In mathematics and physics, an anyon is a type of projective representation of a Lie group. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Inter. ... Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

In detail, there are projective representations of SO(2,1) which don't arise from linear representations of SO(2,1), or of its double cover, Spin(2,1). These representations are called anyons. In mathematics, in particular in group theory, if G is a group and ρ is a vector space over a field K, then a projective representation is a homomorphism from G to Aut(ρ)/Kx where Kx here is the normal subgroup of Aut(ρ) consisting of multiplications of vectors... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

The topological reason behind the phenomenon is this: the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is Z (infinite cyclic). This means that Spin(2,1) is not the universal cover: it is not simply connected. On the other hand, for n > 2, for SO(n,1) and Poincaré(n,1), it's only Z₂ (cyclic of order 2); meaning that the spin group is simply connected. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...

Actually, this concept also applies to nonrelativistic systems. The relevant part here is that the spatial rotation group is SO(2), which has an infinite first homotopy group.

This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphite or the quantum Hall effect. In space of three dimensions (or more), elementary particles have tightly constrained quantum numbers and, in particular, are restricted to being fermions or bosons. In two-dimensional systems, however, quasiparticles are observed whose quantum states range continuously between fermionic and bosonic, taking on any quantum value in between. Frank Wilczek coined the term "anyons" in 1982 to describe such particles. In common usage, the dimensions (from Latin measured out) of an object are the parameters or measurements required to define its shape and size, that is, usually, its height, width, and length. ... Graphite (named by Abraham Gottlob Werner in 1789, from the Greek γραφειν: to draw/write, for its use in pencils) is one of the allotropes of carbon. ... The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional systems of electrons subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values where e is the elementary charge and h is Plancks constant. ... In particle physics, an elementary particle is a particle of which other, larger particles are composed. ... A quantum number is any one of a set of numbers used to specify the full quantum state of any system in quantum mechanics. ... Fermions, named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. ... Bosons, named after Satyendra Nath Bose, are particles which form totally-symmetric composite quantum states. ... A quantum state is any possible state in which a quantum mechanical system can be. ... In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Frank Wilczek at Harvard University Frank Wilczek (born May 15, 1951) is an American physicist of Polish and Italian origin. ...

Let's say we have two identical particles on a plane. If we interchange both particles so that each particle travels counterclockwise for half a cycle around the center of both particles, the wave function of the system changes by a factor of e^iθ where θ is an angle which only depends upon the type of particle in question. If θ is zero, we have a boson and if θ is π we have a fermion. For any other value, we have an anyon. If we have two particles a and b, which may or may not be identical, then their mutual statistics is the change in the phase factor, which is picked up after particle b is rotated counterclockwise around particle a for one full cycle. The mutual statistics may be completely unrelated to the interchange angle between two identical particles.

External links

Interview with Frank Wilczek on anyons and superconductivity

See also: plekton. In physics, a plekton is a hypothetical kind of elementary particle, which would obey a different style of statistics with respect to the interchange of identical particles. ...