Generalized Weyl-Wigner-Moyal Formalism and Topological Groups

TL;DR

This paper explores the geometric structure of phase space in quantum mechanics, linking Weyl-Wigner-Moyal formalism with group theory, and extends the framework to locally compact Abelian groups, with applications to crystal lattice dynamics.

Contribution

It introduces a generalization of quantum phase space formalism to locally compact Abelian groups using Pontryagin duality, connecting geometric and algebraic aspects of quantum mechanics.

Findings

Established links between phase space formalism and group geometry

Extended formalism to quantum systems on Abelian groups

Discussed physical implications for crystal lattice quantum dynamics

Abstract

Discussed are some geometric aspects of the phase space formalism in quantum mechanics in the sense of Weyl, Wigner, Moyal and Ville. We analyze the relationship between this formalism and geometry of the Galilei group, classical momentum mapping, theory of unitary projective representations of groups, and theory of groups algebras. Later on we present some generalization to quantum mechanics on locally compact Abelian groups. It is based on Pontryagin duality. Indicated are certain physical aspects in quantum dynamics of crystal lattices, including the phenomenon of "Umklapp-Prozessen"