Talk:Spinor/Archive 7

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Archive 7

Rotation of spinors,again ...

When I wrote "Rotation of spinors" talk, at least I hoped that some obvious errors in the 2-d and 3-d examples could be removed. Since then the article has become more and more abstract, and nothing has been done to correct the errors. How can one write (in GA) that:

$(1/{\sqrt {2}})(1-\sigma _{1}\sigma _{2})(a_{1}+a_{2}\sigma _{1}\sigma _{2})=(1/{\sqrt {2}})[a_{1}+a_{2}+(-a_{1}+a_{2})\sigma _{1}\sigma _{2}]$

is a rotation (45° or whatsoever)ǃǃ

It is true that

$\exp(-B\pi )R=-R$

but that is the end of it. unless you restrain yourself to the spin space. --Chessfan (talk) 17:22, 12 December 2014 (UTC)

Imo the examples should be scrapped and replaced by identifications of the spin Lie algebra representation sitting inside the Clifford algebra in the cases SO(3) (rotation group) and SO(3, 1) (Lorentz group). For reference, in the case of the Lorentz group,

$\sigma ^{\mu \nu }=-{\frac {i}{4}}[\gamma ^{\mu },\gamma ^{\nu }],$

satify

$[\sigma ^{\mu \nu },\sigma ^{\rho \tau }]=i(\eta ^{\tau \mu }\sigma ^{\rho \nu }+\eta ^{\nu \tau }\sigma ^{\mu \rho }-\eta ^{\rho \mu }\sigma ^{\tau \nu }-\eta ^{\nu \rho }\sigma ^{\mu \tau }),$

which is the commutation relations of so(3, 1). (This particular convention requires a complex Clifford algebra. This is not important) Make a choice of gammas, exponentiate the Lie algebra representation and you have the spin representation of the group. Let elements of this group act (by matrix multiplication from the left) on 4×1 columns vectors v ∈ ℂ⁴. Then ℂ⁴ is your space of spinors. It's as simple as that. One can easily devise a path in the original Lie algebra that in the standard representation exponentiates to a full 360 degree rotation in the standard representation of the group, but whose image in the spin representation only manages to get to a 180 degree rotation. This would make the article internally consistent and somewhat self-contained. YohanN7 (talk) 19:12, 12 December 2014 (UTC)

I agree that the examples are not to my taste either, and should be rewritten along the lines you are suggesting. In the GA approach, spinors are thought of as living inside the Clifford algebra itself, and this view serves as more of a hindrance to understanding I think. The GA approach can certainly be mentioned, but it should not be the source of the main examples. Sławomir Biały (talk) 13:16, 28 December 2014 (UTC)

Obviously false statement

"the resulting spinor transformation depends on which sequence of small rotations was used, unlike for vectors and tensors" - This is false; a vector or tensor can transform differently if two finite rotations are applied in different orders: AB != BA in general. Dividing these finite rotations into an infinite number of infinitesimal rotations shows that applying an infinite number of infinitesimal rotations in different orders results in different vectors and tensors (A^epsilon A^epsilon ... B^epsilon B^epsilon... != B^epsilon B^epsilon ... A^epsilon A^epsilon...). Doubledork (talk) 16:44, 22 June 2016 (UTC)

No, the overall transformation of a tensor depends only on the final rotation, not the path in the rotation group. Obviously, if you take some permutation of the sequence of small rotations that don't have the same overall product, the it's not the same final rotation, so neither tensor nor spinors transform the same way. The point here is that even if you have a pair of sequences of small rotations having the same product, the effect on spinors need not be the same. Sławomir Biały (talk) 16:49, 22 June 2016 (UTC)

Encyclopedia material?

Isn't this article in its current form too advanced for Wikipedia? --Mortense (talk) 10:34, 14 August 2016 (UTC)

Is there an aspect of the topic you feel is not covered? Sławomir Biały (talk) 23:59, 14 August 2016 (UTC)

Animations

I've added two new animations to demonstrate the 'belt trick'/720 degree rotation. While the physical objects are not themselves spinors, the animation is created by rotating them using spinors - specifically, the rotation of the fibers from the outside to the inside is an interpolation from an unrotated state to the state that the spinor represents. After the spinor representing the rotation has been rotated to its opposite configuration (causing the cube to rotate 360 degrees) the fibers demonstrate that interpolating toward the new spinor state from identity is a different operation which rotates in the opposite direction. I'm not sure what the best way is to organize these thoughts in order to explain what the animations actually represent, but I am open to any revisions that make it clear what the relationship is between the geometry and spinor mathematical behavior. JasonHise (talk) 02:28, 26 September 2016 (UTC)

Ok, now we have yet one more nice animation not faithfully illustrating what a spinor (as treated in this article) is. I give up, let these in (they are kind of cool), but please please please:

• Make it possible to turn animation OFF. It's like having a stroboscope flashing in your eyes making it impossible to read the article. (I'll remove all animations if this is not done, and will keep removing them.)
• Make it possible to choose speed. At least reduce current speed (in all of them ) with a factor of ten.

YohanN7 (talk) 09:25, 28 December 2014 (UTC)

It's unclear what your first sentence is meant to imply. Several paragraphs of the article (the lead and introduction) discuss lack of simple-connectivity of the rotation group as allowing one to define a notion of spinor. But I too find the new animation to be a bit baffling, and really doesn't illustrate anything clearly. It moves too quickly for me to see any difference between the 360 and 720 degree rotation.

I don't think the ultimatum is helpful. Is there an example of the kind of behavior on some other Wikipedia page, so I can see how it would need to work on a technical level? The second animation must stay, regardless, as it is essential to the content of that section. Sławomir Biały (talk) 13:00, 28 December 2014 (UTC)

My first sentence: It is clear to me what your (second) animation represents (I think). It represents a homotopy faithfully, but not a spinor. These have no spatial extensions (being n-tupes of numbers associated to a point in space, ok, I'll go as far as "arrow"). The Möbius strip does not represent a spinor faithfully. It could be made into something cool though. Make an animation of it and keep the tail of the vector fixed and attached to a point on the Möbius strip. Then somehow "rotate" the strip, while keeping the vector (tail fixed on background and strip, tip only attached on strip) attached to it. After one turn, the vector would be pointing the other way. YohanN7 (talk) 13:42, 28 December 2014 (UTC)

You're wrong about the Möbius strip. The spin group of three Lorentzian dimensions is $SL(2,\mathbb {R} )$, whose fundamental (spin) representation is on the homogeneous vector bundle $O(1)$ over the real projective line. This is exactly the Möbius bundle. Sławomir Biały (talk) 14:59, 28 December 2014 (UTC)

I stand corrected. Maybe update the article with this? But there is something I don't get here. I thought $SL(2,\mathbb {R} )$ is not simply connected. Can you give me a pointer where to read up on this? YohanN7 (talk) 15:22, 28 December 2014 (UTC)

Spin groups are not always simply connected: they are double covers of the connected component of the identity in the pseudo-orthogonal group. The pseudo-orthogonal group has the homotopy type of its maximal compact subgroup, so if the maximal compact subgroup has a factor of the circle group SO(2), then the fundamental group contains an infinite cyclic group, which cannot be resolved by passing to a double cover. A more familiar example is probably the group SU(2,2), which is the spin group of the conformal group SO(2,4) of spacetime. (The four-component complex spinors are sometimes called "twistors", but a little care I think is needed because twistors are usually associated with a four-fold cover rather than a two-fold cover: twistors feel an additional discrete invariant called the Grgin index, which I don't really understand.) This spin group SU(2,2) is not simply connected: its maximal compact subgroup is $S(U(2)\times U(2))$, which has an additional $U(1)$ charge. Regarding the interpretation of spinors as sections of a line bundle, there is a fairly high brow approach to this in Baston and Eastwood, "The Penrose transform and its interaction with representation theory". They cover the complex case, but the split case over the reals is "morally" the same. There is probably a more pedestrian account somewhere, but I don't know where offhand. Sławomir Biały (talk) 16:27, 28 December 2014 (UTC)

As far as turning animation off, I thought that the software used to produce the animations supported this. YohanN7 (talk) 13:58, 28 December 2014 (UTC)

I used mathematica to make the animations. The source code is in the file description. Mathematica supports interactive applets, that allow animations to be turned on and off, but I do not know if this can be imported into Wikipedia. Maybe ask at WP:PUMP/T? Sławomir Biały (talk) 14:59, 28 December 2014 (UTC)

In Wikipedia:WikiProject Mathematics (tesseract) someone found a solution. I can't figure out by reading the source what is done exactly. YohanN7 (talk) 14:26, 28 December 2014 (UTC)

Well, I would not object to someone implementing something similar here. But the animation must stay. Sławomir Biały (talk) 14:59, 28 December 2014 (UTC)

Don't take my ultimatum too seriously. I have been complaining about this several times w/o any responses. That's why I formulated it that way. If it is too hard (time-consuming) to do this, then it is just that way. YohanN7 (talk) 15:22, 28 December 2014 (UTC)

Figure 41.6 (chapter on spinors) in MTW Gravitation is clearly related to the new animation. YohanN7 (talk) 14:07, 28 December 2014 (UTC)

Yes, I agree that the animation has something to do with orientation entanglement. But it is not a good illustration of that, because it is impossible to discern what the "something" is (at least, for me). It's too busy. Sławomir Biały (talk) 14:59, 28 December 2014 (UTC)

The new animation does seem too busy, but would be much better if it just stopped for half a second or more at the start/end. That would not only make the extent of the animation much clearer but give the viewer time to take it in, hopefully dealing with the busy aspect of it. Perhaps the uploader JasonHise could look at this?--JohnBlackburne^words_deeds 15:33, 28 December 2014 (UTC)

I think two pauses would be helpful, one at the beginning and one halfway through the animation. I would suggest that these two milestones should be presented as stills in lieu of the animation. Also, the caption should explain what it is we are meant to be looking for. But even so, I do not think that this image is suitable for the lead to this article. It should be added to orientation entanglement, and possibly worked into a subsection here. Sławomir Biały (talk) 16:52, 28 December 2014 (UTC)

Revised the image to make it less busy, thanks for the feedback. Feel free to move to whichever section makes the most sense. JasonHise (talk) 05:48, 29 December 2014 (UTC)

Simple intuitive introduction needed

This article reminds me of online courses that consist entirely of material on why you should take the course but never get down to teaching you anything, if you know what I mean. Is there any good reason for this article to not start off with a simple intuitive definition of spinor that could be understood by the lay reader? Something like, "Spinors are mathematical objects that represent rotation dilations (rotations with accompanying scaling). They can thus be considered abstractions of complex numbers and of quaternions which represent rotation dilations in 2 and 3 dimensions respectively." (If this is new to you or you disagree with it, you don't understand spinors!!!) 197.234.164.85 (talk)

That's not what a spinor is. It is only in two and three dimensions that rotation-dilations are realized in the manner you describe. (Indeed, this is true on dimensional grounds alone, as well as for other reasons.) Sławomir Biały (talk) 14:28, 18 September 2017 (UTC)

It pretty much is what a spinor is, but agreed the wording should perhaps be "Spinors are mathematical objects that represent rotation-dilations (rotations with accompanying scaling) in 2 and 3 dimensions and generalizations of such in higher dimensions. They can thus be considered abstractions of complex numbers and of quaternions which represent rotation dilations in 2 and 3 dimensions respectively." The effect of applying a spinor in higher dimensions is still a generalization of the rotation-dilation that occurs in 2 and 3 dimensions, but from dimension 4 upwards the effects of higher order even blades kick in making it something more than just a rotation and dilation. 197.234.164.85 (talk)

A spinor isn't "applied". Vectors are applied to spinors, via gamma matrices, not the other way around. Sławomir Biały (talk) 23:00, 24 September 2017 (UTC)

I came to this article looking for a simple rigorous introduction to spinors. A mathematical definition.

But after scanning the article for too long, I can report that it is not easy to find. (I suspect it *is* somewhere in the article, though I am not sure where.)

Suggestion: Can someone who knows about the mathematics of spinors please include a clearly marked section titled something like "Mathematical definition" so that it is easy to find. Important: This section (or subsection) this is in should be short, so that the definition is not buried among other words.108.245.209.39 (talk) 22:40, 19 March 2018 (UTC)

A better article for the mathematics is spin representation. I just put three definitions of "Spinor" into this article, though, since you're right that they are not easy to find. The whole "Overview" section should be rewritten. Sławomir Biały (talk) 23:23, 19 March 2018 (UTC)

Clarify: are spinors related to topological quantum mechanics?

The Möbius approach might motivate some explanation as to whether they're related. --Daviddwd (talk) 17:39, 18 September 2018 (UTC)

Multiple dubious and incorrect statements removed.

I'm editing this article, as it seems to contain multiple false statements. I removed mention of gamma matrix from the intro - the gamma matrices only apply to 4 dimensions, whereas that section is talking about V or arbitrary dimensions. I'm also removing this:

<ref group="nb"> Although there are several more intrinsic constructions, the spin representations are not functorial in the quadratic form, so they cannot be built up naturally within the tensor algebra. </ref>

The tensor algebra construction is textbook-standard, see e.g. Jurgen Jost, Riemannian Geometry. The word "natural" also has a very specific, precise meaning, viz. natural transformation, and I don't see what the intended meaning of "functoriality" is, in this context, or why this affects representations ... all the more, as this is in the introduction, whereas functors and representations are "advanced" concepts. 67.198.37.16 (talk) 18:17, 8 May 2019 (UTC)

Also, this statement from the introduction:

Although the Clifford algebra can be defined abstractly in a coordinate-independent way, its particular realization as a specific algebra of matrices depends on which orthogonal axes the matrices represent. So what precisely constitutes a "column vector" (or spinor) also depends on such arbitrary choices.

I find this to be indecipherable. It seems to be saying that the component entries in a matrix depend on a choice of basis. But of course they do; this is basic linear algebra; it's got nothing to do with Clifford algebras or spinors. I'm removing this too, but leaving a note here, in case it gets contested. 67.198.37.16 (talk) 18:43, 8 May 2019 (UTC)

This statement is patently false:

The most typical type of spinor, the Dirac spinor,[4] is an element of the fundamental representation of Cℓp+q(C), ...

Dirac spinors are plane-wave solutions of the Dirac equation. The Dirac equation holds for Minkowski space. There is no Minkowski space, or other base manifold, in the Clifford-algebra construction of spinors; the construction from Clifford and/or SO(n) representations applies to zero-dimensional spacetime, not 4-dimensional spacetime. To get Dirac spinors, one needs to build a spin structure on Minkowski space, i.e. build the spin group on top of the tangent manifold of Minkowski space. It's a category error to equate these rather completely different constructions in this way. I don't see any particular easy fix, because whoever wrote that text got it all tangled together very elegantly, defying easy surgery... 67.198.37.16 (talk) 21:47, 8 May 2019 (UTC)

I just now fixed this. 67.198.37.16 (talk) 23:02, 8 May 2019 (UTC)

This statement is patently false:

It is not known whether (spin-1/2) Weyl spinors exist in nature.

That's just-plain wrong; Weyl spinors are deeply embedded in the Standard Model and are given mass by the Higgs mechanism. The result, after symmetry breaking, are physical Dirac spinors. It's even worse than that, because the so-called "Weyl spinors" of physics are plane-wave solutions built on the tangent bundle (see comment above) which is distinct from the "Weyl spinors" discussed here, which apply to zero-dimensional spacetime. I don't know how to dis-entangle the correct and the incorrect statements, such as this, they are deeply tangled into the text. 67.198.37.16 (talk) 22:02, 8 May 2019 (UTC)

I fixed this just now. 67.198.37.16 (talk) 23:02, 8 May 2019 (UTC)

Third sentence is poorly placed

"However, when a sequence of such small rotations is composed (integrated) to form an overall final rotation, the resulting spinor transformation depends on which sequence of small rotations was used. "

This is true in general of rotations, so it doesn't serve to distinguish spinors from the other examples cited just before (vectors and tensors), so it doesn't add anything to our understanding of spinors. Better to just remove this sentence and get right to the fourth sentence, which does actually explain something important about spinors:

"Unlike vectors and tensors, a spinor transforms to its negative when the space is continuously rotated through a complete turn from 0° to 360° (see picture). "

--24.5.180.247 (talk) 16:26, 16 September 2020 (UTC)

Fraktur

https://en.wikipedia.org/wiki/Spin_group#

And other pages within Wikipedia, use fraktur script, when talking about some Lie algebra, is there a reason why it's not consistent?

Please don't take it as a criticism of a "B" rating article, we all know Wikipedia is the most accessible loci for research. I know there is no timeline for Wikipedia edits, and it is probably just as easy for a senior editor to search for the term "so(" and change to fraktur, than it is for them to review my edits...

But if you want, I'll change all the instances of spin groups within Lie algebras pages?

Signed - Abbot Johann Heidenberg 49.185.41.142 (talk) 17:02, 1 May 2022 (UTC)