Lowest Landau level field theory for the fractional quantum Hall effect

We develop a field theoretical formalism for the description of a system of planar electrons moving under the influence of a strong magnetic field, with emphasis on its relevance to the physics of the fractional quantum Hall effect. For excitation energies much smaller than the cyclotron gap the states can be described in terms of their lowest Landau level content, neglecting the possible mixing between Landau levels induced by the potential terms. The LLL approximation, however, leads to a paradox resulting from the apparent impossibility of describing charge transport in a system of electrons for which the kinetic energy has been frozen. We solve this paradox by a careful treatment of the dynamics of the electrons, within the LLL approximation, that takes into account the constraints of the system, and show the self-consistency of the approximation by the explicit construction of non-solenoidal LLL current operators. We review the description of the low energy physics of the FQHE in terms of a Chern-Simons effective field theory, and use our methods to give a detailed account of the construction of one such effective action based on a bosonic order parameter describing the binding between electrons and q quasi-holes observed in the Laughlin state at $\nu$ = 1/q. Finally, we construct well defined second quantized representations of the Laughlin quasiparticle operators, and find that they can be written in terms of the generators of the W$\sb\infty$ algebra of canonical transformations within the LLL.U of I OnlyETDs are only available to UIUC Users without author permissio