Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

TL;DR

This paper explores the properties of the Kirkwood-Dirac quasiprobability distribution in quantum mechanics on second countable locally compact abelian groups, linking it with phase space quantization and characterizing positive states.

Contribution

It introduces a generalized framework for Kirkwood-Dirac distributions on LCA groups, characterizes positive pure states, and describes the classical fragment of quantum mechanics in this setting.

Findings

Positive pure states are Haar measures on closed subgroups, up to Weyl-Heisenberg action.

The classical fragment exists only if the group has a compact identity component.

Complete geometric description of the classical fragment for connected compact abelian groups.

Abstract

We define the Kirkwood-Dirac quasiprobability representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups. We discuss its link with the Kohn-Nirenberg quantization of the phase space $G\times \widehat{G}$. We use it to argue that in this abstract setting the Wigner-Weyl quantization, when it exists, can still be interpreted as a symmetric ordering. Then, we identify all generalized (non-normalizable) pure states having a positive Kirkwood-Dirac distribution. They are, up to the natural action of the Weyl-Heisenberg group, Haar measures on closed subgroups. This generalizes a result known for finite abelian groups. We then show that the classical fragment of quantum mechanics associated with the Kirkwood-Dirac distribution is non-trivial if and only if the group has a compact identity component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.