The importance of the tenfold way in physics was only recognized in this century. Simply put, it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when the topologist C. T. C. Wall classified the associative real super division algebras and found ten of them. The three 'purely even' examples were already familiar: the real numbers, complex numbers and quaternions. The rest become important when we classify representations of groups on super Hilbert spaces. I explain this classification, its connection to Clifford algebras, and some of its implications for quantum physics.

You can see the slides here, and also a video:

You can also watch another version, where I explain this material to my friend James Dolan:

I like the idea of being able to watch an official talk but also watch
the speaker chatting about the talk with a friend. It gives another
view of the material. I skim over stuff Jim already knows, explain
things I didn't have time to get into in the actual talk, and
emphasize the things I *don't* understand. And he points out
lots more patterns lurking in the tenfold way!

For more details, read this:

- John Baez, The tenfold way.

© 2023 John Baez

baez@math.removethis.ucr.andthis.edu