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

TL;DR

The paper constructs a nonrelativistic $D=2$ model that implements the $D=2$ Galilean algebra with two central charges, interpreting the second charge as space noncommutativity. It achieves this via a Lagrangian with a Chern-Simons-like term and analyzes quantization through both Ostrogradski-Dirac and Faddeev-Jackiw methods, revealing a clean separation between external motion in a noncommutative plane and internal oscillator modes. The study extends to two-body interactions and a noncommutative harmonic oscillator, showing how to preserve Galilean invariance while maintaining a positive physical spectrum after handling ghost states with a subsidiary condition. The results illustrate the utility of higher-derivative formulations and noncommutative geometry for encoding central extensions and offer potential links to string-theoretic substructures and 0-branes.

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 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.