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 = i z, 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 $${{\mathcal {M}}}$$ .