The Renormalization Group and Quantum Hall Edge States

The role of edge states in phenomena like the quantum Hall effect is well known. In this paper we show how the choice of boundary conditions for a one-particle Schr\"odinger equation can give rise to states localized at the edge of the system. We consider both the example of a free particle and the more involved example of a particle in a magnetic field. In each case, edge states arise from a non-trivial scaling limit involving the boundary conditions. Second quantization of these quantum mechanical systems leads to a multi-particle ground state carrying a persistent current at the edge. We show that the theory quantized with this vacuum displays an ``anomaly'' at the edge which is the mark of a quantized Hall conductivity in the presence of an external magnetic field. We also offer interpretations for the physics of such boundary conditions which may have a bearing on the nature of the excitations of these systems.