On the classical limit of the hyperbolic quantum mechanics

Abstract

We demonstrated that classical mechanics have, besides the well known quantum deformation, another deformation -- so called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit $h\to 0$ not only of the ordinary Moyal bracket, but also hyperbolic analogue of the Moyal bracket. Thus there are two different deformations of classical phase-space: complex Hilbert space and hyperbolic Hilbert space (module over a so called hyperbolic algebra -- the two dimensional Clifford algebra). To prove the correspondence principle we use the calculus over the hyperbolic algebra similar to functional superanalysis of Vladimirov-Volovich. Ordinary (complex) and hyperbolic quantum mechanics are characterized by two types of interference perturbation of the classical formula of total probability: ordinary $\cos$-interference and hyperbolic $\cosh$-interference.