Magnetic translation groups in an n-dimensional torus

Abstract

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.

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

References in corpus (3)

• Physical Instances of Noncommuting Coordinates
• Spontaneous breaking of the C, P, and rotational symmetries by topological defects in two extra dimensions
• Spontaneous breaking of the rotational symmetry induced by monopoles in extra dimensions

Cited by in corpus (6)

• Shifted orbifold models with magnetic flux
• Dirac Operator Zero-modes on a Torus
• An extension of Fourier analysis for the n-torus in the magnetic field and its application to spectral analysis of the magnetic Laplacian
• Landau problem with a general time-dependent electric field
• On quantum mechanics with a magnetic field on R^n and on a torus T^n, and their relation
• Quantization on a torus without position operators