Talk:Spinor/Archive 4

Plate trick
Something that makes me uneasy about the plate trick and other such claimed demonstrations or analogies is that spinors in 3d or 4d relate to 3d or 3+1d Clifford algebras, which in turn relate to rotation about the point at the origin; whereas the plate trick appears to involve rotation about a distant point. It then seems to be argued that two-valuedness for a finite length of arm requires two-valuedness in the limit as the length of the arm is made microscopically small. But is this necessarily true? After all, it's also for two solutions to exhibit a fold bifurcation, annihilating each other with only one solution where they join. It's also said that spinors can model the two-valuedness, but seemingly the details of the model are never explicitly made clear.

As it happens, there is a very tidy way to represent rotations about distant points in the general style of Clifford algebra type approach, called Conformal geometric algebra, which rests on a mapping from Rn to Rn+1, 1 in a particular projective way, that allows translations in Rn to be modelled as rotations in the higher-dimensional space; and therefore rotations about a distant point, being the composition of a translation, a rotation, and another translation. It might be interesting to look at the plate trick in CGA, and trace exactly what happens using rotors (the GA equivalent of spin matrices); and then look at what happens as the length of the arm tends to zero.

One can imagine that such a model in Clifford algebra form might show something like an X-shaped scissor-like crossing of two solutions at x=0 for some quantity x; whereas an alternative modelling might be like modelling in terms of x2, with the change in independent variable turning the previous scissor-like crossing into a C-shaped fold bifurcation, with only one solution at the origin. Jheald (talk) 00:27, 22 August 2014 (UTC)


 * Comments


 * "Plate trick": I think you're probably overthinking the role of translational symmetry in the plate trick if you find yourself wanting to muck about in a five-dimensional space to understand it. Perhaps the game of tangloids is a better illustration (this is what is directly illustrated by the animation in a previous section).  Using arclength along the two strings of the tangloid as the parameter, a solution of the game is then an explicit 1-parameter homotopy of the tangloid to the identity.
 * "It's also said that spinors can model the two-valuedness, but seemingly the details of the model are never explicitly made clear." Yes, I agree that this is something that the article should describe, probably later on once spinors have been "constructed" or whatever, but still.  Sławomir Biały  (talk) 01:42, 22 August 2014 (UTC)

Geometric Algebra
I think it would be nice to do more in the article that presented spinors using Geometric Algebra. GA is an approach introduced by David Hestenes that makes Clifford Algebras accessible to engineers, computer scientists, never-quite-made-it-physicists like myself, etc, by going really really concrete (but coordinate-free). Thus Clifford Algebra elements 1, e1, e1e2, e1e2e3 etc are taken to represent a scalar, a particular vector, a particular bivector, a particular trivector, etc.

What stands out as particularly disctinctive about spinors, compared to other objects, in GA, is that spinors systematically contain factors like (1/2)(1 + e1), an idempotent, since
 * ((1/2)(1 + e1))2 = (1/4)(1 + 2 e1 + e12) = (1/4)(2 + 2 e1) = (1/2)(1 + e1)

Now adding together objects of different grades isn't unknown in GA -- the lecture that first introduced me to the subject memorably included a slide with the title "But how the hell can I add a scalar to a bivector?" (back in the days when slides were hand-written), analysing the idea that in GA rotations are represented by operations of the form A → R A R-1, where a rotor R has a form like $$\scriptstyle{\cos \theta/2 + \mathbf{e}_1 \mathbf{e}_2 \sin \theta/2}$$, analogous to $$\scriptstyle{\cos \phi + i \sin \phi}$$.

But what is to be made of a quantity like (1/2)(e2 + e1e2), that appears to add a vector to a bivector? This was a question that confused me for a very long time -- if these are spinors, then what on earth do they represent?

The trick is to realise what a factor like (1/2)(1 + e1) means. The key is to realise that, for this factor,
 * $$\tfrac{1}{2} (1 + \mathbf{e}_1) \mathbf{e}_1 = \tfrac{1}{2} (\mathbf{e}_1 + 1) =  \tfrac{1}{2} (1 + \mathbf{e}_1) 1$$

so the effect of this factor is to make any dependence on e1 in the expression it multiplies indistinguishable from dependence on a scalar. So if one takes this factor, and premultiplies the entire Clifford Algebra by it, it has the effect of annihilating any relevance, in any term, of the dimension corresponding to the vector e1 -- any such dependence on such a term becomes indistinguishable from just dependence on a scalar. So one is essentially projecting down the entire Clifford Algebra of Rn into something isomorphic to the Clifford Algebra of Rn-1, in a systematic way throwing away any dependence (whether in a vector, a bivector, a trivector or whatever) on one particular direction. This is exactly what one wants to do, if one wants to analyse general rotations in terms of lower dimensional primitives, that only act on a particular subspace.

This can be related to the "classic" picture of Clifford Algebras in terms of complex matrices and column vectors as follows:

Typically, when working on matrix representations of groups, we consider the effect of the matrices on vectors, as objects to keep track of what the matrices are doing, eg to build a group table. But there's no reason we can't just as well how consider those matrices acting on matrices. This is essentially what we're doing when we consider the effect of the Clifford Algebra (each element of which can be represented as a matrix) on itself.

But, if we're physicists, then following Wigner (1927), if we're interested in particle physics then we're interested in the irreducible representations of the group corresponding to the algebra.

But the action of a whole algebra on a matrix is immediately reducible into the action of the algebra on its separate columns. These are what are first projected out by pre-multiplying by algebra elements like (1/2)(1 + e1) and (1/2)(1 - e1). Any element A of the algebra can be split into two parts,
 * (1/2)(1 + e1) A and (1/2)(1 - e1) A

which is similar to splitting the matrix into two halves, that remain split under the action of the algebra.

This is as far as we can go for 3d space, where the matrices that represent the Clifford algebra are 2x2 complex matrices built from the Pauli matrices.

But for Minkowski space, one can split again, to give a sub-space of the full original Clifford algebra that can be represented by a column vector containing 4 complex numbers -- the familiar spinor. However, we can now recognise that this spinor simply represents an object isomorphic to a lower-dimensional Clifford Algebra.

(For more details see the book by Lounesto, summarising Hestenes).

It seems to me that when the article tries to answer "What are spinors?", this is the kind of thing it should be presenting, showing how the whole topic hangs together, and actually giving the reader some sense of what spinors are (physically), and why they are what they are. Jheald (talk) 00:27, 22 August 2014 (UTC)

... I still do not understand why you are thinking that expressions like (1+ e) are spinors. Chessfan (talk) 12:22, 1 September 2014 (UTC)

Comments (GA)

 * General comment on the GA approach: I have yet to see a really convincing explanation that geometric algebra, in the style of Hestenes and his followers, actually offers any advantages over more traditional approaches involving Clifford algebras and central division rings.  Indeed, it seems to me that the entire approach is basically a repackaging of things that are either well-known in the traditional approach or completely obvious.  But perhaps this is my own mathematical upbringing.  The observation that spinors can be represented (up to sign!) by certain rank one elements of the algebra is itself quite old, and is not terribly helpful as a definition of a spinor.  To me, the idea of a representation of the algebra is a much more conceptually rewarding notion on which to base the idea of a spinor.  This also seems to be the "modern" approach that most treatments now take (although I am also partial to Weyl and Brauer's original approach, which nowadays seems little used, in that case the spinors are just column vectors acted on by tensor products of 2x2 matrices&mdash;so there's no mystery there).
 * That having been said, the construction of spinors in the article currently is a bit convoluted. In my opinion, the "best" way to construct spinors is just to say what the Clifford algebras are.  These are subject to algebraic Bott periodicity (the Brauer-Wall group of graded Morita equivalence classes over the ground field), so they are rather trivial to describe completely as matrix algebras.
 * Still mulling. Have to go.   Sławomir Biały  (talk) 01:42, 22 August 2014 (UTC)


 * Well, yes, I suspect what I've written about GA above is just a repackaging of more traditional ideas involving Clifford algebras and central division rings. It's sort of reassuring if you find all that I've written above either well-known or completely obvious, albeit dressed up in particular language.
 * But I think it is a repackaging that (IMO) can really help engineers, computer scientists, people like me who've never taken a course in ring theory, because of its focus on taking what can otherwise be quite abstract and remote, and presenting it in a way in which a consistent concrete geometric interpretation shines through. (And it's quite handy for actually doing geometry).
 * No, what I've written above isn't a "definition" of a spinor. But for me, what it does do, is to help me feel I understand what a spinor actually is -- what it represents, and why it is the way it is.  By making a spinor just another object in a Geometric Algebra, it means I can think about it without leaving a system which is comfortable and familiar.  For me, writing the projector that characterises the spinor space in the form (1/2)(1 + e1) means that I can directly see how that projection immediately relates to a geometrical thing -- namely essentially making irrelevant every instance of e1 in what follows, reducing a system of n physical dimensions into one of n-1 physical dimensions.
 * Yes, the Dirac equation can be considered just an equation for a column vector of complex numbers acted on by some particular matrices. But I think you're wrong to say "no mystery there", because for generations of students the mystery has been, what are these column vectors?  Where have they come from?  What do they actually mean, physically?  GA I think helps answer these questions, in a way that is accessible to somebody meeting this equation for the first time.  This is what the objects mean (or at least one way to think about what they mean); this is where they have come from, physically.  And so if the Dirac equation becomes no longer an equation of mysterious matrices acting on a mysterious complex column vector, but instead one of geometrically-interpretable operators acting on a certain geometrically-interpretable rank-one element of the algebra, I think that that is something valuable to understand.   Perhaps you're right, that such things were always implicit in the numbers.  But for me, until now, the rungs on the ladder had always been just too far apart.   Jheald (talk) 08:12, 22 August 2014 (UTC)


 * I probably shouldn't comment about the GA approach because I don't know it really. But on the surface of things, it appears just to complicate matters needlessly. YohanN7 (talk) 10:10, 22 August 2014 (UTC)


 * I think it is misleading to insist on thinking of spinors as elements of the geometric algebra. There is no canonical way to do this (apart from what amounts to a Segre embedding, which is non-linear and 2:1.) The approach to construct the spin representations as minimal ideals in the Clifford algebra is already described in the "Minimal ideals" section of the article.  However, the problem with this approach as I see it, is that there are many minimal ideals, but really only one spin representation.  (I don't know if the minimal ideals construction is what your post is about.  It's hard for me to extract any precise meaning, but it does have that overall vibe.)  At any rate, the basic intuition conveyed by this approach is wrong: spinors cannot be canonically associated with elements of the geometric algebra.   Sławomir Biały  (talk) 12:59, 22 August 2014 (UTC)


 * I guess in my language what you're objecting to is that there is more than one way that the Clifford algebra could be projected to a space of spinors -- one might use either (1/2)(1 + e1) or (1/2)(1 - e1), corresponding to choosing different columns of the matrix representation of the Clifford algebra. But I'm not sure that that is important; the relevant thing, I think, is that whichever of the possible projectors one chooses, the resulting set of elements is in 1:1 correspondence with the elements of the Clifford algebra of a lower physical dimension.


 * So I would say that this is the key property that I am asserting that spinors possess:
 * Spinors are members of a subset of a Clifford alegbra, the subset being in isomorphism with a Clifford algebra corresponding to a lower physical dimension,
 * such that the (one-sided) action of the whole algebra on the subset corresponds to the (one-sided) action of lower-dimensional projections of the algebra on the lower dimensional Clifford algebra.


 * That may or may not be enough to define spinors; but I think it is a property that is (a) true, and (b) can be helpful for thinking about them.
 * (If you prefer to think of spinors as columns of complex numbers which are in isomorphism to such CA members, then that's a detail, but doesn't seem to me to much change the essentials).


 * As for geometric algebra, that Yohan says he doesn't know really, the essence of geometric algebra is really just to identify the gamma matrices of a Clifford algebra, such as the gamma matrices of the Dirac algebra, with unit vectors, and their antisymmetric products with bivectors, trivectors etc. See eg Section A of Hestenes's 1967 paper, ie pp 2--6, for a concise presentation.  It often means that an element of a Clifford algebra can be given a very direct geometric interpretation.


 * Hestenes then goes on to identify spinors with unit-modulus members of the even part of a Clifford algebra. (A claim for -- or perhaps redefinition of -- the word spinor that is often made in the GA community, which can be confusing). Hestenes shows this isomorphism works for the Dirac equation.   It's not clear that the claim is generally true -- the two types of things would seem usually to have different dimensions.  On the other hand, the even part of a Clifford algebra (where rotations live) is consistently isomorphic to a Clifford algebra built over one fewer physical dimensions; so it's conceivable that a spinor may always be isomorphic to the even part of some Clifford algebra.


 * Despite the cold water poured on this that's been the first reaction above I do think this is worth persuing, because otherwise merely presenting spinors as a column of complex numbers really gives very little physical intuition as to how this complex vector relates to something physical or geometric in the underlying space of physical dimensions. Jheald (talk) 12:13, 23 August 2014 (UTC)


 * Of course it is just wrong to identify spinors as "unit-modulus members of the even Clifford algebra". This conflates spinors with elements of the spin group.  In the paper you linked to, Hestenes declared spinors to be elements of the even Clifford algebra, at the bottom of page 7, which seems to be rooted in the same kind of confusion.  So this perspective seems rather unhelpful to us.  Regarding the assertion "Spinors are members of a subset of a Clifford alegbra, the subset being in isomorphism with a Clifford algebra corresponding to a lower physical dimension."  This is also wrong, for at least two reasons.  First, although it is certainly possible to obtain spin representations by looking at linear subspaces of the Clifford algebra, picking a single such linear subspace involves considerably more structure than one is entitled to.  So it is unclear in this approach to thinking about spinors what properties one can ascribe to the spinor-ness of the objects under consideration, and what properties are artifacts of the choice of linear subspace.  This mistake has the same character as insisting that, instead of the concept of a vector space, we should just confine our attention to $$\mathbb R^n$$ (which has considerably more structure, namely a preferred linear coordinate system).  The second reason is that it conveys the mistaken idea that it makes sense to multiply two spinors and obtain another spinor.  This is also not true.  In fact, the irreducible components of the tensor product of two spinors consists entirely of tensor representations; there isn't a spinor in sight.  Now, none of this means that it is impossible to do calculations using the GA approach to spinors, and thus "feel" that one has some confidence that the approach is a correct one.  But it really is not strictly correct to think of spinors as belonging to the Clifford algebra.
 * Also, looking at Hestenes' book Space time algebra, he is a little bit clearer on how he identifies spinors with elements of the even subalgebra of the Clifford algebra. This involves picking a fixed spinor and acting on it with the even subalgebra.  Apart from depending sensitively on this initial choice of spinor (which presumably must be covariantly constant, itself a nontrivial holonomy constraint on the spacetime), this trick only works in certain low dimensions and signatures because of "accidental" isomorphisms (actually, possibly just in the Lorentzian 4-dimensional case).  I'm sure there are other ad hoc constructions that can be successfully implemented in other particular signatures, but I disagree strongly that the emphasis in a general article on the spinor concept should be rooted in such ad hoc constructions.   Sławomir Biały  (talk) 13:52, 23 August 2014 (UTC)


 * This is a valuable perspective. GA (or Clifford algebra) seems to have useful correspondences and operations in this context, but we must take care not to ascribe structure to spinors that happens to be attached to the algebra. So far, the structure of spinors that I am interpreting is that they are elements of a set upon which a particular group action is defined, and that the interesting aspect is really in the group. As such, they might even be thought of as a Klein geometry, and this suggests that spinors could be defined in a way that does not involve representations or vector spaces. Also, it might be worth noting that in physics, additional structure seems to be defined: in essence, whatever is needed to support the bra–ket constructions with linear operators on a manifold. Is this a reasonable view? —Quondum 16:21, 23 August 2014 (UTC)


 * I'm going to leave Quondum's questions to others.
 * In respect of User:Sławomir Biały's post, I should first underline that I too am uncomfortable with association made by Hestenes, and often glibly repeated in GA books, eg the recent major work by Dorst et al, using the word "spinor" for a general unit element of the even Clifford algebra. This, at least on the face of it, is not what spinors as described in this article most obviously correspond to, even if an isomorphism exists in the most physically important case, that of the Dirac algebra.
 * Sławomir's point about multiplication is a fair one.
 * However, given the difficulty experienced developing intuition as to what it is that spinors actually do represent in terms of the underlying physical space, the systematic, non-accidental, 1-to-1 correspondences between spinors and the projections of the alegebra under multiplication by various idempotent projectors -- which are geometrically interpretable -- are surely not something to ignore.
 * Sławomir is of course right that the linear subspaces that result from these projections have more structure than spinors. But looking for what the different linear subspaces resulting from each of the projectors share in common takes away structure, and should get us closer to the notion of what it is that the spinors are capturing.
 * And yes, a Clifford algebra assumes we can start with an orthonormal frame (though we are free to chose its directions to match whatever happens to be convenient). But if we can understand what physical intution spinors reflect in such a setting, that's not a bad place to understand how much depends on the setting; compared to how much generalises.


 * I should note that above I very much identified the idempotents with quantities like (1/2)(1 ± ei), so that the spinors were in 1-to-1 correspondence with elements of the algebra in which various directions had been made irrelevant.
 * I still think the truth may be very close to that, but it's possible I may not have got it exactly right. For example, consider a Clifford algebra with an 8x8 matrix representation, representing a Geometric Algebra over n physical dimentsions.  One property of the spinors is that if we consider how an element of the algebra (one-sidedly) maps 8 basis spinors, that must fully determine its matrix representation.  But if those basis spinors only reflect the properties of (n-m) of the physical dimensions, should tracking only of what happens to those really be enough to determine a full representation?
 * So it might be worth looking at such a Clifford Algebra, and just re-checking exactly what in the geometric algebra the idempotent matrices
 * $$\left(\begin{smallmatrix} 1 & 0 & ... & 0 \\

0 & 0 & & \vdots \\ \vdots & & \ddots & 0 \\ 0 & ... & 0 & 0 \end{smallmatrix} \right) $$, $$\left(\begin{smallmatrix} 0 & 0 & ... & 0 \\                               0 & 1 &  & \vdots \\ \vdots & & \ddots & 0 \\ 0 & ... & 0 & 0 \end{smallmatrix} \right) $$, etc.
 * do in such a case in fact correspond to. Jheald (talk) 20:18, 23 August 2014 (UTC)


 * The general way to do this is as follows. There is a way to define "rank" for self-adjoint idempotents in any *-algebra.  In detail, define a partial order on the set of self-adjoint idempotents, by declaring that $$x\le y$$ if $$xy=x$$.  The rank of a self-adjoint idempotent is its height in that partially ordered set.  If x is a rank one self-adjoint idempotent, then the principal left ideal generated by x in the Clifford algebra is isomorphic, as a module over the Clifford algebra, to a spin representation.  This is a minimal ideal of the kind already covered in the article, for what it's worth.   Sławomir Biały  (talk) 20:56, 23 August 2014 (UTC)


 * Separate question about the Weyl and Brauer construction (which I think has an article of its own): why does it work? what is it that leads one to these particular matrices? is it obvious?  Jheald (talk) 08:12, 22 August 2014 (UTC)


 * The Weyl-Brauer approach works by manifestly representing the relations of the Clifford algebra. As for "why", the starting point is that it is easy to represent Clifford algebras in dimension two.  Over the complex, the two dimensional Clifford algebra is isomorphic to the algebra of 2x2 complex matrices (this can be also given a compatible grading).  Over the reals, it is either the algebra of Pauli matrices or the algebra of 2x2 real matrices (again, with some grading).  This is actually enough to get the theory going, because there is a decomposition of a Clifford algebra of an orthogonal sum of two spaces into the tensor product of the Clifford algebras of the individual spaces.  That is, $$C\ell(V_1\oplus V_2)=C\ell(V_1)\widehat{\otimes}C\ell(V_2)$$.  (Here $$\widehat{\otimes}$$ is the "graded" tensor product.)  This identity allows an arbitrary Clifford algebra to be broken into simpler pieces, and this is exactly what Weyl and Brauer do.   Sławomir Biały  (talk) 11:47, 22 August 2014 (UTC)


 * The "magic" with spinors have nothing to do with the spinors themselves. They are just boring n-tuples of numbers, and this holds for any description of them, whether they are elements of a GA or not (just fix a basis, and this is what they become). The key to the "magic" lies in the underlying group being doubly connected. The lead captures this, but it gets lost in the rest of the article. (Edit: Quondum observes this too in a post above.) YohanN7 (talk) 23:06, 23 August 2014 (UTC)
 * For me the "magic" of spinors is not the two-valuedness, ie the difference corresponding to a rotation of 360°. I can get that two-valuedness in the two different spin matrices (rotors) that can be used to express a rotation, and I'm comfortsble that, in geometric algebra form, I can see exactly why a rotor does what it does, and why it has the structure that it has.
 * No, for me the magic of spinors is that they are the objects of the Dirac equation; or, in other geometries, they are the objects that irreducible representations of the rotation group act on, so per Wigner (1927) they are the objects that would describe quantum-mechanical particles with that symmetry in those geometries. That is why I would like to understand what they represent in geometric and physical ways, not just algebraic ways, and at least as transparently and as well as I can understand rotors. Jheald (talk) 06:20, 24 August 2014 (UTC)
 * To be honest, I understand very little of what you are saying here. But I have seen this tendency before, i.e. that GA is supposed to "explain" things. But these things don't always really seem to be there. The things that truly are there are there without GA. When it comes to spinors it is exactly the path-dependent two-valuedness that comes from the topology of the group in question that is special. The solutions of the Dirac equation are, by the way, just 4-tuples of complex numbers (depending on spacetime) called bispinors. YohanN7 (talk) 07:12, 24 August 2014 (UTC)
 * Jheald, nice description; it seems to distinguish the role of the spinors in an action of a group. YohanN7, yes, the 2-valued nature of the topology is important, but that is possibly the aspect of spinors best understood by all of us. Here, it would possibly be more useful to consider any way of representing a spinor as equivalent, e.g. as an element of a GA or and element of a complex vector space. Like Jheald, I think better in GA. However, I'd suggest that determining what structure of each is not of relevance should help. Spinors seem to form a homogeneous space (correct me if I'm wrong). This implies that every spinor can be transformed into any other, e.g. into the spinor that we would represent in GA as the scalar 1, so this "nonspecialness" of scalars for this use in GA we must keep in mind. Similarly, if we use a complex 4-vector to represent a spinor, multiplication by a complex number does not correspond to some identifiable action distinguishable from say some other linear transformation (a cyclic rotation of three components?). As such spinors in 4-space can be thought of as "merely" points of $S^{7}$ (or so I guess). —Quondum 14:41, 24 August 2014 (UTC)
 * Spinors form a vector space, with a faithful linear representation of the spin group. In (1,3) signature, there is an invariant complex structure on the spin representation, so it does make sense to multiply by a complex number.  That's not true in some other signatures, like the (2,3) signature in 5 dimensions (the spinors then carry an invariant real symplectic structure).  There is a host of possibilities, the cases over the reals being summarized in the last section of the article on spin representations.  In all cases, the invariant structure is very concretely describable in terms of elements of the Clifford algebra, basically involving the product of all of the gamma matrices (which does not actually depend on the choice of basis, suitably normalized).  For example, in the (1,3) case, the complex structure is just given by $$J=\gamma_1\gamma_2\gamma_3\gamma_4$$.  In any case, regardless of the signature, spinors are not usefully thought of as a homogeneous space.   Sławomir Biały  (talk) 15:03, 24 August 2014 (UTC)
 * Spinors form a vector space: this topic went a little quiet, with me unsure of a couple of points. The lead says: In the language of group theory, spinors are ("projective") representations of the rotation group, which is to say that they are vector-like objects that transform under rotations.  This wording suggests to me that elements of the vector space fall into equivalence classes (elements of which are vectors related by a scalar multiplication), and that we are not interested in the vector magnitude (so that vector magnitude and vector addition only have relevance within the construction of $GL(V)$) – I'm throwing this out to have it shot down, before I go further. —Quondum 18:02, 24 August 2014 (UTC)
 * I think the confusion in the first sentence of the second paragraph comes from the fact that the word "vector" has two different meanings. One is as an element of a vector space, and one as an object that transforms in a certain way (covariance and contravariance of vectors).  It is the latter meaning that the word vector is being used in the first sentence.  At the moment, I don't know the best way to address this.  (Here the link to "projective" means that the representation actually goes in to the projective general linear group of V, not the general linear group.  But there is still a vector space V involved.)   Sławomir Biały  (talk) 19:47, 24 August 2014 (UTC)
 * Re: projective, Sławomir was very patient with me, took a lot of his time last year to sort me out on this point -- see this thread, where I was very very slow to get it. The conclusion was as follows:
 * The representation of SU(2) on C2 is a not a representation of SO(3):
 * You can arbitrarily choose half of SU(2) to map SO(3) into, but then the set of elements you've chosen isn't closed. For the algebra to be closed, you need the whole of SU(2). But then there's no longer a 1-to-1 mapping from SO(3) to the whole of the target algebra.
 * You can make a mapping if you declare elements R and -R of SU(2) to be equivalent. But then you're not mapping into GL(V) any more, you're mapping into PGL(V). And that means it's not a bona fide representation, only a projective one.
 * Hope that helps. (It helped me!)  Jheald (talk) 20:44, 24 August 2014 (UTC)
 * Sławomir, I think we're saying the same thing, with different words. Where you say as an object that transforms in a certain way, I am saying a space that is acted on by the group, but which does not carry (or at least we are not interested in) the structure of the vector space beyond this.
 * Jheald, you seem to be focused (here and in the linked thread) on issues relating to double covers, which are tricky, but I'm not too concerned about those as those relate to the group structure itself; I'm more concerned about getting clarity on the terminology itself, the concepts on which the very definitions are based. We'll be talking at cross purposes, but I'm afraid I'm taxing everyone's patience before we even get started on the mathematics built thereon. —Quondum 01:34, 25 August 2014 (UTC)
 * I want to emphasize one thing. I don't have anything against GA. What I mean is that GA is a huge subject of its own and creates an additional layer of abstraction that is not needed to describe spinors. It even seems that GA introduces misunderstandings (because of the additional complexity).
 * On the side of terminology, there is probably a difference between the physicists view on what a projective representation is and the mathematicians. The physicist can accept the two-valued thing (of the orthogonal group, not the spin group) as being a representation, while the two-valued thing isn't a representation in the mathematicians view. The physicist gets away with this because of the non-importance of phase factors in the Hilbert spaces representing particle states. YohanN7 (talk) 05:09, 25 August 2014 (UTC)
 * Agreed, GA brings with it a lot of structure that is incidental, and this can confuse things. For example, it is easy to confuse a rotor and a spinor in GA. I find that most of the confusion that I have to deal with in mathematics is of this nature, especially in communication with others. Once one has identified the exact mathematical structure one is dealing with and understands that some of the operations possible in GA are "disallowed" in the context, GA can be great for understanding, though. It is with trying to identify the defined structure that I've been bogging down here. Use of matrix representations carries with it the same type of issue. (This, curiously, seems to be something that even many mathematicians do not seem to "get": in communicating a definition, it is at least as important to communicate what axioms are not part of a category's definition as the ones that are.) —Quondum 14:08, 25 August 2014 (UTC)

Abstract spinors
In this section something has been lost in the last paragraph. The letter W is used to denote two things it seems. I don't know the workings of this and ask somebody who knows them to fix it. YohanN7 (talk) 13:30, 1 September 2014 (UTC)


 * I think throughout the section W denotes a fixed choice of maximal isotropic subspace of V (or, if we are over the reals, then of the complexification of V). But I think pretty much the whole section from Abstract spinors onwards needs to be rewritten for clarity.  I may do something about it eventually, unless someone else wants to take it in hand in the mean time.   Sławomir Biały  (talk) 12:56, 2 September 2014 (UTC)
 * I turns out that there is a prime (or whatever it is) on the second $W$ in each pair of them. This thing simply doesn't show in my browser. I'll change it to a '*'. YohanN7 (talk) 13:16, 2 September 2014 (UTC)