The exotic Galilei group and the "Peierls substitution"

TL;DR

This work derives a principled, first-principles framework for the Peierls substitution in a two-dimensional nonrelativistic setting by using the two-parameter central extension of the planar Galilei group. The authors construct an exotic planar particle, identify an effective mass $m^*$ in a perpendicular magnetic field, and show that when $m^* = 0$ the dynamics reduce to a two-dimensional Hall motion space, which upon quantization yields Laughlin/Bargmann–Fock-type wavefunctions. The approach unifies geometric quantization, noncommutative coordinates, and magnetic translation symmetries to connect exotic Galilean symmetry with the Fractional Quantum Hall Effect, offering a minimal classical route to the quantum states observed in the lowest Landau level. The results suggest experimental relevance for the invariant $\kappa$ via band-structure–induced effects and illuminate topological aspects of Hall physics in terms of a reduced phase-space geometry.

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.