Probably the first people to realise that the number of generations might have something to do with idempotents and Jordan algebras were Dray and Manogue in 1999, in *The Exceptional Jordan Eigenvalue Problem*, which was pointed out to me by kneemo. On pages 10 and 11 they discuss how the usual Dirac equation comes from the 9+1 dimensional one, which is written as a simple eigenvalue problem using a 2×2 octonion matrix, or again as a nilpotent equation using the Freudenthal product. The three generations fit into the Moufang plane, which are Jordan elements satisfying

$M \circ M = M$, tr$M = 1$

so the matrix components lie in a quaternion subalgebra of the octonions. These elements are **primitive idempotents**.

Naturally we should improve upon the reliance here on a higher dimensional Dirac equation, for which we see no real physical motivation. Brannen’s idempotents are a big step forward in this regard. But we can also reinterpret the higher dimensions in a categorical context, where they are not naively taken to mean spatial dimension.

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## CarlBrannen said,

February 21, 2007 @ 6:17 am

My Java programming effort to find the charged lepton masses in the mass (scalar) spectrum of 3×3 non Hermitian primitive idempotent matrices has completed.

Sadly, one cannot obtain that danmned number. Instead, the only solutions give delta = pi/12, which are the solutions that give a zero mass electron.

With Hermitian primitive idempotents, there is only one solution, which is spins relatively oriented by 90 degrees. Non Hermitian PIs allow a lot more solutions, but they all have delta = pi/12, just like the single Hermitian solution.

I should have seen this coming. We can’t understand the electrons without simultaneously understanding the neutrinos. And I bet it’s going to be a mixing between the two things, just like the Z versus the photon.

Well, at least now I’ve got some nice tools for doing this sort of thing.

## Kea said,

February 21, 2007 @ 8:37 pm

We can’t understand the electrons without simultaneously understanding the neutrinos.Yeah, we can only keep trying! Maybe we need to solve the Riemann hypothesis before we can derive delta.