On the classical limit of the hyperbolic quantum mechanics

TL;DR

This paper explores hyperbolic quantum mechanics as a deformation of classical mechanics, showing it arises alongside the well-known quantum deformation and analyzing its mathematical structure and interference phenomena.

Contribution

It introduces hyperbolic quantum mechanics as a new deformation of classical mechanics and develops the mathematical framework using hyperbolic algebra and calculus.

Findings

Classical Poisson bracket is obtainable as a limit of hyperbolic Moyal bracket.

Hyperbolic quantum mechanics features hyperbolic cosine interference.

Two types of interference perturbations distinguish hyperbolic from ordinary 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.