In quantum mechanics, and some classical stochastic systems,
indistinguishable particles have the property that exchanging the states of particle with particle (symbolically {{tmath|\psi_i\leftrightarrow\psi_j \text{ for } i \ne j}}) does not lead to a measurably different many-body state. In a quantum mechanical system, for example, a system with two indistinguishable particles, with particle 1 in state and particle 2 in state , has state in
Dirac notation. Now suppose we exchange the states of the two particles, then the state of the system would be . These two states should not have a measurable difference, so they should be the same vector, up to a
phase factor: : \left|\psi_1\psi_2\right\rangle = e^{i\theta}\left|\psi_2\psi_1\right\rangle. Here, {{tmath|e^{i\theta} }} is the phase factor. In space of
three or more dimensions, the phase factor is or . Thus,
elementary particles are either fermions, whose phase factor is , or bosons, whose phase factor is . These two types have different
statistical behaviour. Fermions obey
Fermi–Dirac statistics, while bosons obey
Bose–Einstein statistics. In particular, the phase factor is why fermions obey the
Pauli exclusion principle: If two fermions are in the same state, then we have : \left|\psi\psi\right\rangle = -\left|\psi\psi\right\rangle. The state vector must be zero, which means it is not normalizable, thus it is unphysical. 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 the case of two particles this can be expressed as : \left|\psi_1\psi_2\right\rangle = e^{i\theta}\left|\psi_2\psi_1\right\rangle, where {{tmath|e^{i\theta} }} can be other values than just or . It is important to note that there is a slight
abuse of notation in this shorthand expression, as in reality this wave function can be and usually is multi-valued. This expression actually means that when particle 1 and particle 2 are interchanged in a process where each of them makes a counterclockwise half-revolution about the other, the two-particle system returns to its original quantum wave function except multiplied by the complex unit-norm phase factor . Conversely, a clockwise half-revolution results in multiplying the wave function by . Such a theory obviously only makes sense in two dimensions, where clockwise and counterclockwise are clearly defined directions. In the case we recover the Fermi–Dirac statistics () and in the case (or ) the Bose–Einstein statistics (). In between we have something different.
Frank Wilczek in 1982 explored the behavior of such quasiparticles and coined the term "anyon" to describe them, because they can have any phase when particles are interchanged. Unlike bosons and fermions, anyons have the peculiar property that when they are interchanged twice in the same way (e.g. if anyon 1 and anyon 2 were revolved counterclockwise by half revolution about each other to switch places, and then they were revolved counterclockwise by half revolution about each other again to go back to their original places), the wave function is not necessarily the same but rather generally multiplied by some complex phase (by in this example). We may also use with particle
spin quantum number
s, with
s being
integer for bosons,
half-integer for fermions, so that : e^{i\theta} = e^{2i \pi s} = (-1)^{2s}, or : |\psi_1 \psi_2\rangle = (-1)^{2s} |\psi_2 \psi_1\rangle. 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. In a three-dimensional position space, the fermion and boson statistics operators (−1 and +1 respectively) are just 1-dimensional representations of the
permutation group (S
N of
N indistinguishable particles) acting on the space of wave functions. In the same way, in two-dimensional position space, the abelian anyonic statistics operators () are just 1-dimensional representations of the
braid group (
BN of
N indistinguishable particles) acting on the space of wave functions. Non-abelian anyonic statistics are higher-dimensional representations of the braid group. Anyonic statistics must not be confused with
parastatistics, which describes statistics of particles whose wavefunctions are higher-dimensional representations of the permutation group.
Topological equivalence The fact that the
homotopy classes of paths (i.e. notion of
equivalence on
braids) are relevant hints at a more subtle insight. It arises from the
Feynman path integral, in which all paths from an initial to final point in
spacetime contribute with an appropriate
phase factor. The Feynman path integral can be motivated from expanding the propagator using a method called time-slicing, in which time is discretized. In non-homotopic paths, one cannot get from any point at one time slice to any other point at the next time slice. This means that we can consider
homotopic equivalence class of paths to have different weighting factors. So it can be seen that the
topological notion of equivalence comes from a study of the Feynman path integral. In 2020, two teams of scientists (one in Paris, the other at Purdue) announced new experimental evidence for the existence of anyons. Both experiments were featured in
Discover Magazines 2020 annual "state of science" issue. In July, 2020, scientists at Purdue University detected anyons using a different setup. The team's interferometer routes the electrons through a specific maze-like etched nanostructure made of
gallium arsenide and
aluminium gallium arsenide. "In the case of our anyons the phase generated by braiding was 2π/3", he said. "That's different than what's been seen in nature before." As of 2023, this remains an active area of research; using a superconducting processor, Google Quantum AI reported on the first braiding of non-Abelian anyon-like particles in an arXiv article by Andersen
et al. in October 2022, later published in Nature. In an arXiv article released in May 2023, Quantinuum reported on non-abelian braiding using a trapped-ion processor. In May, 2025, researchers at the
University of Innsbruck have observed anyons in a one-dimensional quantum system. == Non-abelian anyons ==