General Introduction

Since its discovery the quantum Hall effect has been an intensively studied subject. Until today there is an ongoing discussion on how to set up a quantum field theory that describes the fractional quantum Hall states. There is some consensus on what the approximate form of the low energy effective action is. All authors agree it will have a Chern-Simons form, but there is some debate on the precise form. In particular they disagree on the number of Chern-Simons gauge fields one needs to introduce. These differences become apparent when one studies the edge of such states. Wen [28] was the first to recognize that the effective theory of the edge of a quantum Hall sample is a realization of the 1D electron state known as the Luttinger liquid. The effective action for the edge can be derived from the bulk action for the corresponding bulk state. The edge states have some fascinating properties, which can be studied by experiments in the laboratory.

In the first chapter of this thesis we will introduce some general concepts needed to understand the quantum Hall effect and the problem we are dealing with. Then we will study some features of the Chern-Simons term and see what it has to do with flux attachment (chapter  3) and quasiparticles. In chapters 4, 5 and 7 we will study two theories that attempt to describe fractional quantum Hall states. As stated, the edge lends itself very well for a comparison of the two theories. That is why in chapter 6 we study the edge of a Chern-Simons theory. Then we will go on and compare the two theories and try to draw some conclusions in chapter 8.

My own contributions include filling in some minor calculations not found in the literature (by me that is) and of course the conclusions drawn in chapter 8.

One subject in quantum Hall physics that in this thesis does not get the attention it probably deserves is that of impurities. We motivate this by saying we only study quantum Hall states as opposed to the entire quantum Hall effect. We mean by this, that we do not include impurities and hence do not want to explain the plateaus and such. Also we want to mention the two theories under study are not the only theories around. Two other theories that deserve mentioning are one by Shankar and Murthy [26,15,14], and a theory by Pruisken and $\check{\text{S}}$koric [8], that focuses on impurities and has the promise to be also valid in the regions between quantum Hall states.