On the Constraint Equation for the Lowest Landau Level in Fractional Quantum Hall System

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Abstract
In the framework of collective field theory, We apply the Chern-Simon field theory treatment to the constraint equation for the lowest Landau level to investigate the generic properties for the quasi-particles of the FQH system. It shows a transparent connection to the Laughlin's wave functions. If we take an average over the wave functional for the constraint equation, the resulted equation can be interpreted as the vortex equation for the fractionally charged quasi-particles. Introducing a generalized ρ (density)-ϑ (phase) transformation, not only the fractional statistics and the hierarchy scheme can be drawn from the constraint equation, it also gives rise an interesting picture that vortices condense as a Halperin like wave fuction on a Laughlin like background condensate of ν=1/m.

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