The exceptional Jordan algebra (2020) | Hacker News Reader

The exceptional Jordan algebra (2020)

cp4space.hatsya.com

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I wonder if there is any connection of this being on the front page today with this paper [0] being uploaded on arXiv today or if it's just pure coincidence.

[0]: https://arxiv.org/abs/2503.10744

Hi! Wanted to ask you a question in another thread, but it was an old one and already closed. Glad I could catch you posting.

It was one of your posts regarding von Neumann, where you listed books about him. What I wanted to ask is I remember reading one or two comments by his friends from after his death (one for sure, but the second one might be from the same person, I didn't know) in which they talk about his hidden parameter proof. What I remember is that they were like "well, sadly after his death he was attacked for his proof not being absolute, but of course he knew that!" I'm looking for this quote or a similar one.

The usual suspects are of course Ulam or Birkhoff, but it could have been someone else, Wigner I don't think so, but maybe. I can't find it.

So that's what I thought maybe you could help me with.

What i think is super cool about h3(O) is that you have scalars, a vector and two spinors in one object with the jordan-product giving you all the products between them that have to do with triality. See section 3.4 of John Baez' paper on the octonions.

How does one formally define a spinor? I've seen the definition of a spinor field as "things that transform like a spinor", and a spinor as a "representation of the spin group" (which representation), but I would like to know a canonical mathsy definition of what the heck a "spinor" is! May I please have one? :)

How come "representation of the spin group" is an insufficient starting point?

Spin group seems like they either have a specific group (from Algebra) in mind or that spins are at least defined by choosing a specific group (a set with a binary operation satisfying the group axioms/definition).

A "representation" also has a definition in Algebra with regards to groups. There are group homo-morphisms between two groups. This means you have a mapping that preserves group structure. Representation theory is about mapping groups into the set of matrices or a subset of matrices "with numbers in the matrices." Then there are group actions (don’t care for the name) - basically/conceptually a set of functions that behaves like a specific group under composition, but way more notation around that. Finally, category theory looks at "groups of groups" with the binary operation being homo-morphisms between the "inside/smaller/contained/internal" groups thus forming a larger "outside" group called a category. Because this involves talking about sets of sets you end up also needing the term "class" from set-theory.

It's not that "representation of the spin group" is undefined, but that there are too many of them for it to pin things down uniquely. (In this case, fortunately, it's not hard to say which representation it is (see https://news.ycombinator.com/item?id=43388052), but just saying "a representation" isn't enough.)

While we're talking about representations, there's something I've always wondered. Why are the objects that the maps which are the representations act on also called representations? Spinors don't act as the spinor group, S ⊂ Hom(Spinor,Spinor) does.

That's the 'regular representation' of a group. https://en.wikipedia.org/wiki/Regular_representation

> While we're talking about representations, there's something I've always wondered. Why are the objects that the maps which are the representations act on also called representations? Spinors don't act as the spinor group, S ⊂ Hom(Spinor,Spinor) does.

Physicists and mathematicians speak differently, but I think that mathematicians usually avoid this language. For us, spinors are elements of the spinor representation, and, more generally, the things on which a representation acts are called generically "vectors in the representation", not representation themselves.

(That said, one will often see in math language like "let V be a representation of G", meaning more formally "let G \to GL(V) be a representation", which probably is the sort of abuse of language you mean.)

Actually in math parlance, the map G \to GL(V) is the representation while V is the underlying module, and yeah its elements are vectors.

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.

> 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).

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).

I slightly edited my post while you were writing yours. I'm aware of the article but I still think it could be clearer. I'm a big fan of constructive proofs but that does not seem the case for most mathematicians, so i find a lot of stuff in math very unsatisfying and have to rediscover it myself via different routes. Educational for sure but not always so easy.

And yes, of course there are many possible matrix reps, sorry i was not being precise. The ones i was referring to are the ones given here: https://en.wikipedia.org/wiki/Classification_of_Clifford_alg...

> How does one formally define a spinor? I've seen the definition of a spinor field as "things that transform like a spinor", and a spinor as a "representation of the spin group" (which representation), but I would like to know a canonical mathsy definition of what the heck a "spinor" is! May I please have one? :)

For Spin(8), three of the four fundamental representations are conjugate, and so we can use any one of them to define spinors.

Intriguing! I remember when Jordan algebras were the talk of the theoretical math town. Time to revisit the scene!

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