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Non-commutative mechanics, in mathematical & in condensed matter physics

P. A. Horvathy
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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.

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P. A. Horvathy. 2006-12-15. Non-commutative mechanics, in mathematical & in condensed matter physics. https://arxiv.org/abs/cond-mat/0609571

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