99 9 Oslo - Tp 2 - 99 Spin and Statistics for Quantum Hall Quasi - Particles

In two space dimensions the possibilities of fractional spin as well as fractional statistics exist. I examine the relation between fractional spin and statistics for Laughlin quasi-particles in a two-dimensional electron system with spherical geometry. The relevance of this for quasi-particles in a planar system is discussed. 1 Statistics and spin in two dimensions I would like to begin by reminding you of the fact that in two space dimensions there is a richer set of possibilities than in higher dimensions as far as statistics and spin of particles is concerned. Quantum statistics is determined by the symmetry of the wave function under interchange of particle coordinates, and in three and higher dimensions the corresponding symmetry group is the permutation group. However, when particle interchange is viewed as a continuous process under which the coordinates are changed, then the symmetry group in two dimensions is larger, it is the two-dimensional braid group rather than the permutation group [1]. An element of this group does not only specify the permutation of the particles, but also the windings of the particle trajectories under the interchange of the positions. In dimensions higher than two these windings can be disentangled, since only interchanges corresponding to different permutations of the particles are topologically distinct. This is not possible in two dimensions. For particles on the plane the coordinates can be written as complex variables, z = x + iy, and for two particles the symmetry under interchange of the particle To appear in the proceedings of the conference Orbis Scientiae 1998, Fort Lauderdale, December 18 – 21.