Talk:Spinor

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)

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)