## Saturday, April 17, 2010

### M Theory Lesson 313

The transformed quark matrices did not look like braids, but they are nonetheless familiar. If we use the $Z$ boson triplet to define three cycled Fourier matrices, the quark boson becomes a sum. Moreover, the tribimaximal mixing matrix now appears in the form $QF_2$, where $F_2$ is the usual two dimensional Fourier operator. This is now a symmetric representation with regard to the electroweak basis. The tribimaximal form is remarkably robust, also resulting from the other quark boson matrices. Observe how this form associates a colour index with three out of nine possible Fourier matrices.

1. Interesting! It seems that the electric charge is proportional to the total imaginary content of the 3x3 matrices. And this implies a new way of parameterizing unitary matrices which I will attempt to tex into another comment.

2. I can't preview latex, but the way I've been parameterizing the 3x3 unitary matrices is:
$\exp( i\left[\begin{array}{ccc} \theta_{12}+\theta_{13}&-\theta_{12}+i\theta_{123}&-\theta_{13}-i\theta_{123}\\ -\theta_{12}-i\theta_{123}&\theta_{12}+\theta_{23}&-\theta_{23}+i\theta_{123}\\ -\theta_{13}+i\theta_{123}&-\theta_{23}-i\theta_{123}&\theta_{13}+\theta_{23} \end{array}\right])$

The coefficient for theta_123 amounts to the difference between the two neutrinos in your paper. But maybe there's another way to split up the imaginary degrees of freedom (which basically encode the degree to which the unitary matrix is anti-symmetric).

3. Carl, please use dollar signs for in line latex ... matrices are too ambitious for comments.