The Landau problem is discussed in two similar but Still different non-commutative frameworks. The "standard" one where the coupling to the gauge field is achieved using Poisson brackets, yields all Landau levels. The "exotic" approach where the coupling to the gauge field is achieved using the symplectic structure. only yields lowest-Landau level states. as advocated by Peierls and used in the description of the ground states of the fractional quantum Hall effect. The same reduced model also describes vortex dynamics in a superfluid He-4 film. Remarkably. the spectrum depends crucially on the quantization scheme. The system is symmetric w.r.t area-preserving diffeomorphisms. (C) 2002 Elsevier Science (USA). [References: 41]
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