  Quantum Hall Dynamics is formulated on von Neumann lattice representation
where electrons in Landau levels are defined on lattice sites and are treated
systematically like lattice fermions. We give a proof of the integer Hall
effect, namely the Hall conductance is the winding number of the propagator in
the momentum space and is quantized exactly as integer multiple of ${e^2}/h$ in
quantum Hall regime of the system of interactions and disorders. This shows
that a determination of the fine structure constant from integer quantum Hall
effect is in fact possible. We present also a unified mean field theory of the
fractional Hall effect for the in-compressible quantum liquid states based on
flux condensation and point out that the known Hofstadter butterfly spectrum of
the tight binding model has a deep connection with the fractional Hall effect
of the continuum electrons. Thus two of the most intriguing and important
physical phenomena of recent years, the integer Hall effect and fractional Hall
effect are studied and are solved partly by von Neumann lattice representation.
