Supmech: the Geometro-statistical Formalism Underlying Quantum Mechanics

As the first step in an approach to the solution of Hilbert's sixth problem, a general scheme of mechanics, called `supmech', is developed integrating noncommutative symplectic geometry and noncommutative probability theory in an algebraic framework; it has quantum mechanics (QM) and classical mechanics as special subdisciplines and facilitates an autonomous development of QM and satisfactory treatments of quantum-classical correspondence and quantum measurements (including a straightforward \emph{derivation} of the von Neumann reduction rule). The scheme associates, with every `experimentally accessible' system, a symplectic superalgebra and operates essentially as noncommutative Hamiltonian mechanics incorporating the extra condition that the sets of observables and pure states be mutually separating. The latter condition serves to smoothly connect the algebraically defined quantum systems to ilbert space-based ones; the rigged Hilbert space - based Dirac bra-ket formalism naturally appears. The formalism has a natural place for commutative superselection rules. Noncommutative analogues of objects like the momentum map and the Poincar$\acute{e}$-Cartan form are introduced and some related symplectic geometry developed.