Talk:Spinor/Archive 6

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The first paragraph

I think we have made good progress with the lead. My concern now is again the first paragraph. Although this is now better than what I had written in some frustration yesterday (the aforementioned "high brow" approach), I think it needs to be brought down a peg. I think we need to consider how we should explain spinors to someone with almost no mathematical background, and perhaps a basic scientific literacy. Currently it assumes too much background. Spinors are widely used in physics, chemistry, and presumably even in some engineering contexts. The first paragraph needs to be written to convey a sense of what spinors are and what they are good for that basically anyone can understand, and I don't think we're there yet. Sławomir Biały (talk) 14:14, 29 September 2014 (UTC)

I've gone ahead and done this, consuming most of the last paragraph in the process. Sławomir Biały (talk) 14:46, 29 September 2014 (UTC)

Slawomir, I really don't like what you are doing here. Stop behaving as if this page is yours! I cannot and donot want to spend all day editing this page and I have not said I was finished. I am going to see which of your edits I think are an improvement and the rest I am going to revert. Frankly, by now you should have realised that what you think is right not necessarily is. What you write is regularly mathematically incorrect and your taste in making things easier to understand are debatable. At the very least let other people give their opinion on a change. RogierBrussee (talk) 16:29, 29 September 2014 (UTC)

Rogier, I have tried to keep the improvements that you made, whole clarifying some aspects of them without sacrificing technical accuracy. It would be helpful I think if more novices could participate in the process though. Even the idea of a vector space cannot necessarily be assumed of someone reading the article, and it is completely appropriate to have a first paragraph that conveys an intuitive sense of what a spinor is, to be clarified later. I disagreed with the level of detail devoted to spinors for the rotation and Lorentz groups, finding that sort of information perhaps more appropriate for the end of the intorduction section. I also removed specific statements concerning the irreducibility if the different spin representations. These depend not only on the dimension, but also the signature (assuming we are looking at real representations, or representations in the algebraic group setting) as well as whether by "spin representation" we mean the representation of the Lie or Clifford algebra. The lead is not a good place to deal with these nuances, and I think the statement does rather more harm than good. Otherwise, I don't see how my edits are that controversial, and I think your "vehement opposition" is clearly an over-reaction. Sławomir Biały (talk) 16:57, 29 September 2014 (UTC)

More input

See also: Wikipedia talk:WikiProject Mathematics § Spinors

In case these comments are helpful (for the current lead in general):

• First paragraph. It may be useful to state from the outset that spinors themselves can't be visualized like vectors (this is not explicitly written in the lead). This geometric interpretation is not the same as the topological origin of the belt trick. The current lead says "vector-like objects", but since they become negative after a rotation of 2π, if there is away to geometrically interpret them, it can't be as simple as an arrow. I know - I'm duplicating what Jheald has said in an earlier thread (and elsewhere).
• Second paragraph:

• The statement

"Unlike tensors, however, spinors do not transform coherently"

means little to typical readers who will not know what "transforms coherently" means. I can't find the exact older version, but vaguely remember the lead said at one point something like "spinors transform linearly under infinitesimal rotations, but not finite rotations", which was more compact, before mentioning finite transformations can be obtained by integration.

• The statement

"Unlike for vectors and tensors, however, the final transformation depends on the choice of the one parameter family, or more precisely on its homotopy class. This leads to a sign ambiguity that makes it impossible to define a consistent linear transformation for any final rotation. For example, a spinor changes to its negative when the system is subjected to a complete turn over an angle from 0 to 2π."

for a typical reader reads a bit convoluted (and repeats the negative-after-2π-rotation concept already said first thing), since it isn't clear or obvious here how the sign ambiguity arises. This version reads better:

"However, spinors are unlike vectors (and other tensors) in the sense that, whereas a vector (or tensor) undergoes the same rotation as the frame of reference, spinors account for an additional sign ambiguity in the continuous rotation. This sign ambiguity is called the class of the particular continuous rotation. This class actually has a topological origin, famously illustrated in the belt trick puzzle (shown), which demonstrates two different continuous rotations having the same final configurations but different classes."

since it explicity states how spinors differ from tensors, and relates the concept of a spinor to the belt trick neatly.

• Overall:

• Correct me if wrong: spinors are "beyond tensors" and can be used where tensors are used, but not the other way round? And that a rank-2s spinor can be used to obtain a rank s tensor (s = 0, 1/2, 1, 3/2, 2, ...)? The lead does not say this plainly. The lead could also mention:

• Both 3d Euclidean vectors in non-relativistic physics and Lorentzian four vectors (and tensors) in relativistic physics can be expressed as spinors. Excellent sources include Electrodynamics and Classical Theory of Fields & Particles by Barut, as well as Evgeny Lifshitz (et al) volume 4 part 1 Relativistic Quantum Theory of the course of theoretical physics. Niether are cited in this article. (Cartan's theory of spinors is of course, for priority and historical importance).
• Electromagnetism and general relativity can be described using spinorial formulations, a source might be Theory of Spinors: An Introduction by Moshe Carmeli and Shimon Malin, but no doubt in other places (Rindler and Penrose and others have researched in this area).

• The lead no longer says that spinors are used more widely than quantum theory. If anything the lead now biases quantum theory.

There may be more to points to add, these jump at me.

Thanks to everyone for trying and apologies if I'm pulling editors in conflicting directions, and wish Sławomir a quick recovery from flu. Best regards, M∧Ŝc²ħεИ_τlk 20:02, 29 September 2014 (UTC)

To update here for the record, my above post may already be dated. The lead is changing very fast (no doubt in good faith, but not necessarily for the better). If it helps, I think the best lead was this one, others are welcome to disagree. Splitting off some material into an introduction section is good for keeping the length of the lead down though. We just need agreement on what the 3-way split lead-intro-intuitive understanding should be. Good luck to those writing. M∧Ŝc²ħεИ_τlk 21:34, 29 September 2014 (UTC)

I found Rogier's sequence of edits to the lead, and the general gutting of the "Introduction" section, to be quite counterproductive. Our goal here is to present the article in good introductory style, not batter our poor reader with an endless web of blue links from which they will never understand anything. We must explain in a careful and accessible style what it is that we mean. I have reverted the destruction of the lead and introduction section, and made some further edits along these lines to the second paragraph. I think it all reads fairly well now, and addresses at least some of the points you (and others) have raised, while incorporating most of the things that Rogier put there. I would suggest that Rogier should play to his strengths and not attempt to write material intended to be introductory on this topic. It would, I think, be more constructive, and a better use of his considerable expertise in this particular area to comment here on precisely what the problems are with the current state of the earlier sections of the article. Sławomir Biały (talk) 22:26, 29 September 2014 (UTC)

I reverted to one edit after my last edit which was indeed an improvement. After that things got worse than what they were in my opinion. More to the point I had asked Slawomir to back off after he spent days improving things without getting to a satisfactory result and let me have a go at it to which he agreed, and I would let it be known that I was finished. I object to Slawomir taking of on a new editing spree just hours after my first edits and before I said I was finished. I wrote that i would revert his edits and I did. After that I moved the two more technical paragraphs to the overview section and deleted what was there. In my opinion that is an improvement. Slawomir obviously disagrees and reverted everything. There seems to be this mistaken idea that spinors are just like vectors and one just has to use some magic explanatory tool like the non simply connectedness of the Rotation group. That is just not true. Root systems or Clifford algebra's will get you the existence of spinor representations, and with a lot of explanation you can relate that to representation theory but it is just not trivial. Therefore it is pointless to try to explain things for the laymen. I am all in favour making things as simple as possible but not simpler. For example saying that the Clifford algebra is generated by the gamma matrices misses the point, because what you have to do is to construct a representation of the abstract Clifford algebra (which is constructed from the vector space) to a concrete matrix algebra. So every choice of orthonormal basis and every choice of gamma matrices gives a different, albeit isomorphic representation of the Clifford algebra and a different but isomorphic representation of the Spin group. See? Seems like a trivial difference at first but now start reading Weinbergs (otherwise excelent) book on quantum field and notice how he starts writing down explicit gamma matrices on page 3 or so, which is horrible because now what depends on the choice of gamma matrices and what does not. The worst thing about this whole affair is that all this energy would be better spent on other sections. I particularly hate the example section which seems to be written by someone from geometric algebra people that want every thing inside the Clifford algebra. It would be so much better if the different constructions were run through and compared in dimension 3 and 4 (and perhaps dimension 2). RogierBrussee (talk) 23:05, 29 September 2014 (UTC)

Let me add to what I said about not catering to laymen. There are obviously laymen and laymen. The reasonable assumption if you get to this page is that you either get there because of quantum mechanics or because of relatively sophisticated mathematics most probably either involving representation theory of Lie groups and or Lie algebras or through a subject like global analysis or topology. in the latter case we can thus assume a level of sophistication that goes well beyond first year university including having a passing familiarity with things like a homotopy class. The former case is what I would like to take as the "laymen" level. If it is a bright highschool student and he/she somehow stumbled on spinors the first paragraph should tell him/her: It has something to do with vectors in a way that he/she doesn't quite understand yet and the quantum mechanics of electrons and spin-1/2 that I have heard of. The next paragraph or the intro will tell him its over his head and not for him. I would say this perfectly fine because that is just the way it is, but it would suggest taking the more technical paragraph for the intro. The second case is that of a university student getting a course in quantum mechanics. Now chances are that he or she gets component spinors in relation to electrons.Then he or she has most likely learned about Pauli matrices and if (s)he is lucky about rotations of the frame of reference. Again most likely this is taught through making an infinitesimal rotation perhaps integrated to exponentiated to elements of SU(2) through exponentials of the Pauli matrices. Chances are he or she is confused but just slowly gets familiar. Here I think that above all, the "laymen" should not get the feeling that the subject is incomprehensible because it is somehow mysterious. So the last thing we want to do is not be precise by using confusing weasel words (like vector-like), seemingly intuitive but on second thought multi interpretable words (gradual rotation although much better than continuous rotation), being overly verbal and mixing up related but different concepts because that is only going to add to the confusion. What this point to however is noting that Pauli matrices are gamma matrices. Unfortunately, as gamma matrices go they are rather confusing because their _commutation_ rather than anti commutation relations are stressed. This is the same "confusion" that obscures rank 2 anti symmetric tensors in 3 dimensions because you can use the cross product. Moving the technical section to the introduction is not a real problem but has the disadvantage that the lead does not actually contain anything resembling a definition. The reason why infinitessimal rotations don't get lifted to consistent transformation (maybe consistent is better than coherent) should be clear and exposure to the idea that group representations exist is not wrong. I don't think a mention of the homotopy class should really scare off, he or she can but need not click the link. I am not against something like "topological equivalence class (homotopy class)" but I doubt if it adds value. Anyway I am tired of this. I have work to do, a wife and children to pay attention to. It takes effort to actually make edits. Directing them through Talk pages takes too much effort. — Preceding unsigned comment added by RogierBrussee (talk • contribs) 07:05, 30 September 2014 (UTC)

• "The reasonable assumption if you get to this page is that you either get there because of quantum mechanics or because of relatively sophisticated mathematics most probably either involving representation theory of Lie groups and or Lie algebras or through a subject like global analysis or topology." This would be a more reasonable assumption if we were writing a book introducing spinors, but it is not a reasonable assumption for a general encyclopedia article. Many laymen with some mathematical sophistication encounter and are interested in the notion of a spinor, without necessarily being prepared to devote themselves to a serious study of quantum mechanics, representation theory, global analysis, etc. Even absolute laymen, with no mathematical sophistication, may stumble upon this article because of something they read somewhere, or by following a link. The article should strive at least to convey a sense of what it is about to such a person.
• "[T]he lead does not actually contain anything resembling a definition" If you read it carefully, the lead actually contains four definitions. One definition is in the first sentence. Although somewhat vague, and obviously unsuitable as a proper mathematical definition, it does convey a sense of what spinors are to an absolute layman. Twp more definitions come in the second paragraph, as elaborated in the footnotes. One of these concerns the representations of the Lie algebra, and the other of the spin group. The fourth definition appears in the third paragraph.
• "Directing them through Talk pages takes too much effort" I'm sorry you feel that discussion is a waste of your time, but this is one of the most important ways in which Wikipedia articles are written, through discussion and consensus-building on the talk pages. Since my first edits on August 14, there have been reams of discussion here by multiple editors of varying interests and skill-levels and the lead of the article has been written and rewritten as a result of these discussions. You yourself do not seem to have read or understood the context of most of this discussion (e.g., you commented "seconded" in places referring to a much older revision). I know it is hard to follow, and I thought it would be helpful to take stock of things by providing some links to various milestones. User:Maschen provided yet another milestone. But to eschew discussion completely because your time is too valuable is not going to work under these circumstances. Sławomir Biały (talk) 09:58, 30 September 2014 (UTC)

I appreciate the effort for addressing most of the points I made. However, in the lead:

• for the bit on finite transformations - it does not mention integration anymore (but the rest of the wording is very good),
• it doesn't mention the applications like it used to:

"General spinors were discovered by Élie Cartan in 1913. Soon after, spinors turned out to be essential in quantum physics, and currently enjoy a wide range of applications. Spinors in three dimensions are needed to describe non-relativistic electrons and other fermions which have spin-½. Dirac spinors, spinors of the Lorentz metric in dimension 4, are required to describe the quantum state of the relativistic electron via the Dirac equation. In quantum field theory, spinors describe the state of relativistic many-particle systems. In mathematics, particularly in differential geometry and global analysis, spinors have also found broad applications in algebraic and differential topology, symplectic geometry, gauge theory, complex algebraic geometry, index theory, and special holonomy."

• it does not mention explicitly that spinors can be used where tensors can, and the converse is not true.

I would tweak along these lines, but the lead would become longer (too long?), and it is constantly changing so I don't want to form edit-conflicts. Also I have only scanned through much of the discussion and edit history of the article so apologies if I missed anything. M∧Ŝc²ħεИ_τlk 10:12, 2 October 2014 (UTC)

References for the above:

360 degrees

I can see that this misleading formulation that a spinor returns to minus itself upon a 360 degree rotation is back. Obviously, things that are rotated 360 degrees, everything else kept equal, return to themselves. Spinors aren't ghosts. You just haven't rotated them 360 degrees, when you in the standard representation have rotated its objects 360 degrees. (Easiest way to see this is to follow a path in the Lie algebra (standard rep), and follow the corresponding events in the standard rep of the group, and the spin rep.) On the other hand, the misconception is so widespread in the literature that it is almost true. YohanN7 (talk) 14:00, 30 September 2014 (UTC)

I think a careful reading of the first sentence is needed here. When the physical system is rotated through 360 degrees, a spinor attached to that system transforms to its negative. It is not the spinor that is being rotated, but the system. I was toying with a possibly useful illustration of this, motivated by one of the images at orientation entanglement (an article which overall I think gets things pretty wrong). But I think for the purposes of illustration, the Möbius band is actually pretty good. Take a single vector, that is a nonzero point of the Möbius bundle over the circle. Then rotate the circle continuously through 360 degrees. Parallel-transporting the vector as we go, it will have transformed to its negative after this operation. The "physical system" in this scenario is the circle, and the "spinor" is the point of the Möbius bundle. I think this illustrates nicely what the first sentence should convey on a first reading. The trouble with this illustration is that it is unclear (to me at least) what precisely it has to do with the way mathematicians think about spinors. Really it involves a holonomy representation of the fundamental group of the circle, rather than anything specifically spinorial. If I can manage to clarify this (at least to myself), then I think this should be added as an image to the lead.

You might find it instructive to compare with Penrose's treatment of spinorial objects in "The Road to Reality". There he actually does say that we "rotate the spinor through 360 degrees" and arrive at the negative! He then "justifies" the absurdity of this statement by saying that we are to imagine that the spinor is somehow connected by invisible strings to its surroundings (in the style of orientation entanglement). This justification seems to be begging the question though, and I think we have done better here. Sławomir Biały (talk) 14:25, 30 September 2014 (UTC)

What I'm trying to say is that the 360 degrees are 360 degrees as a rotation in the standard representation, but not in a spin representation. But I'll not get stubborn about this. The "360 degrees rotation" is everywhere in the literature and would be right for Wikipedia even if it is wrong. YohanN7 (talk) 14:35, 30 September 2014 (UTC)

But I think we agree about this. If you like the "physical system" is another word for the standard representation. Sławomir Biały (talk) 14:38, 30 September 2014 (UTC)

Hopefully this settles the terminological ambiguity. Sławomir Biały (talk) 14:44, 30 September 2014 (UTC)

(EC) How about this: If e^X = I in the standard representation, then e^π(X) = +I or −I in a spin representation. (π is here a spin Lie algebra rep and X is an element of the Lie algebra in its standard rep.) I thinks this is correct. Provided it is correct, it is quite illuminating. This is what I mean by "following a path in the Lie algebra and see what happens". I have not yet seen the latest edit. YohanN7 (talk) 15:00, 30 September 2014 (UTC)

Yes, that's right. This is what the second paragraph tries to explain. Sławomir Biały (talk) 15:08, 30 September 2014 (UTC)

Thanks, yes, I think can live with the 360 degrees rotation now. I see better what is meant by "the system". YohanN7 (talk) 17:33, 30 September 2014 (UTC)

Out of interest - I had a pseudo-3d illustration of the Möbius band (not as good as the 3d one in the lead) as a possible heurustic to interpreting a spinor, about two years ago, and with Quondum's advice decided it would be wrong/misleading. It was compared with an arrow rotated in a plane to sweep out a disc, while the spinor sweeps out a Möbius band.

Is the Möbius band actually a correct interpretation in 3d? Does a spinor literally turn around the circle continuously (gradually, smoothly, whatever)? And the choice of the two topologically distinguishable paths correspond to rotating in a clockwise/anticlockwise sense along the circle turns the spinor into its negative in two ways? M∧Ŝc²ħεИ_τlk 11:37, 2 October 2014 (UTC)

The Mobius band picture is associated to the real spinors in three dimensions. The spin group is SL₂(R). The vector representation is the action by conjugation on symmetric matrices (the determinant defines a quadratic form of signature (1,2).) The spinors are given by the fundamental representation of SL₂(R). In this dimension and signature, a vector can be factorized into a symmetric tensor product of two spinors. In particular, the null vectors are those that have rank one (they can be factored as a product of the same spinor with itself). If you already know about such things, this is the standard presentation of Weyl spinors in relativity theory, just one dimension lower (and real instead of complex). So, in the picture, I am imagining specifically that the circle corresponds to the space of null rays, and the arrow representing a spinor pointing along that null ray. These have the property when the null cone (which is a literal cone in three dimensions) is rotated through a full turn about its axis, the spinor returns to its negative. So, yes, the picture is "accurate" provided "spinor" is interpreted in the sense meant in the nb footnote. Although a version of the same picture would also be true for the usual Lorentz group, it is confounded by the fact that the spinors are complex. I don't know to what extent there is a picture like this associated to the ordinary group of rotations in three dimensions (we have "Wick rotated" into the split signature to visualize it). Sławomir Biały (talk) 12:24, 2 October 2014 (UTC)

Thanks, I like this explanation, all familiar but I still don't know Lie groups and representation theory (yet have had loads of exposure especially in the relativistic QM literature) or vector bundles. The idea of spinors as "half-integer bases" is given for example in Rotations, Quaternions, and Double Groups by Altmann and I intend to read and work through it (recently bought it, could be a reference for the article). Also hadn't realized the picture is in 2+1 spacetime (which is certainly familiar and easy to draw), not 3d space... Out of interest since the higher-dimensional analogue of the Möbius band is the Klein bottle, for the 3+1 spacetime case I wonder if the arrow is normal to the surface of the bottle...

Very interesting, need to read far more before posting any more here... M∧Ŝc²ħεИ_τlk 19:47, 2 October 2014 (UTC)

Spinors are connected with the algebraic geometry of quadrics, so it is actually the quadric in three dimensional projective space that governs spinors in four dimensions. So it wouldn't be the Klein bottle that governs the four dimensional case, but rather a product of two conics, or some real slice of this. Sławomir Biały (talk) 00:11, 3 October 2014 (UTC)

I noticed the reappearance of the Möbius band depiction, and was going to mention that in some sense it can be understood that way (in a sense retracting my earlier comment). I can understand it as a spinor as a fibre bundle over the space of rotations. It could be taken as a section of each space, or even as a spinor in a 2-d space, I guess. That the circle and vector belong to distinct spaces (i.e. that it is a fibre bundle) might be made clearer, perhaps my mention in the caption. To interpret the vector as a vector in real space, as might be suggested by the geometry, might introduce confusion. —Quondum 13:29, 2 October 2014 (UTC)

In the low dimensions, the bundle picture is fairly straightforward. The "circle" can be thought of as the projective line, and the spinors are the points of the two-dimensional vector space of which the circle is the projective line. Now, a (nonzero) spinor can be regarded in two ways: one is as just a point of this two-dimensional vector space, and the other is as a point in the tautological bundle of the projective line (fortuitously, this is actually just the Mobius band). The action of the group comes from rotations (more generally, projective transformations) on the projective line. The tautological bundle has non-trivial first cohomology, which measures the failure of a spinor to return to its original position after a full rotation. In higher dimensions, spin representations can be associated to certain sections of line bundles over (parabolic quotients) of the group (these are the moduli spaces that I already alluded to). The general theory concerns the Borel-Bott-Weil theorem, although I haven't really thought too hard in general about how to obtain the spin representations via that avenue, and it probably isn't that helpful in general. Sławomir Biały (talk) 13:52, 2 October 2014 (UTC)

You seem to be missing my point. A hint about interpretation (of the vector as a bundle) would be valuable with the picture in the article, not here on the talk page. This might help to counteract the intuition to interpret the Möbius band geometrically. —Quondum 15:36, 2 October 2014 (UTC)

There are now changes edited into a fairly technical footnote on the caption. Would it make sense to change the visible caption from A spinor visualized as a vector pointing along the Möbius band to something like A spinor visualized as a vector bundle over a space of rotations, forming a Möbius band (or whatever would be most correct)? —Quondum 16:58, 2 October 2014 (UTC)

But how is the technical footnote any different from the description you seek? A spinor is a point of a line bundle over a conic in the projective plane. That line bundle is the Möbius band. I don't think moving it into the caption text would be very helpful. Somewhere in the article, this can possibly be explained further. Sławomir Biały (talk) 19:30, 2 October 2014 (UTC)

Yes, lets not overload captions, keep the explanation in the text. M∧Ŝc²ħεИ_τlk 19:47, 2 October 2014 (UTC)

I agree that we want to avoid overloading captions. The provided detail in the footnote is of course too involved for the general audience, and so keeping it in a footnote (or article body) is appropriate. All I was looking for was some way of avoiding making the wrong associations from the diagram, since it is so geometric. What I missed is that there is a parenthesis added to "the circle", which essentially does the clarification that I was looking for, so, really I should retract what I was saying. —Quondum 22:26, 2 October 2014 (UTC)

New edits

I don't think this sequence of edits clarifies the subject matter for likely readers of the article. I believe we had the lead at just about the right level of detail. User:Mark viking, for instance, commented quite favorably on the prior version, before the renewed editing spree, and most of the final edits have been made in response to specific comments made here and elsewhere. So I take this as evidence that major structural changes to the lead are no longer required, at least not until more discussion is had. I think we should go back to the earlier consensus version, possibly clarifying some of the finer techincal points in nb footnotes. I don't find Rogier's rather haphazard efforts at editing without discussion to be very constructive at this point. Sławomir Biały (talk) 22:43, 5 October 2014 (UTC)

Sławomir, I don't entirely see what you are getting at. Are you sure you're not being a little quick to judge Rogier's edits, as seems to me to have possibly been the case in the past? Before focussing on exact wording though, it might make sense to first consider what technical detail will be reasonably moved into the body, since the lead is pretty large and challenging as it is. For example, I see the first sentence of the last paragraph of the lead as it stands now as illuminating in the lead, whereas the rest of that paragraph might be too technical for the lead. —Quondum 00:08, 6 October 2014 (UTC)

Quondum, the latest sequence of edits has rendered the third paragraph all but incomprehensible to someone who is not already familiar with Clifford algebras. Even before the reader is introduced to the basic examples of Clifford algebras, he is already being bludgeoned with marginalia about how they can be naturally associated to general Euclidean spaces with inner products and how the spin group and algebra sit inside them. But is that not all already covered in the fourth paragraph? Did Rogier even read past the third paragraph when he undertook the latest bout of edits? I suggest, as I have before, that clarifying mathematical ideas for general consumption is not something that Rogier is good at, and that further edits to the lead should be discussed. I am restoring the lead to its prior, better state. 00:30, 6 October 2014 (UTC)

So here is, as I see it again, the basic structure of the lead. The first paragraph should contain basic information about the subject of the article. It is intended to be read (and understood!) by all likely readers of the article, which may even include children with just a passing familiarity with scientific ideas (they know what an electron is, and what a rotation is, with possibly a vague idea what a vector is). The second paragraph concerns the geometrical origin of spinors as objects that detect different homotopy classes of rotations. I believe that this paragraph belongs in the lead since it holds everything else together, bridging between the first paragraph which just contains a general idea, and the later paragraphs concerning the specific construction via Clifford algebras. The third paragraph should concisely say what the Clifford algebra is in the case of three dimensions and how the spinors actually appear. Besides the first transition sentence, I don't see this dwelling on marginalia. Remember, the goal here is to communicate ideas to people without mathematical sophistication. There is room for nuances in the article, clarified in nb's if necessary, but it should not dominate the text. The fourth paragraph concerns the general construction, but I think it should still be readable by someone for whom the language of representations and homomorphisms are unfamiliar. Realizing the Clifford algebra as some specific algebra of matrices is the classical way of doing things, and although it is inelegant by modern mathematical standards, it is more likely to serve a wider audience. Sławomir Biały (talk) 00:53, 6 October 2014 (UTC)

Well, much of what is in the lead, and has been there at any point in the edit history, is IMO beyond the target audience that you describe. Mere mention of Clifford algebras, gamma matrices, unitary matrices, complex vectors, spin groups, linear representations and homotopy classes in the lead is already a challenge. But as I can see that my input is having very little impact, I think perhaps I should stay out of this entirely. —Quondum 01:23, 6 October 2014 (UTC)

While it's true that the mere mention of these things is already a challenge for a lot of readers, I don't see this as an excuse to make the lead so high brow it can only be read by people who already have a detailed knowledge of the relevant mathematics. The attitude that we should not care about readers lacking the requisite mathematical background has already been articulated by Rogier.

To specifics, this revision introduces a Clifford algebra as follows:

Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra. The Clifford Algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the algebra is often easier to work with than the group. A choice of convention for the gamma matrices, gives an explicit faithful matrix representation of the Clifford algebra, and hence of the Spin group, for every choice of orthonormal basis of Euclidean space.

This is impenetrable to anyone who does not already know what a Clifford algebra is, not to mention repeating some of the same points raised in the last paragraph. It also needlessly obfuscates what the spinors actually are. It is much better introductory style to introduce the basic example of a Clifford algebra first, before dealing with the general theory:

Spinors, although they can be defined purely as points of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra, an auxiliary mathematical associative algebra that encodes the relevant features of the geometry and is often easier to work with in applications. In three Euclidean dimensions, for instance, the Clifford algebra is the system of matrices that is generated by the Pauli spin matrices that correspond to angular momenta about the three coordinate axes. These are 2×2 matrices with complex entries, and the spinors are just the two-component complex column vectors on which these matrices act by ordinary matrix multiplication.

Here we have, in just three sentences, stated what the Clifford algebra is and what the spinors are. The last paragraph concerns the general case. Sławomir Biały (talk) 05:24, 6 October 2014 (UTC)

No, you have most definitely not. As I pointed out earlier, you merely confused the (note the not a) Clifford algebra with a (note a not the) matrix representation. This is the source of a great deal of confusion because this is the heart of the problem why there is no such thing as an invariant construction of spinors. The matrices act on column vectors, but the Clifford algebra does not without making further choices. Why do you think people want more abstract constructions of spinors? Why do you think the first thing people do when talking about spinors in general relativity is to work with vierbeins i.e. a choice of orthonormal basis in every point (while silently assuming a convention of gamma matrices to have been chosen )? Why do you think that things like Spin and Spin^c structures exist (Hint: can one always choose a global vierbein?).

As for structure I for one would favor using the second, third and fourth paragraph as the introduction. That would probably require a slight extension of the first paragraph with the remark that spinors transform under infinitessimal transformations but not quite under full rotations, and that spinors are often constructed and used through Clifford algebra's. But overall the article would get shorter. I just didn't feel like starting of on another edit war. That would leave room for the final paragraph in the August 14 version or any improvement that has gotten on the way because I do think it is a pity that it is now no longer clear what spinors are used for apart from quantum mechanics, and does not mention the Dirac equation anymore.

Oh and finally, please stop treating my edits as vandalism, reversing most of them within an hour. And please stop justifying that by saying that I donot care about laymen. I have explained my position in detail. In particular I mentioned there that we should move the technical paragraphs out of the lead in that explanation, because it is too technical in the lead.

RogierBrussee (talk) 08:29, 6 October 2014 (UTC)

Pointing out that there are abstract constructions of spinors is indeed something important. But I don't think it is necessarily helpful to focus the lead in that direction. You have consistently been of the opinion that spinors are just too hard to even try to explain in a way that is suitable for general consumption. I disagree, and think that every effort should be made to do this, even if it means de-emphasizing some important details. Sławomir Biały (talk) 11:44, 6 October 2014 (UTC)

From a physics background, I have to say that I find Rogier's paragraph easier and more informative. Jheald (talk) 10:46, 6 October 2014 (UTC)

So, I'm getting from this that Rogier's third paragraph is preferred, and that the fourth paragraph should be moved to the introduction section. I have restored most of Rogier's third paragraph edits, moving part of it and the entire fourth paragraph into the introduction section, together with some rearrangement and tightening of content. Is everybody happy with this? Sławomir Biały (talk) 12:09, 6 October 2014 (UTC)

I have restored the message that gamma matrices generate a representation of the Clifford algebra, and reintroduced Pauli matrices. Overall the lead seems to be finally converging. Although I agree that it is still rather high level for a lead. As a side remark, the second, topological, paragraph is now,in my opinion, the most advanced one. The question now becomes what to do with the intro (I will leave that to Slawomir for now) and what to do with the material in the last paragraph in pre edit wars version that contained useful info on how spinors are used. RogierBrussee (talk) 18:53, 6 October 2014 (UTC)

I have also moved the fact that spinors transform under infinitesimal transformations but that for full rotations the trafo depends on the path, without explaining why (i.e. the dependence on the homotopy class) to to the first paragraph. This should be understandable to the proverbial bright high school student. I also deleted the word vector-like which seems to be widely discussed but which I donot like. — Preceding unsigned comment added by RogierBrussee (talk • contribs) 19:19, 6 October 2014 (UTC)

Reducibility

I think some clarity is needed on the business of irreducibility. It's certainly true that the spin representations in even dimensions decompose if we tensor everything with the complex, but this does not seem to be true over the reals. (Compare the three signatures in four dimensions.) I've put the dependence on the metric signature back in for now, since what was written was misleading on a naive reading of things. It is true that the complexification of a reducible algebra is reducible, but the converse is not true (e.g., $M_{2}(\mathbb {C} )$ is simple, but its tensor product over the reals with $\mathbb {C}$ decomposes as a sum of two copies of $M_{2}(\mathbb {C} )$, which is what "complexification" means in this setting). So I think it is probably better for readers to be told that the reducibility can depend on metric signature, even if we regard all spinors as "over the complex" this is not necessarily true. Sławomir Biały (talk) 19:39, 6 October 2014 (UTC)

I think that is solved now. In the lead we deal with complex representations and only mention real representations in the nb'sRogierBrussee (talk) 21:24, 6 October 2014 (UTC)

I really don't understand why you (Slawomir) keep deleting that the Clifford algebra representation will decomposes as two irreducible representations of the Spin group for even dimensions and is irreducible for odd dimensions, and insists on the word "may" and wants to get rid of the Spin group. That only creates unnecessary uncertainty. RogierBrussee (talk) 21:58, 6 October 2014 (UTC)

At that point in the article, it is unclear whether we are discussing the representation of the spin group or the Clifford algebra. A reader has no reason to think these would be different. So it is better left as a slightly vague statement with a sharper statement given in an NB. Discussing the distinction in the text of the lead is not worth the explanatory overhead. Sławomir Biały (talk) 22:39, 6 October 2014 (UTC)

And precisely because it is unclear, it is much, much better to be explicit than to be vague, because that actually increases the cognitive load. RogierBrussee (talk) 07:04, 7 October 2014 (UTC)

I'm not sure what the goal is in including this information anyway. A reader looking for specific information on how the representations decompose is much better served by the footnote than a statement in the lead of the article, since such a reader may not even be aware of the difference. Presumably we should also have a decent section on this issue in the article itself. But on a casual reading, the "cognitive load" of understanding "as a spin rep" in that context is certainly greater than not having it there. The lead should not dwell on these nuances. You yourself said that we should make things as smile as possible, and this is obviously one of those cases. Sławomir Biały (talk) 07:32, 7 October 2014 (UTC)

I still feel that the original issue that I raised is not "solved" by the nb. Even in the case of three dimensional Euclidean space, which that paragraph already mentions as an example, there is a mismatch between a reading of that example and the statement that irreducibility is independent of metric signature. The spinors are the two-component complex column vectors. These are irreducible regardless of whether they are regarded as over the spin group or the Clifford algebra. And the spinors are already complex, so a naive reading of "complex" does not help our poor reader here, who is not told that he must first complexify the complex representations. It's only when one is used to obtaining spinors first from the complex and then examining their behavior under the real that the latter can sensibly be called reducible. I think we should go back to the old version of the statement, that said that their reducibility depends on both the dimension and the metric signature. Sławomir Biały (talk) 12:56, 7 October 2014 (UTC)

Sorry, but I really don't understand what your point is. The reducibility or irreducibility of the complex reps has nothing to do with the signature, and we only talk about the complex reps. In three dimensions the complex Clifford representation is irreducible as a complex representation of the spin group. That's why it said "depending on whether the dimension is odd or even" and "as a complex representation of the spin group". But you deleted that and put back your old version. It is actually much more confusing. RogierBrussee (talk) 17:31, 7 October 2014 (UTC)

In three dimensions, Cl(3,0) is simple, but Cl(1,2) is not. At the level of spin reps in four dimensions, Cl_0(1,3) is simple but Cl_0(2,2) is not. Sławomir Biały (talk) 18:12, 7 October 2014 (UTC)

That is true, but it has nothing to do with the irreducibility of the complex spin representation. See the article on the spin representation which is indeed excellent.RogierBrussee (talk) 21:50, 7 October 2014 (UTC)

Ok, I'm just being thick. By "complex spin representation", I think we mean (effectively) the representations of the complexified algebra restricted to the real. I've self-reverted. Sławomir Biały (talk) 23:15, 7 October 2014 (UTC)

Second paragraph

So, I think we have achieved some form of consensus on the first paragraph. The second paragraph, unfortunately, was pillaged of its context, so that now it feels out of place. The spin group and the idea of a linear representation need to be brought in somehow, and this is what the second paragraph does. But it is not easy to read, and it lacks a clear topic sentence. Sławomir Biały (talk) 20:57, 6 October 2014 (UTC)

I think the whole lead is now in reasonably good shape. Others may disagree of course, in particular about the choices what is in or out (the Dirac equation???, spinors in topology and global analysis?). I did try to give a topic sentence for the second paragraph. So Slawomir, I think something good came out of this after all. Cheers! RogierBrussee (talk) 21:24, 6 October 2014 (UTC)

I am sorry to say, I think the second paragraph is in slightly worse shape now (it is not actually bad though).

Version of Oktober 6 (the nb's should probably be moved to the end of the sentence and)

The dependence of the spinor transformation on the way an overall rotation is decomposed as a sequence of small rotations starting at the identity is only through two topologically distinguishable equivalence classes. They correspond to two different homotopy classes of paths, with fixed initial and final points,[nb 5] in the rotation group. In particular, a spinor changes to its negative when the system is subjected to a complete turn. The set of all such inequivalent homotopy classes of paths from the identity to a final overall rotation itself[nb 6] forms a group called the spin group. It doubly-covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors.[nb 7]

Version of Oktober 7

Spinors are characterized by the specific way in which they behave under rotations. However, their behavior exhibits path dependence: they change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group). For any overall final rotation, there are two topologically distinguishable classes (homotopy classes) of paths through rotations that result in the same overall rotation, as famously illustrated by the belt trick puzzle (below). These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class.[nb 5] It doubly-covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class.[nb 6]

links are put in by hand because I am lazy and did not actually copy the source, so some of them are wrong. RogierBrussee (talk) 07:27, 7 October 2014 (UTC)

Well, I think we should give others the chance to comment. Sławomir Biały (talk) 07:35, 7 October 2014 (UTC)

Once more on topological analogies through illustrations/animations

An old file retrieved and uploaded. Needs to be redrawn much, much nicer, cleaner, and clearer, if people think it helps this article. Now modified

Sorry to repeat the section above, but on my talk page is a request from Sławomir Biały for a diagram (or possibly animation) for plate trick. Presumably it is to show orientation entanglement, since the diagram may be used here also?

Drawing/animating an arm holding a plate/cup etc. rotating around would take time, but the same principle can be shown with the "belt" attatched to an arrow. (There are youtube videos which show the rotating palm for the plate trick e.g. this. I know we should not rely on youtube videos, but it could be in the external links section.)

Is the animation above along the lines of what people would like to see (cleaned up and slowed down of course)?

P.S. I haven't read through the lead in detail but it looks like a lot of good work has been put into it recently, will see it later. M∧Ŝc²ħεИ_τlk 07:53, 19 August 2014 (UTC)

Yes, we should use this image. Thanks! Sławomir Biały (talk) 13:22, 20 August 2014 (UTC)

Very sorry for the belated activity, thanks for the feedback and compliments. I didn't realize untill now the original (and crinkly) animation (here on the right) had been added. A newer and hopefully clearer version (particularly for colour blind people) has been made as promised:

and will be smoothened further. Feel free to take or leave as required. Best, M∧Ŝc²ħεИ_τlk 23:39, 23 August 2014 (UTC)

Actually I like the old one better. The new image lacks the same three dimensionality. Also I like that the axis of rotation is visible in the old one. I do not think color is a significant issue, since it is not used (at present) to convey any necessary information. Sławomir Biały (talk) 00:07, 24 August 2014 (UTC)

OK, I redrew the older one more tidily. If there are suggestions for improvements please say. Thanks, M∧Ŝc²ħεИ_τlk 00:54, 25 August 2014 (UTC)

I hate to say this, but the illustration doesn't work. A 2π rotation with only two strings can be untangled using the operations illustrated. We need to replace the two strings with a ribbon, in which the two sides could be coloured differently for clarity. —Quondum 01:46, 25 August 2014 (UTC)

Yes, just checked this physically... The belt image above is correct, but will be drawn better and more like the original. M∧Ŝc²ħεИ_τlk 06:39, 25 August 2014 (UTC)

Good point. Perhaps if I explain the way it is meant to work, that can help to draw an image that illustrates the concept. Each point on the surface of a ribbon has a tangent, a normal, and a binormal vector (see, for example, Frenet-Serret formulas). So a twisted ribbon, having one rotation for each point of its length, defines a one-parameter family of rotations (what we are here calling a "continuous rotation"). An operation of moving the ribbon through space whilst fixing the ends is a homotopy of continuous rotations; the homotopy parameter actually being "time". In principle, it seems to me like it should be possible to illustrate this with a pair of cords, although perhaps the apparatus describing this in the article is incomplete. When we do this in practice, the 2π twist actually just gets shoved into a curling of the cord around the fixed bar at the end, and to undo this curl requires pivoting the cord around the bar. Since nothing has been said that would preclude this (mathematically, the "bar" is modeled as a line segment, having zero cross-sectional breadth), it would seem to be possible to continuously undo the 2pi; rotation. So, this having been said, I think we probably ought to illustrate this with the ribbon shown in the new animation rather than the tangloids shown in the old one. I have gone ahead and done this. Sławomir Biały (talk) 12:11, 25 August 2014 (UTC)

Request: would it be possible to make the animation a little bit slower, and have it pause a little at the 2π point? Sławomir Biały (talk) 12:16, 25 August 2014 (UTC)

I think tangloids solves the problem by using three strings instead of two, but the ribbon/belt seem more appropriate here. I found Penrose's use of a familiar object very nice because it is so easy to try and experiment with (unlike the flexibility needed for the plate trick): a belt (as suggested in Sławomir's choice of words) with one end clamped inside a book and the other end anchored. In the illustration the arrow is merely an object with a distinguishing feature to track orientation, but in fact the illustrated concept works for all rotations, including around the axis of the arrow. Should the arrow not rather be replaced by some 3-d object that we can easily orient mentally? A book (with a clearly distinguishable front and spine) seems to be a particularly suitable object, though there may be other good examples. —Quondum 13:47, 25 August 2014 (UTC)

I think there may still be a problem with the current animation. When I do the belt trick, the belt only needs to be brought around the end one time. But in the animation it appears to take two times. Also, the second pass in the last frames of the animation appears to undo a 2π rotation which, as we know, should be impossible. Sławomir Biały (talk) 11:27, 26 August 2014 (UTC)

Hi everyone, yes, the current one still looks like two chords anyway so both of these animations may as well be scrapped.

I have found a way to show the belt trick clearer, in a way that also resembles the "rotating palm" for the plate trick at the same time, and can include an extended object being attatched to a belt. Would a sphere with an arrow through it's centre along a diameter in a random direction be OK? This would not bias the rotated object to be "perpendicular" to the belt.

It's on the drawing board now, but will take some time (sorry about that), should be ready by tommorow or thenabouts.

Thanks for the feedback, M∧Ŝc²ħεИ_τlk 00:11, 27 August 2014 (UTC)

I'd suggest something that is mentally more easily orientable. I'd suggest a book (as simplified as you like, e.g. a rectangular solid with three dark faces for the cover), with a belt clamped off-centre and a distinguishable front cover (e.g. a circle/rectangle/blob high on the front cover). This also allows people to easily translate it to a model that they can physically assemble and check from everyday materials (a book and a belt are typically available, and are easy to attach and manipulate without glue etc.). A sphere as you suggest does not seem to have enough to "grab onto" mentally, IMO. Also, slow movement is good. —Quondum 02:21, 27 August 2014 (UTC)

Eccchhh... It looks bad to keep delaying everyone like this after all the feedback and effort for improving the article, so apologies once again (and forever in advance) since I seem to not be pulling my weight here (and elsewhere).

The original was developed on the 27 August, but subsequent edits became very complicated, so I stripped it down to the absolute basics like this:

File:Spinor orientation entanglement animated 3.gif

Before smoothening and detail, are the topological deformations of the belt clear enough (from the 4π to the untwisted state)? Any better choice of colours? etc.?

I understand about the book, and can and will change the arrow to the book, but for now an arrow is much more general and I would think any potential reader is capable of mentally substituting the arrow for an object like a book/plate/cup/whatever. Also, I know it looks better to see the belt twisting around before undoing it, but left this out for now since it is irrelevant to the untwisting from the 4π twist to the untwisted state (arguably - the most important part). M∧Ŝc²ħεИ_τlk 00:29, 1 September 2014 (UTC)

Sorry - just noticed an error, fixing now >_< M∧Ŝc²ħεИ_τlk 01:23, 1 September 2014 (UTC)

I really like the belt, colours and all. For now, can you slow things down a bit? To no more than half the speed, so we can follow it? It might also make sense to keep the object stationary while the belt is stretched around to untwist it, since movement of the object could be confused with some critical aspect of the untwisting. And hey, you're doing something I wouldn't even know how to start putting together graphically, it clearly involves effort, there is no timetable and all contribution is strictly voluntary. An apology is not called for. —Quondum 01:35, 1 September 2014 (UTC)

I would prefer something where the belt moves, with the endpoints fixed. Otherwise, I think one tends to focus too much on the role of translations of the end of the belt, but these are not involved in an essential way. Sławomir Biały (talk) 13:12, 1 September 2014 (UTC)

File:Spinor orientation entanglement static.svg

Belt trick: the belt is wrapped once around the arrow attatched to an untwisted belt to obtain the equivalent of a clockwise 4π twist. Unwrapping in the opposite way returns to the untwisted state. The arrow can be replaced by any object fixed to the belt.

In the mean time, for the purposes of having an image that is actually correct, I had to demote the graphics to a static version which should get the point across. It is certainly inferior to an animation, but the stumbling block in every case occurs when untangling around the arrow (book, etc.) - which tends to occur (in practice) very quickly and it's hard to animate that segment. (I have tried rotating a book with a slack belt, keeping one end of the belt fixed and the book not translated through space (as much as possible due to the slightly limited size of the belt) and sketching by hand in small incremental steps, but animating seems unexpectedly awkward)...

Sorry about that, the animation will be back at some point, I just don't want to drag everyone's time on this for much longer... M∧Ŝc²ħεИ_τlk 23:43, 5 September 2014 (UTC)

New animations

A new animation of the belt trick

A new animation of the belt trick

Here's a pair of animations I made in Mathematica. Sławomir Biały (talk) 14:27, 6 September 2014 (UTC)

These look brilliant! Thanks and great work answering your own request and sorry for being totally useless... Fwiw I don't have Mathematica right now, even so would not be able to program an animation like this... M∧Ŝc²ħεИ_τlk 19:25, 8 September 2014 (UTC)

Yeah, these are really neat, Sławomir. Easy to follow the motion too. —Quondum 23:26, 8 September 2014 (UTC)

Is there a way to start the animation "manually" (or alternatively stopping it)? It is neat, but also distracting. YohanN7 (talk) 16:37, 12 October 2014 (UTC)

(Almost) commutative diagram

Ugly

Any use of something like this? It is particular to the Lorentz group for now, but can be generalized. The Π is the physicists projective rep (generally 2-valued, but can locally be made single-value using a local section of the covering map), and the Π* is what the mathematician means.

It looks like ****, but can perhaps be fixed to look okay. Unfortunately, everything is converted to paths. YohanN7 (talk) 15:13, 7 September 2014 (UTC)

It would be fairly easy to do it in LaTeX. I wonder if there is a way to output to a vector image? Sławomir Biały (talk) 17:15, 7 September 2014 (UTC)

Better image

Much better. I wonder if it is really necessary to involve Clifford algebras here. It might suffice to say that Im(Φ) is a faithful representation of the covering spin group (regardless of how it is constructed). It might be useful to point out that p is 2:1. This can be made in the figure caption, but we should probably generalize the image from SO(3, 1) to SO(p, q) (at least) or SO(n, ℂ) and from SL(2, ℂ) to (at least) Spin(p, q) or Spin(n, ℂ). YohanN7 (talk) 20:21, 7 September 2014 (UTC)

Also, the Clifford algebra could go in as an ingredient in the diagram, containing (at least) a Lie algebra rep. YohanN7 (talk) 16:42, 7 September 2014 (UTC)

@Sławomir Biały: Could you copy the LaTeX to my sandbox? I've tried doing things like this in LaTeX before, but stuff (like fonts) have been lost in the process. YohanN7 (talk) 15:16, 8 September 2014 (UTC)

I've put the source at File:Commutative_diagram_SO(3,_1)_latex.svg. You won't be able to run it on Wikipedia, because we don't support xypic. (Incidentally, Wikipedia also just generates PNG images rather than SVG images from the LaTeX. I wonder why that is.) Sławomir Biały (talk) 21:48, 8 September 2014 (UTC)

Thanks. I actually got it to work now on my old machine. YohanN7 (talk) 00:02, 9 September 2014 (UTC)

Isn't it still not possible to use xy-pic in WP? M∧Ŝc²ħεИ_τlk 19:25, 8 September 2014 (UTC)