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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)

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

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)

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 $\pi ^{0}$ is self-conjugate is because its $({\overline {u}}u-{\overline {d}}d)/2$ 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)

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)

And I fourth! I still don't have a clue what a spinor is, or why it's related to the recently discovered/found Majorana particle.JustinCase27 (talk) 20:06, 9 May 2012 (UTC)

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)

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)

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)

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)

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)

---

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).

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.

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!

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.

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)

Assessment comment

The comment(s) below were originally left at Talk:Spinor/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

The Clebsch-Gordan stuff is probably too detailed for this article: suggest summary style. One or two sections are a bit stubby (e.g. the history) and could be expanded. Geometry guy 16:08, 7 September 2008 (UTC)

Last edited at 16:08, 7 September 2008 (UTC). Substituted at 22:05, 3 May 2016 (UTC)