Anyon

Particle statistics
Maxwell-Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statistics
Parastatistics
Anyonic statistics
Braid statistics
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In mathematics and physics, an anyon is a type of particle that only occurs in two-dimensional systems. Particle statistics refers to the particular description of particles in Statistical mechanics. In Statistical mechanics, Maxwell–Boltzmann statistics describes the statistical distribution of material particles over various energy states in Thermal equilibrium In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Quantum mechanics and Statistical mechanics, parastatistics is one of several alternatives to the better known Particle statistics models ( In Mathematics and Theoretical physics, braid statistics is a generalization of the statistics of Bosons and Fermions based on the concept Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. It is a generalization of the Fermion and Boson concept. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein

1 In physics
2 Topological basis
3 References
4 See also
5 External links

In physics

This mathematical concept becomes useful in the physics of two-dimensional systems such as sheets of graphene or the quantum Hall effect. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Graphene is a one-atom-thick planar sheet of sp2-bonded Carbon atoms that are densely packed in a honeycomb crystal lattice The quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems

In space of three or more dimensions, particles are restricted to being fermions or bosons, according to their statistical behaviour. In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Particle physics, bosons are particles which obey Bose-Einstein statistics; they are named after Satyendra Nath Bose and Albert Einstein Fermions respect the so-called Fermi-Dirac statistics while Bosons respect the Bose-Einstein statistics. In Statistical mechanics, Fermi-Dirac statistics is a particular case of Particle statistics developed by Enrico Fermi and Paul Dirac that In Statistical mechanics, Bose - Einstein statistics (or more colloquially B-E statistics determines the statistical distribution of In the language of quantum physics this is formulated as the behavior of multiparticle states under the exchange of particles. This is in particular for a two-particle state (in Dirac notation):

$\left|\psi_1\psi_2\right\rangle = \pm\left|\psi_2\psi_1\right\rangle$ (where the first entry in $\left|\dots\right\rangle$ is the state of particle 1 and the second entry is the state of particle 2. Bra-ket notation is a standard notation for describing Quantum states in the theory of Quantum mechanics composed of angle brackets (chevrons and Vertical 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 both particles being Bosons and the "-" to both particles being Fermions (composite states of Fermions and Bosons are not possible).

In two-dimensional systems, however, quasiparticles can be observed which 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^[1]. In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The University of Oslo (Universitetet i Oslo Universitas Osloensis is the oldest and largest University in Norway, situated in the Norwegian capital In our above example of two particles this looks as follows:

$\left|\psi_1\psi_2\right\rangle = e^{i\,\theta}\left|\psi_2\psi_1\right\rangle$

With "i" being the imaginary unit from the calculus of complex numbers and $θ$ a real number. Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, the real numbers may be described informally in several different ways Recall that $| e i θ | = 1$ and $e 2 i π = 1$ as well as $e i π = - 1$ . So in the case $θ = π$ we recover the Fermi-Dirac statistics (minus sign) and in the case $θ = 2π$ the Bose-Einstein statistics (plus sign). In between we have something different. For these types of particles Frank Wilczek coined the term "anyons"^[2] to describe such particles, since they can have "any" phase when particles are interchanged. Frank Anthony Wilczek (born May 15, 1951) is an American theoretical physicist and Nobel laureate.

Topological basis

In dimensions greater than two, the spin-statistics connection states that any multiparticle state has to obey are either Bose-Einstein or Fermi-Dirac statistics. The spin-statistics theorem in Quantum mechanics relates the spin of a particle to the statistics obeyed by that particle This is related to the first homotopy group of SO(n,1) (and also Poincaré(n,1)) with n>2, which is $Z 2$ (the cyclic group consisting of 2 Elements). In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an Therefore only two possibilities remain. (the details are more involved than that, but this is the crucial point)

The situation changes in two dimensions. Here the first homotopy group of SO(2,1) (and also Poincaré(2,1)) is Z (infinite cyclic). In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime This means that Spin(2,1) is not the universal cover: it is not simply connected. In Mathematics, a covering group of a Topological group H is a Covering space G of H such that G is a topological In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In detail, there are projective representations of the special orthogonal group SO(2,1) which don't arise from linear representations of SO(2,1), or of its double cover, the spin group Spin(2,1). In the mathematical field of Representation theory, a projective representation of a group G on a Vector space V over a In Mathematics, the indefinite orthogonal group, O( p, q) is the Lie group of all Linear transformations of a n = p In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a covering space is a Topological space C which "covers" another space X by a Surjective Local homeomorphism In Mathematics the spin group Spin( n) is the double cover of the Special orthogonal group SO( n) such that there exists a Short These representations are called anyons.

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 fact is also related to the Braid group well known in Knot theory. In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces The relation can be understood when one considers the fact that in 2 Dimensions the group of permutations of 2 particles is no longer the symmetric group $S 2$ (2-dimensional) but rather the Braid group $B 2$ (infinite dimensional). In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

References

^ J. M. Leinaas, and J. Myrheim, "On the theory of identical particles", Nuovo Cimento B37, 1-23 (1977).
^ F. Wilczek, Phys. Rev. Lett. 49, 957 (1982).

External links

Interview with Frank Wilczek on anyons and superconductivity

In Physics, a plekton is a theoretical kind of Elementary particle, which obeys a different style of statistics with respect to the interchange of Identical particles The fractional quantum Hall effect (FQHE is a physical phenomenon in which a certain system behaves as if it were composed of particles with charge smaller than the Elementary charge

Dictionary

-noun

(physics) Any particle that obeys a continuum of quantum statistics, only two of which are the standard Bose-Einstein and Fermi-Dirac statistics.

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