Talk:Spinor

From Wikipedia, the free encyclopedia

WikiProject Mathematics
Mathematics Portal
This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating: A-BB+ Class High Priority  Field: Geometry
WikiProject Physics This article is within the scope of WikiProject Physics, which collaborates on articles related to physics.
??? This article has not yet received a rating on the assessment scale. [FAQ]
??? This article has not yet received an importance rating within physics.

Help with this template

Comments:

Contents

[edit] Spin group reps

Um, spinors are PROJECTIVE reps of SO(p,q) but (linear) reps of spin(p,q). In the section on spinors in various dimensions, you keep writing spin(N)/Z_2, at least for a group isomorphic to spin(N). Phys 02:42, 7 Aug 2004 (UTC)

[edit] Needs expansion

I think this article could use some expanding, particularly in the overview, to make it more understandable by people who don't already know about spinors. It sometimes helps to explain the same thing a couple of different ways, so that people who are approaching the topic from different backgrounds can get it. It's a very good article otherwise though. Hope this helps.... —Preceding unsigned comment added by Spiralhighway (talk • contribs) 17:01, 18 November 2004

[edit] Fraktur

Who uses \mathfrak{so} instead of SO? I've never seen this before. _R_ 21:01, 1 Feb 2005 (UTC)

Mathematicians. Specifically, \mathfrak{so}(n) refers to the Lie algebra and SO(n) refers to the Lie group. Physicists often blur the distinction and use the latter for both. -- Fropuff 21:40, 2005 Feb 1 (UTC)
I beg to differ. Physicists may call \mathfrak{so}(3) \mathbb{R}^3, but certainly not SO(3)! The notation \mathfrak{so}(n) used in the article is therefore highly confusing, as my ill-advised modification of \mathfrak{so} to SO certainly demonstrates. _R_ 02:50, 2 Feb 2005 (UTC)
They do sometimes. Actually I think it's more a case of blurring the distinction between groups which have the same Lie algebra (e.g. calling spinors representations of SO(n) instead of Spin(n)). I have many physics books which say SU(2) and SO(3) are isomorphic (without using the word locally).
I should add that using lower case gothic letters (or at least lower case) for Lie algebras is a very standard practice in mathematics. -- Fropuff 03:28, 2005 Feb 2 (UTC)

Ditto with Fropuff. The gothic lower case is standard notation. Most texts on differentiable manifolds will write, for instance, "Let \mathfrak{g} be the Lie Algebra of a Lie Group G."--Perkinsrc008 21:59, 8 June 2006 (UTC)

[edit] Majorana particle

I think that Majorana particle should be its own page, rather than just a redirect to this particularly mathematical page. If there are no objections I will create a Majorana particle page at some point. --Flying fish 05:10, 5 Jun 2005 (UTC)

Hm, I'll object. as far as I know, there's no such thing as a "majorana particle" per se; there have been searches for physical particles that might behave like Majorana spinors, but I don't know that any were ever found. Can you elucidate what can be said about a "majorana particle" that this page doesn't say? linas 14:20, 6 Jun 2005 (UTC)
Ok, this prompted an interesting discussion in the lab... A common definition given for Majorana particles is "a particle which is identical to its antiparticle". Under this definition photons and neutral pions are Majorana particles. Since Majorana was working on generating masses with fermion fields, he clearly had fermions in mind. So the question is one of the exact definition of "Majorana particle". Does it apply only to fermions or to bosons as well? (The interesting part, Majorana mass terms, only exist for the fermions of course). For what it's worth the most senior person I've talked to says that you should not apply the term to bosons.
So Instead of asking if Majorana particle should be its own page I'll instead ask if it makes sense to create a page devoted to "Majorana mass" or "Majorana neutrinos" that focusses on the difference between Dirac and Majorana mass?--Flying fish 15:27, 6 Jun 2005 (UTC)

Right. Gosh, I'd forgotten all this ... the reason that the Majorana spinor is "its own anti-particle" is precisely because it is a *real* representation; it doesn't have a distinct complex conjugate representation. So there's two ways of handling this. Amend the article to state something like a physical particle transforming under a real representation has the curious property that it is it's own antiparticle. Another possibility would be to start an article on the majorana particle that would detail some of the history of experimental searches for such beasties.

The reason that π0 is self-conjugate is because its (\overline uu -\overline dd)/2 so again its "real"; conjugating it is a no-op. Alternately \overline {SU_2} \otimes SU_2 \equiv SO_3\oplus U_1 and SO(3) is "real", whereas SU(2) is complex (a doublet times a doublet is a triplet plus a singlet, fancy math talk for basic undergrad clebsch-gordon math) ...but this now walks down the slippery slope of having to explain what real and complex reps imply for physical properties of particles, and that's a whole nother mess ... linas 00:27, 7 Jun 2005 (UTC)

[edit] Understandability - please simplify this page!

Is there any way this page can be made more understandable to the ordinary engineering graduate like me. It really is written only for the (math?) specialst at the moment. Can someone try to simplify please!--Light current 01:32, 2 October 2005 (UTC)

I agree! I would like to see a more intuitive explanation of what the heck a spinor is (how can we visualise it, describe it in less technical terms than presently given ?). There should also be some more on the use of spinors in physics (including relativity theory ). ---Mpatel (talk) 13:41, 17 October 2005 (UTC)
I third! At the moment the 'overview' is utterly impermeable to myself and all of the physics undergraduates that I've shown it to. While credit goes to the author(s) for doing it in the first place, there's really no point unless it can be explained to someone who doesn't already understand spinors. They're obviously not a simple topic, but they're not as hard as they seem at the moment. -- drrngrvy 15:11, 30 November 2005 (GMT)

[edit] Sorry, but...

...this article needs a cleanup. The intro. is ok at present, but more could be done to provide a better intuition of spinors (are there any diagrams, graphics that would help ?). The examples section is waaay too long and the example containing the long list towards the end is particularly ugly and definitely needs to be placed in a new article, perhaps spinors in relativity ? The content of the article is fine, just that it needs a better explanation of some of the maths to people who don't know too much about spinors (like me). ---Mpatel (talk) 13:55, 17 October 2005 (UTC)

[edit] Pronunciation necessary?

The IPA pronunciation given in the article appears to be from British english (the lack of an 'r' gave it away to me; I'm American). Clearly "spinor" is pronounced with a hard 'r' at the end ;-) What I propose is that either the American pronunciation is added in addition to the British one, or the pronunciation simply be removed (I think it is pretty obvious how one should pronounce it in one's native accent). This is not the name of a person or something where the native pronunciation of the subject would be important. - Gauge 04:25, 24 October 2005 (UTC)

It is almost never clear how to pronounce an english word, though one can guess. Most people here first pronounce it to rhyme with minor. My current knowledge is that it should be pronounced to rhyme with the two words "spin, or" instead of "spine, or".--MarSch 13:46, 24 October 2005 (UTC)
I think it's much less than obvious that spinor should be pronounced as it should be, so the pronunciation should stay, IMHO. However, with the single (English) pronunciation it becomes obvious how to alter the pronunciation to your accent or country. Having an 'American' pronunciation just seems like too much, I think.--Drrngrvy 15:18, 30 November 2005 (UTC)

[edit] Pronunciation: a poor example

Surely it is not wise to give as an example of pronunciation perhaps the only word ('Linux') whose pronunciation, while correctly similar to spinor, is more hotly debated, and for the same reasons. Could we not say that the 'i' is pronounced like the 'i' in 'windows'? —Preceding unsigned comment added by 69.63.49.155 (talk • contribs) 08:37, 22 November 2005

The whole point, really. You can say 'spinnor' or spine-or. (I think the latter is older-generation, but I know it exists.) You can say Linux either way. You are not going to be misunderstood, and it is exactly the same point about English pronunciation. Charles Matthews 10:44, 22 November 2005 (UTC)
So, someone cut it out under a minor edit, and now the status is being queried. Shouldn't people lighten up about this? Charles Matthews 11:00, 16 December 2005 (UTC)

I thought the joke was funny, but now it just looks stupid in the article. Let's get rid of the whole pronunciation thing all together. -- Fropuff 17:28, 16 December 2005 (UTC)

Huh? Here in Texas, we call it whine-ders, doesn't everybody? linas 21:55, 16 December 2005 (UTC)
I like the "spinners" and "winners" suggestions in the current article. - Gauge 03:14, 18 December 2005 (UTC)
Humph. See Wikipedia:Humour police. Charles Matthews 21:42, 5 January 2006 (UTC)

[edit] Mathematics vs Physics

The history of the study of spinors displays part of the intricate interplay between mathematics and physics over the last century. I believe that this article could benefit from teasing out this tangle. Give a potted history first, then the mathematical definitions and properties, then finally applications to physics.

As far as the mathematics goes, I believe that the study of spinors is eased somewhat by studying their relationship to Clifford algebras as well as Lie groups and Lie algebras. After Élie_Cartan [1], you could mention works by Claude Chevalley [2] ([3] 1954); Marcel_Riesz[4]([5] 1957, 1958, 1993); Michael Atiyah [6], Raoul_Bott [7][8] and Arnold Shapiro ([9] 1963); Ian Porteous [10] and Pertti Lounesto [11]. Porteous' two books "Topological Geometry" ([12] 1969, [13]1981) and "Clifford Algebras and the Classical Groups" ([14] 1995) explain the relationship between Clifford algebras and Lie groups with great care. Lounesto's book, "Clifford Algebras and Spinors" (1997, [15] 2001) makes the link between Clifford algebras and spinors very explicit.

See also Representations_of_Clifford_algebras. Leopardi 00:45, 14 February 2006 (UTC)

Please note that WP articles need to serve multiple audiences; these include young students learning the topic for the first time, as well as old timers who are trying to refresh thier memory. While I find history to be interesting, many others will want to cut to the facts, and are looking for a plain-old undergrad ntro-level presentation of 2-d rep spinors for su(2), and little more. A rarified few will be interested in the 10-d spinor rep of so(32). Finding that balance is tricky. linas 23:10, 14 February 2006 (UTC)
You seem to assume that there are such things as "the topic" and "the facts", whereas there are multiple points of view depending on which discipline you are talking about. I assume that by "plain-old undergrad intro-level" you mean "plain-old undergrad *physics* intro-level". What about undergraduate mathematicians? Leopardi 01:04, 15 February 2006 (UTC)

[edit] no definition

There is no definition to be found of what a spinor is, not even in the Mathematical details section, which is written as if spinor has already been made clear and ONLY the details need to be treated in isolation. Unfortunately my own understanding is limited and ungeneral. --MarSch 11:23, 13 April 2006 (UTC)

[edit] The article does define spinors, but in a rather mathematical way

Actually, the very first sentence of the article is a quite good, but rather mathematically inclined, definition of what spinors are:
In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects similar to spatial vectors, but which change sign under a rotation of 2π radians.
I suspect the difficulty is more that while the terminology of the definition is mathematically precise, it assumes familiarity with a lot of non-intuitive terms. And even then, phrases such as "changes sign" don't convey much of the quite interesting physics implications of such mathematics.
Not entirely. One of the difficulties with that "definition" (and with the page in general) is it refuses to say exactly what spinors *are* in any very clear way. So the definition above says they are "certain kinds of mathematical objects", but what kinds of objects? Are they vectors in some vector space? What does it mean to say they change sign under a rotation? Rotation of what? There must be a story about a connection between elements of an orthogonal group (here the rotations) and a vector space which has automorphisms that relate to those rotations. Or something. A definition should say a "spinor is a(n)..." or something Francis Davey 21:14, 24 July 2007 (UTC).

[edit] A very informal definition of a spinor

Here's a less formal definition: A spinor is an object with two main properties. The first is that has a definite orientation in ordinary space, much like a vector. Think, for example, of an arrow embedded within a clear ball, so that it can be set up to point in any arbitrary direction. The second property is a lot harder to picture, but it is also the behavior that makes spinors interesting to physicists: If you take such a spinor ball and turn it around 360°—one full circle—something about it gets "twisted" in a way that makes it into a negative version of itself. By that I mean that if you place such a rotated sphere at the same location as an otherwise identical one that was never rotated, the two cancel each other out—both disappear! To get a spinor back into its "real" original form, you have to turn it around another 360° for a total of two full turns.
The second property sounds really confusing. How can you claim that you have rotated the object for 360° if you in fact just reversed its orientation? The "rotation" operation mentioned here is as mysterious as spinor itself.

[edit] Spinors, probabilities, and ordinary matter

All of this is important in physics because spinors or their mathematical equivalents (the Pauli spin matrices) are used to calculate the probability of certain types of identical "spinorial" particles being in the same place at the same time. Rotation is a funny thing, since it turns out that a lot of scenarios in which two particles get close to each other actually include an implied 360° rotation relative to each other. That's geometry, not spinors, but it's a fairly subtle form of geometry that is not terribly intuitive until you get more precise about just what a "rotation" really is.
In any case, the result of such implied rotations is that when two otherwise identical spinorial objects try to get too close to each other, they turn into each other's negatives, and both go Poof!—which means, mathematically, that the likelihood of finding both of them in that spot in space drops pretty close to zero.
An easier way of saying that is this: Two identical spinorial particles really, really hate being in the same place, and so will repel each other, powerfully. The more common name for this effect is the Pauli exclusion principle, and the "spinorial particles" in questions are more commonly called fermions—that is, the electrons, protons, neutrons, and others that make up ordinary matter. The odd math of spinors thus is important for a very simple reason: Without it, you would not be reading this, because you would be part of a marble-sized black hole that would represent the mass of what earth would have been, assuming that in such a universe that anything at all of interest could have formed at all!

[edit] Spinors vs. Pauli matrices

So why then don't spinors get more press? Mostly because Pauli provided an elegant and computationally effective alternative with his equivalent spin matrices, and those have pretty much ruled the roost every since. For example, while Richard Feynman does mention spinors occasionally in his older and more technical works (see for example the middle of page 61 of Quantum Electrodynamics - A Lecture Note and Reprint Volume, W.A. Benjamin Inc., 1961, ISBN 0-8053-2501, he largely deemphasizes or ignores them in later and more popular works. I don't believe he uses the term at all in his 1986 Dirac Memorial Lecture writeup. That is a bit surprising considering that his lecture contains a vivid description of how to turn around twice in distinct cycles while carrying a glass of water, which is mathematically identical to the book-and-ribbon example of how a spinor behaves.

[edit] Quaternions: Spinors, basically... but way, way, way before their time

[The four paragraphs below were updated extensively by Terry Bollinger 16:30, 1 January 2007 (UTC)]
There is a final intriguing twist on the history of spinors, and that is this: The mathematical construct that most directly and intuitively represents them is something called a quaternion. Quaternions were invented in a flash of insight by the great mathematician Sir William Rowan Hamilton on October 16, 1843, as he was walking with his wife across what is now called the Broom Bridge in Dublin, Ireland. (One can't get much more historically specific on the timing of an insight than that!)
Hamilton's insight came more than half a century before spinors, spin matrices, or for that matter quantum mechanics in general were even contemplated. He arrived at them not through physics, but by way of his interest in a more abstract problem: How to extend the extremely useful concept of complex numbers from two dimensions to higher dimensions such as three or more. His quaternions were successful in doing that, but in an oddly limited way: They only worked for three dimensions (and to a lesser degree four), and could not be generalized further.
Considering the extensive intellectual effort Hamilton and others spent trying to apply quaternions to the real world, surprisingly little came of it. For one thing, Hamilton's insight was so early that it predated and to some degree anticipated vector algebra, which came along shortly thereafter and stole much of the mathematical thunder away from quaternions. Compared to vectors, quaternions suffered competitively by being locked into three or four dimensions.
All this is a bit sad. If quantum mechanics had been an issue at the time Hamilton had his insight, quaternions would have very likely been recognized as an amazing match up between mathematics and one of the more profound principles of quantum mechanics: Why like particles repel each other, and in doing so provide all of the volume of matter and complexity of chemistry that allow all of us to exist. Hamilton died still looking for that special application that he was convinced was out there for his quaternions, still decades away from the arrival of the physics issues for which they were best suited.

Terry Bollinger 06:22, 31 December 2006 (UTC)

And spinors relate to quaternions how exactly? Are quaternions a kind of spinor or what? Francis Davey 21:16, 24 July 2007 (UTC)

[edit] Spinors and Pauli Matrices/Dirac Matrices

Okay, I typed in "Spinors of the Pauli Spin Matrices" back in June and no one deleted it, I'm going to tempt fate and type in "Spinors of the Dirac Algebra" and see what happens. I also added the general solutions for spin in the (a,b,c) direction for spin-1/2 Pauli particles and also for Dirac particles and antiparticles.

As an aside, there are several versions of spinors based on operator theory that probably belong somewhere on wikipedia. The basic idea is to define spinors entirely through the projection operators themselves. The Cambridge geometry group calls this "density operator" theory: http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/hd_density.html

These methods allow all the usual spinor calculations without having to specify a representation. It's far more elegant than the usual methods.

Carl BrannenCarl Brannen 09:15, 25 January 2007 (UTC)

Sorry, but I moved your edits out to spinors in three dimensions and dirac spinor, respectively. The Dirac spinors edit needs to be adjusted to fit in that context, I think, but it looks to me as though it belongs there more than here. Besides, that page needs all the help it can get. Please have a look, and see if you approve. Regards, Silly rabbit 15:56, 17 April 2007 (UTC)

[edit] Article too large already

From the discussion on the talk page it appears that at least some people want to further expand this article. Maybe my comments will go against the flow, but I feel very strongly that this article is already way too big and (a related issue) diluted. I think it can be improved a lot by removing much of inessential material, then adding some important points that are not adequately addressed (eg Dirac equation or spinor fields). If I had not known what the spinors were already, I would never even come close to finding my way to their definition through the infinite ramblings of this article! This may be one of those cases, where giving a definition and a couple of examples straight up would serve the reader much better than vague and convoluted preliminary comments about their "meaning". (Curiously, the stub of an article spinor bundle seems to be better in this regard.) There should be a paragraph close to the beginning to the effect that "Spinors are elements of the spinor representation of orthogonal group G, which is constructed from the Clifford algebra as follows...", then the example of SO(3) worked out with as little notation as possible. Normally, I would have done it myself, but given the size of the article and apparently enormous amount of effort invested in it, I wanted to discuss it first. Also, there are a couple of wrong or, at least, misleading statements right at the beginning of the article, which raise the issues of the consistency, hence not easily fixable.

  • Quantization — this is not the most common use of the quantization, and certainly not the best way to define spinors (aside from the role of spinors in the Dirac equation, which indeed may be said to quantize something).
  • The introduction specifically refers to inner products and Spin(n), making the impression that the space is real and the form is positively-definite. But then the article almost immediately jumps into the discussion of indefinite orthogonal groups and later, isotropic vectors and various reality issues. Arcfrk 13:08, 10 March 2007 (UTC)
I basically agree. And the 'to do' comments, with more physics planned, will only make matters worse. Material should be ruthlessly moved into specialised sub-articles. By the way, tensor is no better, after being multiply forked. Charles Matthews 12:27, 20 March 2007 (UTC)
Good job cleaning it up, Charles! There is obviously still a lot of room for improving this article, but giving it more structure is an excellent first step. Are you going to move out "higher dimensional spinors" as well? At the moment, its length and multitude of details contrasts rather sharply with the minimalistic nature of "spinors in three dimensions". It also seems to me that "history" and several sections that immediately follow should be moved closer to the beginning. I am going to wait a bit until you stabilize the article, then make my own contributions. Arcfrk 21:58, 20 March 2007 (UTC)
Ugh, big job. We always (on the physics/mathematics boundary) run into these pedagogical issues, in other words 'what is X?' versus 'how to introduce X?'. The Bourbaki-flavoured mathematician can deal with the first quite well. The theoretical physics input can deal with the second, but is inevitably multi-valued. Charles Matthews 08:11, 22 March 2007 (UTC)
Right, the article is now more in the 'concentric' style, with an Overview covering all main aspects, and more algorithmic discussion separate. Charles Matthews 08:52, 22 March 2007 (UTC)
Phantastic job! A+. Arcfrk 09:52, 22 March 2007 (UTC)

[edit] Two ways to think about ψ M ψ~

It occurs to me that there are two ways to think about the quantity ψ M ψ~ -- two viewpoints which can clash, unless the article explains them. But at the moment it doesn't say a word.

The first way to think about ψ M ψ~ is to think of M as a bilinear operator, and ψ as "half a vector" (or half a multivector) -- so it and its twin ψ~ are just the passive objects being operated on.

Thus one can write

\psi M \tilde{\psi} = \phi \tilde{\phi}\,

and then

\phi \tilde{\phi}\, = N

because, if you like,

M^{1/2} : \tilde{\psi} \rightarrow \tilde{\phi}

The other way is to think of ψ as the operator, and M as the passive object on the receiving end -- ie so that between them ψ and ψ~ define a transformation, which has the effect on M of mapping it to N:

\psi M \tilde{\psi} = N\,

because, if you like,

\psi : M^{1/2} \rightarrow N^{1/2}

Now the first way is the point of view from which much of the article is written. (Indeed it seems to correspond to both of the approaches described in "Two basic approaches"!). It's also the way most of us probably first get conditioned to thinking about spinors via the wavefunction |ψ> in quantum mechanics.

The second way, with the spinor as the operator, is eg the way spinors are used to do rotations in computer algebra. It's the point of view now taken in the article by the sections on spinors in 2D and spinors in 3D. It's also possible to think of the wavefunction in this way, something that boosts an initial pure state |ej><ej| into a density matrix |ψ>|ej><ej|<ψ|, and any operator M into the corresponding matrix |ψ>M<ψ|.

Unless some sort of bridge is added, at the moment I fear there is a real dissonance between the different sections. Jheald 23:07, 26 March 2007 (UTC)


ψ is easier to see as an operator when it looks like a matrix. It is more difficult to think of ψ acting like an operator when it looks like a column.

When ψ looks like a column, the way it does in the last sections of the article, what is the hidden machinery in the formalism that allows it to act like an operator? Jheald 23:13, 26 March 2007 (UTC)

[edit] Aaagh

'My' lead section was completely redone, with physics buzzwords to excess. Charles Matthews 21:13, 18 April 2007 (UTC)

Ahh... I found your version of the intro in the edit history. Someone had gone in and mangled it, and I think I must have missed yours. Yours was very good. I'll see if I can merge it with the new one. As for the 'buzzwords', I don't see any. It would be shameful not to mention the physical applications of spinors, and the discussion is quite minimal. Silly rabbit 21:25, 18 April 2007 (UTC)

Thanks for the edits. The pain has gone. Charles Matthews 09:42, 20 April 2007 (UTC)

[edit] Aaagh from me, too

It's not just the lead section. There are whole paragraphs full of buzz words, such as 'polarization' used left and right without explaining anything.

Once upon a time, after Charles had gone over the article and cleaned it up, for a while, it was actually possible to find out what spinors *are*, and have a good view of the topic: first a short overview, then the details. Now a long 'motivation' section replaced the overview, the new overview is just a rambling. The old overview survives in the commented out form with a cryptic remark:

already covered somewhat above

I am sorry if it sounds a bit harsh, since it looks like someone invested a lot of effort in editing the article, but while certain parts have been somewhat improved, the overall quality of article has gone way down. I definitely appreciate the time and effort, but it has to lead us in the right direction, I think.

Some specific comments:

  • Sexy pictures of coffee cups are fine and dandy, but this is an article about spinors, not the fact that the special orthogonal group SO(3) isn't simply-connected. Pare down, or summarize and move out the bulk to a separate article.
  • Far from everyone can grasp the meaning of 'entanglement' (if indeed there is any precise meaning to it at all). This can be very annoying.
  • It's ridiculous to have to scroll down 3 pages to find a first approximation to a definition of a spinor.
  • In the positive-definite case, there are no isotropic subspaces, therefore, all talk about polarizations isn't helpful. And this is one of the most important cases for applications!
  • The new overview section is largely pseudo-scientific. Here is how it starts:
In higher dimensions, spinors can also be introduced as geometrical objects generalizing the notion of a vector in order to allow for the orientation entanglement of rotations. These spinors must allow for the fact that orientation entanglement occurs not just for vectors, but also for linear subspaces (such as lines, planes, and so on). Hence a spinor defines a sort of polarized multivector.[9]
There are several approaches for defining and describing the properties of spinors. In one approach, one looks closely at the notion of polarization. By factorizing each vector into isotropic components, we achieve a polarization of the space by consistently selecting one isotropic component of every vector. The spinors are formed by taking exterior products of this isotropic component of different vectors.[10] Usually, the exterior product can be thought of as a way to encode the subspace spanned by a collection of vectors, and hence spinors are (roughly speaking) polarized isotropic subspaces.[11]

This is (1) vague, (2) not particularly insightful (unless one happens to already possess suitable intuition), (3) mathematically meaningless, for the most part. For example, if spinors are geometrical objects, what can I do with them? Can I add them? Do they have length? How about angles? What does 'factorizing a vector' mean? How can a multivector be 'polarized'? And so on, and so forth. The next few paragraphs about things acting on each other are particularly unclear. In addition, some of the links, such as

polarized multivectors

are rather confusing (wrong link, in this case? I coudn't even be sure!).

Some suggestions:

  • Restore the basic structure of the article, with short summaries and detailed explanations elsewhere, at least for the first half.
  • State things more precisely, so that at least people who know mathematics, but may not know what spinors are or what they could be used for, could follow the flow of the article.
  • Try not to talk about many possible vague interpretations of something that has not even been defined (Ok, the last one is just out of utter frustration).
  • If you feel like you can say a lot about spinors using your favourite approach, then write a wikibook! As a reference for many other articles dealing with spinors, such as spin structure, it would be preferable to have a more precise and concise version of this article.

Arcfrk 01:22, 28 April 2007 (UTC)

Noted. The page has been reverted. Cheers. Silly rabbit 01:41, 28 April 2007 (UTC)
Ok, I've calmed down somewhat now, and can respond in more detail to your comment.
  • I'm not married to the idea of spinors being polarized multivectors. In direct response to your criticisms: Yes, it was the right link. No it isn't mathematical nonsense as you claim. Yes, I see what you mean about buzzwords. But that's all a moot point, since as you point out I probably should avoid talking about things which haven't yet been formally introduced (at least not until much later in the article).
  • When I returned to the article, it was in need of some global improvements. Firstly, about 1/3 of the article seemed to be of the opinion that spinors were elements of the Clifford algebra, rather than its representations. Further, the introduction had been mangled, and now restored and expanded per Charles Matthew's indication.
  • The article (even Charles Matthew's version) still failed to define spinors correctly. Apparently subsequent editors realized this, but incorporated their views in inappropriate places (such as the introduction).
  • Not a lot of motivation was presented, and I think that many readers would find a bit of elementary (sexy coffee cups) discussion helpful. Since most of the important content (largely written by myself) had been moved (by C.M.) out of the article, I was under the impression that he wanted this to be a light introduction to spinors with the real content contained elsewhere.
So, I'm sorry if I've made you angry. Let me assure you that my edits were done in good faith, with the honest intention of improving the article. Silly rabbit 03:13, 28 April 2007 (UTC)
I've made some changes to correspond approximately to your suggestions. The points you made were all good ones. You don't need to be antagonistic to get your point across. Silly rabbit 04:22, 28 April 2007 (UTC)
I've been told that my criticism, while intended to be constructive, often appears antagonistic. I should learn to express myself better, since my comments were also in good faith. Arcfrk 07:00, 28 April 2007 (UTC)

[edit] What is a spinor

To give some flavour of why definitions of spinors are often next to useless and at best highly confusing, consider the following from the current edit. I read: "a spinor must belong to a representation of the double cover of the rotation group SO(n,R)". So I think, aha, I know what a spinor is if I know what a representation is. Reading representation, I learn that a representation is a homomorphism from a group G to an automorphism group. So what that means is that a spinor is a group homomorphism?

No. That's doesn't sound right. Ah, the problem must be this phrase "belong to", what does "belonging to a representation" mean. No idea. No definition of *that* anywhere.

A similar problem occurs later in "In this view, a spinor is an element of the fundamental representation of the Clifford algebra Cℓn(C) over the complex numbers". What is an "element" of a representation?

When I was at college I was taught a representation was just as wikipedia defines it (a hom) but now I suspect that maybe we are talking about a module over the algebra and the spinor is a vector in that module. Is that right? Who can say?

I completely, 100% understand what a clifford algebra is -- and clifford alegbra is pretty clear and lucid. I *think* I know what a representation is, and I could do character theory at college. I feel that I ought to understand or be able to deduce what a spinor is. But I'm afraid this page leaves me as confused as ever. Will someone help?

By the way, its no use giving an example. I'm not sure if my efforts to generalise an example will work. Eg, I am quite familiar with (say) the first few chapters of spinors and space-time by Penrose and Rindler. Great fun, but no definition of what a spinor is in general. I can manipulate different kinds of objects (called spinors) when calculating interaction cross-sections from Feynman diagrams in QED. I have seen examples. What *exactly* am I looking at? Anyone care to help? Francis Davey 21:34, 24 July 2007 (UTC)

I'm going to restore Adam's previous answer, which I thought was also quite revealing: -- Jheald (talk) 08:18, 18 December 2007 (UTC)
Good question, Francis, and I hope to be able to answer you (and the rest of the WP community) in a helpful manner, but first I'd like to ask what is meant by the question "What is an X?" I'll take as an example "What is a vector?"
Vague: A vector is an object with magnitude and direction. Good for intuition, persuades you that they are useful things, certainly that they are "independent of basis", whatever that means at this point, but not much use for maths. Actually, you can formalise this and use it to do maths, and Euclid got remarkably far, but it's painful.
Example: This arrow thingy is a vector, and so is this 3d arrow, and also this function from the reals to the reals, in a weird infinite dimensional way. Often helpful if done in the right way. Rarely definitive enough to be useful as the only definition.
Representation: A vector is an n-tuple of reals (or maybe members of some field K). The physicist might add that the components transform in a certain way under "change of basis", which is painful to a mathematician because a basis is a set of vectors.
Intrinsic: If you add two vectors, you get a vector, and if you multiply a vector by an element of the field you get a vector. The computer science term for this kind of definition is duck typing. This is based on the idea that these seem to be the only properties of vectors that you ever really need, so anything which also quacks like this we'll agree to call a vector as well. If we needed anything else, we'd be quite happy to add it in. See Hilbert space for example. Notice that this actually define a vector space, and a vector is just an element of this.
I think the problem is that we have a vague definition - lots of them in fact, depending on how much intuition you derive from words like "quantum" and "polarised" - and we have two different representations, but we have no intrinsic definition. The two representations are (a) an element of Spin(p,q) which is a subset of a clifford algebra and (b) a member of a set of explicit matrices and rules for manipulating them which are derived from specific complex representations of specific spin groups, and totally rely on the classification of real and complex clifford algebras, or at least the representations of Spin(1,3). The fact that there are 2 competing representations and no consensus means that some people (mostly mathematicians) think that a spinor 'is' (a) and some people (mostly physicists) think that a spinor 'is' (b). Also, for spinors, tensors, and many other mathematical objects, there's another kind of definition:
Formal: A spinor is an element of the connected double cover of SO(n) for n at least 2. The classic example here is that a tensor space is the universal object for bilinear maps. At least the spinor formal definition is constructive, whereas the tensor isn't even that.
These are all useful types of definition to have in an encyclopedia, and we should have all of them for the important subjects, including spinors.
So, what are you looking for?
Adam1729 10:53, 24 September 2007 (UTC)
I've just realised that there's another and more important confusion with spinors, so I've removed my previous reply. If anyone objects to me editting history in this way, I can easily put it back.
Maybe the natural mathematical definition is an element of Spin(p,q), but the natural physical definition is not the image of this element under some representation, it's a vector from the vector space on which the representation acts. This really is a vector in the mathematician's sense, but we need a new word, because it's not like the (p+q)-dimensional space-time vectors. For SO(p,q), there's a really obvious representation to use, so vectors are the obvious thing. For Spin(p,q), this is not true, so we end up with a variety of different things being called spinors. The thing that they have in common is that they are elements of vector spaces on which a representation of Spin(p,q) acts. The word 'spinor' is not strictly meaningful to apply to a single element, but it implies a set in which the spinor lives. This set is a normed vector space, and it has an extra property, namely that there is an (irreducible? faithful?) action of Spin(p,q) (for a particular p,q of interest) on this space.
Does this sentence (in the article) already cover it?
One can remove this sign ambiguity by regarding the space of spinors as a (linear) group representation of the spin group Spin(n).

Adam1729 10:51, 25 September 2007 (UTC)

[edit] Lack of definition

This article lacks easily findable definition (which should be backed with motivation, informal explanation, etc...).

Somebody fix this, please! Thank you!--83.131.0.38 (talk) 14:37, 26 November 2007 (UTC)

The article more lacks a simple understandable definition. This from the lead: "spinors can be defined as geometrical objects constructed from a given vector space endowed with a quadratic form by means of an algebraic[1] or quantization[2] procedure". is a pretty good definition. It's just not so simple, but I don't quite know what to do about that. Martijn Hoekstra (talk) 08:23, 18 December 2007 (UTC)
I disagree that the above sentence is a good definition as it doesn't say how to define a spinor but just that is can be defined in some way.
Shouldn't the definition be something like: A vector of an vectorspace V is called a Spinor, if there is an irreducible representation (rho, V) of a Spin-group. Meaning, like a position vector is a vector of a vectorspace V, that is part of a representation of SO(n), a Spinor is a vector of a vectorspace representing this further information encoded in the larger group Spin(n). -- JanCK (talk) 10:54, 18 December 2007 (UTC)
Yes, and this definition is already (effectively) in the article:
These missing representations are then labeled the spin representations, and their constituents spinors. In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n,R), or more generally of the generalized special orthogonal group SO(p, q,R) on spaces with metric signature (p,q).
There are also two different explicit constructions of the spin representations, which could be called "definitions" in the sense that they refer to more or less concrete things. There is also a definition using Clifford algebras. I believe that one difficulty is that it is very difficult to define spinors in complete generality without bringing in a whole bunch of other things in the process. Either you need to be willing to accept the existence of certain representations of a simply connected double-cover of the special orthogonal group, or you need to be willing to get your hands dirty with Clifford algebras. The article currently follows a "pyramid" design: first starting out with vague statements which should (hopefully) be at least understandable to most people, and then ending with the details that make everything work, with various levels of detail in between. The grandparent poster was, apparently, not willing to go past the first few paragraphs in the search for a suitable definition. 72.95.241.69 (talk) 16:08, 22 December 2007 (UTC)