Classical and quantum mechanics from the universal Poisson-Rinehart algebra of a manifold

The Lie and module (Rinehart) algebraic structure of vector fields of compact support over C∞ functions on a (connected) manifold M define a unique universal noncommutative Poisson *-algebra ΛR(M). For a compact manifold, a (antihermitian) variable Z∈ΛR(M), central with respect to both the product and the Lie product, relates commutators and Poisson brackets; in the noncompact case, sequences of locally central variables allow for the addition of an element with the same rôle. Quotients with respect to Z*Z-z2I, z ≥ 0, define classical Poisson algebras and quantum observable algebras, with z = ħ. Under standard regularity conditions, the corresponding states and Hilbert space representations uniquely give rise to classical and quantum mechanics on M.