On the uniqueness of the Weyl Correspondence as an Invariant Quantization

In this article we propose a new set of requirements for a quantization of the Poisson algebra of a manifold, based on Hamiltonian group actions. The conditions are analogous to those used in deformation quantization to define an invariant *-product, cast in a gauge theory-like form. We discuss the quantization of the plane equipped with Poisson brackets, without assuming any further structure, and show that the symplectic affine group and certain subgroups of it yield the Weyl correspondence as a unique quantization satisfying the new conditions. These are the only Hamiltonian group actions to yield a unique quantization within a large class of Hamiltonian actions. Thus, there is a natural choice of both group action and quantization for the plane. This is a possible solution to tile classical problem of arbitrariness in the choice of a quantization of a Poisson algebra (in this case, for the plane). We also analyze the most general form of so(3)-invariant quantizations of the functions on S2. (author) 16 refs