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

We consider a newD=2 nonrelativistic classical mechanics model providing via the Noether theorem the (2+1)-Galilean symmetry algebra with two central charges: massmand the coupling constantkof 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 ofkas describing the noncommutativity ofD=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 noncommutativeD=2 spaces 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 inD=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.