Derivation of the Symmetry Postulates for Identical Particles from Pilot-Wave Theories

The symmetries of the wavefunction for identical particles, including anyons, are given a rigorous non-relativistic derivation within pilot-wave formulations of quantum mechanics. In particular, parastatistics are excluded. The result has a rigorous generalisation to n particles and to spinorial wavefunctions. The relation to other non-relativistic approaches is briefly discussed. 1 Quantum indistinguishability It is well known that identical particles in standard (non-relativistic) quantum mechanics are characterised by symmetry conditions on the wavefunction. For spinless wavefunctions, these are given by ψ(x1, . . . ,xi, . . . ,xj , . . . ,xn, t) = e ψ(x1, . . . ,xj , . . . ,xi, . . . ,xn, t), (1) where γ = 0 (mod 2π) for bosons (symmetry), and γ = π (mod 2π) for fermions (antisymmetry). In two dimensions, it is possible for γ to be arbitrary (anyons), where the value depends on the homotopy class of the path along which the particles are exchanged, and the wavefunction is multi-valued. For spinorial wavefunctions (which we discuss only in three dimensions) one requires total symmetry or antisymmetry under simultaneous exchange of both spatial and spinorial indices (bosons or fermions). ∗Department of Philosophy, University of California at Berkeley, 314 Moses Hall, Berkeley, CA 94720-2390, U.S.A. (e-mail: [email protected]). The possibility of anyons was first pointed out by Leinaas and Myrheim (1977). They were described in detail by Goldin et al. (1981) and by Wilczek (1982), and appear to be relevant to the explanation of the fractional quantum Hall effect (Arovas et al., 1982, Laughlin, 1983, Tsui et al., 1982).