Galilean-Invariant (2 + 1)-Dimensional Models with a Chern-Simons-Like Term and D = 2 Noncommutative Geometry

We consider a new D = 2 nonrelativistic classical mechanics model providing via the Noether theorem the (2 + 1)-Galilean symmetry algebra with two central charges: mass m and the coupling constant k of a Chern-Simons-like term. In this way we provide the dynamical interpretation of the second central charge of the (2 + 1)-dimensional Galilean algebra. We discuss also the interpretation of k as describing the noncommutativity of D = 2 space coordinates. The model is quantized in two ways: using the Ostrogradski-Dirac formalism for higher order Lagrangians with constraints and the Faddeev-Jackiw method which describes constrained systems and produces nonstandard symplectic structures. We show that our model 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 descrobed by local potentials in D = 2 noncommutative space. We treat, in detail, the particular case of a harmonic oscillator and describe its quantization. It appears 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. Department of Mathematical Sciences, Science Laboratoires, University od Durham, South Road, Durham DH1 3LE, England On leave of absence from the Institute of Theoretical Physics, University of Wroc law, pl. Maxa Borna 9, 50-204 Wroc law, Poland Faculty of Physics, University of Bielefield, Universitätsstr.25, 33615 Bielefeld, Germany