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, $R_{ab}=-3\zeta\,\eta_{ab}$. The Laplacian, defined in the noncommutative frame formalism, is not hermitian and gives nonunitary evolution. When symmetrically ordered, it has the usual quadratic form $\Delta=\Pi_a\Pi^a$ (when acting on functions in representation space, $\Psi\in {\cal H}$): we find its eigenstates and discuss its spectrum. This result is a first step in a study of the scalar field Laplacian, $\Delta = [\Pi_a, [\Pi^a,\ ]]$, and its propagator.