Classical Implications of the Quantum Statistical Interpretation

Abstract

Regarding Quantum Mechanics as the fundamental theory we derive, from its underlying geometry, a complete version of Classical Mechanics, including: the interpretation of the phase space as the manifold of classical states; the Wigner-Moyal correspondance rule, now obtained from first principles; the Jordan and Lie structures of the algebra of classical observables, given by the point-wise multiplication and the Poisson bracket of functions on the cotangent bundle of the configuration manifold. The geometric formulation of the Dirac problem followed here allows to extend the solution which we propose to cases where the configuration space is an homogeneous Riemannian manifold. In case this manifold is not flat, the theory provides a natural distinction between the momenta and the Darboux coordinates relative to the symplectic form associated to the Poisson bracket. For instance, the momentum map is explicitly derived for the case where the configuration space is a Riemannian manifold of constant negative curvature. Finally, the connection between our approach and the general geometric quantization programme is discussed.