The Renormalization Group and Quantum Hall Edge States

The role of edge states in phenomena like the quantum Hall e(cid:11)ect is well known. In this paper we show how the choice of boundary conditions for a one-particle Schr(cid:127)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 (cid:12)eld. 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 (cid:12)eld. We also o(cid:11)er interpretations for the physics of such boundary conditions which may have a bearing on the nature of the excitations of these systems.