THE DYNAMICAL NOETHER SYMMETRIES OF A BOSONIC q-OSCILLATOR

The classical dynamical (phase space) Noether symmetries which correspond to the quantum, one-dimensional, bosonic, deformed "Biedenharn–Macfarlane q-oscillator" as defined by V. I. Man'ko and others, are given for small values of the parameter q by considering the model as a nondeformed theory with a highly nonlinear but of small strength interaction. For this nonconstrained one-degree of freedom system and by applying Noether's procedure in the form established by Katzin and Levine for velocity dependent transformations, we found the corresponding two functionally independent phase space first integrals. These classical integrals, as we explicitly prove, lead to a finitely generated infinite Poisson bracket dynamical algebra of first integrals which generalizes a recently obtained Noether dynamical algebra of the nondeformed harmonic oscillator system. We also show that a subalgebra of that infinite dynamical algebra, after quantization of the small-q classical model here proposed, corresponds exactly to the small deformation limit of the deformed quantum spectrum generating algebra su q(1,1) previously obtained for the q-oscillator system, on purely quantum grounds, by Kulish and Damaskinsky.