  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
Schr\"odinger 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 $|\alpha| \not=1$ wave functions.
