Can Star Products be Augmented by Classical Physics?

TL;DR

This paper critically examines the possibility of augmenting star products in phase-space quantization with classical effects, concluding that such augmentation is generally not feasible with Hermitian star products.

Contribution

It provides a rigorous analysis showing that augmented quantization via classical effects cannot be achieved with Hermitian star products, and explores the limitations of local transition operators.

Findings

Augmentation of star products with classical effects is impossible with Hermitian products.

Known examples of classical augmentation do not generalize to a consistent classical limit.

Local transition operators are limited in validity and practicality for augmented quantization.

Abstract

It has been suggested that star products in phase-space quantization may be augmented to describe additional, classical effects. That proposal is examined critically here. Two known star products that introduce classical effects are: the generalized Husimi product of coarse-grained quantization, and a non-Hermitian damped star product for the harmonic oscillator. Following these examples, we consider products related by transition differential operators to the classic Moyal star product. We restrict to Hermitian star products, avoiding problems already pointed out for the original damped product. It is shown, however, that with such star products, augmented quantization is impossible, since an appropriate classical limit does not result. For a more complete study, we then also consider generalized, or local, transition operators, that depend on the local phase-space coordinates, as well as their derivatives. In this framework, one example of possible physical interest is constructed. Because of its limited validity and complicated form, however, it cannot be concluded that augmented quantization with local transition operators is practical.