Quantum Theory on Lobatchevski Spaces

TL;DR

This paper develops a formalism for quantum theories on Lobatchevski spaces with negative curvature, using a basis of eigenfunctions of the Laplace-Beltrami operator, simplifying previous complexities and enabling applications to cosmological models.

Contribution

It introduces a specialized basis of plane waves for quantum mechanics on Lobatchevski spaces, adapting mathematical group theory tools for physical applications.

Findings

Constructed a basis of eigenfunctions for the Laplace-Beltrami operator.

Simplified the analysis of quantum theories on negatively curved spaces.

Applied the formalism to Milne and de Sitter universes.

Abstract

In this paper we set up a general formalism to deal with quantum theories on a Lobatchevski space, i.e. a spatial manifold that is homogeneous, isotropic and has negative curvature. The heart of our approach is the construction of a suitable basis of plane waves which are eigenfunctions of the Laplace-Beltrami operator relative to the geometry of the curved space. These functions were previously introduced in the mathematical literature in the context of group theory; here we revisit and adapt the formalism in a way specific for quantum mechanics. Our developments render dealing with Lobatchevski spaces, which used to be quite difficult and source of controversies, easily tractable. Applications to the Milne and de Sitter universes are discussed as examples.