On the Associativity of Star Product in Systems with Nonlinear Constraints

The noncommutative star product of phase space functions is, by construction, associative for both non-degenerate and degenerate case (involving only second class constraints) as has been shown by Berezin, Batalin and Tyutin. However, for the latter case, the manifest associativity is lost if an arbitrary coordinate system is used but can be restored by using an unconstrained canonical set. The existence of such a canonical transformation is guaranteed by a theorem due to Maskawa and Nakajima. In terms of these new variables, the Kontsevich series for the star product reduces to an exponential series which is manifestly associative. We also show, using the star product formalism, that the angular momentum of a particle moving on a circle is quantized.