Star products on symplectic vector spaces: Convergence, representations, and extensions

We briefly survey the general scheme of deformation quantization on symplectic vector spaces and analyze its functional analytic aspects. We treat different star products in a unified way by systematically using an appropriate space of analytic test functions for which the series expansions of the star products in powers of the deformation parameter converge absolutely. The star products are extendable by continuity to larger functional classes. The uniqueness of the extension is guaranteed by suitable density theorems. We show that the maximal star product algebra with the absolute convergence property, consisting of entire functions of an order at most $2$ and minimal type, is nuclear. We obtain an integral representation for the star product corresponding to the Cahill–Glauber $s$-ordering, which connects the normal, symmetric, and antinormal orderings continuously as $s$ varies from $1$ to $-1$. We exactly characterize those extensions of the Wick and anti-Wick correspondences that are in line with the known extension of the Weyl correspondence to tempered distributions.