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The exotic Galilei group and the “Peierls substitution”. (English) Zbl 1050.81568
Summary: 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.

MSC:
81S10Geometric quantization, symplectic methods (quantum theory)
53D05Symplectic manifolds, general
53D50Geometric quantization
70S05Lagrangian formalism and Hamiltonian formalism