Exotic galilean symmetry and non-commutative mechanics

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.

Review paper. Published version

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• Emergence of Geometric phase shift in Planar Non-commutative Quantum Mechanics
• Chiral decomposition in the non-commutative Landau problem
• Thermodynamics of Quantum Phase Transitions of a Dirac oscillator in a homogenous magnetic field
• On non-local representations of the ageing algebra
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• Triply Extended Group of Translations of ${\mathbb{R}}^4$ as Defining Group of NCQM: relation to various gauges
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• Aspects of perturbative quantum field theory on non-commutative spaces
• Analytical and numerical analysis of a rotational invariant D=2 harmonic oscillator in the light of different noncommutative phase-space configurations
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• Symmetries of the Free Schrödinger Equation in the Non-Commutative Plane
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• Field Theory with Coordinate Dependent Noncommutativity
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• Group Theoretical Construction of Planar Noncommutative Systems
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