Two-dimensional boson and W-symmetry in the quantum Hall effect

Abstract

We perform consistently the Gupta–Bleuler–Dirac quantization for a two-dimensional boson with parameter (α) on the circle, the boundary of the circular droplet. For α=1, we obtain the chiral (holomorphic) constraints. Using the representation of Bargmann–Fock space and the Schrodinger picture, we construct the holomorphic wave function. In order to interpret this function, we construct the coherent state representation by using the infinite-dimensional translation (W∞) symmetry for each Fourier (edge) mode. The α=1 chiral wave function explains the neutral edge states for integer quantum Hall effect very well. In the case of α=−1, we obtain a new wave function which may describe the higher modes (radial excitations) of edge states. The charged edge states are described by the |α|≠1 wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed.