Chiral boson, $W_{\infty }$-coherent state and edge states in the quantum Hall effect

Abstract

We perform consistently the Gupta-Bleuler quantization combined with Dirac procedure for a chiral boson with the parameter ($\alpha$) on the circle, the boundary of the circular droplet. For $\alpha =1$, we obtain the holomorphic constraints. Using the representation of Bargmann-Fock space and the Schrödinger equation, we construct the holomorphic wave functions. In order to interpret these functions, we introduce the $W_{\infty }$-coherent state to account for the infinite-dimensional translation symmetry for the Fourier (edge) modes. The $\alpha =1$ wave functions explain the neutral edge states for $\nu =1$ quantum Hall fluid very well. In the case of $\alpha =-1$, we obtain the new wave functions which may describe the higher modes (radial excitations) of edge states. Finally, the charged edge states are described by the $|\alpha |\ne 1$ wave functions.

arXiv comment19 pages, RevTeX, the invited paper for Nuclear Science and Techniques