  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.
