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Particles obeying unconventional forms of quantum statistics. For many years it was believed that only two possible forms of quantum statistics, Bose-Einstein and Fermi-Dirac statistics, were possible, but in fact a continuum of possibilities exists. Elementary excitations (quasiparticles) in the fractional quantum Hall effect are anyons.
In quantum mechanics, in the behavior of identical particles there are important dynamical effects that have no classical analog. Thus, in the case of two indistinguishable particles A and B, the amplitude for the process that leads to A arriving at point x while B arrives at point y must be added to the amplitude for the process that leads to A arriving at y while B arrives at x—the so-called exchange process—because the final states cannot be distinguished. Actually the recipe of adding the amplitude for the exchange process is appropriate only for particles obeying Bose-Einstein statistics (bosons); for particles obeying Fermi-Dirac statistics (fermions), this amplitude must be subtracted. See Fermi-Dirac statistics
The definition of anyons posits other possible recipes for adding exchange processes, refining the analysis of exchange to take account of the direction in which the exchange takes place. These more general possibilities can be defined only for particles whose motion is restricted to two space dimensions. However, many important materials are effectively two-dimensional, including microelectronic circuitry and the copper oxide layers of high-temperature superconductors. The quantum statistics of the quasiparticles in these systems is under investigation, but the fractional quantized Hall states are known to be anyons. See Hall effect, Quantum statistics, Superconductivity