Inspired by a recent interpretation of the Husimi function in terms of Gaussian semigroups and its connection to the Wigner distribution, we begin the talk by presenting a generalized approach for constructing positive representations of quantum states with respect to a set of quantum observables. The main focus of the talk is to then view each representation as a smearing of its corresponding Wigner distribution, in complete analogy with the Gaussian smearing that gives rise to the Husimi function. A key step towards this is studying the existence of a Weyl symbol for the Gaussian semigroup that would give rise to such a smearing, hence the title of the talk. Relevant concepts that will be covered include the Weyl functional calculus for unbounded self-adjoint operators and infinitesimal representations of Lie groups.
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