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by aap_
458 days ago
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Unfortunately there are slightly different but related notions of what spinors are. One key idea is indeed how they transform. A spinor ψ transforms with a transformation S ∈ Spin(n) in a one-sided way: ψ -> Sψ. A vector v in contrast transforms in a sandwich way (with the inverse on one side): SvS^-1. Intuitively this explains why spinors transform "half as much" as vectors, e.g. the 720° vs 360° rotational symmetry that shows up in physics. So for any S ∈ Spin(n) the sandwich-action (S|S^-1) gives you the corresponding element of SO(n). Because a negative sign on S squares away in that case, S and -S map to the same SO(n) action, and therefore Spin(n) is said to be the double-cover of SO(n) (personally I think it would be better terminology to call SO(n) the half-cover of Spin(n)). So a spinor could be said to be anything whose symmetries are a Spin group. Spin groups are easily constructed in clifford algebras and it turns out that they have matrix representations. Whenever you have matrices (linear maps) you may wonder what the vector space is that they act on (i'm now using the term "vector" abstractly, not in contrast to spinors as above). Well, those are the spinors (technically pinors)! Another definition of spinors is that they live in a minimal left ideal of a clifford algebra. This does not sound very intuitive at first, but it can be understood easily as simply taking the matrices with only one non-zero column. These are really not very different from colunm vectors then.
There seems to be some confusion about pinors and spinors in that perspective though...it just seems to be a somewhat confusing concept in general. The spinors/vectors relevant to the article are those of Spin(8), which has something to do with triality (still need to understand all of this better myself). The basic idea is that in Cl(8) the vectors and spinors are both 8-dimensional and the algebra can be generated by left-multiplication of octonions. So there are some interesting symmetries occurring. The Baez-article goes into that too but it could have been a bit more explicit for my taste. I hope some of that made sense, i don't know your background. I'm still trying to wrap my head around this topic myself and have been for about 2 years now. Maybe check out the "spinors for beginners" series on youtube. It's very good and quite extensive. |
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Spin(8) itself doesn't have much to do with triality; it's just that triality describes an unusual symmetry among representations of Spin(8), due to an unusual outer automorphism. (Of course, from some perspectives, that means that Spin(8) has everything to do with triality, but I hope my meaning will be clear.) The best accessible mathematical explanation of triality I know is from Baez: https://math.ucr.edu/home/baez/octonions/node7.html.
> So a spinor could be said to be anything whose symmetries are a Spin group. Spin groups are easily constructed in clifford algebras and it turns out that they have matrix representations. Whenever you have matrices (linear maps) you may wonder what the vector space is that they act on (i'm now using the term "vector" abstractly, not in contrast to spinors as above). Well, those are the spinors (technically pinors)!
One has to be a little careful here, because algebras have lots of representations, and there's no one representation that a priori may be said to be "the vector space on which they act" ("the" rather than "a"). For example—though it's a bad example because it's not a faithful action—Spin(8) naturally double covers SO(8), but we don't want to take the resulting 8-dimensional orthogonal representation (the "vector representation"). Instead, we want to take one of the three fundamental representations permuted by the triality automorphism (the V_1, V_2, V_3 in Baez's article).