# Talk:Spinor

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## 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)

## 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 ${\displaystyle SL(2,\mathbb {R} )}$, whose fundamental (spin) representation is on the homogeneous vector bundle ${\displaystyle O(1)}$ 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 ${\displaystyle SL(2,\mathbb {R} )}$ 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 ${\displaystyle S(U(2)\times U(2))}$, which has an additional ${\displaystyle U(1)}$ 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?--JohnBlackburnewordsdeeds 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)

## 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)