On the Usefulness of Modulation Spaces in Deformation Quantization

TL;DR

This paper explores how modulation spaces, especially weighted Sjöstrand classes, are useful in deformation quantization, focusing on their application to the Moyal star-product and star-exponential in quantum physics.

Contribution

It highlights the relevance of modulation spaces in deformation quantization, a connection not widely recognized in physics, and analyzes their suitability for pseudo-differential operators.

Findings

Modulation spaces are effective symbol classes for pseudo-differential operators.

They are particularly well-suited for studying the Moyal star-product.

The paper bridges a gap between time-frequency analysis and quantum physics.

Abstract

We discuss the relevance to deformation quantization of Feichtinger's modulation spaces, especially of the weighted Sjoestrand classes. These function spaces are good classes of symbols of pseudo-differential operators (observables). They have a widespread use in time-frequency analysis and related topics, but are not very well-known in physics. It turns out that they are particularly well adapted to the study of the Moyal star-product and of the star-exponential.