Formalism for the quantum Hall effect: Hilbert space of analytic functions

Abstract

We develop a general formulation of quantum mechanics within the lowest Landau level in two dimensions. Making use of Bargmann's Hilbert space of analytic functions we obtain a simple algorithm for the projection of any quantum operator onto the subspace of the lowest Landau level. With this scheme we obtain the Schr\"odinger equation in both real-space and coherent-state representations. A Gaussian interaction among the particles leads to a particularly simple form in which the eigenvalue condition reduces to a purely algebraic property of the polynomial wave function. Finally, we formulate path integration within the lowest Landau level using the coherent-state representation. The techniques developed here should prove to be convenient for the study of the anomalous quantum Hall effect and other phenomena involving electron-electron interactions.