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 semi-finite spectral triple with non-flat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a simplified local index formula (even case) that is similar to the commutative setting.