Talk:Spinor/Archive 5

Lead edits
To me, this edit was a massive step backwards in quality. It is not well-written, and fails to convey any intuitive sense for the concept. It is also wrong. Spinors don't transform under individual rotations, only one-parameter families of rotations ("continuous rotations"). In the misguided attempt to instill mathematical rigour in the first paragraph, the whole point of spinors has been removed! Sławomir Biały (talk) 16:03, 21 September 2014 (UTC)


 * The current version has improved slightly over my last objection. The added content is mostly good, but I think that the lead structure needs to be revisited. One thing I do like is the discussion of the ambiguities and ways in which the spin representation is constructed.  But I think now the first paragraph tries to do too much, and we should try to form some agreement on what overall structure the lead should have.


 * In my conception, the first paragraph is intended to convey an intuitive idea for what a spinor is. This paragraph is intended to be read by someone who has limited knowledge of mathematics.  The general rule is "one level down".  Spinors are introduced in most university physics curricula, so in my view the first paragraph should be readable possibly by someone who is mathematically literate at an (advanced) high-school level.  In particular, my intention was to have the first paragraph as free as possible from language like "projective", "vector space", and "representation".  With the exception possibly of "vector space", terms like these are not usually encountered in fact until considerably after most people have already learned a little about spinors.  So I think having the first paragraph parasitic on "more advanced" ideas is rather a wrong-headed approach.


 * The second paragraph was intended to introduce a few more of the details of what is meant by "spinor". A mathematically literate reader will at this point want to know that spinors form a vector space.  Some who are comfortable with the idea of a representation of a group should certainly be told that they are representations.  Clifford algebras should also be mentioned.


 * I had not thought much about the last paragraph. And it remains the domain of miscellany.


 * So, in light of these I suggest that the lead can be restructured so that the first paragraph is again purged of technical detail. The second paragraph can contain a few of the important concepts, like vector spaces and representations, but still be a discussion on a fairly conversational level.  Here it can be mentioned also that the "rotations" can be Lorentz boosts, and perhaps the word "metric" can also be used somewhere.  The third paragraph can contain a few details of the construction, discussing Clifford algebras and Pauli spin matrices, as well as the Dirac equation.   Sławomir Biały  (talk) 21:02, 21 September 2014 (UTC)


 * Having more or less implemented this change, it now seems like the third paragraph really goes into too much detail. I think, modulo keeping two things from it in the lead more or less where they are, the entire paragraph could serve much more usefully as the first paragraph of the Overview section.  Here, we start getting into details.  The two things it says that should be kept in the lead, I think, are (1) that spinors are not naturally constructable from vectors, (2) the mention of the Dirac equation (although I would mention the fundamental nature of this in physics as well).  But it does not take a separate paragraph detailing the constructions to do this.  It would be nice to get some outside thoughts on the changes thus far.   Sławomir Biały  (talk) 21:17, 21 September 2014 (UTC)


 * Looking only at the first paragraph, the first obstacle I see is the phrase "in a certain way". This kind of placeholder requires quite an amount of mental energy to parse due to its unresolved (abstract and uninformative) nature and the way the brain works.
 * The reference "transforms" also seems to make implicit reference to a representation; as I understand it such a characterization will not work even for the coordinate-free definition of a tensor (let alone for spinors), and so either explicit reference to what (i.e. the representation with respect to some basis) is transforming is needed, or a definition that does not rely upon transformation should be sought.
 * It may need working on the first paragraph a bit to fill in what I see as gaps in the characterization of what constitutes a spinor. It still comes across to me as though one is needs to understand fundamentally what a spinor is to make sense of the lead, which in a way suggests that the lead does not have it right. I, for example, cannot distinguish between rotors of GA (as a space of objects being transformed by rotations as described in the lead, not as formulations of objects that act on others to represent a rotation) from spinors. As a linear representation, rotors (or at possibly the identity components of the manifold of rotors) seem to give me a more solid basis for intuition than I can get from the article. It would be nice if I could at least determine from the lead whether this intuition of mine is bollocks. —Quondum 22:40, 21 September 2014 (UTC)
 * Does this change address that point adequately?  Sławomir Biały  (talk) 00:09, 22 September 2014 (UTC)
 * It addresses my first point. My second point still largely stands though: the mention of a frame of reference tells me that you are dealing with a definition of tensors and spinors that is in terms of a frame of reference, but it would be nice if the type of definition upon which this wording relies were made explicit, since this is not the only way of defining tensors. Perhaps I'm being a bit picky, but in a subject like this, being able to approach it from a more mathematically abstract direction, or at least an indication that it can be defined in a more abstract (and to me cleaner) way would make sense. —Quondum 00:59, 22 September 2014 (UTC)
 * Thinking about it a bit, would it make sense to make the following wording change, or something like it?:
 * Spinors are vector-like objects in geometry and physics that, like vectors and other tensors, may be represented by components that transform linearly when the system containing them is subjected to a continuous rotation of its frame of reference.
 * —Quondum 05:03, 22 September 2014 (UTC)
 * Unfortunately that gives the wrong impression. See below. RogierBrussee (talk) 15:32, 26 September 2014 (UTC)

There have been a few changes since last I posted here for feedback (here is the current version). One of the participants in those edits seems to have foresworn talk pages, but I think it is time to solicit another round of input. I have attempted to address Quondum's last comment. I have also included a better link to what is meant by "vector" in the first paragraph. The first paragraph also now mentions "vector spaces" in a way that will hopefully not be too confusing to readers. Also, I have left the third paragraph as it is. I still wonder if it is too detailed and technical for the lead, but I think there are some important points in there that should be kept. What remains to be done? Sławomir Biały (talk) 13:06, 23 September 2014 (UTC)


 * I'm focusing on only one point at a time to avoid confusion. The lead has the statement
 * Spinors can be associated, as an auxiliary mathematical object, to any vector space equipped with a quadratic form
 * and the statement
 * Spinors cannot be constructed from just vectors without making certain choices or using some kind of extra structure.
 * The first makes sense to me, but the latter does not and feels like it contradicts the former: it sounds a bit like saying "tensors cannot be constructed without choosing a basis". Obviously the space of spinors is an auxiliary space, but this does not seem to be what is meant by "some kind of extra structure". Perhaps the word "cannot" does not belong? I would have thought that given a vector space with a nondegenerate quadratic form, there would naturally be an associated space of spinors, just as there is an associated space of bivectors. —Quondum 05:18, 24 September 2014 (UTC)
 * I have tried out a new wording.  Sławomir Biały  (talk) 12:04, 24 September 2014 (UTC)
 * As a suggestion for this, I removed the sentence that seems to say something that does not need saying, or else I'm misunderstanding what it is trying to communicate. Construction, in some component form, always needs choices like this, and we do not need to make it sound like spinors might be more special in this regard. —Quondum 13:38, 24 September 2014 (UTC)
 * I've tried again. What I am trying to communicate is that there is no actual construction of spinors using only the data available from vectors.  That is, if you ask me to "construct a spinor" (that is, to construct something that transforms like a spinor under continuous rotations), then by necessity I am forced to introduce some extra structure in order to exhibit such a thing at all.  We know from abstract nonsense that they exist, but we cannot actually explicitly construct them.  The space of spinors is something that exists only up to isomorphism, in some sense.  There are no distinguished representatives of the isomorphism class that are "natural".   Sławomir Biały  (talk) 14:12, 24 September 2014 (UTC)
 * Okay, so the crux of what you are trying to communicate seems to be what you say here: that some necessarily arbitrary choice is forced in construction of the space of spinors, and it is worth trying to make this thought clearer in the article. Does this include the choice of replacing every spinor with its negative (like being able to replace $i$ with $−i$ without any effect)?  My intuition is that the choice of Cartesian coordinates itself does not make this choice, leaving it still to be made, e.g. via the choice of association of axes with some elements of the representation (Pauli matrices).  It would be nice if we could capture what the arbitrary choice actually chooses, so that one has a sense of its degree of freedom (your mention of association with orthonormal axes suggests that this would be the same as for $O(n)$, which I think is $n(n − 1)/2)$ real degrees, plus I suppose some "gauge freedoms"), which would relate to the chosen representation.  But since you mention up to isomorphism, would it be possible to construct a spinor modulo this arbitrary choice at an abstract level, thus making it natural?
 * I suppose I'm trying to distinguish between a spinor as a concrete representation and a spinor as a core abstract object that embodies the necessary properties. This might not be standard, e.g. the term spinor might only apply to the former. —Quondum 16:30, 24 September 2014 (UTC)

There is something here called a moduli space of different spin representations. It turns out that the spin representations of SO(Q) can be put into a natural correspondence with the maximal isotropic subspaces for the quadratic form Q. So this moduli space is an algebraic variety. (In fact, it has dimension $$n(3n-1)/2$$ at least when Q is in odd dimensions 2n+1; even dimensions is a similar story but with half-spin nuances.) This variety is of a very special kind, a generalized flag variety, roughly an algebraic variety equipped with an action of the group SO(Q), and such things have many nice properties. For instance, there is a tautological vector bundle on the moduli space whose fiber over a point is just the total space of that spin representation. No choices beyond the data of the quadratic form are required to construct this moduli space and bundle. The choice is precisely a choice of point in the moduli space, and the associated spin representation is the fiber of this vector bundle over that point. All choices are on equal footing in this construction.

TLDR. The bottom line is that this construction shows that the set of all spin representation is a $$n(3n-1)/2$$ parameter set, with no distinguished elements! (E.g., in three dimensions, 3=2n+1 gives n=1, and hence $$n(3n-1)/2=1$$, so there is a 1-parameter family of different spin representations in three dimensions.)  Sławomir Biały  (talk) 21:52, 24 September 2014 (UTC)


 * That paragraph reads very naturally to the novice now. And thanks for the details here – this gives me a lot to review and consider, and answers quite a bit of my uncertainty. —Quondum 02:59, 25 September 2014 (UTC)
 * Spinors form in a very fundamental way auxillary additional vector spaces. The basis of spinor space cannot in a natural way be constructed in a natural way from those of ordinary space, unlike tensors of a vectorspace which automatically inherit a basis once you choose a basis of the vectorspace. Put another way, tensors are functorial: given a linear map V \to W there is a linear map V\tensor ...\tensor V \to W\tensor ... \tensor W (same number of tensors) and W^\dual \to V^\dual.  Spinors are not in this sense functorial even if they preserve some quadratic form and even if you have chosen a specific concrete representation of spinor space  as the sign ambiguity shows. It is exactly what Slawomir says, "the" spin representation space exists only up to isomorphism: there is only a irreducible representation &Delta; of the Clifford algebra which is unique _up to isomorphism_. None of those representations are prefered, (which is why the article talks about a space of spinors, at least the parts I wrote) although the abstract representation that involves choosing an isotropic subspace is a bit more elegant than others and indeed is parametrised by an algebraic variety  (in even dimensions, it is the space of maximal isotropic subspaces which is in an obvious way is a subvariety of the Grassmannian of half dimensional subspaces). But even the isomorphism is _not_ unique. Over the complex numbers  the space of intertwiners between two irreducible representations is  1 dimensional. Thus the  isomorphisms of the representation is the space of non zero complex numbers.


 * You can get a better feeling for the problems if you try to do define spinors for Riemannian or Lorentzian manifolds. Given the metric, you can now define a Clifford algebra bundle by just doing the usual abstract construction of the Clifford algebra fibre by fibre (i.e. Cl(M, g) = Tensor^* (TM) / (ideal generated by v \tensor v - g(v, v)). Locally you can now construct a spinor bundle by taking an orthogonal frame (aka Vierbein) and your favorite choice of Gamma matrices. However there is no natural way to glue these locally defined spinor bundles. You cannot, however, always construct a spinor bundle, and if you can do it, there are several different possibilties (so called Spin of Spin^c structure depending on whether you want to lift the frame bundle to the spin group  or just want to define spinors with a multiplication of Cl(M,g)).

RogierBrussee (talk) 15:32, 26 September 2014 (UTC)

Convenient break

 * Although I can see that there are valuable elements in the work that Slawomir has put in and I wanted to wait till he could finish it (not to mention had work to do) I must admit that I still don't like the lead.
 * The first sentence really does not give any idea of what Spinors are good for. This is a real problem because even the name suggests they have something to do with spin. However you have to read all the way down to find that out.
 * The use of vector-like suggests that spinors are somehow like tensors. See how Quondum interpreted that statement. That is just not true.
 * The concept "continuous rotation" is absolutely non standard and and I find it confusing. The explanation of the animated gif doesn't make it any clearer either. Yes I know you want to say homotopy class of path in the rotations, but you don't, and for the good reason that it would lead too far in the lead.  I know I also tried to phrase things in these terms, but the more I think about it the more I am of the  opinion it is a dead end. IMNHO the right starting point are infinitesimal rotations, that is Lie algebra representation. Slawomir is of the opinion that somehow misses the point, but I disagree: infinitessimal rotations/Lie algebras are standard and it is a  honest and simple way of introducing the subject.  It is certainly desirable to  mention the difficulty you encounter when you exponentiate them to honest representations of the group (i.e. the sign ambiguity corresponding to non homotopic paths), but it not so easy to do without either saying something wrong or making it complicated,  and maybe  it is even better left to the mathematical literate part and the main article.
 * The Pauli matrices don't form an algebra, they generate one (in fact, the whole algebra of 2 times 2 matrices) together with a representation of the Clifford algebra Cl(2, C).. SU(2) is not contained in the Pauli matrices but in the algebra generated by them. Takes more effort to see that SU(2) is the covering group of SO(3).
 * I don't like the animated gif. It would be a fine addition to the belt trick  and/or  Orientation entanglement  articles or perhaps even to  the Spinor section but in the leader it is just highly distracting.


 * RogierBrussee (talk) 15:32, 26 September 2014 (UTC)


 * I've made a few edits along the lines of your suggestion. I disagree very strongly with the proposal of cleansing the lead of the geometrical details intended to explain what a spinor is to someone with no mathematical background, which seems to be the overall tenor of these comments.  The approach via Lie algebras, while certainly more suitable for understanding spinors in a deep way, are manifestly unsuitable for explaining spinors to someone with just a basic mathematical background.  I really don't think this point needs any elaboration.
 * I would like to add that your contention about continuous rotations being "absolutely non standard" is just wrong. A mathematician might call it an element of the obvious path groupoid, but such things are one of the basic tools in any discussion of any covering space.  That's about as standard as it gets.  It also has the virtue of something that can be explained to just about any one using (relatively) common everyday expressions in essentially their everyday meanings.  But, with your feeling that the term was confusing, I have tried to make it a bit clearer what I am getting at.   Sławomir Biały  (talk) 21:43, 26 September 2014 (UTC)
 * There are so many differences between the two versions that it is difficult to know where to start. Rogier's introductory sentences are easier to follow, but has aspects that I still find problematic.
 * -- Which ones? RogierBrussee (talk) 07:13, 27 September 2014 (UTC)
 * I see that your intro sentences were merely a rearrangement of what was there, so what I labelled problematic aspects are not specific to your edit. see the comment that I just made below for one. —Quondum 15:23, 27 September 2014 (UTC)
 * In general, both versions seem to gloss over the non-subtle differences in transformation properties of tensors of different orders; this is partly what confuses me. If I am to contribute constructively here (as a guinea pig layman), I'm probably going to have to express my understanding and perception of each statement, and then you can maybe try to synthesize something from the pieces that feel natural and understandable to me.  So far this approach has only led to incremental changes, with a lot not really dealt with, but I'm not sure what to suggest that might be reasonably efficient. —Quondum 05:03, 27 September 2014 (UTC)
 * The leader has much improved, but I still think continuous rotations are confusing as hell. Rotations are linear transformations and so always continuous (at least in finite dimensions) I know you dont mean this. In laymans terms a continuous rotation is a rotation that does not stop, I know you don't mean that either. It


 * Topologists call this is called a one parameter family or a path of rotations. Spinors (or rather their components with respect to  a frame of reference AND a fixed choice of gamma matrices) transform well under an infinitessimal  or small rotation, but if you try to integrate this transformation along a one parameter family  (i.e in a continuous process starting from the identy) you can get two different answers differing by a sign if the two paths are not homotopic.  See, no deep Lie algebra theory, just using the bog standard notion of infinitesimal rotation found in virtually every physics textbook, and quite understandable for everyone with a bit of math literacy. Anyway I am away all day.  RogierBrussee (talk) 07:13, 27 September 2014 (UTC)


 * The chief objection here seems to be to the term "continuous rotation". Although we do explain what is meant by this, you seem to feel that the explanation is inadequate, and that there is still some residual confusion over the use of the term "continuous" in this context.  Would calling it a "gradual rotation" resolve this objection?  This also conveys the right idea to the layman, and does not use a mathematically familiar word in an unfamiliar way.  I have implemented this change.   Sławomir Biały  (talk) 11:51, 27 September 2014 (UTC)

In light of the recent discussions here, I have tried out a new configuration of the lead that I'm hoping everyone likes. There is now a new "Introduction" section with scope for expansion. The animation has also been moved there. Ideally, I see it consuming the existing "Overview" section, which is not very good as it currently stands although parts of that section are useful as an outline. The general body of the article needs a good deal of work, since although it does cover many important things, it does none of them well, but that's going to be a long row to hoe. Sławomir Biały (talk) 14:35, 27 September 2014 (UTC)


 * If I can make an analogy, it feels to me like someone is explaining what an insect is: "An insect is an organism with a faceted eye", and then showing an enlargement of a facet of an ant's eye.
 * Switching to active rotations from coordinate transformations for clarity of expression, it seem to me that, for example, spinors encode everything about a rotation. By which I mean that, given two spinors, the second being the result the action of a rotation on the first, one can determine the action of that rotation on every tensor and on every spinor. Not so? This is not a property of tensors. The homotopy class is one of the further aspects that are encoded. —Quondum 15:19, 27 September 2014 (UTC)
 * I see that it says Spinors [...] provide a linear representation of the group of rotations in a space with any number ... of dimensions .... Aside from that this description omits any mention of the homotopy, and might be a little inaccurate (they evidently don't directly represent rotations, and they don't have a group operation between them, as such), it is presumably sufficiently accurate to describe them as: Spinors are objects (elements of a vector space upon which the double cover of the group of rotations acts) that track the an overall rotation as well as the homotopy. Such a description might be an approximation an intuitive layman's introductory sentence (needs work, but it relates to geometric concepts). The homotopy aspect needs to be expanded on, which is already there, but what I'm trying to say is that we should have a solid description before expanding on the plate trick property.  From my perspective, it adds what gets lost when focusing only on the binary homotopy property too early, and grounds the picture in familiar concepts: rotations, vector spaces, group actions. (I know I've rephrased it in terms of active rotations, not coordinate rotations, but this is more intuitive.) —Quondum 20:19, 27 September 2014 (UTC)


 * How about now?  Sławomir Biały  (talk) 21:18, 27 September 2014 (UTC)


 * There are some things that I guess are a matter of taste. The first paragraph is reasonably interpretable, so here are a few detail comments:
 * The phrase "vector-like" could be dropped; it does not tell me anything that the next part ("like geometric vectors and other tensors") does not already tell me. Lorentzian bivectors are vector-like in that bivectors can be represented as a pair of 3-vectors in a spacelike section of four-space, but it would be meaningless to call them vector-like without this additional information. I'm guessing that a spinor can be associated with a vector in a similar way, but where the spinor changes sign with a 2π rotation, the associated (spin) vector is unchanged in direction. The rest of the first sentence seems fine (though I might quibble with the phrase "objects in geometry and physics" – it might be better as "In geometry and physics, spinors are objects").
 * You clearly would like to stay with the rotation of frame of reference approach (we each have our own preferences ;-) ). Nothing wrong with this, but it does not give as much purchase to the layman's mind, especially someone who "knows" that an electron an be represented as a spinor field, which can have its spin axis (both its angular momentum and its magnetic moment) rotated through 2π in a magnetic field, whereupon it will interfere destructively with its unrotated self.  There is no rotation of the frame of reference in this picture.
 * "over the complex numbers" – Is this necessary? In physics (and in particular, quantum mechanics), this is generally how they are regarded. I expect that spinors can be defined over other fields. I'm not convinced that the complex structure is natural, and people using real Clifford algebras might take issue with this description.
 * In most dimensions and signatures there are no real spinors (aka Majorana spnors). If you think about the construction of choosing a maximally isotropic subspace you will appreciate that this does not work over the reals and Euclidean signature. You need an agebraically closed field. RogierBrussee (talk) 08:35, 28 September 2014 (UTC)


 * "this seemingly paradoxical property" – nothing is gained by using a phrase like this. The space (or manifold) of rotations is not simply connected with all the implications that this produces. It is a simple geometric fact, albeit surprising. Spinors do not introduce this, but merely serve as good way to encode this. We should be demystifying this, not the reverse. Spinors are not vectors (in the sense of belonging to the space of 1-vectors), but then neither are nonvector tensors.
 * Seconded RogierBrussee (talk) 08:35, 28 September 2014 (UTC)
 * "encoded by a representation of the spin group" – while this looks good, the wording does not make clear the connection between the spin group action and the spinor (I think what is missing is that the action is faithful), expressed by me above as the spinor tracking the action. The spin group is also being introduced within a sentence that uses it.  It might make sense to start a new sentence to describe spin group description, and then to relate this to its action to spinors (probably a minor reordering of phrases).
 * It now says the representation does not factor and so is not trivial.RogierBrussee (talk) 08:35, 28 September 2014 (UTC)
 * "discovered that spinors were an essential part of the physical universe because they describe the intrinsic angular momentum" – Is this accurate? AFAIK, Dirac discovered a vector-space field quantity that worked in a partial differential equation to describe the behavior of the electron, and this quantity come to be known as (or merely identified as?) a spinor. Spinors are classical, and we should not create the impression that they arose or belong exclusively with quantum mechanics. Would it not make more sense to say that fermions cannot adequately be described without spinors, and leave angular momentum out of this statement?
 * —Quondum 23:26, 27 September 2014 (UTC)
 * They really are fundamentally part of the quantum description. What Dirac did was introduce the gamma matrices (or the alpha and beta matrices) which satisfy the canonical anticommutation relations and so generate a representation of the Clifford algebra. I actually think that Paui's work to explain the Zeeman and Goudsmid's work on the Zeeman effect in the non relativistic approximation (hence rank 2 spinors) predates the relativistic Dirac equationRogierBrussee (talk) 08:35, 28 September 2014 (UTC)

Ok, sorry about all of the edits (I have the flu and have been doing Wikipedia in a semi-addled state, avoiding my real work). I have tried now to say, as clearly and concisely as possible, what a spinor actually is in the first paragraph of the lead. Sławomir Biały (talk) 00:46, 28 September 2014 (UTC)


 * The first sentences work pretty well now:
 * In geometry and physics, spinors are objects that form a vector space, usually over the complex numbers, that are needed to encode basic information about the topology of the group of all rotations. The group of rotations is not simply connected.  There is a (simply connected) group known as the spin group that doubly covers the group of rotations, meaning that for every rotation there are two elements of the spin group that represent it.
 * It serves to give a framework (In ..., a vector space, purpose) in the first sentence into which to slot the coming concepts. I'm tempted to suggest moving the first mention of the spin group into the first sentence to pigeonhole the mechanism (as the action of the spin group):
 * In geometry and physics, spinors are objects that form a vector space, usually over the complex numbers, that are needed to encode basic information about the topology of the group of rotations through the action of the spin group. The group of rotations is not simply connected, but the simply connected spin group is its double cover, meaning that for every rotation there are two elements of the spin group that represent it.
 * My suggestion here does not yet connect why a simply connected covering group is needed (i.e. connecting this property to the encoding of the topology), which would be nice to squeeze into the second sentence. This would segue into the explanation of homotopy in the next sentences, which still need some work to avoid losing the readers.  I like the final sentences of the paragraph (... there is no "natural" construction ... and A notion of spinors can be associated ...) as they add helpful information and context.  I would delete the parenthetical comment as such an auxiliary mathematical object.
 * I would want to keep the start and the end of the first paragraph without much change as giving an easy-to-understand nutshell, but the middle will need work as it uses words (homotopy class, group representation) that can't be glossed over to get a first understanding of what is being said; they should not be avoided, but this part must be shortened and made more accessible. I would also want to sneak in a mention somewhere that spinors encode everything about the action of the spin group (not only the motivating topological detail), thus doing what you suggested above: more completely what a spinor is, rather than focusing primarily on why we use it. —Quondum 03:36, 28 September 2014 (UTC)

Length
One other thing, which may not be the most positive comment to make, and seems unfair given all the good work and discussion being made, but probably needs to be said anyway, is that IMO the lead is now too long. On my widescreen laptop, it really shouldn't be more than about four paragraphs of 6 lines; but paragraph 2 is about three times that length, and IMO that makes the lead as a whole too intimidating.

It's maybe a premature comment, while we're still trying to work out exactly what line we want to take, and how to express it. Which is why I've held off from making a comment like this until now. But a typical way forward would be to try to make the lead more terse, to give a quite short tight overview of what the article will contain, while moving some of the more explanatory material into something like an Introduction section. Jheald (talk) 07:32, 27 September 2014 (UTC)


 * Something like this maybe? Is that good?   Sławomir Biały  (talk) 12:34, 27 September 2014 (UTC)


 * Seconded. not sure I know which version Quondam and heald were talking about but I personally think that in the rightful push for clarity and precision we have thrown out the child with the bathwater, and make the first sentence too high brow.  Overall the section has been improving, so thanks for Slawomir for that. However, I also just read the introduction as it was on August 14. It is a pity of all the hard work but, and not that it cannot be improved but I do think that the  August 14 version is easier to understand than what we have now for casual reader and contains a lot of good information for the mathematically literate.  It probably just needs a sentence or two that Spinors are representation of the spin group which we can think of as homotopy classes of  continuous one parameter  families of rotations, that genuinely depend on the homotopy class and so do not descend to the rotation group (for which the new picture is quite good). Perhaps one can also use some of the material we have now on the Clifford algebra and probably a few other lessons learned from the current edit bonanza  but it did have about the right size. Oh and Slawomir best wishes for getting well soon.  RogierBrussee (talk) 08:35, 28 September 2014 (UTC)

Well, I feel that I've been getting pulled in different ways here. In addition to the August 14 revision, as well as the current revision, I would encourage people to look at this revision which I think was something of a highlight, before I tried out the new "high brow" in response to Quondum's apparent request for a clearer precise description of spinors.

Also, I'm not so sure the first paragraph of the old revision was any less high brow than the current one. The first sentence in particular is already very difficult to read because of all of the subordinate clauses. But it also fails to explain what a "linear representation" is, and I think this is part of the confusion of a number of readers of the article and something that I had sought to address. It seems more natural to me to think of things at the group level, rather than infinitesimally, especially for the purposes of exposition. Sławomir Biały (talk) 12:30, 28 September 2014 (UTC)


 * Hi Slawomir, I sympathise with your feeling of being pulled in different directions. I agree that the https://en.wikipedia.org/w/index.php?title=Spinor&oldid=627391743 current revision] is actually quite decent, it is just overly verbal and leaves out a lot of useful information in trying to get to grips with the topological description of things. Maybe you should just leave things alone for a while and get some rest.  OK?  RogierBrussee (talk) 15:50, 28 September 2014 (UTC)


 * Ok, I'll leave it in your hands. Things are not too bad for me as they are right now, actually.  I am not enthusiastic about the current lead, but I think that now most of the good stuff is in the new "Introduction" section.  A lead that is brief and clear to all likely readers is obviously an important goal, and I still have the sense that we haven't reached that in any version.  The demands of brevity and of accessibility are in conflict, I think.  But I'll let things as they stand for now, awaiting further input.   Sławomir Biały  (talk) 16:05, 28 September 2014 (UTC)
 * yes, brevity and precission are in conflict here and it should cater for different audiences. I will give it a try anyway. I will let it be known when I think things are good. Thanks RogierBrussee (talk) 17:52, 28 September 2014 (UTC)

Is this right?
The lead at some point currently states:
 * "The spin group is defined as the group of homotopy classes of such paths in the rotation group..."

I may be reading this wrong, but to me this sentence seems to say that the spin group is the first homotopy group of the rotation group. However $$\pi_1(SO(3)) = Z_2 \neq Spin(3).$$. Again I may be reading this sentence wrong.TR 15:58, 30 September 2014 (UTC)


 * The fundamental group is the group of homotopy classes of loops based at the identity, that is paths starting and ending at the identity. The spin group is the group of homotopy classes of paths starting at the identity and ending at the same final overall rotation.  (These are called relative homotopies: homotopies with fixed endpoints.)  It is true that the fundamental group sits inside the spin group when the final rotation coincides with the identity. This is just the kernel of the double cover, as mentioned in an nb.  Sławomir Biały  (talk) 16:22, 30 September 2014 (UTC)
 * I used this particular description (or something very similar) of $SL(2, C)$ in Representation theory of the Lorentz group. YohanN7 (talk) 17:01, 30 September 2014 (UTC)
 * Slawomir is correct but I changed the wording to make this more explicit because it is easy to get confused. I had to read twice as well. By the way, desepite all the sniping I do like the way things are progressing now. RogierBrussee (talk) 08:27, 1 October 2014 (UTC)

The ambiguity
The last part of the lead suggests in a way that I don't understand that there is an extra measure of ambiguity due to "arbitrary choices" that is supposedly not there for ordinary representations. This presumably leads to an equivalence class of representations (this I would understand). What is the difference as compared to "ordinary" representations? You can change basis there too at will, and you don't have a fixed set of matrices until you have fixed a basis. That an arbitrary choice of gamma matrices would add "more" ambiguity is, I guess, a mirage. You might pass from uncountably many equivalent representations to uncountably many squared (= uncountably many). They are all equivalent in that they are related by similarity transformations (after choices are made). It appears to be business as usual. (The choice of gammas is of practical importance of course, for instance Weinberg's choice in his QFT series is designed for the ultra-relativistic limit.)

The special thing, AFAIK, is that spin representations don't occur naturally buried in the tensor algebra of the standard representation (regardless of basis), at least not without some extra structure. The standard representation is, I suspect, implicitly referred to as the vector representation, and - worse - as "vector". This causes confusion, but also explains why some people have been disgusted at the thought of calling spinors vectors.

I may be wrong, but I think this part of the lead needs a citation. At least, point me in the direction of a reference. I'll need to read up YohanN7 (talk) 07:14, 2 October 2014 (UTC)

Ok, I understand now, but I think the formulation is decidedly cryptic. The (a posteriori) fact that extra arbitrary choices need to be made does probably imply that the spin reps could not have been found naturally in the tensor algebra of the standard rep (and it should be formulated that way instead of messing around with "vectors"). YohanN7 (talk) 07:51, 2 October 2014 (UTC)

Terminology
Let me first say that I think the present version (upper half of the article) is a good read for me personally. However, there are potential problems left, in particular problems that have been discussed at unusual length above.

One particular such problem was whether the term "representation" should be used very far up. Now it is used far up. And it is used with the deepest form of the ubiquitous terminological abuse:
 * The space of spinors is a linear representation of the spin group &hellip;

Firstly, it is not standard to include "linear" and, secondly, the confusion between a representation proper (a homomorphism) and its representation space (a vector space of suitable dimension) is present from the outset without comment. This confusion prevails throughout the article. YohanN7 (talk) 18:15, 1 October 2014 (UTC)

Suggestion:
 * A space of spinors is a vector space of suitable dimension that is endowed with a representation of a spin group. With a choice of basis in the vector space, it can be identified with $C^{n}$, and the spin representation can be taken as square matrices, one for each element in the spin group, acting on column vectors in $C^{n}$ by matrix multiplication from the left.

This doesn't solve all problems, but it is a start. YohanN7 (talk) 18:31, 1 October 2014 (UTC)


 * I don't see the abuse as very bad there. We already explain parenthetically what we mean by "representation". I think attempts to clarify further in the second paragraph are likely to do more harm than good. Paragraph 3 already discusses what this all means concretely in terms of matrices.  Sławomir Biały  (talk) 18:52, 1 October 2014 (UTC)


 * I'd agree totally if the intended audience is expected to know, what a representation is. We don't, at least not to the extent that they know both the formal definition, and the "standardized abuse of notation". To introduce both of these things the very first sentence that mentions representations will be confusing. Again, the reading is good for me and the parenthetical remark explains to me that the author will be using the "standard abuse" (so that I can adapt my interpretation of what I read further down immediately). I doubt that people unsure of representations will interpret the sentence the intended way. YohanN7 (talk) 19:22, 1 October 2014 (UTC)


 * I agree with YohanN7 here that it would be particularly helpful to at least use the phrase representation space in place of the abuse representation where appropriate. At the cost of a word, the cognitive load on the novice for figuring out what is meant can be reduced substantially. —Quondum 23:07, 1 October 2014 (UTC)


 * Looks better now. Thanks! YohanN7 (talk) 08:37, 2 October 2014 (UTC)

For another example of abuse of professional slang (this time from physics), please see the next section and my last post. It took me hours to catch that one - even though I am aware of the abuse. The standard representation (space) is simply called "vector". YohanN7 (talk) 08:37, 2 October 2014 (UTC)

Aside on terminology for "spin-tensors"
An aside on terminology and to ressurect a post that went quiet after this revert happened while the editing is active: could we somewhere clarify in the lead/intro/overview about "spin-tensors"? Yes - they are separate quantities used in physics, but I think a spinor is sometimes mis-called a "spin-tensor" (maybe in popular science books like Penrose but I don't have a copy right now to check)? Even just a hatnote to point away the confusion may help. M&and;Ŝc2ħεИτlk 10:25, 2 October 2014 (UTC)
 * The appropriate terminology is (too?) concisely described in the relevant section (5.6) in Weinberg vol I. There are spinors, tensors and spinor-tensors. (A tensor is "allowed" to be a "vector".) These aren't necessarily all irreducible. Their construction generally involves reduction (involving Clebsch-Gordan decomposition and more) of tensor products of (direct sums of) the irreducible ones. A big subject of its own. I can dig up a good freely available paper given time. Have planned to include a section in the Lorentz rep article. Long-term project. YohanN7 (talk) 11:53, 2 October 2014 (UTC)

Other articles treating spinors
While this article has some (plenty!) attention of experts, it would be nice if some sort of sanity check could be made on other articles also dealing with spinors. One such article is spin representation. This one seems to be in pretty good shape (though it isn't easy). Another one is Clifford algebra, and then there is bispinor.

My personal concern is representation theory of the Lorentz group. It makes fairly detailed attempts at describing spinors, at least spin (projective) representations. In particular, the approach is to define things "along a path", then come to the intermediate conclusion that the homotopy class of the path is what counts (for projective representations). Finally, the homotopy class of the path is disposed of and replaced by a Lie algebra element (in the standard representation) to "represent" the homotopy class. As far as I can see, this should be equivalent to keeping the complete path (class), since the Lie algebra certainly is simply connected, and a path there from the origin to any element can be deformed freely, and only the endpoint in the Lie algebra of the path "counts".

A sanity check would be appreciated. YohanN7 (talk) 08:32, 1 October 2014 (UTC)


 * I think spin representation is actually one of the best references on the real spin representations I have seen. I have been through that one myself fairly carefully, and only found one small mathematical error.  Other articles that I have noticed are orientation entanglement, belt trick, spinors in three dimensions, Weyl-Brauer matrices (which is presumably the same thing as Higher-dimensional gamma matrices), Dirac spinor, spinor bundle, spin structure.  These are all in fairly bad shape, but I think a comparatively low priority.  I'm sure there are others.   Sławomir Biały  (talk) 10:00, 1 October 2014 (UTC)


 * Agreed, spin representation is (apart from some terseness) a very nice technical article on a mathematically nontrivial subject.
 * As far as my (whimsically stated) claims go about the Lorentz reps: I have found a proof that applies to the compact classical groups (at least for loops). Here the path homotopy class is determined by the "final" point in the Lie algebra. The proof (due to Weyl, presented in Rossmann's book) is far from trivial. I can only hope it extends to $SO(3, 1)$, but I guess this is so because the "root" of the double connectedness of the Lorentz group (by purely topological arguments, e.g. Weinberg vol I) is the compact classical subgroup $SO(3) ⊂ SO^{+}(3, 1)$. I don't remember from where I got the claim in the first place, but it is decidedly how Lorentz transformations of spinors are done in physics (no visible continuous parameter), and I rarely write anything (in articles) without a solid reference. Still worried. YohanN7 (talk) 12:59, 2 October 2014 (UTC)