The exotic Galilei group and the "Peierls substitution"

TL;DR

This paper constructs a non-relativistic particle model in the plane using the extended Galilei group, revealing non-commuting coordinates and connections to the Fractional Quantum Hall Effect through quantization.

Contribution

It introduces a novel non-relativistic particle model based on the extended Galilei group, linking non-commutative geometry to quantum Hall phenomena.

Findings

Coordinates do not commute in the model.

Effective mass depends on magnetic field and group parameters.

Wave functions match Laughlin's proposal for FQHE.

Abstract

Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective ``Hall'' motions and justifies the rule called ``Peierls substitution''. Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect.