Exotic galilean symmetry and non-commutative mechanics

TL;DR

This paper reviews the exotic Galilean symmetry linked to non-commutative mechanics, exploring its properties, extensions, and applications in field theory and condensed matter phenomena like the Hall effects.

Contribution

It introduces and analyzes the exotic Galilean symmetry, its non-commutative structure, and its relevance to various physical models and phenomena.

Findings

Non-commuting position coordinates in exotic particles

Existence of supersymmetric extensions

Application to Moyal field theory and Hall effects

Abstract

Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.