Quantum Euclidean spaces with noncommutative derivatives

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semifinite spectral triple with nonflat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a local index formula in our setting via the computation of the Connes-Chern character of the corresponding spectral triple.