Global Gauge Invariance in two-Dimensional Quantum Mechanics

Abstract

In two-dimensional non-relativistic quantum mechanics, there is no a priori reason why the angular momentum of a given system should be quantized in integral units. Equivalently, there is no obvious basic principle that forces the quantum mechanical wavefunction of a two-dimensional system to be single-valued. Quite remarkably, one finds that the physics underlying the description of a two-dimensional system in terms of a single-valued wavefunction may be very different from the physics that follows for a multivalued approach. In particular, the single-valued wavefunction approach to the problem of scattering of a charged particle off an impenetrable solenoid leads to Aharonov-Bohm (A-B) scattering and the Aharonov-Bohm effect,1 while in the multi-valued wavefunction approach, there is no A-B scattering, although the A-B effect still remains.2 furthermore, particles with multivalued wavefunctions may carry fractional spin and obey fractional statistics3 in sharp contrast with standard nonrelativistic quantum mechanics!