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

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.

Talk given by the first author at the 40th Symposium on Mathematical Physics, Torun, June 25-28, 2008

References in corpus (4)

• Quantization Methods: A Guide for Physicists and Analysts
• Topological Factors Derived From Bohmian Mechanics
• Quantum mechanics on manifolds and topological effects
• The Noncommutative Poisson Algebra of Classical and Quantum Mechanics

Cited by in corpus (1)

• The Physical Principles of Quantum Mechanics. A critical review