Non-relativistic anyons, exotic Galilean symmetry and noncommutative plane

We show that the Lukierski et al. model, invariant with respect to the two-fold centrally extended Galilei group, can be decomposed into an infinite number of independent copies (differing in their spin) of the “exotic” particle of Duval et al. The difference between the two models is found to be sensitive to electromagnetic coupling. The nature of the noncommutative plane coordinates is discussed in the light of the exotic Galilean symmetry. We prove that the first model, interpreted as describing a non-relativistic anyon, is the non-relativistic limit of a particle with torsion related to relativistic anyons. As it has been known for some time, the planar Galilei group admits an “exotic” twoparametric central extension. Recently, two classical systems have been presented that exhibit this extended Galilean symmetry. One of them, put forward by Lukierski, Stichel and Zakrzewski [1], uses the second-order Lagrangian LLSZ = 1 2 m~̇x 2 + κ 2 εijẋiẍj , (1) where m, the mass, and κ, the “exotic” parameter, label the central extension [2]. This system requires a 6-dimensional phase space. The other model [3, 4, 5] is derived from the “exotic” Galilei group following Souriau [6], who identifies classical “elementary” systems with the coadjoint orbits of the group, endowed with their canonical symplectic structures. These orbits are 4-dimensional, and depend on 4 parameters denoted by s, h0, m and κ. Their symplectic structure induces on Permanent address: Laboratoire de Mathématiques et de Physique Théorique, Université de Tours (France). E-mail: [email protected]. Also: Institute for High Energy Physics, Protvino (Russia). E-mail: [email protected]