M Theory Lesson 206

Carl Brannen's new post on 1-circulant and 2-circulant operators extends his previous analysis to the remainder of the fundamental fermions and their quantum numbers. He works with $6\times 6$ circulants of the form for $\left(1\right)$ a 1-circulant and $\left(2\right)$ a 2-circulant. Just as for the $2\times 2$ case with numerical matrix entries, we can think of $\left(1\right)\pm \left(2\right)$ as the eigenvalues of the $6\times 6$ operator. Notice that the idempotents obtained have simple 2-circulants $\left(2\right)$ of democratic form, which means that adding or subtracting them from $\left(1\right)$ results in another 1-circulant. For example, for the $e_R^{+}$ quantum numbers one finds that which is a unitary 1-circulant since all entries have norm $\frac{1}{3}$. The same matrix results from $\left(1\right)+\left(2\right)$ for ${\overline{\nu }}_R$. The democratic matrix with all values equal to $\frac{1}{3}$ comes from, for instance, the ${\overline{d}}_L$ quark idempotent. Tony Smith, who likes to think of the Higgs as a top quark condensate, might like this correspondence between Higgs numbers and quark operators.