Frame transforms, star products and quantum mechanics on phase space

TL;DR

This paper introduces a new mathematical framework using frame transforms and star products to reformulate quantum mechanics on phase space, linking operator products with functions on groups.

Contribution

It develops a class of isometries called tight frame transforms from Hilbert-Schmidt operators to functions on group products, establishing a phase space formulation of quantum mechanics.

Findings

Range of transforms are reproducing kernel Hilbert spaces

Star product mimics operator multiplication at the function level

Links with Wigner and wavelet transforms are established

Abstract

Using the notions of frame transform and of square integrable projective representation of a locally compact group $G$, we introduce a class of isometries (tight frame transforms) from the space of Hilbert-Schmidt operators in the carrier Hilbert space of the representation into the space of square integrable functions on the direct product group $G\times G$. These transforms have remarkable properties. In particular, their ranges are reproducing kernel Hilbert spaces endowed with a suitable 'star product' which mimics, at the level of functions, the original product of operators. A 'phase space formulation' of quantum mechanics relying on the frame transforms introduced in the present paper, and the link of these maps with both the Wigner transform and the wavelet transform are discussed.