The Constraint for the Lowest Landau Level and the Effective Field Theory Approach for the Fractional Quantum Hall System

By applying the Dirac quantization method, 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 and show that the constraint can be transmuted from hierarchy to hierarchy. For a finite system, we derive that the action for each hierarchy can be split into two parts: a surface part provides the action for the edge excitations while the remaining part is precisely the bulk action for the next hierarchy. And the action for the edge could be decoupled from the bulk only at the hierarchy filling.