Derivation of an Operator-Based Spatial Noncommutativity Parameter

The idea that the operators defining spacetime could be noncommuting has gained popularity in recent years. The formulation in which the commutators themselves are a set of commuting numbers has been applied to a number of quantum phenomena to determine what effects it might have. Most of the work overall has focused on this set of commuting numbers, but some earlier theories of noncommutative coordinates established the coordinate commutator as having an operator-based matrix value involving the angular momentum operator, and Döplicher et al. obtained a very similar algebra without mention of the angular momentum. Beginning from the relativistic geometry used to derive this result, we hope to explore an alternative foundation for a noncommutative geometry of space and determine its effects on some well-studied quantities in atomic physics.