3 Noncommutative Planar Particle Dynamics with Gauge Interactions

We consider two ways of introducing minimal Abelian gauge interactions into the model presented in [1]. These two approaches are different only if the second central charge of the planar Galilei group is nonzero. One way leads to the standard gauge transformations and the other one to a generalized gauge theory with gauge transformations accompanied by time-dependent area-preserving coordinate transformations. Both approaches, however, are related to each other by a classical Seiberg-Witten map supplemented by a noncanonical transformation of the phase space variables for planar particles. We also 1 formulate the two-body problem in the model with our generalized gauge symmetry and consider the case with both CS and background electromagnetic fields, as it is used in the description of fractional quantum Hall effect.