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

Abstract

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 spaces as well as the``internal'' oscillator modes. We add a suitably chosen class of velocity-dependent twoparticle interactions, which is described 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.

Noncommutative mechanics and exotic Galilean symmetry

“…More recently, similar features has been found in 2D systems for the so called exotic particles, i.e., ones which are associated with the kinematical two-parameter central extension of the planar Galilei group [12,13]. These models are, once again, instrumental in describing the Fractional Quantum Hall Effect.…”

A note on noncommutative Chern–Simons theories

“…see [12]. However, before a new re-motivation from string theory [13], it was not studied extensively.…”

N = 2 supersymmetric radiation damping problem on a noncommutative plane

“…In [21] the authors have introduced a nonrelativistic classical mechanics for the free particle model which is quasi-invariant under D = 2 Galilei symmetry as…”

“…The organization of this paper is: in section 2 we will carry out a very brief review of the mechanical model with a Chern-Simons term developed in [21] and its Galilean-symmetric version, i.e., the LSZ model. In section 3 we will present a symplectic structure for the model in order to introduce the noncommutativity through the variables used in [22,23].…”