Timeline for Can spinors be explained or understood without group or representation theory?
Current License: CC BY-SA 4.0
24 events
| when toggle format | what | by | license | comment | |
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| S Dec 20, 2023 at 22:53 | history | bounty ended | Jagerber48 | ||
| S Dec 20, 2023 at 22:53 | history | notice removed | Jagerber48 | ||
| Dec 20, 2023 at 7:11 | answer | added | Joe Schindler | timeline score: 1 | |
| Dec 20, 2023 at 7:08 | comment | added | akhmeteli | @Jagerber48 : "I would like to know if we can at least mathematically define them without reference to group/representation theory." I would think my answer qualifies. | |
| Dec 20, 2023 at 4:45 | history | edited | Jagerber48 | CC BY-SA 4.0 |
added 580 characters in body
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| Dec 19, 2023 at 8:35 | answer | added | akhmeteli | timeline score: 0 | |
| Dec 16, 2023 at 14:28 | comment | added | Jagerber48 | @Valac can you please post that comment as an answer? | |
| Dec 16, 2023 at 11:40 | comment | added | Xenomorph | A spinor field is a section of a clifford module bundle over a spin manifold. At each point of the spin manifold, the fiber is understood as a left-module of the clifford algebra containing the Lie group $Spin(p,q)$. For a physicist, this can be understood as the vector space where Dirac's gamma matrices act on. But again, the definition of a clifford module requires representation theory, because $Spin$ is a covering space of the Lorentz group. As long as one is considering products of spinors in the Lagrangian formalism, one should consider the wedge product of the spinor bundle. | |
| Dec 13, 2023 at 14:08 | answer | added | AccidentalFourierTransform | timeline score: 7 | |
| Dec 13, 2023 at 10:51 | comment | added | Andrew Steane | To get intuition about why $4\pi$ and not $8\pi$ you can observe what happens when an ordinary object connected to something else by a flexible string is rotated through $2\pi$ and $4\pi$. The string tangles at $2\pi$ and untangles at $4\pi$. | |
| Dec 13, 2023 at 5:29 | answer | added | Toyesh Jayaswal | timeline score: 2 | |
| S Dec 13, 2023 at 0:12 | history | bounty started | Jagerber48 | ||
| S Dec 13, 2023 at 0:12 | history | notice added | Jagerber48 | Canonical answer required | |
| Dec 13, 2023 at 0:08 | history | edited | Jagerber48 | CC BY-SA 4.0 |
Change title to include group and representation theory. Add a paragraph with more detail on why I want to avoid "spinors are things that transform like spinors"
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| Dec 10, 2023 at 16:51 | comment | added | Jagerber48 | @printf yeah sure, an explanation without group theory would be great. Again, vectors can be understood without group theory (elements of the tangent space of a differentiable manifold). | |
| Dec 10, 2023 at 15:52 | comment | added | printf | @BrianBi A very valid point! Ordinary classical objects, of course, return to themselves when rotated by $2\pi$. (Imagine that you, the observer, rotate by $360^\circ$: you expect everything to be the same. It would be strange if you rotated through $360^\circ$ and everything changed sign! But that is exactly what happens with spinors, since they are quantum objects, and a spinor and its negative belong to the same ray.) Spinors return to themselves after a rotation by $4\pi$, but why? Why not $8\pi$, or something else? Group theory, and homotopy theory, must be invoked to explain that. | |
| Dec 10, 2023 at 15:38 | comment | added | printf | While it is possible to discuss spinors without explicitly mentioning representation theory, the fact is that all mathematics is interconnected; representation theory can be seen as a branch of group theory. So, would you like a description of spinors that makes no mention of group theory at all? Or a description of spinors that makes no mention of Lie groups and Lie algebras? What about such notions as homotopy and the fundamental group of a manifold? | |
| Dec 10, 2023 at 9:04 | answer | added | bolbteppa | timeline score: 11 | |
| Dec 8, 2023 at 22:47 | comment | added | Brian Bi | I'm not sure if you can even understand vectors without representation theory, so I'd be even more pessimistic about spinors. There are other ways of getting intuition about spinors, but they don't explain why we don't have additional kinds of objects, e.g., that only return to their original configuration after a rotation of $8\pi$ compared with $4\pi$ for spinors and $2\pi$ for vectors. | |
| Dec 8, 2023 at 17:46 | history | became hot network question | |||
| Dec 8, 2023 at 12:58 | answer | added | The Tiler | timeline score: 6 | |
| Dec 8, 2023 at 11:02 | answer | added | Nullius in Verba | timeline score: 27 | |
| Dec 8, 2023 at 10:35 | comment | added | gandalf61 | It is possible to conceptualise spinors as geometric objects, although some aspects of their behaviour may be less intuitive from this point of view. See math.stackexchange.com/q/2269714 | |
| Dec 8, 2023 at 9:46 | history | asked | Jagerber48 | CC BY-SA 4.0 |