represents the boost, not angular momentum. For O(2), a single set of creation and annihilation operators forms a ladder representation for the allowed oscillator states for any s, and the degeneracy of energy states is always finite. However, in three dimensions, the integer and half-integer eigenstates are qualitatively different: the former can be expressed as finite dimensional irreducible tensors under O(3) or O(2,1) while the latter exhibit infinite degeneracy. Creation operators that produce the allowed integer states by acting on the non-degenerate ground state are constructed as irreducible tensor products of the fundamental vector representation. However, the half-integer eigenstates are infinite-dimensional, as expected for the non-compact O(2,1), and no Fock space description is found for this case. For all s ≠ 0 solutions, the SU(N) symmetry of the harmonic oscillator Hamiltonian recently discovered by Bars is spontaneously broken by the ground state. The connection of this symmetry breaking to the non-separability into one-dimensional Cartesian solutions is demonstrated. We conclude with a discussion of the O(2,1) solutions, and the problem of timelike ghost excitations.