Magnetic translation groups in an n-dimensional torus

Details Magnetic translation groups in an n-dimensional torus Magnetic translation groups in an n-dimensional torus

2002-05-07

arxiv.org/abs/hep-th/0205053v5

J.Math.Phys. 43 (2002) 5926-5948

Physics

High Energy Physics

High Energy Physics - Theory

29 pages, LaTeX2e, title changed, re-organized, to be published in Journal of Mathematical Physics

Scientific paper

10.1063/1.1513208

A charged particle in a uniform magnetic field in a two-dimensional torus has a discrete noncommutative translation symmetry instead of a continuous commutative translation symmetry. We study topology and symmetry of a particle in a magnetic field in a torus of arbitrary dimensions. The magnetic translation group (MTG) is defined as a group of translations that leave the gauge field invariant. We show that the MTG on an n-dimensional torus is isomorphic to a central extension of a cyclic group Z_{nu_1} x ... x Z_{nu_{2l}} x T^m by U(1) with 2l+m=n. We construct and classify irreducible unitary representations of the MTG on a three-torus and apply the representation theory to three examples. We shortly describe a representation theory for a general n-torus. The MTG on an n-torus can be regarded as a generalization of the so-called noncommutative torus.

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