The N-particle Wigner Quantum Oscillator: non-commutative coordinates and physical properties

After introducing Wigner Quantum Systems, we give a short review of the one-dimensional Wigner Quantum Oscillator. Then we define the threedimensional N-particle Wigner Quantum Oscillator, and its relation to the Lie superalgebra 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 commutation 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 example, 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 oscillator). For the three-dimensional N-particle WQO, a solution related to the Lie superalgebra sl(1|3N) is considered. For N = 1 we investigate the operators corresponding to coordinates, momentum and angular momentum of the particle in a class of representation spaces. Remarkable properties include the discrete spectrum of coordinate operators and their noncommutativity (and the same for the momentum operators). This has some unconventional consequences for the particle localisation. We compare some of these properties with those of the canonical quantum oscillator (CQO). Finally, the case for general N is described. Let Ĥ be the Hamiltonian of a system expressed in terms of the coordinates and momenta. This system is a WQS [1, 2] if all postulates of ordinary quantum mechanics hold, i.e. P1 The state space W is a Hilbert space. To every physical observable O there corresponds a Hermitian (self-adjoint) operator Ô acting in W .