Se p 20 06 Non - commutative mechanics in mathematical & in condensed matter physics ∗

Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics. Souriau's construction applied to the two-parameter central extension of the planar Galilei group lead 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. 1 " Exotic " Galilean symmetry and mechanics in the plane Central extensions first entered physics when Heisenberg realized that, in the quantum mechanics of a massive non-relativistic particle, the position and momentum operators did not commute. As a consequence, phase-space translations act up-to-phase only on the quantum Hilbert space. In more mathematical terms, it is not the [commutative] translation group itself, only its [non-commutative] 1-parameter central extension, the Heisenberg group, which is represented unitarily. Similarly, Galilean boosts act, for a * Talk given at the O'Raifeartaigh Symposium on Non-perturbative and symmetry methods in field theory. Budapest, June 2006.