The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

Abstract

The Lie–Rinehart algebra of a (connected) manifold \({\mathcal {M}}\) , defined by the Lie structure of the vector fields, their action and their module structure over \({C^\infty({\mathcal {M}})}\) , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra \({\Lambda_{R}({\mathcal {M}})}\) , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact \({{\mathcal {M}}}\)) \({Z}\) which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z = iz, z = 0 and \({z = \hbar}\) , respectively; canonical quantization uniquely follows from such a general geometrical structure. For \({z =\hbar \neq 0}\) , the regular factorial Hilbert space representations of \({\Lambda_{R}({\mathcal{M}})}\) describe quantum mechanics on \({{\mathcal {M}}}\) . For z = 0, if Diff(\({{\mathcal {M}}}\)) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on \({{\mathcal {M}}}\) .