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, $\Psi \in \mathcal{H}$): 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.

17 pages

References in corpus (2)

• Fuzzy de Sitter Space
• Discrete fuzzy de Sitter cosmology

Cited by in corpus (1)

• QFT on Fuzzy AdS Spaces: Classical Limit and Boundary Correlation Functions