  The Lie-Rinehart algebra of a manifold M, defined by the Lie structure of the
vector fields, their action and their module structure on the infinitely
differentiable functions on M, is a common, diffeomorphism invariant, algebra
for both classical and quantum mechanics. Its (noncommutative) Poisson
universal enveloping algebra contains a central variable Z which relates the
commutators to the Lie products; classical and quantum mechanics are its only
factorial realizations, corresponding to the indentification of Z with i times
the Planck constant. In this form, canonical quantization appears therefore as
a consequence of such a general geometrical structure. The regular factorial
Hilbert space representations are, for nonzero values of Z, unitarily
equivalent, apart from multiplicity, to one of the irreducible quantum
representations, which are locally Schroedinger and in one to one
correspondence with the unitary irreducible representations of the fundamental
group of M. For Z = 0, if Diff(M) is unitarily implemented, they are unitarily
equivalent, up to multiplicity, to the representation defined by classical
mechanics on M.
