The non-commutative Landau problem

Abstract

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 ${}^4$He film. Remarkably, the spectrum depends crucially on the quantization scheme. The system is symmetric w. r. t. area-preserving diffeomorphisms.

Published with shortened title as Ann. Phys. (N. Y.) 299, pp. 128-140 (2002) 14 pages, LaTex, no figures

References in corpus (7)

• Exotic galilean symmetry in the non-commutative plane, and the Hall effect
• Testing Non-commutative QED, Constructing Non-commutative MHD
• Two phases of the noncommutative quantum mechanics
• Quantum Mechanics on the Noncommutative Torus
• Path Integral Formulation of Noncommutative Quantum Mechanics
• Exotic plasma as classical Hall Liquid
• Deformed Chern-Simons interaction for nonrelativistic point particles

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