Talk:Anyon

message: The use of Spin(2,1) as notation here seems not to be consistent with that at spinor.
Charles Matthews 09:19, 29 Nov 2003 (UTC)
replies: text: Spin(p,q) is really the double cover of SO(p,q), not its universal cover in general... Of course, for Spin(3,1) and Spin(3), it makes no difference. Phys 13:25, 2 Dec 2003 (UTC)

message: I qualified the Stonybrook experiment as being controversial (which it is... very!).  And removed reference to anyons in quantum optics which are very tangential to the field, if relevant at all.  — Preceding unsigned comment added by 194.3.129.221 (talk) 23:04, 29 October 2011 (UTC)
thread_title:  Attempts at self promotion ?

message: The article notes 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 eiθ."  
A rotation of each particle about the other by a half revolution does not exchange their positions.  Is this meant to be a rotation of each particle about the center of the line joining them?  (talk • --Luriol (talk) 22:04, 3 May 2023 (UTC)
thread_title:  Particle Exchange 

message: I added some missing references
138.246.7.141 (talk) 12:49, 27 February 2008 (UTC)
thread_title:  Missing references 

message: The 23:04, 1 February 2009 Maayanh version of this article is translated into Chinese Wikipedia.--Wing (talk) 14:22, 1 March 2009 (UTC)
thread_title:  Translation into Chinese Wikipedia 

message: It is great to see how the article developed since I first contributed quite a while ago. Is it still appropriate to have it be "start class"? I think in relation to the importance of the article, it is quite developed. There is a historical explanation of the term "anyon", it's relevance for condensed matter physics is stressed, there is a comprehensible physics explanation of anyonic statistics, there is one part on the Mathematics and there is plenty of references. What else do we need? --MKlaput (talk) 17:50, 18 August 2010 (UTC)
thread_title:  Start Class 

message: I'm surprised to see after all this time that the only mention of non-abelian anyons is in the references.  I'll try restructuring the article a bit to include them in the body. Teply (talk) 21:44, 9 April 2012 (UTC)
thread_title:  non-abelian 

message: Needless to say this article is badly in need of diagrams. Here are some, feel free to criticize, take, or leave. 
The reference used was the popular Springer monograph on the Aharonov–Bohm effect:
: 
(I don't have the book right now so can't give page numbers, but it's a short, thin, concise one easy to look through, so it's not really that essential in this case). Thanks. M∧Ŝc2ħεИτlk 06:31, 9 May 2013 (UTC)
thread_title: Initial diagrams

message: Here's two more for the 2+1 spacetime case, showing the clockwise and anticlockwise senses. I'll add them and the other 3d one above if approved, to this article (anyon) and exchange operator. M∧Ŝc2ħεИτlk 07:11, 26 May 2013 (UTC)
replies: text: Could you make the animation slower? It is REALLY difficult to see what is happening at such speed... I had to look at it like 10 times, just because the dynamics of it were so fast.  — Preceding unsigned comment added by 143.107.229.249 (talk) 11:18, 4 December 2014 (UTC)
replies: text: Hi IP, saw your comment earlier and tried looking for the original file but it seems to be lost. I'll create another slower one. Best, M∧Ŝc2ħεИτlk 05:52, 5 December 2014 (UTC)
thread_title: Animations

message: Wouldn't it be better if the first sentence started- "In physics, an anyon is a type of quasiparticle..."? Bhny (talk) 00:36, 24 June 2014 (UTC)
replies: text: Since there were no objections I made the change. Bhny (talk) 17:18, 24 June 2014 (UTC)
thread_title:  Particle or quasiparticle? 

message: There is no explanation of why "abelian anyons" are abelian. The lead says is that it is explained below but it isn't. The word abelian doesn't even appear in the Anyon#Abelian_anyons section.
Let me know if this is helpful: Abelian anyons are particles who change their phase when moved. In the most general case, these particles are one-dimensional representations of braid groups. Gaugerigour (talk) 11:15, 4 January 2020 (UTC)
thread_title:  No explanation of abelian 

message: create Fractional statistics page  — Preceding unsigned comment added by 2A02:587:4110:4100:3408:7924:A4A:F99B (talk) 19:32, 16 November 2016 (UTC)
thread_title:  create [[Fractional statistics]] page 

message: Hello fellow Wikipedians,
I have just modified one external link on Anyon. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20150619152736/http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf to http://quantum.physics.sunysb.edu/MyPapers/PDF/prbFracStat.pdf
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thread_title:  External links modified 

message: April 2020 experimental evidence:  Rolf H Nelson (talk) 04:30, 19 May 2020 (UTC)
thread_title:  April 2020 experimental evidence update request 

message: I was not sure which type of Anyon they detected, but I was reading the article in some news and I was searching, for more reliable source and found article at Phys.org (added to the article).
The Wiki article say "Abelian anyons have been detected" I'm not sure if article at Phys.org is confirmation of that or that is for the other type. jcubic (talk) 18:33, 10 July 2020 (UTC)
replies: text: A very good new article from Discover Magazine got added today. It emphasizes the 2020 saw two important experiments about anyons -- one from Paris and the other from Purdue. Currently, the lead links to Paris-but-not-Purdue, while "Experiment" section describes Purdue-but-not-Paris. That mismatch needs to be fixed. HouseOfChange (talk) 02:10, 13 December 2020 (UTC)
thread_title:  Evidence 

message: : "Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to Dirac" [BLSW90]
Three independent perspectives on anyons
The discovery of anyons and intermediate quantum statistics in two spatial dimensions may be attributed to three independent research groups, working from three distinctly different perspectives. Chronologically, these are:
1. Geometric: In 1977 Jon Magne Leinaas and Jan Myrheim at the University of Oslo (Norway) used geometric methods from gauge theory, fiber (line) bundles, and extensions of the kinetic energy operator via boundary conditions (also known as "Schrödinger quantization") to arrive at the possibility for "intermediate statistics" in 2D (as well as in 1D), and worked out some of its detailed consequences for two particles [LM77].
: "The two possibilities, corresponding to symmetric and antisymmetric wave functions, appear in a natural way in the formalism. But this is only the case in which the particles move in three- or higher-dimensional space. In one and two dimensions a continuum of possible intermediate cases connects the boson and fermion cases." [LM77]
They further pointed to the role of the fundamental group for arbitrary numbers of particles and the possibility for higher-dimensional fibers (necessary for nonabelian representations), but they did not explicitly mention the braid group or its representation theory in their work.
2. Algebraic: Starting out from the representation theory of the Lie algebra of local currents on \R^d and the related group of diffeomorphisms (this is more closely related to "Heisenberg quantization", or the canonical quantization scheme as outlined by Dirac), Gerald A. Goldin, Ralph Menikoff and David H. Sharp at the Los Alamos National Laboratory (USA) found in 1980-'81 a kinematical derivation of the possibilities for quantum statistics with an unexpected dependence on the dimension d [GMS80,GMS81]:
: "Bose and Fermi N-particle systems were recovered as unitarily inequivalent induced representations of the group [...] In two-dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper." [GMS81]
In [GMS81] they also explored connections to the Aharonov-Bohm effect (in 3D). In 1983 the same research group highlighted the role of the one-dimensional unitary representations of the braid group [GMS83], and in 1985 they pointed out that higher-dimensional braid group representations induce inequivalent representations of the current algebra [GMS85]. Such particles were later termed "plektons" (see e.g. [KLS99]) or simply "nonabelian anyons" (nowadays the more common term).
Goldin, Menikoff and Sharp also pointed out in 1985 that exchange phases in 2D do not require indistinguishability; the colored braid group serves as the fundamental group for configurations of distinguishable particles [GMS85]. The general algebraic (and potentially nonabelian) approach to quantum statistics was further developed in the field theory direction by Tsuchiya and Kanie, as well as Fröhlich, Gabbiani, Kerler and Marchetti, and Fredenhagen, Rehren and Schroer [TK87,F89,KLS99].
3. Magnetic: Motivated by gauge theory examples in higher dimensions (theta-vacua and dyons), in 1982 Frank Wilczek at the University of California, Santa Barbara, considered a composite of a 2D magnetic flux and a charged particle and coined for it the name "anyon" [W82a,W82b]. 
: "Since interchange of two of these particles can give any phase, I will call them generically anyons." [W82b]
He was first to predict the fractional spin of such composites (a subsequent rigorous demonstration was provided by Goldin and Sharp [GS83]). The same line of investigation was continued by Wu in 1984, who connected it to path integrals, extended it to arbitrary numbers of particles and independently invoked the braid group and its one-dimensional representations explicitly [W84a,W84b]. Further, Arovas, Schrieffer, Wilczek and Zee considered in 1985 the statistical mechanics of a gas of such abelian anyons by means of its lowest-order virial expansion [ASWZ85].
These three groups were initially unaware of each other's earlier contributions and their initial works therefore lack the respective citations. There have subsequently been attempts to remedy this in the literature [BLSW90].
The first concrete application of intermediate/fractional statistics was to the fractional quantum Hall effect [L99], in which the emergent fractionally charged quasiparticles/holes were proposed by Halperin to have such properties [H84] (taking the magnetic perspective). Arovas, Schrieffer and Wilczek subsequently verified this (under tacit assumptions of adiabaticity [F91]) by computing the corresponding Berry phase for quasiholes [ASW84] (thus also invoking the geometric perspective).
Precursor ideas
As is evident from the above, anyons and intermediate ("fractional") quantum statistics are the convergence of many central ideas and concepts in mathematical physics. The former notion (anyon) is more closely tied into the concepts of exchange phases and symmetry classes of wave functions while the latter (fractional statistics) to that of the exclusion principle and permissible probability distributions. It has been and still is a non-trivial task to rigorously connect these two notions in the strictly intermediate case (see e.g. the review [CJ94]). Exploration of intermediate statistics from the latter approach of exclusion with a finite occupation number of one-body states was done by Gentile already in 1940 [G40] and 1942 [G42] (inspired by anyons, another, dimensionally independent, approach allowing for fractional exclusion was suggested by Haldane in 1991 [H91]).
The interpretation of the braid group as the fundamental group of a configuration space was rediscovered by mathematicians in 1962. [wiki:braid group] Ideas of charged particles circling regions of magnetic flux were developed by Ehrenberg, Siday [ES49], Aharonov and Bohm [AB59] and the more general concept of geometric phases by Berry [B84] and Simon [S83]. Methods of geometric quantization were developed by Souriau around 1967-'70, and he pointed out that if the configuration space is not simply connected then more than two group characters could arise and this "would lead to prequantizations of a new type" [S67,S70]. Around the same time, homotopy classes of Feynman paths brought another topological perspective on quantum statistics [S68,LdW71,D72] (cf. e.g. [M21]). For example, Laidlaw and DeWitt remark that in two space dimensions, possibilities are not limited to bosons and fermions; but they did not elaborate further [LdW71].
Foundations for nonabelian anyons were laid much earlier in the theory of parastatistics (nonabelian representations of the symmetric group) [G53,MG64]. Further, the central idea employed by Leinaas and Myrheim that equivalent configurations of indistinguishable particles need to be identified goes back all the way to Gibbs [F89].
The above shows that the essential ideas that were necessary for the discoveries of anyons were floating around at the time, and it is also interesting to remark that while it took about half a century from the discovery of bosons and fermions to that of anyons, it took another half a century to see their experimental confirmation.
Some remarks concerning one-dimensional anyons
Although this article concerns anyons in the 2D setting, it is unavoidable to note the influence of ideas from generalized notions of quantum statistics in 1D (i.e. 1+1 spacetime dimensions). One of the first works in which generalized commutation relations for field operators can be found is Klaiber [K69], followed by Streater and Wilde [SW70]. Transmutation of 1D hard-core bosons into fermions and vice versa was considered by Girardeau [G60], who also noted topological and dimensional consequences for the configuration space [G65]. The Lieb-Liniger model [LL63] for point-interacting bosons in 1D constitute the intermediate statistics found by Leinaas and Myrheim in 1D, however exchange phases may also be added to that model [K99]. Leinaas and Myrheim later also considered a different approach to statistics in 1D (and in higher dimensions) akin to "Heisenberg quantization" [LM93], and found a relationship to the Calogero-Sutherland model [C69,S71,P89]. 2D anyons that are forced into the lowest Landau level by a strong magnetic field may also be understood using such 1D concepts [O07].
References
[G40] G. Gentile (1940), Osservazioni sopra le statistiche intermedie. Nuovo Cimento 17, 493.
Abstract: "Si studiano le statistiche quantistiche, intermedie tra quella di Bose-Einstein e l'altra di Fermi-Dirac, quelle cioè in oui il numero massimo di occupazione di una cella nello spazio delle fasi è un numero finito d. Si discutono brevemente le proprietà corrispondenti d'un gas perfetto."
[G42] G. Gentile (1942), Le statistiche intermedie e le proprietà dell'elio liquido. Nuovo Cimento 19, 109.
[ES49] W. Ehrenberg and R. E. Siday (1949), The Refractive Index in Electron Optics and the Principles of Dynamics. Proceedings of the Physical Society B. 62 (1): 8–21.
From the abstract: "An expression for the optical path difference is given in terms of the magnetic flux enclosed."
[G53] H. S. Green (1953), A generalized method of field quantization. Phys. Rev. 90, 270-273.
Abstract: "A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose or Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined."
[AB59] Y. Aharonov and D. Bohm (1959), Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115, 485-491.
Abstract: "In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials."
[G60] M. Girardeau (1960), Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension. J. Math. Phys. 1, 516.
From the abstract: "A rigorous one-one correspondence is established between one-dimensional systems of bosons and of spinless fermions."
[LL63] E. Lieb and W. Liniger (1963), Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State. Phys. Rev. 130, 1605.
From the abstract: "A gas of one-dimensional Bose particles interacting via a repulsive delta-function potental has been solved exactly."
[MG64] M. L. Messiah and O. W. Greenberg (1964), Symmetrization postulate and its experimental foundation. Phys. Rev. 136, B248-B267.
From the abstract: "The symmetrization postulate (SP) that states of more than one identical particle are either symmetric or antisymmetric under permutations is studied from the theoretical and experimental points of view. The theoretical analysis is carried out within the framework of particle quantum mechanics; the field-theory approach to identical particles using Bose, Fermi, para-Bose and para-Fermi quantization is not considered in this article. Particles not obeying SP can be accommodated in quantum mechanics, provided some modifica- tions are made in the usual quantum-mechanical formalism. The main modification is to replace the usual ray by a many-dimensional "generalized ray" as the representative of a physical state."
[G65] M. D. Girardeau (1965), Permutation symmetry of many-particle wave functions. Phys. Rev. 139, B500-B508.
From the abstract: "The symmetrization postulate (SP) states that wave functions are either completely symmetric or completely antisymmetric under permutations of identical particles. It is shown by one-dimensional counter-examples that SP is not demanded by the usual physical interpretation of the mathematical formalism of wave mechanics unless one makes use of further physical properties of real systems; the error in a standard proof of SP which ignores these properties is pointed out. It is then proved that SP is true for those systems of spinless particles which have the following properties: (a) probability densities are permutation-invariant, (b) allowable wave functions are continuous with continuous gradient, (c) the 3n-dimensional configuration space is connected, (d) the Hamiltonian is symmetric, and (e) the nodes of allowed wave functions have certain properties. The counterexamples show that SP is not a necessary property of those systems which do not have property (c)."
[S67] J.-M. Souriau (1967), Quantification géométrique. Applications. Annales de l’I. H. P., section A, tome 6, no 4, p. 311-341.
http://www.jmsouriau.com/Quantification_geometrique_1967.htm
p 338-339:
"admettant V^P privé de sa diagonale comme revêtement universel, et par conséquent le groupe symétrique \Gamma comme groupe d’homotopie."
[S68] L. S. Schulman (1968), A path integral for spin. Phys. Rev. 176, 1558-1569.
[C69] F. Calogero (1969), J. Math. Phys. 10, 2191, 2197.
[K69] B. Klaiber (1969), The Thirring model. Presented at the Theoretical Physics Institute, University of Colorado, Summer 1967.
"The relation between "spin" and statistics is thus maintained and (VI.1) is a continuous interpolation of it."
[S70] J.-M. Souriau, Structure des systèmes dynamiques. Dunod, Paris (1970)
http://www.jmsouriau.com/structure_des_systemes_dynamiques.htm
English translation by R. H. Cushman and G. M. Tuynman, Progress in Mathematics (1997), Vol. 19.
p 343 (p 331 in English version):
"A system of n identical particles without interactions [..] admits exactly two inequaivalent prequantizations", "We shall call them the prequantization of Bose-Einstein and Fermi-Dirac repectively."
"As before, prequantization is only possible if twice the spin is an integer. But geometry does not provide the relation between the spin and the character x as suggested by experiments."
p 386 (p 373 in English version): 
"the particles are bosons (fermions, respectively). In fact, these are the only ones encountered in nature."
"In the case of nonelementary particles, it could happen that Un is not simply connected. Then the homotopy group of Un no longer equals Sn and could have more than two characters. This would lead to prequantizations of a new type."
[SW70] R. F. Streater and I. F. Wilde (1970), Fermion states of a boson field. Nucl. Phys. B 24, 561-575.
From the abstract: "The Bose-Fermi alternative does not hold: instead, we find a continuum of representations which interpolates between the Bose and Fermi sectors."
[LdW71] M. G. G. Laidlaw and C. M. DeWitt (1971), Feynman functional integrals for systems of indistinguishable particles. Phys. Rev. D 3, 1375-1378.
Abstract: "The theory of path integration is extended to include systems whose configuration space is multiply connected, and it is seen that there are as many distinct propagators as there are scalar representations of the associated fundamental group. It is shown that the configuration space for a system of indistinguishable particles is multiply connected. There are only two propagators for this system, giving bosons and fermions, and showing that the Feynman formalism excludes parastatistics."
"Y(n,2) is multiply connected"
[S71] B. Sutherland (1971), J. Math. Phys. 12, 247, 250.
[D72] J. S. Dowker (1972), Quantum mechanics and field theory on multiply connected and on homogeneous spaces. J. Phys. A: Gen. Phys. 5, 936-943.
[LM77] J. M. Leinaas and J. Myrheim (1977), On the theory of identical particles. Nuovo Cimento 37B, 1-23.
Abstract: "The classical configuration space of a system of identical particles is examined. Due to the identification of points which are related by permutations of particle indices, it is essentially different, globally, from the Cartesian product of the one-particle spaces. This fact is explicitly taken into account in a quantization of the theory. As a consequence, no symmetry constraints on the wave functions and the observables need to be postulated. The two possibilities, corresponding to symmetric and antisymmetric wave functions, appear in a natural way in the formalism. But this is only the ease in which the particles move in three- or higher-dimensional space. In one and two dimensions a continuum of possible intermediate cases connects the boson and fermion cases. The effect of particle spin in the present formalism is discussed."
"We follow the Schrödinger quantization scheme and assume that the state of the system is given by a quadratically integrable function defined on the configuration space. The problem is then to define the free-particle Hamiltonian, by taking properly care of the physical effects of the singular points. An interaction between the particles is to be described in the usual way, by adding a potential to the free Hamiltonian. To exhibit the special nature of three-dimensional, physical space in this context, we want to examine the three cases of two identical particles moving in either one, two or three dimensions. We consider here only Euclidean one-particle spaces."
"The generalization involved is that [the local Hilbert space] h_x, in general, will no longer be one-dimensional."
"We have seen that, for N = 2, n = 2, the fundamental group is the cyclic group of infinite order, that is, the addition group of the integers, Z, which is much larger than S_2. It has a continuum of different representations by phase factors."
[GMS80] G. A. Goldin, R. Menikoff and D. H. Sharp (1980), Particle statistics from induced representations of a local current group. J. Math. Phys. 21, 650.
From the abstract: "The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations."
"It is interesting that the topology of the N-particle orbit depends critically on the number of space dimensions s. The nontrivial connectedness properties of the orbit are a consequence of the fact that configurations in which two or more points coincide are not included. For s >= 3, the coordinate space [...] is simply connected, and hence it may be identified with the universal covering space of \Delta^{(s)}_N. For s = 2, the coordinate space [...] is multiply connected. In carrying out the inducing construction in Sec. IV, we are led to functions on the universal covering space of the orbit. Consequently, for s = 2 there may be representations other than the usual ones corresponding to wave functions on R^{2N} \ D. These are not discussed further in this paper."
[GMS81] G. A. Goldin, R. Menikoff and D. H. Sharp (1981), Representations of a local current algebra in nonsimply connected space and the Aharonov-Bohm effect. J. Math. Phys. 22, 1664.
From the abstract: "In two-dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov-Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of R^s. The vector potential plays no role in this description of the Aharonov-Bohm effect."
[W82a] F. Wilczek (1982), Magnetic Flux, Angular Momentum, and Statistics. Phys. Rev. Lett. 48, 1144.
From the abstract: "The unusual statistics of flux-tube-charged-particle composites is discussed."
"In intermediate cases, the composites cannot be described as fermions nor as bosons."
[W82b] F. Wilczek (1982), Quantum Mechanics of Fractional-Spin Particles. Phys. Rev. Lett. 49, 957.
"Since interchange of two of these particles can give any phase, I will call them generically anyons.
In this paper some elementary examples in the quantum mechanics of anyons are worked out. Description of these particles requires some widening of a wave function. Also, we will see that the energy levels of a system of two noninteracting anyons are not in general simply related to the one-anyon levels.
Although practical applications of these phenomena seem remote, I think they have considerable methodological interest and do shed light on the fundamental spin-statistics connection. Some related work has been done previously."
"The universal covering space seems very awkward to parametrize and I have not made much progress with it. It is certainly an intriguing mathematical problem to see how the statistical mechanics of many free anyons interpolates between bosons and fermions."
[GMS83] G. A. Goldin, R. Menikoff and D. H. Sharp (1983), Diffeomorphism Groups, Gauge Groups, and Quantum Theory. Phys. Rev. Lett. 51, 2246.
Abstract: "Unitary representations of the infinite-parameter group Diff(R^3) are presented which describe particles with spin as well as tightly bound composite particles. These results support the idea that Diff(R^3) can serve as a "universal group" for quantum theory."
"The induced representations of S_N correspond to bosons or fermions. If R^2 replaces R^3 in the above development, G becomes the braid group B_N. Its one-dimensional unitary representations can describe not only bosons or fermions, but recently discussed particles obeying unusual statistics."
"The above results support the idea that Diff(R^3) can function as a kind of universal group for quantum theory, with its infinitesimal generators corresponding to noncanonical local quantum fields describing a wide variety of particles in inequivalent representations."
[GS83] G. A. Goldin and D. H. Sharp (1983), Rotation generators in two-dimensional space and particles obeying unusual statistics. Phys. Rev. D 28, 830.
Abstract: "We describe systems of particles obeying unusual statistics in two-dimensional space, as well as solenoid—charged-particle composites, in terms of a complete set of local gauge-invariant currents."
"Distinguishable as well as indistinguishable particles can be described by means of the rich class of inequivalent representations existing for the Lie algebra of local currents."
[S83] B. Simon (1983), Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase. Phys. Rev. Lett. 51, 2167.
Abstract: "It is shown that the "geometrical phase factor" recently found by Berry in his study of the quantum adiabatic theorem is precisely the holonomy in a Hermitian line bundle since the adiabatic theorem naturally defines a connection in such a bundle. This not only takes the mystery out of Berry's phase factor and provides calculational simple formulas, but makes a connection between Berry's work and that of Thouless et al. This connection allows the author to use Berry's ideas to interpret the integers of Thouless et al. in terms of eigenvalue degeneracies."
[ASW84] D. Arovas, R. Schrieffer, and F. Wilczek (1984), Fractional statistics and the quantum Hall effect. Phys. Rev. Lett. 53, 722.
Abstract: "The statistics of quasiparticles entering the quantum Hall effect are deduced from the adiabatic theorem. These excitations are found to obey fractional statistics, a result closely related to their fractional charge."
"As emphasized recently by Berry and by Simon (see also Wilczek and Zee and Schiff), given a Hamiltonian H(z_0) which depends on a parameter z_o, if z_o slowly transverses a loop, then in addition to the usual phase [...] an extra phase \gamma occurs in \psi(t) which is independent of how slowly the path is traversed."
[B84] M. V. Berry (1984), Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 1984 392, 45-57.
From the abstract: "A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters in the Hamiltonian \hat{H}(R) will acquire a geometrical phase factor \exp{i\gamma(C)} in addition to the familiar dynamical phase factor."
"It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor."
[H84] B. Halperin (1984), Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States. Phys. Rev. Lett. 52, 1583.
From the abstract: "Quasiparticles at the fractional quantized Hall states obey quantization rules appropriate to particles of fractional statistics."
"The appearance of fractional statistics in the present context is strongly reminiscent of the fractional statistics introduced by Wilczek to describe charged particles tied to "magnetic flux tubes" in two dimensions. As in Ref. 6, the quasiparticles can also be described by wave functions obeying Bose or Fermi statistics, the various representations being related by a "singular gauge transformation"."
"However, the boson or fermion descriptions require, in effect, a long-range interaction between quasiparticles which alters the usual quantization rules. The transformation between representations is analogous to the well-known transformation between impenetrable bosons and fermions in one dimension."
[W84a] Y.S. Wu (1984), General theory for quantum statistics in two dimensions. Phys. Rev. Lett. 52, 2103-2106.
Abstract: "Because of complicated topology of the configuration space for indistinguishable particles in two dimensions, Feynman's path-integral formulation allows exotic statistics. All possible quantum statistics in two-space are characterized by an angle parameter 
\theta which interpolates between bosons and fermions. The current formalisms in terms of topological action of multivalued wave functions can be derived in a model-independent way."
"Especially the \theta statistics in those models exhaust all exotic possibilities. Second, mathematically there is a very close analogy of all this to \theta worlds in gauge theories."
[W84b] Y.S. Wu (1984), Multi-particle quantum mechanics obeying fractional statistics. Phys. Rev. Lett. 53, 111-114.
Abstract: "We obtain the rule governing many-body wave functions for particles obeying fractional statistics in two (space) dimensions. It generalizes and continuously interpolates the usual symmetrization and antisymmetrization. Quantum mechanics of more than two particles is discussed and some new features are found."
[ASWZ85] D. P. Arovas, R. Schrieffer, F. Wilczek and A. Zee (1985), Statistical mechanics of anyons. Nucl. Phys. B 251, 117-126.
Abstract: "We study the statistical mechanics of a two-dimensional gas of free anyons - particles which interpolate between Bose-Einstein and Fermi-Dirac character. Thermodynamic quantities are discussed in the low-density regime. In particular, the second virial coefficient is evaluated by two different methods and is found to exhibit a simple, periodic, but nonanalytic behavior as a function of the statistics determining parameter."
"For the two-particle problem with no external potential and no interparticle interactions (free particles), the situation becomes eminently tractable."
[GMS85] G. A. Goldin, R. Menikoff and D. H. Sharp (1983), Phys. Rev. Lett. 54, 603. Comments on [W84a].
"It has been shown that quantum mechanics in R^s can be described by unitary representations of Diff(R^s), the group of diffeomorphisms of R^s which become trivial at infinity. Quantum statistics arises from certain induced representations of this group."
"There are, however, quantum theories obtained as representations of Diff(R^s) induced by higher-dimensional representations of \pi_1(\Delta), corresponding to parastatistics (for S_n) or "unusual parastatistics" (for B_n)."
"It should also be noted that even systems of distinguishable particles can be described by quantum theories with unusual phase shifts in two-dimensional space"
[TK87] A. Tsuchiya and Y. Kanie (1987), Vertex Operators in the Conformal Field Theory on P^1 and Monodromy Representations of the Braid Group. Lett. Math. Phys. 13, 303-312.
From the abstract: "The commutation relations of vertex operators induce monodromy representations of the braid group on the spaces of vacuum expectations of compositions of vertex operators."
"Our aim here is to give rigorous mathematical foundations to the work of [Knizhnik and Zamolodchikov], and to reformulate and develop the operator formalism in the conformal field theory on the complex projective line P^1."
[F89] J. Fröhlich, Quantum Statistics and Locality. Proceedings of the Gibbs Symposium, Yale University, May 15-17, 1989. American Mathematical Society, 1990, pp. 89–142.
"The notion of identical particles and their statistics has played a key role in the study of physical systems since the early days of statistical mechanics and quantum theory."
"This is the first instance where the notions of identical particles and their indistinguishability have had important physical consequences. For a review of rigorous results in the context of Gibbsian classical (and quantum) statistical mechanics (thermodynamic functions, equilibrium states, thermodynamic limit, etc.) we refer to Ruelle's book"
"We now claim that Gibbs's idea that configurations of identical particles which differ from each other only by a relabelling of the particles must be identified and the theory of quantization in §1.2 give a natural theory of particle statistics in quantum mechanics."
"\theta-statistics has been invented by Leinaas and Myrheim in 1977, rediscovered by Goldin, Menikoff, and Sharp, and further analyzed by Wilczek, who has coined the name "anyons" for particles with \theta-statistics, Zee, Dowker, Wu, and others. The first application of \theta-statistics to real physics concerns the theory of the fractional quantum Hall effect and is due to Laughlin, with extensions due to Halperin and others. There are tentative applications of \theta-statistics in the theory of certain highly anisotropic high-T_c superconductors."
"We would like to know what representations can appear [...], what the connection between spin and statistics looks like, whether nonabelian braid statistics can be reduced to, e.g., abelian braid statistics by introducing additional degrees of freedom (the answer turns out to be "no"), and whether partices with nonabelian statistics have been observed in nature (the answer is a likely "no", again).
In order to find answers to these questions, we shall present a general analysis of statistics which is based on the fundamental principles of local quantum theory (in the framework of Haag and Kastler and Doplicher, Haag and Roberts). This analysis has been worked out in collaboration with F. Gabbiani, T. Kerler, and P.-A. Marchetti and is fairly recent"
[P89] A. P. Polychronakos (1989), Non-relativistic bosonization and fractional statistics. Nucl. Phys. B 324, 597-622.
Abstract: "We demonstrate that a quantum system of N identical particles in one space dimension, interacting with a 1/x^2 two-body potential is unitarily equivalent to a system of N free particles. This system can be interpreted as a system of N particles obeying fractional statistics of order l, where l(l - 1) is the strength of the potential. For l an even (odd) integer, in particular, the system is equivalent to N free bosons (fermions) respectively. As a byproduct, a set of operators in the phase space of the interacting system generates a nontrivial realization of the Virasoro algebra without central extension."
[BLSW90] Larry Biedenharn, Elliott Lieb, Barry Simon and Frank Wilczek (1990), The Ancestry of the 'Anyon'. Phys. Today 43(8), pp. 90-91.
http://dx.doi.org/10.1063/1.2810672
"The interesting news story "Bosons Condense and Fermions 'Exclude,' but Anyons...?" by Anil Khurana (November 1989, page 17), though quite detailed in its attributions, actually omitted some significant work on the mathematical and physical foundations for fractional statistics of particles in two dimensions prior to Frank Wilczek's 1982 articles."
"These papers included several of the fundamental facts about "anyons"—the fact that the angle parameter interpolates between Bose and Fermi statistics, the shifted angular momentum and energy spectra, and the connections with the topology of configuration space and with the physics of a charged particle making a circuit about a solenoid."
"Like so many deep ideas in physics, the topological underpinnings of anyons can be traced back to Dirac (who in fact originally interpreted the Pauli Verbot as a symmetry principle). Dirac explained the double covering of the three-dimensional rotation group by means of a string construction, which was in 1942 analyzed by M. H. A. Newman using the braid group—to our knowledge, the first use of the braid group for a physical problem."
[F91] S. Forte (1991), Berry's phase, fractional statistics and the Laughlin wave function. Mod. Phys. Lett. A 6, 3153-3162.
Abstract: "We determine the path-integral for particles with fractional spin and statistics in the adiabatic limit, and we discuss the identification of the spin-changing term with a Berry phase. We show that the standard proof that the Laughlin wave function describes excitations with fractional statistics holds only on the basis of a tacit assumption of questionable validity."
"It is actually apparent that the Berry phase argument used to prove fractional statistics of the Laughlin quasiparticles is, at best, incomplete."
"It seems thus impossible to establish the statistics of a system by looking only at the wave function, without knowledge of the Hamiltonian."
"This implies that there is no way of establishing whether or not the Laughlin wave function carries fractional statistics, without knowledge of the effective interaction between quasiparticles"
"Whether the Laughlin wave function is of this kind, or rather it does carry fractional statistics as claimed by Arovas et al., is a question that can only be answered by an understanding of the dynamics which underlies the Laughlin theory. Whether this is relevant to the fractional quantum Hall effect rests ultimately with experiment."
[H91] F. D. M. Haldane (1991), "Fractional Statistics" in Arbitrary Dimensions: A Generalizaiton of the Pauli Principle. Phys. Rev. Lett. 67, 937.
"The new definition does not apply to "anyon gas" models as currently formulated."
[LM93] J. M. Leinaas and J. Myrheim (1993), Heisenberg quantization for systems of identical particles. Int. J. Mod. Phys. A 8, 3649-3695.
Abstract: "We show that the algebraic quantization method of Heisenberg and the analytical method of Schrödinger are not necessarily equivalent when applied to systems of identical particles. Heisenberg quantization is a natural approach, but inherently more ambiguous and difficult than Schrödinger quantization. We apply the Heisenberg method to the examples of two identical particles in one and two dimensions, and relate the results to the so-called fractional statistics known from Schrödinger quantization. For two particles in d dimensions we look for linear, Hermitian representations of the symplectic Lie algebra sp(d,R). The boson and fermion representations are special cases but there exist other representations. In one dimension there is a continuous interpolation between boson and fermion systems, different from the interpolation found in Schrödinger quantization. In two dimensions we find representations that can be realized in terms of multicomponent wave functions on a three-dimensional space, but we have no clear physical interpretation of these representations, which include extra degrees of freedom compared to the classical system."
[CJ94] G. S. Canright and M. D. Johnson (1994), Fractional statistics: \alpha to \beta. J. Phys. A: Math. Gen. 27, 3579-3598.
"The concepts of fractional statistics may also be considered to fall into two areas, namely, the by-now well known fractional exchange parameter \alpha (defined by writing the exchange phase as \exp(i\alpha\pi)), and the less well known, but equally novel, fractional exclusion principle (first defined by Haldane in 1991, defined by us below, and symbolized by the 'exclusion coefficient' \beta henceforth). In fact, following Haldane's invention of the latter idea, and our own efforts to test and apply it in the FQHE, we claim that there are now two, in principle distinct, kinds of anyons-which we will call, not surprisingly, \alpha-anyons and \beta-anyons."
[K99] A. Kundu (1999), Exact Solution of Double \delta Function Bose Gas through an Interacting Anyon Gas. Phys. Rev. Lett. 83, 1275.
Abstract: "A 1D Bose gas interacting through \delta, \delta' and double \delta function potentials is shown to be equivalent to a \delta anyon gas, allowing an exact Bethe ansatz solution. In the noninteracting limit, it describes an ideal gas with generalized exclusion statistics and solves some recent controversies."
[KLS99] C. Korff, G. Lang and R. Schrader (1999), Two-particle scattering theory for anyons. J. Math. Phys. 40, 1831.
"Also, the general case, where the finite-dimensional representation of the braid group is not one-dimensional, has been considered. The corresponding particles are called "plektons." The relevance of braid group statistics in conformal quantum field theory was realized by Tsuchiya and Kanie and in algebraic quantum field theory by Fröhlich and Fredenhagen, Rehren and Schroer."
[L99] R. B. Laughlin (1999), Nobel Lecture: Fractional quantization. Rev. Mod. Phys. 71, 863.
"Fractional quantum Hall quasiparticles exert a long-range velocity-dependent force on each other—a gauge force—which is unique in the physics literature in having neither a progenitor in the underlying equations of motion nor an associated continuous broken symmetry. It arises spontaneously along with the charge fractionalization and is an essential part of the effect, in that the quantum states of the quasiparticles would not count up properly if it were absent. This force, which is called fractional statistics (Leinaas and Myrheim, 1977; Wilczek, 1982), has a measurable consequence, namely, the values of the subsidiary fractions 2/5 and 2/7 and their daughters in the fractional quantum Hall hierarchy."
"It was subsequently discovered by Jain (1989) that the sequence of fractional quantum Hall ground states could be constructed by a method that did not employ quasiparticles at all, and thus the obvious conclusion that the occurrence of these fractions proves the existence of fractional statistics was called into question. However, it should not have been. The quasiparticles are quite far apart—about 3l—in the 2/7 and 2/5 states, and the gap to make them is large, so to assume that they simply vanish when these subsidiary condensates form makes no sense. Had the quasiparticles been fermions these densities would have been 10/27=0.370 rather than 2/5=0.40 and 8/27=0.296 rather than 2/7=0.286. The effect of the fractional statistics is therefore small but measurable, about 5% of the observed condensation fraction."
"I similarly acknowledge Bert Halperin’s many outstanding contributions, including particularly his discovery that quasiparticles obey fractional statistics (Halperin, 1984)."
[O07] Ouvry (2007), Anyons and Lowest Landau Level Anyons. Séminaire Poincaré XI (2007), 77–107.
From the abstract: "However, a simplification arises when the anyons are coupled to an external homogeneous magnetic field. In the case of a strong field, by projecting the system on its lowest Landau level (LLL, thus the LLL-anyon model), the anyon model becomes solvable, i.e. the classes of exact eigenstates alluded to above provide for a complete interpolation from the LLL-Bose spectrum to the LLL-Fermi spectrum. Being a solvable model allows for an explicit knowledge of the equation of state and of the mean occupation number in the LLL, which do indeed interpolate from the Bose to the Fermi cases. It also provides for a combinatorial interpretation of LLL-anyon braiding statistics in terms of occupation of single particle states."
"The one dimensional Calogero model is also shown to have the same kind of thermodynamics as the LLL-anyons thermodynamics. This is not a coincidence: the LLL-anyon model and the Calogero model are intimately related, the latter being a particular limit of the former."
[M21] A. Mouchet (2021), Path integrals in a multiply-connected configuration space (50 years after). Foundations of Physics 51, 107.
DOI: 10.1007/s10701-021-00497-y
Abstract: "The proposal made 50 years ago by Schulman (1968), Laidlaw & Morette-DeWitt (1971) and Dowker (1972) to decompose the propagator according to the homotopy classes of paths was a major breakthrough: it showed how Feynman functional integrals opened a direct window on quantum properties of topological origin in the configuration space. This paper casts a critical look at the arguments brought by this series of papers and its numerous followers in an attempt to clarify the reason why the emergence of the unitary linear representation of the first homotopy group is not only sufficient but also necessary."
Doctagon (talk) 23:26, 15 August 2022 (UTC)
replies: text:  Perhaps a standalone article on "History of Fractional Statistics" would be able to use some of this material. Unfortunately, quite aside from the continued SPA efforts to foreground obscure papers by Gerald Goldin, this material represents OR interpretations of primary research papers, rather than summarizing what published RS have said about important events in the development of theories concerning fractional statistics. HouseOfChange (talk) 01:05, 16 August 2022 (UTC)
replies: text: With respect to the algebraic approach, this proposal is repeating what is said in [F89] and [BLSW90], and these are among the most reliable sources one can think of, having been written by the experts and originators of the field. As already pointed out in this discussion, numerous other reliable and respected sources will confirm the same information, however some other authors have either replicated earlier incomplete citations or deliberately omitted such for reasons one can only speculate. In some cases a possible explanation for such omission may be that the algebraic approach is extremely technical as compared to the other approaches and therefore its development only fully grasped by a handful of researchers, including those cited above (thus, authors adding the simple "see also" would typically be those who did not spend much effort trying to understand it). It is therefore reasonable to let these experts' own words summarize this aspect, and to invite other experts to further revise the proposal objectively. Based on our discussion here I am not convinced that HOC is qualified to make such an objective judgment on this topic, and I strongly encourage further participation from the mathematical physics community (and in particular those familiar with the Journal of Mathematical Physics). Glancing through the supplied references it will be clear, also to non-experts, that the information currently in the history and nonabelian section does NOT provide a sufficiently accurate reflection of the history of this topic. Doctagon (talk) 01:08, 17 August 2022 (UTC)
replies: text: This article's "History" section gives a BRIEF account of the most important contributions to the topic of anyons. Fermi–Dirac_statistics#History and Bose–Einstein_statistics#History are similar examples. The level of detail you propose is out of balance with the size of the article, and out of balance with the importance of multiple early contributions to the topic of "anyons." This is why I suggest you create a separate article History of Fractional Statistics for such a level of detail. HouseOfChange (talk) 13:53, 17 August 2022 (UTC)

text: @Doctagon Two references in the Further reading section contain some information about the history of anyons. The algebraic approach by Goldin et. al. which  needs to be mentioned in the article. Since you seem to be knowledgeable about this topic, it would be nice if you can edit the article in a suitable way. ApoorvPotnis2000 (talk) 10:17, 29 December 2022 (UTC)
thread_title:  History 

message: Thanks User:Maschen for creating that diagram of interferometer.  I'm sure the diagram itself is good, but it lacks anchoring in the text.  Could someone who knows what's going on there add a brief description of the 2005 experiment at the top of the "Experiment" section?  The Experiment section currently gives the impression that experiments started in 2020. CyreJ (talk) 13:12, 17 March 2022 (UTC)
thread_title:  Diagram context 

message: The "Further reading" section has 3 scholarly reviews and a short popular piece from Quanta Magazine. Only one of these four items mentions Gerald Goldin: Nayak et al. say "This topological difference between two and three dimensions, first realized by Leinaas and Myrheim, 1977 and by Wilczek, 1982a, leads to a profound difference in the possible quantum mechanical properties, at least as a matter of principle, for quantum systems when particles are confined to 2 + 1 D (see also Goldin et al., 1981 and Wu, 1984)" Similarly, Khare's 1998 book on fractional statistics summed up the 1981 paper as "Few years later, Goldin, Menikoff and Sharp obtained the same results [as Leinaas and Myrheim] by an entirely different approach." 
Per BRD, I reverted the recent addition of a 5th item to this short list, a 1991 article by Gerald Goldin, mostly behind a paywall except for its reference list: about 70 papers with 18 of those 70 having Gerald Goldin as first author. Goldin is not an impartial RS concerning the history of anyon research.  HouseOfChange (talk) 03:52, 31 December 2022 (UTC)
thread_title:  Further reading section