The non-commutative and discrete spatial structure of a 3D Wigner quantum oscillator

Abstract

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 4 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 centered on the origin of fixed, finite radius ̺ 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 two. In certain states in W (2) (resp. in W (1)) the oscillator is "polarized" so that all the nests lie on a plane (resp. 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. †

Harmonic oscillators coupled by springs: Discrete solutions as a Wigner quantum system

“…Consider the eigenvectors ψ r,x,g of the position operatorq r for the eigenvalue x expanded in the w(θ)-basis: 18) and assume that these vectors are orthonormal, i.e.…”

“…This algebraic or representation theoretic approach to quantum systems has revived the interest in WQS's [13,14,15]. The WQS approach has so far been applied to simple systems of free harmonic oscillators, with some interesting and surprising results [16,17,18,19,20]. Here it is, for the first time, applied to a more realistic quantum system.…”

Representations of the Lie superalgebra in a Gel'fand–Zetlin basis and Wigner quantum oscillators

“…In previous papers, an investigation was made of the physical properties of the gl(1|DN ) solutions in the Fock representation spaces W (p), see [5,8] for D = 3, N = 1 (the single particle 3-dimensional WQO) and [7,9] for D = 3 and N arbitrary (the last case corresponding to a superposition of N single particle 3-dimensional WQOs). The most striking properties are as follows: the energy of each particle has at most four different eigenvalues (equidistant energy levels); the geometry is non-commutative, in the sense that coordinate operators do not commute; the position and momentum operators have discrete spectra.…”

Representations of the Lie Superalgebra osp(1|2n) with Polynomial Bases

“…Parabosons were introduced by Green [11] in 1953, as generalizations of bosons. Parabosons have been of interest in quantum field theory [25], in generalizations of quantum statistics [13,2] and in Wigner quantum systems [17,19]. Whereas creation and annihilation operators of bosons satisfy simple commutation relations, those of parabosons satisfy more complicated triple relations.…”