Observables of angular momentum as observables on the Fedosov quantized sphere

Abstract

In this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the two-sphere S2 by implementing the Fedosov ∗-formalism. The Fedosov ∗ is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov ∗-formalism; essentially an undeformation of Fedosov’s algorithm.