An Exact Formulation of Phase-Space Electronic Structure Theory in One Dimension by Exploiting the Zero Curvature Condition

We show that, for the unique case of a system of electrons and nuclei in one dimension, the $Γ$ operator in phase space electronic structure theory has a zero non-Abelian curl. Using this fact, we are able to show that a phase space electronic structure framework in Wigner space has a one-to-one map