Natural star-products on symplectic manifolds and related quantum mechanical operators

•Invariant representations of natural star-products on symplectic manifolds are considered.•Star-products induced by flat and non-flat connections are investigated.•Operator representations in Hilbert space of considered star-algebras are constructed.

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.