## 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.

$\exp( i\left[\begin{array}{ccc} \theta_{12}+\theta_{13}&-\theta_{12}+i\theta_{123}&-\theta_{13}-i\theta_{123}\\ -\theta_{12}-i\theta_{123}&\theta_{12}+\theta_{23}&-\theta_{23}+i\theta_{123}\\ -\theta_{13}+i\theta_{123}&-\theta_{23}-i\theta_{123}&\theta_{13}+\theta_{23} \end{array}\right])$