Star products and their Hilbert space mirrors

Abstract: We discuss the Cahill-Glauber (CG) $s$-parameterized representation, a framework in the phase space formulation of quantum mechanics. We present the differential form of the hatted star product $\mathbin{\widehat{\star}_s}$, which emerges when we consider the inverse Cahill-Glauber (iCG) transform of a product of phase space functions. It appears to be the Hilbert space mirror of the usual star product $\mathbin{\star_s}$, and supports the interpretation of classical mechanics as the irreducible deformation of quantum mechanics by decoherence. Additionally, alternative forms of the evaluation of $\mathbin{\star_s}$ and $\mathbin{\widehat{\star}_s}$ are considered.