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Quantum Hydrodynamics of Fractional Hall Effect: Quantum Kirchhoff Equations

We argue that flows of the quantum electronic liquid in the Fractional Quantum Hall state are comprehensively described by the hydrodynamics of vortices in the quantum incompressible rotating liquid. We obtain the quantum hydrodynamics of vortex flow by quantizing Kirchhoff equations for vortex dynamics. We demonstrate that quantized Kirchhoff equations capture all major features of FQH states including subtle effects of Lorentz shear force, magneto-roton spectrum, Hall current in a non-uniform electromagnetic field, thus providing a powerful framework to study FQHE and superfluids. The results are also useful for classical vortex flows.