Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

• URL: http://arxiv.org/abs/2507.23628v1
• Date: Thu, 31 Jul 2025 15:08:39 GMT
• Title: Characterizing the Kirkwood-Dirac positivity on second countable LCA groups
• Authors: Matéo Spriet,
• Abstract summary: We define the Kirkwood-Dirac representation of quantum mechanics associated with the Fourier transform over second countable locally compact abelian groups.<n>We 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 connected component.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 connected component. Finally, we provide for connected compact abelian groups a complete geometric description of this classical fragment.

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