Supmech: the Geometro-statistical Formalism Underlying Quantum Mechanics

TL;DR

Supmech is a comprehensive algebraic framework combining noncommutative geometry and probability theory to unify quantum and classical mechanics, providing new insights into quantum measurements and the quantum-classical transition.

Contribution

It introduces a novel formalism called supmech that integrates noncommutative symplectic geometry with probability theory, unifying quantum and classical mechanics within an algebraic approach.

Findings

Derivation of the von Neumann reduction rule

Smooth connection between quantum and classical systems via superalgebras

Inclusion of superselection rules and noncommutative symplectic objects

Abstract

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.