Exotic galilean symmetry in the non-commutative plane, and the Hall effect

TL;DR

This paper explores quantum mechanics on a non-commutative plane, revealing an exotic Galilean symmetry and deriving the Hall effect through a reduction process linked to Chern-Simons mechanics.

Contribution

It demonstrates the presence of exotic Galilean symmetry in non-commutative quantum mechanics and connects it to the Hall effect via Faddeev-Jackiw reduction and Chern-Simons theory.

Findings

Identification of exotic Galilean symmetry in non-commutative quantum mechanics

Derivation of Hall law dynamics through reduction process

Connection between non-commutative geometry and Chern-Simons mechanics

Abstract

Quantum Mechanics in the non-commutative plane is shown to admit the ``exotic'' symmetry of the doubly-centrally-extended Galilei group. When coupled to a planar magnetic field whose strength is the inverse of the non-commutative parameter, the system becomes singular, and ``Faddeev-Jackiw'' reduction yields the ``Chern-Simons'' mechanics of Dunne, Jackiw, and Trugenberger. The reduced system moves according to the Hall law.