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

SIGMA 6 (2010), 060, 26 pages      arXiv:1002.4772      https://doi.org/10.3842/SIGMA.2010.060
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Exotic Galilean Symmetry and Non-Commutative Mechanics

Peter A. Horváthy a, Luigi Martina b and Peter C. Stichel c
a) Laboratoire de Mathématiques et de Physique Théorique, Université de Tours, Parc de Grandmont, F-37200 Tours, France
b) Dipartimento di Fisica - Università del Salento and Sezione INFN di Lecce, via Arnesano, CP. 193, I-73100 Lecce, Italy
c) An der Krebskuhle 21 D-33 619 Bielefeld, Germany

Received March 23, 2010, in final form July 19, 2010; Published online July 26, 2010

Abstract
Some aspects of the ''exotic'' particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.

Key words: noncommutative spaces; Galilean symmetry; dynamical systems; quantum field theory.

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