Field Theory in a Strong Magnetic Field and the Quantum Hall Effect

Abstract

We present a proof of exactness of the integer Hall effect and a mean field theory for the fractional Hall effect. We apply a representation in which wave functions are well localized and develop multi-pole expansion of the charge operator and of its commutation relation for two dimensional electrons in the perpendicular magnetic field. They enable us to derive a Ward-Takahashi identity, a topological of the Hall conductance, and a low energy theorems of the Hall conductance. A localization due to impurities are easily studied in our representation since the base functions are localized. A mean field theory for the fractional Hall effect is also developed. A gauge field is introduced and acts as an order parameter.