Non viability of hyperbolic quantum mechanics as a theory of Nature

TL;DR

This paper investigates hyperbolic quantum mechanics, which uses split-complex numbers, and demonstrates that it cannot serve as a valid physical theory because only quantum and classical mechanics remain consistent under composition.

Contribution

The paper introduces a generalization of C*-algebras for hyperbolic quantum mechanics and proves its incompatibility as a physical theory, highlighting the uniqueness of quantum and classical mechanics.

Findings

Hyperbolic quantum mechanics is not invariant under tensorial composition.

Only quantum and classical mechanics are consistent under composition.

Hyperbolic structures do not yield physically viable theories.

Abstract

Quantum and classical mechanics share a common algebraic formalism which is expressed naturally in the language of category theory. A third realization of this formalism is the so-called hyperbolic quantum mechanics where split-complex numbers replace the usual complex numbers. We introduce and explore the corresponding generalization of C*-algebras and prove that hyperbolic quantum mechanics is not a viable candidate for describing Nature. Quantum and classical mechanics are the only acceptable theories of Nature which are invariant under tensorial composition.