A system of $N$ 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 $k$-th coordinate and the momenta of $α$-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.
A system of N 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 k-th coordinate and the momenta of α-th is ΔR^αkΔP^αk≤pℏ/∣N−3∣(N=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.
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.
The properties of a noncanonical 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 \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. The rotational invariance of the system is also discussed.
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.
In quantum mechanics, physical observables are represented by operators on a certain Hilbert space. The question of how such operators commute, has been a matter of discussion. In the standard perspective, the operators corresponding to the position and momentum of a system are assumed to satisfy the canonical commutation relations. It is known that these relations imply that the Hamilton and Heisenberg equations of motion are compatible as operator equations. However, Wigner showed that the inverse statement is not true. Therefore, it is a much weaker constraint to impose the compatibility of the equations of motion. For any physical system, this results in a set of compatibility conditions, which form the core of Wigner quantization. The key to finding operators that are subject to the compatibility conditions is provided by Lie superalgebras and their representations. Lie superalgebras can be defined as algebras generated by odd elements satisfying particular superbracket relations. Using these defining relations, Lie superalgebra generators can be found that obey the compatibility conditions. The Lie superalgebra elements act as operators on a vector space if we consider Lie superalgebra representations. Various physical systems are investigated in this thesis in the context of Wigner quantization. For each of these systems solutions are found in terms of Lie superalgebra generators, after which specific representations are examined. In such representations, the focus lies on determining particular physical properties of the system. Most of the studied systems are harmonic oscillator models. First, we examine a set-up of coupled harmonic oscillators, for which the interaction is represented by an interaction matrix. Then we focus on the angular momentum content of a 3N-dimensional harmonic oscillator. Finally, our attention goes to two one-dimensional systems, namely the free particle and the Berry-Keating-Connes Hamiltonian. The latter of these Hamiltonians is notorious for its possible connection with the Riemann hypothesis. All of the aforementioned Hamiltonians have been extensively investigated in the context of canonical quantization, so that our results can be compared to the well-known canonical case.
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.