Quantum scalar field on fuzzy de Sitter space. Part I. Field modes and vacua

Abstract

Abstract We study a scalar field on a noncommutative model of spacetime, the fuzzy de Sitter space, which is based on the algebra of the de Sitter group SO(1, d) and its unitary irreducible representations. We solve the Klein-Gordon equation in d = 2, 4 and show, using a specific choice of coordinates and operator ordering, that all commutative field modes can be promoted to solutions of the fuzzy Klein-Gordon equation. To explore completeness of this set of modes, we specify a Hilbert space representation and study the matrix elements (integral kernels) of a scalar field: in this way the complete set of solutions of the fuzzy Klein-Gordon equation is found. The space of noncommutative solutions has more degrees of freedom than the commutative one, whenever spacetime dimension is d > 2. In four dimensions, the new non-geometric, internal modes are parametrised by S 2 × W, where W is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de Sitter space.

Keywords

• de Sitter space
• Non-Commutative Geometry
• Scale and Conformal Symmetries