Geometric Models of the Quantum Relativistic Rotating Oscillator

A family of geometric models of quantum relativistic rotating oscillator is defined by using a set of one-parameter deformations of the static (3+1) de Sitter or anti-de Sitter metrics. It is shown that all these models lead to the usual isotropic harmonic oscillator in the non-relativistic limit, even though their relativistic behavior is different. As in the case of the (1+1) models [1], these will have even countable energy spectra or mixed ones, with a finite discrete sequence and a continuous part. In addition, all these spectra, except that of the pure anti-de Sitter model, will have a fine-structure, given by a rotator-like term. In the general relativity the relativistic (three-dimensional isotropic) harmonic oscillator (RHO) is defined as a free system on the anti-de Sitter static background [2, 3, 4, 5, 6]. There exists a metric [3] which reproduces the classical equation of motion of the non-relativistic (isotropic) harmonic oscillator (NRHO). Moreover, the corresponding quantum system represented by a free scalar field on this background has an equidistant energy spectrum with a ground state energy larger than, but approaching 3ω/2 in the non-relativistic limit (in natural units, h̄ = c = 1) [7]. Thus, the static anti-de Sitter classical and quantum geometric models reproduce all the properties of the NRHO