nostr:nprofile1qy2hwumn8ghj7un9d3shjtnddaehgu3wwp6kyqpqgyewnrnvhqg0dlqmn5mfp0wjcvpkgxycjat9527w8cmf0c6cxwqs9njev0 wrote: "But how do we proceed from there - how do we decide which of these constructions is the best/most interesting/most promising for physics?"

Nobody has a fucking clue, which is why particle physics has been completely stuck since 1980, except for the experimental revelation that neutrinos have mass.

Well, "completely stuck" is a bit of an exaggeration. People are trying lots of things, none of them have worked, but we are still perhaps learning things in this frustrating process. Different approaches succeed to different extents, and the full answer to your question would outline all of what's been done - too long for a post here!

I am trying to follow a particular approach now, which may be wrong but deserves to be tried. Namely, to take the octonions, exceptional Jordan algebra, and so on seriously. This means, for example, taking seriously the fact that a Euclidean Jordan algebra serves as both an algebra of observables and a kind of "spacetime", with a lightcone structure. The consequences of this fact are well-understood for 𝔥₂(ℂ): we get the connection between qubits, spinors, the Riemann sphere, and Minkowski spacetime, some of which is nicely explained in Penrose and Rindler's books.

Maybe we should think about how all this works for 𝔥₃(𝕆).

The fact that the Standard Model gauge group arises from the interplay of 𝔥₃(𝕆) and 𝔥₂(ℂ) feels incredibly suggestive to me.

But I hadn't known where to go with it.