Arcadian Functor

occasional meanderings in physics' brave new world

My Photo
Location: New Zealand

Marni D. Sheppeard

Monday, July 14, 2008

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 $(1)$ a 1-circulant and $(2)$ a 2-circulant. Just as for the $2 \times 2$ case with numerical matrix entries, we can think of $(1) \pm (2)$ as the eigenvalues of the $6 \times 6$ operator. Notice that the idempotents obtained have simple 2-circulants $(2)$ of democratic form, which means that adding or subtracting them from $(1)$ 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 $(1) + (2)$ 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.


Blogger CarlBrannen said...

You might enjoy Tony Smith's short story on various things that one would not necessarily expect in a single story.

July 15, 2008 9:47 AM  

Post a Comment

<< Home