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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:
is a rotation (45° or whatsoever)ǃǃ
It is true that
but that is the end of it. unless you restrain yourself to the spin space. -- Chessfan ( talk) 17:22, 12 December 2014 (UTC)
"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)
Isn't this article in its current form too advanced for Wikipedia? -- Mortense ( talk) 10:34, 14 August 2016 (UTC)
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:
YohanN7 ( talk) 09:25, 28 December 2014 (UTC)
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)
The Möbius approach might motivate some explanation as to whether they're related. -- Daviddwd ( talk) 17:39, 18 September 2018 (UTC)
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:
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:
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:
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)
This statement is patently false:
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)
"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)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | ← | Archive 5 | Archive 6 | Archive 7 |
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:
is a rotation (45° or whatsoever)ǃǃ
It is true that
but that is the end of it. unless you restrain yourself to the spin space. -- Chessfan ( talk) 17:22, 12 December 2014 (UTC)
"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)
Isn't this article in its current form too advanced for Wikipedia? -- Mortense ( talk) 10:34, 14 August 2016 (UTC)
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:
YohanN7 ( talk) 09:25, 28 December 2014 (UTC)
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)
The Möbius approach might motivate some explanation as to whether they're related. -- Daviddwd ( talk) 17:39, 18 September 2018 (UTC)
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:
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:
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:
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)
This statement is patently false:
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)
"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)
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)