The dispersion relation for matter waves in a two-phase vacuum

Abstract

The cosmological constant Λ of general relativity is a natural consequence of embedding Einstein's theory in a five-dimensional (5D) theory of the type needed for unification. The exact 5D solution for Λ<0 shows waves in ordinary three-dimensional (3D) space with properties similar to those of de Broglie or matter waves. Here the dispersion relation is derived for matter waves in a toy two-phase model, where regions with Λ<0 and Λ>0 average on the large scale to Λ≃0, thus providing in principle a resolution of the cosmological-constant problem. A striking result of the analysis is that the dispersion relation is bimodal, with a well-defined window of high-frequency transmission which effectively defines the speed of light.