Braids may be represented by matrices. Triality gives us three ways to embed a $2 \times 2$ matrix in a $3 \times 3$ one! That is three copies of the massless Bilson-Thompson spectrum, one for each generation pair.

If each entry is a bioctonion, this would be a $24$ (complex) dimensional operator. As a root of unity, this dimension is the neutrino phase $\pi /12$.

6 years ago

A remark on triality:

ReplyDeleteI believe you blogged that the discovery of large CP violation in B mesons would make the "c" parameter in your parametrization of the CKM matrix nonzero, and that this would be a manifestation of triality. But isn't this potential new form of CP violation something that can't be realized through the CKM matrix? Otherwise they wouldn't be calling it a beyond-standard-model effect. Or can you do it in octonionic QCD?

Hi Kea,

ReplyDeleteI agree, and my more pictorial view of things lately have been posted with similar pictures.

I have made some comments about you and other bloggers I hope are not off the mark.

But I hope people enjoy some of it at least from a recreational math viewpoint.

ThePeSla

Mitchell, the CKM matrix is not yet clear, but remember that the complex 3×3 algebra fits into the bioctonion one. ThePeSla, sometimes one cannot help being off the mark.

ReplyDelete