Laplacian on fuzzy de Sitter space

Abstract

We study details of geometry of noncommutative de Sitter space: we determine the Riemann and Ricci curvature tensors, the energy and the Laplacian. We find, in particular, that fuzzy de Sitter space is an Einstein space, R ab = -3ζη ab . The Laplacian, defined in the noncommutative frame formalism, is not Hermitian and gives nonunitary evolution. When symmetrically ordered, it has the usual quadratic form Δ= Π a Π a (when acting on functions in representation space, ψϵH ): we find its eigenstates and discuss its spectrum. This result is a first step in a study of the scalar field Laplacian, Δ= [Π a , [Π a , ]], and its propagator.