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.
Using Connes distance formula in noncommutative geometry, it is possible to retrieve the Euclidean distance from the canonical commutation relations of quantum mechanics.In this note, we study modifications of the distance induced by a deformation of the position-momentum commutation relations.We first consider the deformation coming from a cut-off in momentum space, then the one obtained by replacing the usual derivative on the real line with the h-and q-derivatives, respectively.In these various examples, some points turn out to be at infinite distance.We then show (on both the real line and the circle) how to approximate points by extended distributions that remain at finite distance.On the circle, this provides an explicit example of computation of the Wasserstein distance.