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

We perform consistently the Gupta-Bleuler quantization combined with Dirac procedure for a chiral boson with the parameter ($α$) on the circle, the boundary of the circular droplet. For $α=1$, 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 $α=1$ wave functions explain the neutral edge states for $ν=1$ quantum Hall fluid very well. In the case of $α= -1$, 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 $|α| \not=1$ wave functions.