Quantum Scalar Field on Fuzzy de Sitter Space I. Field Modes and Vacua

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 $S^2\times W$, where $W$ is a discrete matrix space. Our results pave the way to analysis of quantum field theory on the fuzzy de~Sitter space.