Quantum harmonic analysis on locally compact groups

TL;DR

This paper develops a framework for quantum harmonic analysis on locally compact groups, introducing covariant quantization, phase space representations, and localization operators that generalize classical cases like the affine and Heisenberg groups.

Contribution

It extends quantum harmonic analysis to all locally compact groups using new convolution methods between functions and operators, including non-unimodular groups.

Findings

Introduces covariant quantization schemes for locally compact groups.

Generalizes phase space representations and localization operators.

Defines convolutions between functions and trace class operators for these groups.

Abstract

On a locally compact group we introduce covariant quantization schemes and analogs of phase space representations as well as mixed-state localization operators. These generalize corresponding notions for the affine group and the Heisenberg group. The approach is based on associating to a square integrable representation of the locally compact group two types of convolutions between integrable functions and trace class operators. In the case of non-unimodular groups these convolutions only are well-defined for admissible operators, which is an extension of the notion of admissible wavelets as has been pointed out recently in the case of the affine group.