Quantum Hall Effects And Noncommutative Physics

The discovery of the fractional quantum Hall effect (QHE) marked a new epoch in many-body theory and resulted in many new ideas and concepts. Its edge excitations is also an active subject.To describe the fractional quantum Hall edge, a chiral Tomonaga-Luttinger liquid theory is derived from effective Chern-Simons field and hydrodynamic formulation. The electron propagator exhibits a nontrivial power-law correlation, with an universal exponent a. A number of experiments establish the existence of Tomonaga-Luttinger-liquid-like behavior. However, the tunneling exponent measured is different from the prediction.On the other hand, noncommutative Chern-Simons theory (NCCS), derived from microscopic dynamics, is exactly equivalent to the Laughlin theory.We try to pursue whether a better description could be derived from NCCS. The formalism is in Section 4.Considering relabeling symmetry of the electrons and incompressibility of the fluid, which conceals electron-electron interaction, we obtain a constraint. Its solution as well as the action has a total differential form, so we can reduce 2+1 dimensional NCCS to an 1+1 dimensional chiral Tomonaga-Luttinger liquid theory, which contains interaction terms. We calculate one-loop corrections to boson and electron propagators and get a new tunneling exponent. It agrees with experiments.We also studied backward Compton scattering in the accelerator magnetic field, and explored possible effect of space noncommutativity caused by the lowest Landau level.