Characterizing the Kirkwood-Dirac positivity on second countable LCA groups

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.

💡 Research Summary

The paper develops a comprehensive theory of the Kirkwood‑Dirac (KD) quasiprobability representation for quantum systems whose configuration space is a second‑countable locally compact abelian (SCLCA) group (G). Starting from the well‑known finite‑dimensional and (\mathbb R^{n}) cases, the authors extend the definition of the KD distribution to arbitrary SCLCA groups by introducing “generalized operators” acting on the Schwartz‑Bruhat space (S(G)) with distributional kernels in (S’(G\times G)). For a generalized operator (A) with kernel (k_{A}), the KD distribution is defined as a tempered distribution on the phase space (G\times\widehat G): \