Fedosov observables on constant curvature manifolds and the Klein–Gordon equation

In this paper we construct the Fedosov star-algebra of observables on the phase–space of a single particle in the case of all (finite-dimensional) constant curvature manifolds imbeddable in a flat space with codimension one. This set of spaces includes the two-sphere and de Sitter (dS)/Anti-de Sitter (AdS) space–times. The algebra of observables was constructed by DQ techniques using, in particular, the algorithm provided by Fedosov.The purpose of this paper was three-fold. One was to verify that DQ gave the same results as previous analyses of these spaces. Another was to verify that the formal series used in the conventional treatment converged by obtaining exact and nonperturbative results for these spaces. The last was to further develop and understand the technology of the Fedosov algorithm.