ar X iv : h ep - t h / 97 04 18 2 v 3 3 0 A pr 1 99 7 On the Geometry of Relativistic Anyon

A twistor model is proposed for the free relativistic anyon. The Hamiltonian reduction of this model by the action of the spin generator leads to the minimal covariant model; whereas that by the action of spin and mass generators, to the anyon model with free phase space that is a cotangent bundle of the Lobachevsky plane with twisted symplectic structure. Quantum mechanics of that model is described by irreducible representations of the (2+1)-dimensional Poincare' group. 1. Introduction. As is known, in (2+1)-dimensional space-time there can exist anyons, particles with arbitrary spins and statistics [1]. Quantum mechanics of anyons as a spinning particles is described by unitary irreducible representations of the (2+1)-dimensional Poincare' group [2]. However, classical theory of anyons cannot be described within the standard approach based on the introducing of Grassmann variables. For this purpose are use the so-called " minimal " and " extended " approaches (here me employ the classification of Ref.[3]). In the minimal approach, the spin is provided by the nontrivial symplectic structure of phase space [4, 2]. The minimal relativistic-covariant model (in what follows, K−model) of the anyon with spin s is realized on the phase space T * IR 1.2 with the twisted symplectic structure g ab dP a ∧ dQ b + s P