Dynamics of compressible edge and bosonization

We work out the dynamics of the compressible edge of the quantum Hall system based on the electrostatic model of Chklovskii et al. We introduce a generalized version of Wen's hydrodynamic quantization of the dynamics of sharp edge and rederive Aleiner and Glazman's earlier result of multiple density modes. Bosonic operators of density excitations are used to construct a fermion operator and the one-particle Green's function. We also analyze the dynamics starting with the second-quantized Hamiltonian in the lowest Landau level and work out the time development of density operators within the random-phase approximation and recover the classical result for low energy. The scattering due to impurities are considered within the second-order perturbation theory and are shown to be very weak.