a Field Theory for the Fractional Quantum Hall Effect.

In this thesis we derive a field theory approach to the Fractional Quantum Hall Effect (FQHE). The goal is to develop a field theory for a system of interacting electrons moving in a plane in the presence of an external magnetic field, in the hope that the FQHE states will appear naturally as the semiclassical states of the theory. In this framework, the FQHE states are understood as infrared stable fixed points, and the long distance, low energy properties of the system should be described exactly by this theory. We show the existence of an exact equivalence between a system of interacting electrons, and a system of fermions tightly bound to infinitesimal solenoids carrying a statistical magnetic flux. These are the fermionic Chern -Simons gauge field theories. The transformed system can be studied using standard techniques of perturbation theory, because its ground state, at the mean field level, is nondegenerate and has a gap to all the excitations. The power of this theory resides in the fact that it allows for a systematic calculation of the properties of the FQHE states around a mean-field ground state which has a gap. We consider in detail the fluid states in which the mean-field theory replaces the statistical fluxes by an average statistical magnetic field, i.e., the Average Field Approximation. We calculate the electromagnetic response functions within the semiclassical approximation, and determine the spectrum of collective excitations. We derive the ground state wave function directly from the field theory, and, as expected, we find the Laughlin wave function for the states with filling fraction nu = {1over m}. We show that the universal properties of the Laughlin wave functions are a consequence of general principles, i.e., incompressibility, and Galilean and gauge invariance, which determine the analytic structure of the equal-time density correlation functions at long distances. Finally, we generalize our approach to study the FQHE in double-layer systems.