Generalization of the coordinates non-commutativity to a general manifold

Abstract

In the framework of quantum mechanics and based on the non-commutativity between the ‎coordinates in Minkowski space-time, we generalize the geometric non-commutative relation ‎to a space-time other than Minkowski. Using the authority of inserting the unit operator, we ‎exploit the translation operator to derive the Wyle-Moyal star product operator. Up to the first ‎order of translation parameters and by employing the Wyle-Moyal star operator, we find the ‎modified non-commutativity of coordinates relation in terms of geometric structure. The basic ‎premise of this article is that pseudo-Riemannian local homomorphic with Minkowski space-‎time‎‎ are equivalent.