|
In physics, an anyon is a type of quasiparticle that occurs only in ''two''-dimensional systems, with properties much less restricted than fermions and bosons; the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables. Anyons are generally classified as ''abelian'' or ''non-abelian''. Abelian anyons have been detected and play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected although this is an active area of research. == Abelian anyons == In space of three or more dimensions, elementary particles are either fermions or bosons, according to their statistical behaviour. Fermions obey the Fermi–Dirac statistics while bosons obey the Bose–Einstein statistics. In the language of quantum mechanics this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state with indistinguishable particles (in Dirac notation): : (where the first entry in is the state of particle 1 and the second entry is the state of particle 2. So for example the left hand side is read as "Particle 1 is in state ''ψ''1 and particle 2 in state ''ψ''2"). Here the "+" corresponds to the particles being bosons and the "−" to the particles being fermions (composite states of fermions and bosons or distinct particle types are irrelevant since that would make them distinguishable). In two-dimensional systems, however, quasiparticles can be observed that obey statistics ranging continuously between Fermi–Dirac and Bose–Einstein statistics, as was first shown by Jon Magne Leinaas and Jan Myrheim of the University of Oslo in 1977. In our above example of two particles this looks as follows: : with ''i'' the imaginary unit and ''θ'' a real number. This is an application of Euler's formula and can produce any unit complex number (). In the case ''θ'' = ''π'' we recover the Fermi–Dirac statistics () and in the case (or ) the Bose–Einstein statistics (). In between we have something different. Frank Wilczek coined the term "anyon" to describe such particles, since they can have any phase when particles are interchanged. We also may use with particle spin quantum number ''s'', with ''s'' being integer for bosons, half-integer for fermions, so that : or At an edge, fractional quantum Hall effect anyons are confined to move in one space dimension. Mathematical models of one-dimensional anyons provide a base of the commutation relations shown above. Just as the fermion and boson wavefunctions in a three-dimensional space are just 1-dimensional representations of the permutation group (''SN'' of ''N'' indistinguishable particles), the anyonic wavefunctions in a two-dimensional space are just 1-dimensional representations of the braid group (''BN'' of ''N'' indistinguishable particles). Anyonic statistics must not be confused with parastatistics, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「anyon」の詳細全文を読む スポンサード リンク
|