Nonnegativity of Wigner quasi-probability distribution over 2-regular
locally compact abelian groups

The Wigner quasi-probability distribution, or Wigner function, plays a
central role in linking operator-theoretic formulations of quantum
mechanics with phase-space analysis. The celebrated Hudson theorem
states that the Wigner function on Euclidean space is nonnegative (thus
representing a genuine probability distribution) if and only if the
corresponding wave function is Gaussian. In this talk, we introduce a
general construction of the Wigner distribution associated with a
locally compact abelian (LCA) group. Based on recent works
(arXiv:2204.08162, arXiv:2507.13154), we show that Hudson's theorem can
be extended to 2-regular LCA groups that are 2-measure-preserving and
contain a compact open subgroup. This extension in particular suggests
and justifies a natural notion of "gaussianity" over 2-regular LCA
groups. We further discuss how these results may be generalized to
arbitrary 2-regular LCA groups.