Anyons and the Landau problem in the noncommutative plane

The Landau problem in the noncommutative plane is discussed in the context of realizations of the two-fold centrally extended planar Galilei group and the anyon theory. In 2+1 dimensions, Galilei group admits a two-fold central extension [1, 2] characterized by the algebra with the nonzero Poisson bracket relations {Ki,Pj} = mδij, {Ki,Kj} = −κǫij , (1) {Ki,H} = Pi, {J ,Pi} = ǫijPj , {J ,Ki} = ǫijKj , (2) where m and κ are the central charges. The algebra has the two Casimir elements C1 = mJ + κH− ǫijKiPj , C2 = mH− 1 2 P i , (3) which correspond to the (multiplied by the mass m) internal angular momentum (spin) and energy. There are two possibilities to realize this algebra as a symmetry of a free particle on a plane: the minimal realization and the extended one [cf. the two formulations for a free relativistic anyon [3]]. Requiring that the particle coordinate Xi forms a Galilei covariant object with respect to the action of the generators J , Pi and Ki, treating the Galilei generators as integrals of motion and identifying the Pi as the canonical momentum pi, and, finally, putting the spin and internal energy to be equal to zero (C1 = C2 = 0), we arrive at the following realization of the generators: Pi = pi, Ki = mXi − tpi +mθǫijpj , J = ǫijXipj + 1 2 θ~p , H = 1 2m ~ p , (4) θ = κ/m. As a result, the Xi has a usual free particle evolution, Ẋi = 1 m pi. The price we pay for such a minimal realization of the exotic Galilei algebra is the non-commutativity of the coordinate components {Xi,Xj} = θǫij, (5) and the non-canonical form of the associated symplectic structure σ0 = dpi ∧ dXi + 1 2 θǫijdpi ∧ dpj . (6) One can define another sort of the coordinate [4, 5], Yi = Xi + θǫijpj. (7) ∗E-mail: [email protected]