Talk:Anyon

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)


 * 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)

The grammar in the last line of "In physics" is so bad that it's not even clear what is meant. — Preceding unsigned comment added by 128.111.9.44 (talk) 02:24, 20 July 2011 (UTC)

Sub-optimal beginning?
This article now starts by saying that anyons are only possible in two spatial dimensions, ignoring that they are also possible in 1 spatial dimension (including for effective descriptions of edge excitations of quantum Hall systems). There is an extensive literature on this subject, and different approaches to anyons in 1 spatial dimension. This possibility is also described in references already cited in this Wikipedia article, including "Fröhlich, Jürg (1988)" under Non-abelian anyons. I would argue that this Wikipedia article should reflect this. It does not have to go into the details, but it should not explicitly exclude it. I find that misleading. I may be new to editing on Wikipedia, but inclusion, at least in this sense, I thought, was an important value. I tried to edit, as minimally as I could, but this change was reverted.

I would also argue that the current beginning is suboptimal in how it cites references. Why should there be a repeated reference to "Ornes, Stephen (2020)", which seems to me to be a newsletter in a magazine, when there are better references? I would propose a recent addition to the 2nd ed. of the Encyclopedia of Condensed Matter Physics by Leinaas and Myrheim ( https://doi.org/10.1016/B978-0-323-90800-9.00187-6 ), as it is part of a larger recent effort by many authors as described on https://www.sciencedirect.com/referencework/9780323914086/encyclopedia-of-condensed-matter-physics#book-description. I would think such a source is better than a newsletter in a magazine, but maybe I am in the minority. If my specific minimal text edit was not good enough, then I could agree that it could be written better, but the new suggested reference is, I argue, better than repeated references to a magazine, so I don't understand why that was reverted. In any case, perhaps my initial edit was not perfect, but I think it could be (part of) an improvement, which also preserved the source "Ornes, Stephen (2020)" which is used again at the end of the same paragraph. QuantumQuench (talk) 16:49, 11 February 2024 (UTC)


 * The article's lead section summarizes the most important material already in the article. Speculation about one-dimensional anyons has occurred, but it is not (yet) of central relevance to the article. Would you like to create a brief, informative section about one-dimensional anyon theories? You could then cite your encyclopedia article there if it is relevant. When citing sources behind a paywall so that most of our readers can't see them, I find it useful to include in the citation the relevant quote. I tried to improve the lead section to meet your other suggestions. HouseOfChange (talk) 18:30, 11 February 2024 (UTC)
 * As I hope was clear, my concern was that the article shouldn't say something false: In this case limiting the possibilities for anyons. Needlessly. That makes the article not useful. Your new version makes this issue less manifest, although I think some explanatory power was lost. However, I think it is wrong to brush off the 1d case as "speculation". As with your "obscurity" claims before, this comes across as derogatory. I checked the history and this Wikipedia article never called 2d anyons "speculations". It is not an experimental paper on anyons, nor should its uses or purpose be limited to a subgroup of ppl interested in learning about anyons. Main goal should be that it is useful, accurate, and encyclopedic; this implies comprehensive. That said, given that this article (currently) explains the 2d case, of course the focus should be on 2d in the lead. Another option for doing this is to limit the scope in the lead section, acknowledging the 1d case (which is not impossible as this article previously claimed), but then write so to convey what the article (at present) is about.
 * Since you previously jumped to accusing me of conflict of interest, of which I have none (and for which you in principle should apologize), let me just, for clarity, say that it's not "my" encyclopedia article. Maybe you didn't mean to imply this. It is "an" encyclopedia article, but which I would argue summarizes this topic and so can be useful, not only for 1d---that article is even primarily about 2d---nor only for the lead section. Its usefulness is also motivated based on the stated purpose of that entire encyclopedia. Again, it's not "my" encyclopedia. Unfortunately it's behind a paywall, but let's not apply double standards: that is also the case for other references in this Wikipedia article. Of course, having an open access version would be an advantage. QuantumQuench (talk) 11:40, 13 February 2024 (UTC)
 * This talk page exists to promote improvements to the article. I tried to improve it according to your useful suggestion. I also suggested that you create a section about one-dimensional anyons, using material from that encyclopedia article, to which I intended no disrespect. One relevant quote from it, IMO, would be "The two-dimensional case is most familiar, since it finds application to the fractional quantum Hall effect. The one-dimensional case is different, in that the particle statistics is expressed as a boundary condition on the wave function. We do not know of physical applications of the one-dimensional theory." HouseOfChange (talk) 15:29, 13 February 2024 (UTC)

Attempts at self promotion ?
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)

Too technical
Can we get some kind of introduction for lay people? I'm a physics major and I still can't make head or tail of this. Where is this thing on the scale from real to hypothetical? —Keenan Pepper 05:53, 30 March 2006 (UTC)
 * I can say that anyon-related phenomena are observed in solid state physics, so they're real. You might say the article is too technical, but I would say that this article is just not very good, which might explain why it's difficult to understand.  Here's how I think of it: the universal cover of a circle (which is the rotation group in 2d) is the whole real line, and representations of the real line are just real numbers.  Compare with the rotation group in 3d, whose universal cover is compact (it's a 3 sphere).  Compact groups have discrete representations, so spin is discrete.  The universal cover is actually only a double cover, so it takes half-integer values.  The really interesting bit is how the action of interchange of 2d particles is path-dependent; particle exchange is a representation of the braid group in 2d, rather than just the symmetric group as in higher d.  I dunno, if those rambling sentences will be of any use to you.  It would be nice to have a good article on this, I'll put it on my list, but the list has been getting long, so I can't promise anything soon. -lethe talk [ +] 06:25, 30 March 2006 (UTC)


 * I don't understand this either. I think someone needs to take a couple of steps back at the start of the article to explain the context. 81.19.57.146 12:08, 10 May 2006 (UTC)
 * Agreed, this is very possibly the single most baffling little article I've ever read here. And fuller1's explanation above is equally jargon-filled nonsense to a layman like me. I'm sure it all makes excellent sense to some - I can jargonize on certain topics with the best of them - but clearly the only people who could grasp this article are those who already understand its subject. So, here's another vote for some kindly pedagogue to take a good whack at this article....Eaglizard 19:58, 2 May 2007 (UTC)
 * If anyone can spare some 30minutes for this ... just write a comparision with Fermi-Dirac/Bose-Einstein Statistics, with two concrete examples where one can see that anyonic statistics are not completely contradicting to the FD/BE case, but rather a generalization that is possible in 2 dimensions. Maybe some more reference for observable effects would be nice. I think this should suffice to make this understandable --138.246.7.141 (talk) 12:49, 27 February 2008 (UTC)
 * OK, the problem here is not really that the article is too technical, but that it is too mathematical. Specifically, one does not expect even physics majors like Keenan, much less the General Interested Lay Audience (GILA), to know enough topology or even group theory (part of abstract algebra) to make sense of the explanations offered, either in the article itself or equally here on the Talk page by Lethe. Even math majors may not have enough topology to understand this! It follows that Lethe's approach to explanation might succeed if someone could help the GILA imagine what these covers and representations are. yoyo (talk) 01:47, 24 June 2008 (UTC)

I rearranged the whole article and tried to equip it with a pedagocical example. Please give me some feedback if I could get rid of the technicality problem. MKlaput (talk) 13:43, 27 February 2008 (UTC)
 * Please see my comment above. yoyo (talk) 01:48, 24 June 2008 (UTC)


 * You can't expect to be able to understand everything, e.g. representation theory, without some background.  The cleanup has been done, there is now an accessible explanation of how it related to boson and fermion statistics.  So I'm removing the tag.

Particle Exchange
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)contribs) 21:04, 3 May 2023 (UTC)

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

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

Start Class
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)

Could it be given more deteiled explanation of why fundamental group determines number of possible statistics? — Preceding unsigned comment added by 77.46.217.223 (talk) 20:44, 17 August 2011 (UTC)

non-abelian
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)

Initial diagrams








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&and;Ŝc2ħεИτlk 06:31, 9 May 2013 (UTC)

Feedback

 * The bit on Aharonov–Bohm effect is a bit misleading here because it remains a 3-D problem. Anyons are relevant to exchanging particles anywhere within the plane, not just around some solenoid passing through the origin.
 * The spacetime diagram showing rotation in 2 spatial dimensions is better, but still a little confusing. A much more useful activity right now would be to fix the turd that is the current state of the exchange operator article.  I remember someone give a talk that showed an animation of two particles in a 3-D volume get exchanged once, then twice, then topologically deformed back.  A similar kind of animation would do wonders on that page.  An animation (not a static spacetime diagram) showing these exchanges in the plane could be very helpful here.  The exchange operator isn't as well defined in 1-D, though you may be able to show exchange in a network Teply (talk) 17:29, 9 May 2013 (UTC)


 * Thanks for feedback, including the pointer to exchange operator and the paper.
 * The AB effect is in 3d, but I thought the charge-flux composite was the "standard" (?) example of an anyon. The caption can be always be rephrased.
 * In time I'll try to create the animation(s). I can easily produce SWF animations but annoyingly Wikimedia Commons cannot allow SWF files, and I can't yet seem to convert them to animated GIF which would be the best format (without paying for extra software - an obvious refusal). M&and;Ŝc2ħεИτlk 07:45, 10 May 2013 (UTC)


 * As for what is the "standard" example I can't really say as anyonic statistics is largely viewed as exotica. The composite fermion idea has been around the longest to explain the FQHE, and maybe that's what you had in mind?  More recently there's been a lot of excitement over the Majorana fermion as another realization.  See  for starters.  I find it to be a bit more intuitive, but maybe that's just me.
 * Part of the trouble may just be that your first two pictures show all 3 axes. For anyons, you really need to confine everything to a plane.  The other trouble is that they appear to be single-particle pictures.  If you have one particle in isolation, then you aren't really dealing with particle exchange.  You can speak of the particle's rotation properties (e.g. integer vs. half-integer spin) or, as you have drawn it in your second figure, the different possible end states given different possible trajectories.  The statistics comes from forcing particle exchange to occur around the magnetic region.  This is what is really lacking in these articles relating to the spin-statistics theorem, that spin is a single-particle property whereas statistics is a multi-particle property. Teply (talk) 19:03, 10 May 2013 (UTC)


 * Thanks for clarifying the issue with the charge-flux composites - although the charge is drawn in the xy plane, the third axis does make an incorrect suggestion that anyons could be in 3d. The FQHE was not actually the idea, but it does happen to coincide there. Similarly the number of particles should be at least two. Incidentally the FQHE is a many-particle effect.
 * I'll read up more on the Majorana fermion soon also.
 * When you say "The statistics comes from forcing particle exchange to occur around the magnetic region.", presumably this refers to where the B-field is zero while the A-field is non-zero? In the above pics outside and around the solenoid?
 * Without attempting to detract, this article says "spin-statistics connection" while linking to spin-statistics theorem, what does the "connection" term have to do with anything?
 * Thanks, M&and;Ŝc2ħεИτlk 15:13, 11 May 2013 (UTC)
 * Yes, by magnetic region I mean inside the solenoid, where B is nonzero. Anyway, have a look at figure 2 in Camino et al. (see the refs on this page), which shows the setup of an actual interferometer, the magnetic region, and the tunneling junctions that straddle it.  A cartoon version of that could be a useful diagram here.
 * Whoever used the word "connection" probably didn't intend any special, technical definition of the term. There is a connection (relationship) between spin and statistics, as specified by the theorem.  I'll change the wording.
 * If you're interested, another bad article worth improving is exchange symmetry. The identical particles article isn't so bad, and has some of the necessary material at hand. Teply (talk) 19:36, 12 May 2013 (UTC)






 * As promised through a bit of procrastination, here are the Laughlin quasiparticle micrographs in SVG, let me know of any problems/improvements. M&and;Ŝc2ħεИτlk 21:32, 17 May 2013 (UTC)


 * Pretty good for the image based on Fig.2.B. Maybe add some dashed lines to show the tunneling junctions.   Otherwise great. Teply (talk) 06:44, 18 May 2013 (UTC)


 * Aren't those just the blue dots? M&and;Ŝc2ħεИτlk 17:37, 18 May 2013 (UTC)


 * Yes, the new version looks alright. Write a caption and add it to the article.  You don't always have to hide in the talk pages. Teply (talk) 04:50, 22 May 2013 (UTC)

Exchange and orientation entanglement topological analogies


Hi Teply, I can finally produce animated GIFs by the stopframe method in Serif Drawplus X4 (oddly I couldn't get it to work properly before uninstalling and reinstalling the program, maybe just a glitch...), here is a rushed trial animation (not to be used anywhere for anything!!):

So as soon as possible I'll get the animations for the particle exchange done and fix the above pics, but please be aware it'll take a few weeks (exams)... Thanks again for your edits, feedback, and clarification. M&and;Ŝc2ħεИτlk 21:54, 12 May 2013 (UTC)


 * Nooo, your figure shows exactly the opposite of its caption! You have two particles confined to a plane in 2+1 dimensional spacetime, and you are forced to rip one of the particles out of the plane to untie their braided world lines!  I know representing 4 dimensions on a 2 dimensional computer screen is tough, but still...
 * Try this. Instead of a plane, show a translucent cube moving along.  Then at the end, very importantly, keep both endpoints (final particle positions) fixed.  Next, move only the part of the world lines that are within the final cube, to show untying in the third spatial dimension (as opposed to the time dimension).  When that is done, relax the untied world lines back to parallel lines. Teply (talk) 03:17, 14 May 2013 (UTC)


 * I know it's not even shown, never mind implicitly suggested, but in actual intention the diagram is in 3d space + 1d time with only a plane in the 3-space to show exchange in that plane. You could imagine an invisible box, sliced through by the blue plane. I'll repair to this effect later, as said, it's a trial just to get used to GIF and see how well it uploads. M&and;Ŝc2ħεИτlk 17:16, 14 May 2013 (UTC)


 * Not sure if it helps or hurts your understanding, but you can also picture this like the binasuan. Let your torso be the origin.  Point your arms outward, palms up, elbows down.
 * First, let's check that your elbow has rotation characteristics of spin=1/2. With your palm facing up at all times, (counterclockwise) rotate your right forearm under the armpit and back around for a total 2π.  Your elbow should now be pointed up.  Again with your palm facing up at all times, rotate your right forearm over your shoulder and back around for another 2π.  Your elbow should now be pointed down as it was when you started.
 * Now let's see what happens when we exchange arms, so to speak. Rotate your right arm until it is across your chest, pointing left, and rotate your left arm over your head so that it points right.  That's one arm exchange.  Rotate your right arm again (you're allowed to step over it, just don't spill the wine!) so that it returns to the right, and rotate your left arm again so that it returns to the left.  That's two arm exchanges.  Both your elbows should now be pointing down.  In other words, each elbow has picked up a minus sign.  That's 2 minus signs.  Double negative gives the identity.
 * Of course to see that the statistics are fermionic you have to see what happens from just one arm exchange, not two. Since a 2π rotation of an arm inverts the elbow (multiplies by -1), a rotation by π is effectively multiplying by i.  So when you're in that weird position after one exchange when your right arm points left and your left arm points right, each arm picks up a factor of i for an overall factor of -1.
 * The spin=1 case is easier. Let your whole arms held stiff be the vectors, and rotate your entire body. Teply (talk) 06:31, 14 May 2013 (UTC)


 * Here's a few more suggestions if you're feeling ambitious. I noticed a couple other kind of weak articles orientation entanglement and plate trick that try to discuss the same stuff.  In orientation entanglement, the pictures with the mug don't really help me.  Plate trick is also light on content, but a few of the links at the bottom are actually decent as they show both spin 1/2 and fermionic statistics.   If you'd like to try making public domain versions of those or even just link the existing ones more prominently in the appropriate articles, that would be great.  The only drawback I see with them is that they don't really show how 2π rotation or one exchange operation gives you the minus sign.  Instead they show that 4π rotation or two exchange operations give you the identity.  I guess it's left to the viewer to infer that the square root of 1 is supposed to be -1.  Of course for anyonic statistics, the point is that you don't have enough freedom to move those belts around.
 * Of course my opinion is (and I think you'd agree) that every last one of these articles dealing with spin would benefit from just switching to rotations in geometric algebra already. Suppose you've got some object x that you want to rotate using rotor R.  If memory serves me, you can say x has integer spin=s if it transforms as $$R^sxR^{\dagger s}$$ and s is the smallest integer with that property.  For your usual multivector, you of course get spin=1 except if x is entirely scalar+pseudoscalar, in which case R commutes with x and spin=0. You can also say spin=1/2 if x transforms as $$Rx$$.  (Fancier rules for spin=3/2, addition rules, and so on.)  The spin=1/2 case is conceptually easier this way.  Be bold... Teply (talk) 21:51, 14 May 2013 (UTC)


 * I definitely agree that a section on the description of spin particles in terms of GA would be nice. Do you know/have the rule for arbitrary half-integer spin, and addition (composition?) rules? M&and;Ŝc2ħεИτlk 23:47, 21 May 2013 (UTC)


 * Let's not get too off topic. I'll move that discussion to your talk page. Teply (talk) 04:50, 22 May 2013 (UTC)

I quickly redrew the animation. Although the world lines untie in the spacelike region, it may still suggest that the worldlines move into the timelike region. Is it OK? M&and;Ŝc2ħεИτlk 16:52, 17 May 2013 (UTC)




 * Forgot to add about orientation entanglement: the vimeo Dirac belt trick and vimeo belt trick for a two-particle exchange is a bit much for me to reproduce exactly to the same graphical quality (I'll try something later though). As a possible alternative, I drew a while back this crinkly and horrible animation for the orientation entanglement of a spinor, and intended to add it to that article but it was never finished to happen...


 * It's based on the "book and belt" trick in Penrose's The Road to Reality, corresponding to the orientation entanglement for a spin-1/2 particle (no clue for higher spins...). M&and;Ŝc2ħεИτlk 17:22, 17 May 2013 (UTC)

Animations
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&and;Ŝc2ħεИτlk 07:11, 26 May 2013 (UTC)


 * 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)


 * 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&and;Ŝc2ħεИτlk 05:52, 5 December 2014 (UTC)

thermal distribution analogue to fermi-dirac or bose-einstein statistics? do anyons obey Pauli exclusion or not?
I was doing some editing lately on the M-B, F-D, B-E statistics articles and I wondered what is the thermal distribution for anyons? It seems to me that any non-bosonic, indistinguishable particles would have to show Pauli exclusion principle, based on symmetry. In that case, anyons would have to follow identically the Fermi-Dirac thermal statistics. Is this true or am I just thinking about anyons in the wrong way? --Nanite (talk) 07:30, 7 August 2013 (UTC)
 * You are right that for any $θ ≠ 0$ there will be no such state as $ψ ⊗ ψ$, but your further thinking about anyons took the wrong way. For fermions ($θ = 180°$), all $2^{n}$ “multiparticle” states of an $n$-state “particle” are orthonormal. For $θ ≠ 180°$ they are not (and am unsure about how the anyonic Fock space for $θ ≠ 0, 180°$ will look and how many dimensions may it have), and consequently there will be no Fermi–Dirac statistics. Incnis Mrsi (talk) 10:49, 7 August 2013 (UTC)
 * Thanks Incnis Mrsi, I was suspecting that might be the case. I have followup question: Is it possible to speak of a thermal distribution of anyons, at all? Right now this article is listed beside thermal distributions in the Template:Statistical mechanics infobox, under "particle statistics", but there are no statistics (probabilities) mentioned in the article. I suspect there are two meanings of "particle statistics" being conflated here: 1) The phase factor of exchange, which is a really fundamental property, and 2) The correct counting method used in statistical thermodynamics when considering multiple particles falling into the same mode/orbital. Obviously the two are related for fermions/bosons (the latter being a consequence of the former), and that's why people got to calling the exchange factor "statistics". But on the other hand there is nothing statistical (probabilistic) about the exchange factor, since it is a built-in part of the mechanics of pure states. So I'm not so sure that para/anyon/braid statistics really count as statistical mechanics.
 * I found a related issue on the fermion article, where someone had said "all fermions obey Fermi-Dirac statistics" which (as I understand) isn't true since F-D statistics only apply for non-interacting fermions in thermal equilibrium. Again I think they mixed up the "statistics" of exchange with statistics proper. Nanite (talk) 10:39, 15 February 2014 (UTC)
 * While think about the anyonic Fock space, answer please another question: is the concept of particle statistics incompatible with interaction potential between identical particles? You may look at boson where  asserted that it is compatible. Because it is an off-topical question here, post follow-ups to talk:Particle statistics. Incnis Mrsi (talk) 12:51, 15 February 2014 (UTC)

Particle or quasiparticle?
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)


 * Since there were no objections I made the change. Bhny (talk) 17:18, 24 June 2014 (UTC)

No explanation of abelian
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 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)

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

External links modified
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April 2020 experimental evidence update request
April 2020 experimental evidence: Rolf H Nelson (talk) 04:30, 19 May 2020 (UTC)

Evidence
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)
 * 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)

Missing references on "the ancestry of the anyon"
In the following article, co-authored by Frank Wilczek (who coined the name 'anyon') and well-known mathematical physicists Barry Simon, Elliott H. Lieb, and Lawrence Biedenharn: https://physicstoday.scitation.org/doi/10.1063/1.2810672 , they point out two papers (their Ref. 1): https://aip.scitation.org/doi/abs/10.1063/1.524510 and https://aip.scitation.org/doi/abs/10.1063/1.525110 , which are often forgotten or omitted when the history of the anyon is discussed, even though they predate the paper by Wilczek. The same mistake/omission is present also in this Wikipedia article, and as a consequence any reader is likely to make the same omission. I propose to rectify this by adding these two references, between Refs. 6 and 7, where they chronologically belong, and adding an adequate sentence or two to the text. Does anyone object to such an edit? Does anyone support it? If there is support for this edit, I'd be happy to do it: What to write is clear from the Physics Today article (one has quite a challenge if one wants argue that its four authors would not know what they are talking about). However, if someone else wants to rectify this omission, please go ahead. QuantumQuench (talk) 13 September 2021


 * Wikipedia is an encyclopedia; each article exists to inform readers about some notable topic. Goldin et al. was not an "ancestor" of the anyon, nor does the letter to PhysToday signed by Simon et al call it so. None of the people who did major early work on anyons were influenced by, or even aware of this obscure work from an obscure math-phys journal. If we are going to swell out the size of the article by detailing "ancestry", we should start by mentioning every single paper that was influential to and cited by Leinaas and Myrheim 1977 and Wilczek 1982, every one of which is more of an "ancestor" to anyons than Goldin et al.


 * Goldin et al. is mentioned by RS, if it's mentioned at all, as minor footnote to anyons, see for example "Non-Abelian Anyons and Topological Quantum Computation" (Nayak et al. 2008) "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)."


 * A previous enthusiast for memorializing Goldin in the anyon article also cites the same 1990 letter discussing Goldin et al. If Goldin's work were important or influential on anyons, if it were important to helping our readers understand anyons, surely the three decades since 1990 have allowed ample time for review articles on anyons or other RS to support the kindly words of four kindly physicists back in 1990. HouseOfChange (talk) 20:53, 13 September 2021 (UTC)


 * Your answer surprises me. Let me begin by pointing out the problem of calling a paper and an entire journal obscure. That is highly subjective. Anything that one comes across the first time may appear obscure, depending on one's background. For a mathematical physicist, "Journal of Mathematical Physics" is well known, while in certain branches of physics it may be relatively unknown. If one comes from, say, quantum computing, it's quite possible that one has never heard of this journal and depending on how narrowly one views one's research field, may have no reason to know/care about it. Which is OK. But "Wikipedia is an encyclopedia", as you say, and so should not assume that the target audience for a given article is narrowly defined. It skews the article in that direction. I would argue that encyclopedic articles should not give priority to certain research fields, no matter how popular they may be at the moment.


 * As for the second of the two articles that I propose should be cited, it has over 300 citations on Google Scholar. Sure, one may object that this is not so many, but that omits the simple fact that the average number of citations in different research fields varies considerably: Any physics article would be called obscure if measured by the same standard as articles on biological laboratory techniques, and it so happens that the average number of citations in mathematical physics is much lower than, say, condensed matter physics. Now, you asked specifically for review articles citing these works. Checking Google Scholar, I immediately found https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.65.733 and https://doi.org/10.1142/S0129055X90000107 by Fröhlich & Studer and Fröhlich & Gabbiani, respectively, where they cite Leinaas & Myrheim, Goldin, Menikoff & Sharp, and Wilczek all together. Note that Fröhlich is acknowledge in this very same Wikipedia article for his work on non-abelian anyons and so I assume you would agree that his work cannot be called obscure, otherwise I suggest that one should remove also that paragraph; in fact, the section on non-abelian anyons should already be updated to include much earlier references. One more example is this review: https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.64.193, where Goldin, Menikoff & Sharp are given credit for their "equivalent" representation-theoretic way of arriving at the possibility of fractional spin and statistics (see the second paragraph of Sec. III A). Finally, let's take the following PRL article as an example: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.94.166802 by Sarma, Freedman & Nayak, where the above three collaborations are acknowledged in the same way. You yourself mentioned the review by Nayak, Sarma et al. as a reference, so I assume you would not call their work obscure. Feel free to explore any of the other 300+ citations (for the second article) and 200+ citations (for the first article) of Goldin, Menikoff & Sharp using Google Scholar. In any case, the failure by authors to correctly cite articles does not mean that the same mistake should be repeated on Wikipedia.


 * I therefore see no reason why the papers by Goldin, Menikoff & Sharp should not be mentioned in this Wikipedia article: It's not the job of Wikipedia editors to rank the importance of different fields or give priority to one subfield over others (especially not by using recognizability/number of citations/omission in select reviews in one subfield as the only metric) and from a mathematics and mathematical physics perspective, Goldin, Menikoff & Sharp did important original work. Whether or not the average physicist or person on the street knows this or finds it important is besides the point. They, including scientists, even tend to check Wikipedia for such information and so such arguments may even become circular.


 * Now, one can argue that Wikipedia is not the place to give a full historical account of anyons. Unfortunately, the current article pretends to give such an account, even though it does not say so explicitly, and that account is simply not accurate. One could even argue that this entire article would benefit of being rewritten from scratch, so to give it the same appearance of neutrality and objectivity that other similar Wikipedia articles have. However, that the papers by Goldin, Menikoff & Sharp do not appear at all is simply not justifiable given all the other articles that are mentioned: The representation-theoretic approach that these authors took is an important discovery itself, no matter if they did it before or after Leinaas & Myrheim and/or Wilczek. But that they were contemporaries make the current omission even more egregious, and if this omission is intentional could even be argued to be vandalism. QuantumQuench (talk) 11:52, 23 September 2021 (UTC)


 * Wikipedia is not the place to decide if a primary research paper published four decades ago is or is not important. Wikipedia summarizes the most important things that have been reported by reliable sources. If there were RS stating that Goldin et al. did work important to understanding anyons (something the 1990 letter to the editor does not claim) then those RS could be cited to support its inclusion in the "anyons" article. If there were RS stating that that Goldin et al. did work important to some other major line of research (something the 1990 letter to the editor also does not claim), then it could be included in some other article. If Goldin or one of the other authors had a Wikipedia article, it could certainly mention this early work by them. But if their colleagues in whatever fields they are in did not consider this work notable enough to write RS about it, or to give them awards for it or named chairs or whatever, it is inappropriate to ask Wikipedia to contradict experts. Your continued focus on Goldin et al. strongly suggests COI, and I will be asking other editors to lend a hand in dealing with these repeated efforts. HouseOfChange (talk) 13:50, 23 September 2021 (UTC)


 * I would be grateful if other, more expert editors, would join the discussion. Because the COI elements seem clearer to me than do the physics claims, I have started at discussion at Wikipedia:Conflict_of_interest/Noticeboard#Anyon. HouseOfChange (talk) 14:55, 23 September 2021 (UTC)


 * I kindly suggest you reread the second paragraph of my reply. I already provided reliable sources (in accordance with your wishes): I gave links to four published physics articles on anyons where the two papers by Goldin et al are clearly acknowledged and cited for their importance. Are you claiming that those four published articles are not reliable sources? Then I would like to see a motivation. Let me take an excerpt from the review by Forte as an example: "This result can be obtained in several ways; historically, it was first found in a Hamiltonian approach by looking at the most general Schrodinger equation in two dimensions (Leinaas and Myrheim, 1977). An equivalent way of getting at it is to derive all possible representation of the algebra of observables (Goldin et al., 1981)." The speed by which you replied suggests to me that you did not look at the provided references, which I took the time to look up in addition to the Physics Today article by Biedenharn, Lieb, Simon & Wilczek, which by the way is no "letter to the editor" but a regular article in that magazine signed by four experts, including the person who coined the term "anyons". In essence, I am claiming that you are contradicting experts, i.e., all the authors of the sources that I provided, and the only thing I am asking is that you refrain from doing so and instead look at the sources and argue against them if you so want. I would be more than happy if other editors would weigh in: It should be clear from my original post that this is why I wrote here in the first place instead of directly editing the Wikipedia article. As for your unsubstantiated claim of COI, I could equally well say that your insistence on not citing Goldin et al. here, in contradiction with the provided sources that you saw fit to ignore, "strongly suggests COI" from your side. QuantumQuench (talk) 15:19, 23 September 2021 (UTC)


 * Fröhlich wrote possibly the first proper review on anyons (Proceedings of the Gibbs symposium, 1989, https://www.worldcat.org/title/proceedings-of-the-gibbs-symposium-yale-university-may-15-17-1989/oclc/21561250 ), and credited the discovery of anyons by "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", and "The analysis of particle statistics in quantum theory in three space-time dimensions started with the work of Leinaas and Myrheim [...] and was continued in [Goldin et al 1981, Wilczek 1982]." As well as "Our general considerations in $1.3 leave room for the possibility of statistics described by nonabelian, unitary representations, U_n of the braid groups, B_n. (This possibility was first envisaged in [Goldin et al, 1985])". Hence credit for non-abelian anyons is due primarily to this group.


 * Clearly due credit must be given these three independent groups for their important contributions to our understanding of anyons from three different fundamental perspectives: a geometric, a representation theoretic, and a magnetic one. Doctagon (talk) 15:53, 23 September 2021 (UTC)

Physics Today 43:8 (scroll down page linked to see this) clearly lists Biedenham et al under "Letters" published in their August 1990 issue. Letters are not peer-reviewed journal articles nor are they fact-checked professionally written articles. They are RS "evidence" only for the opinion of their authors, which in this case was that it's a shame Goldin et al. didn't get mentioned in an extensive review article about anyons. And several subsequent papers have added brief "see also" type notes concerning their work. Saying that "An equivalent way of getting at it is to derive all possible representation of the algebra of observables" is hardly a ringing endorsement of the importance or relevance of Goldin et al. to an encyclopedia article about anyons. So "credit for non-abelian anyons is due primarily to" Goldin et al. for a 1985 paper? Please cite RS for such expansive claims as you draw from isolated mentions in various antique papers. Please also cite RS for your claim that Goldin et al ca 1981 made "important contributions to our understanding of anyons." Based on your first contribution to Wikipedia, I would also ask you to review our policies on sockpuppetry and/or stealth canvassing. HouseOfChange (talk) 16:21, 23 September 2021 (UTC)


 * The Physics Today paper is clearly in letter format and not a peer-reviewed research article. I did not intend to claim otherwise; I even called Physics Today a magazine, and not a research journal for this reason. Note that PRL papers are called Letters, but in that case they are of course peer-reviewed, while we say Wikipedia articles, without them being peer-reviewed. So to distinguish solely based on if it says Articles or Letters in the corner is not what matters. However, you still continue to claim to know better than the experts that wrote that piece. For no good reason. I proceeded to give you four published physics articles citing Goldin et al., three review articles and one recent PRL. You still have not answered whether you dispute that they are reliable sources or not. That was what you asked for, nothing else.


 * It continues to amaze me why you insist that Goldin et al should not be mentioned in this Wikipedia article. That you mix the words Axion and Anyon in your reply also makes me curious; I suggest that any other editor weighing in on your claim of COI should take note of this "Freudian slip". In fact, you do not even know the details of how I propose to add the Goldin et al. references, nor did you seem to care to discuss that. Why are you convinced that it would not improve this article? Let me try to argue that it would, both in terms of quality and usefulness for the reader, and that this is in fact my motivation for suggesting the edits.


 * I see (at least) the following two reasons:


 * 1) It is useful and of mathematical importance to explain that there is a representation-theoretic approach to anyons based on Representation theory of diffeomorphism groups of R^n, from which it follows that n = 2 and n ≥ 3 lead to different possibilities, and the original works by Goldin et al. are the obvious references here. This is not mentioned at all now. Please explain why this knowledge should be denied the reader, other than it may not interest you (maybe it does interest you, in which case your insistence is even more surprising). I am sure there are enough mathematically oriented readers of these kind of articles to warrant this addition. Do I really have to break down why it is useful to mention connections between different areas of mathematics and/or physics? Moreover, right now there are clickable links to "useful" Wikipedia articles such as Aalto University (why is this in the Introduction?), but for some reason expanding/highlighting the connections with representation theory is objectionable to you.


 * 2) As with many scientific discoveries, their history can be complicated, and different research groups can arrive at the same result about the same time, and sometimes from different perspectives. I argue this should be celebrated, not hidden. Take for instance the Aharonov–Bohm effect; on that Wikipedia page also Ehrenberg and Siday are mentioned. Another example is the Higgs mechanism. Also check List of multiple discoveries. Here I am simply arguing that the history of anyons is not portrayed accurately. Now, adding Goldin et al. would actually not be enough if one wants to go deeper, but it is right now the most striking omission and one for which I have given reliable sources (as you asked for). An even more accurate account would at least start with such initial developments as those by Green, Phys. Rev. 90, 270 (1953) about Parastatistics, and also include works such as, but NOT limited to: Aharonov & Bohm, Phys. Rev. 115, 485 (1959); Klaiber, in Lectures in Theoretical Physics, Vol. X-A: Quantum Theory and Statistical Physics, p. 141 (Gordon and Breach, 1968); Streater & Wilde, Nucl. Phys. B 24, 561 (1970); and Fröhlich, Commun. Math. Phys. 47, 269 (1976). Gathering all of this and writing it up for a Wikipedia article would, however, be too much work and I would agree better suited for a scientific review article on anyons. Having said this, it is still only fair to mention all of the three distinct sets of works by Leinaas-Myrheim, Goldin-Menikoff-Sharp, and Wilczek. I maintain that not doing so diminishes the usefulness of this Wikipedia article to its possible readers, as they are left with an intentionally incorrect account. That these authors arrived to their results in different ways does not mean that one is less important than the other. In fact, that Goldin-Menikoff-Sharp are less cited probably has more to do with that those papers are more mathematical than the others and thus more difficult to read, instead of some nonsensical claim that the entire Journal of Mathematical Physics is obscure. At least I seem to have convinced you otherwise, but please let me know if I rushed my judgement, in which case, for consistency, I suggest that you go over to the Wikipedia article on Luttinger liquid and delete the references to Luttinger and Mattis & Lieb from its bibliography. QuantumQuench (talk) 23:19, 23 September 2021 (UTC)


 * Myrheim ought to be regarded as a credible source on anyons. He writes in his relatively recent review (Anyons, in Topological aspects of low dimensional systems, 1999, http://dx.doi.org/10.1007/3-540-46637-1_4 ): "A third approach leading to the same results is that of Goldin et al. [6, 37–41]. They studied the representations of the commutator algebra of particle density and current operators. This algebra has commutation relations that are independent of the particle statistics, but has inequivalent representations corresponding to the different statistics."


 * The claim concerning non-abelian anyons is substantiated in the Physics Today article: "In 1983 these authors also identified the braid group as the group whose one-dimensional representations describe the statistics of anyons, and in 1985 they noted (in a comment on a paper by Yong-Shi Wu) that higher-dimensional representations of the braid group could describe important quantum systems in two-space."


 * The claim that their work is regarded as important by experts in the field is substantiated in the last paragraph of the same article: "These comments are not intended to detract in any way from the important contributions of those mentioned in the story."


 * and : You asked for the opinion of others and I regard this very discussion on the existence of different routes to anyons, as explored by these three independent groups, as fundamentally important and necessitating clarifications and references in order to begin the work of improving this article on anyons. The above quotes may be of use in such an endeavor. Doctagon (talk) 00:17, 24 September 2021 (UTC)

Maybe a good solution would be for somebody to create an article fractional statistics that includes a history section, where different approaches to fractional statistics are discussed, including the early work by Goldin et al. But that work was not in the main line of anyon research, and nobody, including Biedenham et al 1990, has called it important to understanding anyons. HouseOfChange (talk) 01:36, 24 September 2021 (UTC)

History of fractional statistics
The anyon article doesn't have a "history" section, which means that many early influencers aren't in the article.

Consider some physics articles that do have history sections:
 * Fermi–Dirac_statistics: 4 paragraphs, section mentions 7 researchers
 * Statistical_mechanics: 5 paragraphs, mentions 6 researchers
 * Axion: 2 paragraphs, mentions 6 researchers

One should approach writing a history section by asking, "what is the most important and useful information?" not by asking "Why do I not see the name of Goldin mentioned in this article?" IMO, popping the name of Goldin several times into this article, while continuing not to mention much more influential early work, work cited at length by Khurana and other reviewers, violates WP:BALASP and invites the false inference that Goldin's work was more important than work by all the others whose names are omitted.

If we are going to create a history section or article, we can get help from the history section of a 2005 book by Khare, which describes Goldin et al 1981 as having replicated earlier work using a different method, not as setting the stage for later work. HouseOfChange (talk) 16:58, 24 September 2021 (UTC)


 * Apologies for not answering earlier. I have been very busy, and I still am.


 * I think adding a section about the history in the anyon article is a good idea. I looked at the book by Khare and it looks pretty good. I hope we are converging, but I still do not understand why you interpret Khare's mentioning of Goldin et al as if they did not do important work. The sentence "Few years later, Goldin, Menikoff and Sharp obtained the same results by an entirely different approach [23]." gives credit to those authors and highlights the complementary nature that I was trying to point out from the start. (Possibly, we view words such as "complementary", "equivalent", "entirely different approach", etc., very differently: I find them to be positive and also find the above sentence by Khare to be a positive one.) Please also look at Ch. 11 of Khare's book: Point (1) discusses the rep.-theoretic approach. Khare decided to not go further into this subject, but the same is true for Point (3) about higher-dim. reps. of braid groups, necessary for discussing the non-abelian case, and so this does not come down to importance.


 * I find my approach to this discussion reasonable and justified. These works and the representation-theoretic perspective are omitted and I think adding those reference would benefit the reader and improve the overall Wikipedia article. That does not mean that adding other works would not also be justified. Let me again point out that you did not inquire as to how I proposed to add the references that I argue are omitted (instead I only got some subjective remark about "obscurity"). I think one or two short sentences and pointing to the references would be enough. Usefulness for the reader should be the guide (taking into account that readers do not come from select sub-communities in physics).


 * I also think there are a number of other improvements to be made: (i) That anyons are possible in 1+1 dimensions (i.e., 2-dimensional Minkowski space) is not mentioned at all: They clearly are, and the first reference given in the section on non-abelian anyons deals precisely with 2-dimensional Minkowski space, while the beginning of this Wikipedia article, given the clickable link, limits the discussion to 2-dimensional Euclidean space; the word "only" there even makes that statement false. (ii) I also find quoting "Aalto University" (by having it as a clickable link in the Introduction) unjustified: If smth was used as a reference then it suffices to simply give the reference. (Note: I do not mean to comment on Aalto University; I only mean that that paragraph in this article is sub-optimal.) (iii) The discussion of non-abelian anyons is lacking: E.g., that section does not mention higher-dim. reps. of braid groups. Adding a few sentences, perhaps incl. a cross reference to the Yang-Baxter equation, would be more useful for the reader. These changes should/could of course be discussed elsewhere.


 * I am sure there are other points in this Wikipedia article that also can be improved. Please note: I do not intend to go and undo/redo every other editors' work. I mainly seek to add or improve.


 * In any case, I think we could add a history section, and use Khare's book as one of the references for writing it. QuantumQuench (talk) 15:26, 26 November 2021 (UTC)


 * Modeled on other physics articles with a history section, let's propose 7 names most vital to fractional statistics. HouseOfChange (talk) 03:25, 27 November 2021 (UTC)
 * Wow, I dind't know this was so debated. I started a history section just because some material in "Experiment" was in fact about theory and generic introductory/historical material.  CyreJ (talk) 10:47, 10 March 2022 (UTC)
 * Thanks for creating an excellent history section,, and also thanks to for adding a very apt quote to the top of it. HouseOfChange (talk) 16:38, 10 March 2022 (UTC)


 * Let me propose the following edits on the history section in order to more properly reflect experts' understanding of the development of the topic:

History

 * "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)


 * 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)
 * 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)
 * This article's "History" section gives a BRIEF account of the most important contributions to the topic of anyons. Fermi–Dirac_statistics and Bose–Einstein_statistics 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)
 * @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)

Diagram context
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)

Further reading section
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)