Canonical quantization of a product within the Cahill-Glauber correspondence

The (canonical) quantization of a product of two phase-space functions can be expressed as a commutative mapping between the quantizations of its factors.

In this work, we present the differential form of such a mapping within the Cahill-Glauber $s$-parameterized correspondence framework.

We term this mapping the ``hatted star product'' by virtue of its structural similarity to the star product in phase space, a special case of which is the celebrated Moyal star product.