Generalizing Coordinate Non Commutativity

TL;DR

This paper develops a generalized framework for coordinate non-commutativity on curved manifolds, integrating quantum mechanics and geometric curvature, revealing a mixture of canonical and quadratic formalisms.

Contribution

It introduces a local first-order curvature-based approach to coordinate non-commutativity on general manifolds, bridging quantum mechanics and differential geometry.

Findings

Generalized non-commutativity incorporates both canonical and quadratic formalisms.

The framework applies to four-dimensional pseudo Riemannian manifolds.

The correlation results extend non-commutative geometry to curved spacetime.

Abstract

In this paper, we establish and employ a local framework to the first order of Riemann's curvature tensor in order to develop the corresponding coordinate non commutativity into general manifolds. We also exploit a new translation of function at the level of quantum mechanics to show that the final correlation result of the generalized non commutativity is a mixture of the Canonical and Quadratic formalisms and does not consist only of the Lie algebraic formalism. The basic premise of this article is that the geometry of a four-dimensional pseudo Riemann manifold representing space time, is homeomorphic to Minkowski space time.