Classical and Quantum Mechanics from the universal Poisson-Rinehart algebra of a manifold

TL;DR

This paper constructs a universal algebraic framework that unifies classical and quantum mechanics on a manifold using the Poisson-Rinehart algebra, linking algebraic structures to physical theories.

Contribution

It introduces a universal non-commutative Poisson algebra based on the Lie and module structures of vector fields, unifying classical and quantum mechanics through algebraic quotients.

Findings

Defines a unique universal non-commutative Poisson algebra for manifolds.

Shows how classical and quantum algebras emerge as quotients with specific central variables.

Establishes a correspondence between algebraic states and Hilbert space representations for mechanics.

Abstract

The Lie and module (Rinehart) algebraic structure of vector fields of compact support over C infinity functions on a (connected) manifold M define a unique universal non-commutative Poisson * algebra. For a compact manifold, a (antihermitian) variable Z, central with respect to both the product and the Lie product, relates commutators and Poisson brackets; in the non-compact case, sequences of locally central variables allow for the addition of an element with the same role. Quotients with respect to the (positive) values taken by Z* Z define classical Poisson algebras and quantum observable algebras, with the Planck constant given by -iZ. Under standard regularity conditions, the corresponding states and Hilbert space representations uniquely give rise to classical and quantum mechanics on M.