  Forty-five years after the point de d\'epart [1] of density functional
theory, its applications in chemistry and the study of electronic structures
keep steadily growing. However, the precise form of the energy functional in
terms of the electron density still eludes us -- and possibly will do so
forever [2]. In what follows we examine a formulation in the same spirit with
phase space variables. The validity of Hohenberg-Kohn-Levy-type theorems on
phase space is recalled. We study the representability problem for reduced
Wigner functions, and proceed to analyze properties of the new functional.
Along the way, new results on states in the phase-space formalism of quantum
mechanics are established. Natural Wigner orbital theory is developed in depth,
with the final aim of constructing accurate correlation-exchange functionals on
phase space. A new proof of the overbinding property of the Mueller functional
is given. This exact theory supplies its home at long last to that illustrious
ancestor, the Thomas-Fermi model.
