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

TL;DR

This paper explores the properties of a 3D Wigner quantum oscillator with non-commutative coordinates, revealing discrete spatial structures called nests, and analyzes measurement probabilities, trajectories, and uncertainty relations.

Contribution

It introduces a novel non-canonical 3D Wigner quantum oscillator model based on the Lie superalgebra sl(1|3), detailing its discrete spatial structure and measurement properties.

Findings

Energy levels are limited to four values within each state space.

Particle positions are confined to finite sets of nests on a sphere.

The model exhibits unique polarization states and non-commutative coordinate behavior.

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 \psi_q\in 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 \varrho_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.