Noncanonical quantization of two particles interacting via a harmonic potential

Following the ideas of Wigner a non-canonical quantization of a system of two non-relativistic point particles, interacting via a harmonic potential is studied. The center-of-mass phase-space variables are quantized in a canonical way, whereas the internal momentum and the coordinates are assumed to be operators, generating finite-dimensional representations of the Lie superalgebra A(0, 2). It turns out that the operators of the internal Hamiltonian, the relative distance, the internal momentum and the orbital momentum commute with each other. The spectrum of these operators is finite. In particular the distance between the particles is preserved in time and can have four different values so that the particles are confined. Every coordinate operator can be diagonalized, however, the position of the particles cannot be localized, since the operators of the Cartesian cooordinates do not commute. The angular momentum of the system can be either zero or one (in units h/2π/2)