Generalising the spin concept from quaternions to octonions, any leads?

The spin operators have the same commutation structure as the quaternions.

Can we generalize the 3 spin operators to 7 spin-like operators that behave like the octonions? I'm looking for something matrix-like that would have the same commutation and associative properties as octonions.

Has anyone come across the development of this idea in the literature? I'd be interested to see what people have come up with.

• Two sources that you have probably found already: the geometry of the octonions Tevian Dray p166 and google.ie/url?sa=t&source=web&rct=j&url=https://…. Sorry for the links my phone is rubbish at c & pasting. But you can google the first the second is arxiv. Hope they pan out for ypu
– user108787
Commented Oct 27, 2016 at 10:43
• Note that spin operators are associative, while octonions are non-associative. Commented Oct 27, 2016 at 10:44
• CountTo10: Thanks for the leads. I've come across Tevian Dray, and I've seen research that he's done around this topic, but not an exploration of this topic yet. The paper is new to me, and so far the introduction looks interesting. Commented Oct 27, 2016 at 14:17
• I got the text by Tevian Dray. It's a pretty nice read, just the right level of abstraction. Based on how nice the presentation is, I'm sure I'm going to read the whole text. Commented Nov 11, 2016 at 5:46
• John Baez is in to these things: math.ucr.edu/home/baez/octonions
– JEB
Commented May 22, 2018 at 23:08

1 Answer

In asking for "something matrix-like" that "behaves like the octonions," what you're looking for seems to be a matrix representation of the octonions. Unfortunately, there is no way to represent the octonions as ordinary matrices, since matrix multiplication is always associative while octonion multiplication is not. However, there are some modifications you can make to matrix algebras (e.g. changing the way that matrix multiplication works or giving the matrices themselves a kind of bra-ket structure) that will allow you to get the kind of group action that you want. For more, see https://math.stackexchange.com/questions/96429/matrix-representation-of-octonions.

• That's exactly what I'm hoping to find, some sort of generalization of the matrix concept that could include operators that have both the association rules as octonions, and also could have eigenvalues and eigenvectors like spin-1/2 or spin-1 matrices. Commented May 24, 2018 at 2:03
• @DavidElm Probably the closest thing would be the Zorn vector-matrix algebra, as discussed in the link in my answer. Commented May 24, 2018 at 17:23