Classical and quantum dynamics of a constrained particle on three-dimensional spaces of constant curvature: An algebraic approach on the superintegrable problem

Abstract

In this study, we discuss a particle which is the constraint on a four-dimensional Euclidean space [Formula: see text]. In this respect and under the framework of Dirac’s approach, we analyze the classical and quantum dynamics of a particle constrained on an embedded surface in supersphere [Formula: see text], which has constant curvature and signature equal to [Formula: see text] and [Formula: see text], respectively. In fact, it is presented that the Cartesian components of momentum and position exhibit a proper description for the motion of the particle embedded in a generalized curved space. Thus, quantization of dynamics for a particle on the hypersurface [Formula: see text] derives an appropriate representation of momentum called geometric momentum as it depends on the mean curvature. The invariance algebra generated by the quantization process is an extension of [Formula: see text] with curvature parameters in both classical and quantum dynamics. Moreover, superintegrability appears in the deformed Hamiltonian [Formula: see text] arising from the Dirac brackets, in which [Formula: see text] of the underlying space behaves as the curvature (deformation) parameter.