Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 2 (2006), 090, 9 pages      cond-mat/0609571      https://doi.org/10.3842/SIGMA.2006.090
Contribution to the Proceedings of the O'Raifeartaigh Symposium

Non-Commutative Mechanics in Mathematical & in Condensed Matter Physics

Peter A. Horváthy
Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France

Received September 25, 2006, in final form November 27, 2006; Published online December 14, 2006

Abstract
Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1). Souriau's construction applied to the two-parameter central extension of the planar Galilei group leads to the ''exotic'' particle, which has non-commuting position coordinates. A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non-commutative parameter is momentum-dependent in this case, and can take the form of a monopole in momentum space.

Key words: non-commutative mechanics; semiclassical models; Hall effect.

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