Quantum Euclidean spaces with noncommutative derivatives

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