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

TL;DR

This paper introduces a new nonrelativistic (2+1)-dimensional model with Galilean symmetry, incorporating a Chern-Simons-like term, which reveals insights into noncommutative geometry, central charges, and oscillator quantization.

Contribution

It provides a novel classical and quantum model with Galilean invariance, central charges, and noncommutative geometry, including a detailed analysis of interactions and quantization methods.

Findings

The model exhibits a superposition of free noncommutative motion and internal oscillations.

Quantization of the harmonic oscillator in this framework is achieved with a subsidiary condition.

The model clarifies the role of the second central charge and noncommutativity in D=2 space.

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.