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)

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.

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)

Diagrams

Initial diagrams

Anyon as a charge-flux composite, in the case of a charge q around an infinitely long solenoid confining a "magnetic flux tube" entirely within the solenoid. No B field exists outside of the solenoid in this instance. This is a canonical illustration of the Aharonov-Bohm effect.

Anyon worldline topology in 2 + 1 dimensional spacetime. Interchange of two anyons by a clockwise rotation is inequivalent to an anticlockwise rotation.

In 3 + 1 dimensional spacetime, particle interchanges by clockwise and anticlockwise rotations have equivalent topology, because the third spatial dimensional dimension allows one world line to bend around the other.

Winding number of a charge-flux composite, with positive orientation in the anticlockwise sense, negative in the opposite sense.

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:

Peshkin, M; Tonomura, A (1989). The Aharonov–Bohm effect. Springer-Verlag. ISBN 3-540-51567-4.{{cite book}}: CS1 maint: multiple names: authors list (link)

(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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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 [1][2][3] 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∧Ŝc²ħεИ_τlk 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)

Laughlin quasiparticle interferometer atomic force micrograph, based on Fig.2.A of Camino et al.

Laughlin quasiparticle interferometer scanning electron micrograph, based on Fig.2.B of Camino et al.

As promised through a bit of procrastination, here are the Laughlin quasiparticle micrographs in SVG, let me know of any problems/improvements. M∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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

Worldlines of two identical particles in 3 + 1 dimensional spacetime, spiraling around each other during rotational particle exchange (twice) in the spacelike region. The third dimension allows one worldline to be deformed around the other, returning the system to the initial state, meaning the double exchange is topologically equivalent no exchange.

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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 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 $R^{s}xR^{\dagger s}$ 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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 16:52, 17 May 2013 (UTC)

An old file retrieved and uploaded to be redrawn much, much nicer, cleaner, and clearer.

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 Road to Reality, corresponding to the orientation entanglement for a spin-1/2 particle (no clue for higher spins...). M∧Ŝc²ħεИ_τlk 17:22, 17 May 2013 (UTC)

Animations

Anticlockwise rotation

Clockwise rotation

Exchange of two particles in 2 + 1 spacetime. The rotations are inequivalent, since one cannot be deformed into the other (without the worldlines leaving the plane, an impossibility in 2d space).

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∧Ŝc²ħεИ_τlk 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∧Ŝc²ħεИ_τlk 05:52, 5 December 2014 (UTC)

References

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 2ⁿ “multiparticle” states of an n-state “particle” are orthonormal. For θ ≠ 180° they are not (and I 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 I 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 I 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#Abelian_anyons section.

Let me know if this is helpful: Abelian anyons are particles who change their phase when moved. In the most general case, these particles are one-dimensional representations of braid groups. Gaugerigour (talk) 11:15, 4 January 2020 (UTC)

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)

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April 2020 experimental evidence update request

April 2020 experimental evidence: [4][5][6] 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)

References

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)

QuantumQuenchPhysics 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. Doctagon 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."

HouseOfChange and QuantumQuench: 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)