M Theory Lesson 15

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.


2 Responses so far »

  1. 1

    CarlBrannen said,

    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.

  2. 2

    Kea said,

    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.

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