## Wednesday, August 18, 2010

### M Theory Lesson 343

Recall that the circulant sum form of the neutrino mixing matrix may be expressed as a product $X \cdot Y$ of two $R_2$ factors. This product is a sum of a real $1$-circulant and an imaginary $2$-circulant. If we turn the ordinary matrix product into a Jordan algebra product $X \circ Y$ (much beloved by kneemo) then the $1$-circulant piece will be symmetrised, and the $2$-circulant piece remains unaltered. So for the neutrino mixing matrix, the one circulant piece goes to the norm of a Koide matrix for leptons. Now the nondegenerate mass triplets just require the addition of the appropriate complex phase.

1. Hmm, so it may have been more correct to start with $(1, 1/\sqrt2)$ for the R2 parameters. That is, the choice where both parameters have norm less than or equal $1$.