Theory Update 22

Let us look at one reason why the new table of particle phases is nice. Take an annihilating pair with phases $\pm \pi /12$, such as in the diagram
which can unbraid to the identity braid of the photon. In terms of braid group generators, using fancy matrices, we can write down the product

for a suitable choice of $\phi$. In fact, the choice $\phi = \pi /4$ gives us the matrix relation $A^{12} = 1$. Thus a matrix represents the root of unity, something we see also in the MUB theory that underlies the mixing matrices. Now both the $\pi /n$ phase and the $2/9$ phase are associated to chirality, since annihilation requires a diagram obtained via reflection in the vertical axis.