Quantum Theory on Lobatchevski Spaces

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.