Electric Chern-Simons term, enlarged exotic Galilei symmetry and noncommutative plane

TL;DR

This paper develops an extended exotic planar model with a Chern-Simons-like electric term, revealing noncommutative symmetries and providing two reduction schemes that connect to charged particles in fields and free particles on noncommutative planes.

Contribution

It introduces a novel extended exotic Galilei symmetric model with an electric Chern-Simons term, linking noncommutative geometry and symmetry algebra structures.

Findings

Electric Chern-Simons term induces noncommutativity of boost generators.

Model admits two reduction schemes with distinct physical interpretations.

One scheme reproduces charged particle dynamics; the other yields a free noncommutative plane particle.

Abstract

The extended exotic planar model for a charged particle is constructed. It includes a Chern-Simons-like term for a dynamical electric field, but produces usual equations of motion for the particle in background constant uniform electric and magnetic fields. The electric Chern-Simons term is responsible for the non-commutativity of the boost generators in the ten-dimensional enlarged exotic Galilei symmetry algebra of the extended system. The model admits two reduction schemes by the integrals of motion, one of which reproduces the usual formulation for the charged particle in external constant electric and magnetic fields with associated field-deformed Galilei symmetry, whose commuting boost generators are identified with the nonlocal in time Noether charges reduced on-shell. Another reduction scheme, in which electric field transmutes into the commuting space translation generators, extracts from the model a free particle on the noncommutative plane described by the two-fold centrally extended Galilei group of the non-relativistic anyons.