(2 + 1)-Dimensional models with a Chern-Simons-like term and noncommutative geometry

We consider new D = 2 nonrelativistic model of classical mechanics providing via the Noether theorem the (2 + 1)-Galilean symmetry algebra with two central charges: the mass m and the coupling constant k of a Chern-Simons-like term. We discuss the interpretation of k as describing the noncommutativity of D = 2 space coordinates. We quantize the model and show that it describes the superposition of a free motion in noncommutative D = 2 space as well as the “internal” oscillator modes. We add a suitably chosen class of velocity-dependent two-particle interactions, which is described by local potentials in D = 2 noncommutative space. We show that the indefinite metric due to the third-order time derivative term in the field equations, even in the presence of interactions, can be eliminated by the imposition of a subsidiary condition.