Coordinate Noncommutativity in Strong Non-Uniform Magnetic Fields

Noncommutativity of space coordinates has been much studied from various points of view [1-3]. It arises naturally in string theory, where it is related to the presence of strong backgrounds fields. If these are constant, one obtains the more familiar case where the coordinate noncommutativity [x, x ] is a constant antisymmetric quantity. However, if the background fields depend on the spatial coordinates, one would expect coordinate noncommutativity to be a local quantity. This possibility has been recently studied in the context of noncommutative gauge field theories [4]. On the other hand, coordinate noncommutativity may also arise in more physical situations involving the motion of electric charges in a strong external magnetic field [5,6]. When a charged (e) and massive (m) particle moves on a plane (x, y) in the presence of a strong constant magnetic field B pointing along the z-axis, it has been shown that the noncommutativity of space coordinates is of order of the inverse of the magnetic field: