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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2507.23628 |
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| _version_ | 1866912914304139264 |
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| author | Spriet, Matéo |
| author_facet | Spriet, Matéo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23628 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterizing the Kirkwood-Dirac positivity on second countable LCA groups Spriet, Matéo Quantum Physics Mathematical Physics Functional Analysis Group Theory 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. |
| title | Characterizing the Kirkwood-Dirac positivity on second countable LCA groups |
| topic | Quantum Physics Mathematical Physics Functional Analysis Group Theory |
| url | https://arxiv.org/abs/2507.23628 |