Fedosov Observables on Constant Curvature Manifolds and the Klein-Gordon Equation

TL;DR

This paper constructs quantum observables for a free particle on constant curvature manifolds using Fedosov's formalism, identifies the symmetry algebra as SO(p+1,q+1), and formulates the Klein-Gordon equation in this geometric setting.

Contribution

It extends Fedosov quantization to constant curvature manifolds and derives the algebra of observables and Klein-Gordon equation in this context.

Findings

Algebra of observables is SO(p+1,q+1)

Identifies the symmetry subgroup analogous to Poincare group

Formulates Klein-Gordon equation consistent with previous AdS results

Abstract

In this paper we construct the set of quantum mechanical observables in the Fedosov *-formalism (a coordinate invariant way to do quantum mechanics on any manifold M) of a single free particle that lives on a constant curvature manifold with metric signature (p,q). This was done for most but not all constant curvature manifolds. We show that the algebra of all observables in n=p+q dimensions is SO(p+1,q+1) in a nonperturbative calculation. A subgroup of this group is identified as the analogue of the Poincare group in Minkowski space i.e. it is the space of symmetries on the manifolds considered. We then write down a Klein-Gordon (KG) equation given by the the equation p^2|phi>=m^2|phi> for the set of allowed physical states. This result is consistent with previous results on AdS. Furthermore we lay out the standard scheme for the free KG field from the single particle theory. Furthermore we argue that this scheme will work on a general space-time.