Non Standard Extended Noncommutativity of Coordinates

Abstract

We present in this short note an idea about a possible extension of the standard noncommutative algebra to the formal differential operators framework. In this sense, we develop an analysis and derive an extended noncommutative structure given by $[x_{a}, x_{b}]_{\star} = i(\theta + \chi)_{ab}$ where $\theta_{ab}$ is the standard noncommutative parameter and $\chi_{ab}(x)\equiv \chi^{\mu}_{ab}(x)\partial_{\mu} ={1/2}(x_a \theta^{\mu}_{b} - x_b \theta^{\mu}_{a})\partial_\mu$ is an antisymmetric non-constant vector-field shown to play the role of the extended deformation parameter. This idea was motivated by the importance of noncommutative geometry framework, with nonconstant deformation parameter, in the current subject of string theory and D-brane physics.