The exceptional Jordan algebra (2020)

https://cp4space.hatsya.com/2020/10/28/the-exceptional-jordan-algebra/

by gone35 • 1 year ago

108 15 1 year ago

cp4space.hatsya.com

aap_3• 1 year ago

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.

bollu• 1 year ago

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? :)

dapperdrake• 1 year ago

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.

aap_3• 1 year ago

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.

JadeNB4• 1 year ago

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.

Tewboo• 1 year ago

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

mjreacher• 1 year ago

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

DanceDuett2• 1 year ago

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.

The exceptional Jordan algebra (2020)

Related Stories