The Constraint for the Lowest Landau Level and the Chern-Simons Field Theory Approach for the Fractional Quantum Hall Effect: Infinite and Finite Systems

We build the constraint that all electrons are in the lowest Landau level into the Chern-Simons field theory approach for the fractional quantum Hall system. We show that the constraint can be transmitted from one hierarchical state to the next. As a result, we derive in generic the equations of the fractionally charged vortices ( quasi-particles ) for arbitrary hierarchy filling. For a finite system, we show that the action for each hierarchical state can be divided into two parts: the surface part provides the action for the edge excitations while the remaining bulk part is exactly the action for the next hierarchical states. In particular, we not only show that the surface action for the edge excitations would be decoupled from the bulk at each hierarchy filling, but also derive the explicit expressions analytically for the drift velocities of the hierarchical edge excitations.