Quantum theory of the two-dimensional interacting-boson system

In this paper we expand the microscopic Hamiltonian of a self-interacting boson system in terms of the density and phase operators, which are canonical conjugates to each other. When higher-order terms are neglected, the Hamiltonian can be diagonalized. This method is applied to superfluid helium and the fractional quantum Hall effect. We shall first derive the expression of superfluid density for a two-dimensional (2D) system where canonical treatment has been lacking. We then reproduce the 3D results obtained by the Bogoliubov transformation. Finally, we shall show that the basic phenomenologies of the fractional quantum Hall effect at exact filling numbers can be derived without the introduction of the Chern-Simons term