The Quantum Hall Fluid and Non–Commutative Chern Simons Theory

The first part of this paper is a review of the author’s work with S. Bahcall which gave an elementary derivation of the Chern Simons description of the Quantum Hall effect for filling fraction 1/n. The notation has been modernized to conform with standard gauge theory conventions. In the second part arguments are given to support the claim that abelian non– commutative Chern Simons theory at level n is exactly equivalent to the Laughlin theory at filling fraction 1/n. The theory may also be formulated as a matrix theory similar to that describing D0–branes in string theory. Finally it can also be thought of as the quantum theory of mappings between two non–commutative spaces, the first being the target space and the second being the base space. 1 Fluid Dynamics in Co-moving Coordinates The configuration space of charged particles in a strong magnetic field 1 is a non-commutative space. The charged particles behave unlike conventional relativistic or non–relativistic particles in so much as they are locked in place by the large magnetic field. By contrast, a neutral system such as a dipole [1, 2] does move like a conventional particle although it grows in size with its momentum. Such dipoles are the objects described by non-commutative field theory. If we are considering a system of charged particles in a strong magnetic field we may either describe the system in terms of charge carrying fields such as the electron field or in terms of neutral fields such as the density and current. The former fields correspond to the frozen particles but the latter fields carry the quantum numbers of dipoles. This suggests that the currents and density of a system of electrons in a strong magnetic field may be described by a non-commutative quantum field theory. In this paper we will show that this is indeed the case and that the Laughlin electron theory of the fractional quantum hall states is equivalent to a non-commutative Chern Simons theory describing the density and current. The first six sections of the paper review work done in 1991 with Safi Bahcall [3]. The purpose is to give an elementary description of the fractional Quantum Hall fluid for the case in which the inverse filling fraction is an integer, and to explain how in these cases, the long distance behavior of the Quantum Hall fluid can be described by Chern Simons theory. Both the odd and even integer cases describe quantum hall states, the odd cases corresponding to fermions and the even to bosons. In the remaining sections it is shown that precise quantitative agreement with Laughlin’s theory [4] can be obtained if the ordinary Chern Simons theory is replaced by the non-commutative theory [5]. Alternatively it can be formulated as a matrix theory similar to that describing D0–branes in string theory [6]. We will begin with a description of a dissipationless fluid. Consider a collection of identical non–relativistic particles, indexed by α, moving on a plane with Lagrangian