Neutrinos Again VI
Let's get back to neutrino mixing. Today, Carl Brannen links to some slides by Smirnov on Universality versus Complementarity for quarks and leptons. Complementarity is the observation that for tribimaximal mixing one has the relations
$\theta_{12} + \theta_{12}^{qu} \simeq \frac{\pi}{4}$
$\theta_{23} + \theta_{23}^{qu} \simeq \frac{\pi}{4}$
(and $\theta_{13}$ small) despite the fact that the CKM matrix is very different to tribimaximal. Smirnov then discusses an implied $\nu_{\mu}$, $\nu_{\tau}$ permutation symmetry of the form
But observe that this matrix can also be expressed as the sum of two (Hermitian) $3 \times 3$ circulants, since 2-circulants describe the 2-cycles in the permutation group $S_{3}$. 
As a formal combination of elements of $S_{3}$ (or the braid group $B_{3}$ if appropriate phases are added) we can represent this matrix sum as an element of a diagram algebra on three strands.
$\theta_{12} + \theta_{12}^{qu} \simeq \frac{\pi}{4}$
$\theta_{23} + \theta_{23}^{qu} \simeq \frac{\pi}{4}$
(and $\theta_{13}$ small) despite the fact that the CKM matrix is very different to tribimaximal. Smirnov then discusses an implied $\nu_{\mu}$, $\nu_{\tau}$ permutation symmetry of the form
But observe that this matrix can also be expressed as the sum of two (Hermitian) $3 \times 3$ circulants, since 2-circulants describe the 2-cycles in the permutation group $S_{3}$. 
As a formal combination of elements of $S_{3}$ (or the braid group $B_{3}$ if appropriate phases are added) we can represent this matrix sum as an element of a diagram algebra on three strands.
posted by Kea | 1:22 PM
10 Comments:
Wow! This seems like a clue.
So quarks are dual natured, 1-circulant and 2-circulant?
To fit this into the b-bbar and c-cbar mesons Koide spectrum, one would suppose that the charged leptons are naturally 1-circulant, while the neutrinos are 2-circulant.
So here's the question: How can you distinguish between the eigenvalues of a 1-circulant as compared to a 2-circulant? It should be possible to use either form, so one cannot distinguish between them until one gets them into a combined thing like a quark.
And this gives me new ideas on how I would put together a model for those mesons. I need to have a 3x3 matrix that resonates two different ways.
By the way, when I figured out how to put the weak hypercharge and weak isospin quantum numbers into idempotent form, the result was equivalent to two 3x3 matrices. And I have to suspect that one of them could be put into 2-circulant form.
I'm flying tomorrow to Albuquerque and it will be a great time to think about this.
That was fast. No flight needed. In fact, I know that the weak hypercharge / weak isospin quantum numbers DO fit into a 1-circulant + 2-circulant form.
The permutation group on three elements (R,G,B) has 6 elements. I've called the even permutations I, J, K, and the odd permutations R, G, and B. See dmfound.pdf for the details. The I,J,K subgroup is shown in that paper and it is obviously circulant, and the nature of the R, G, and B are clearly the other type of circulant.
(I hope I am using the right terminology. If not, by "2-circulant" I mean matrices that are reverse circulant.)
Hi Carl, yes that's the terminology I prefer since it's standard in the algebra literature. And I do feel like I'm making progress on the CKM matrix by playing with mixtures of circulant Fourier diagonalisers.
We should start taking bets on how many Standard Model parameters will be left before everybody decides we were always their best buddies.
Oh, Kea, it's probably a lot more fun to do things this way. As soon as they clue in, this field is going to explode and I will move on to things that better befit an old man like chess and porcelain. Did I mention that I now have a pottery wheel?
Meanwhile, I find that the eigenvalues of a (A,B,C) circulant matrix are A+B+C, and the square roots of A^2+B^2+C^2-AB-AC-BC.
For the case of the elementary fermion weak hypercharge and weak isospin quantum numbers, I was able to get away with A=B=C, or B=C=0, either way. Weak isospin was the sum A+B+C.
I crunched along on this while flying today and, assuming that I didn't make any errors, the tribimaximal matrix can also be written as the sum of a 1-circulant and a 2-circulant.
The MNS matrix needs to be maniplated first. You can always subtract a constant from each of the 1-circulant contributions I, J, K, and add the same constant to each of the 2-circulant contributions R, G, B, and the sum will stay the same. You can multiply any row or diagonal of the MNS matrix by a complex phase. Finally, swapping two rows (or columns) of the MNS matrix is equivalent to swapping 1-circulants for 2-circulants.
The calculation for how to write the MNS as 1-circ + 2-circ is kind of involved. I'll write it up in a few days. It's kind of a miracle how the thing fits together, not every matrix can be written this way. Or have you already got this in your voluminous notes or in a post which I didn't understand?
Heh, cool! No, I hadn't done that. I have been thinking more about the CMS matrix.
Marni, if a matrix M is written as M=rR+gG+bB+iI+jJ+kK, where R,G,B,I,J,K are the 3x3 permutation matrices and r,g,...,k are complex numbers, then the matrix M is a magic square (did you cover this already, I suspect you did), with the sum of the elements in any row or column equal to r+g+b+i+j+k. (This is obvious because each permutation touches any row or column exactly once.)
Now the MNS matrix is a matrix which is magic in that the sum of the squared magnitudes of the elements of a row or column always equals 1. And the rows or columns are orthogonal. But its relative phases are arbitrary; you can multiply any row or column by an arbitrary complex phase.
So the first question is "can you choose complex phases so that the MNS matrix becomes doubly magic? The answer is yes, and not only that, but the sums of each row and column is also 1. There is only one way to do this, except for the substitution i -> -i. The MNS matrix is very unusual, I wonder if it is unique as a doubly magic unitary matrix.
I just verified this, but I'm up at my dad's cabin, altitude 8000 feet, so it's at least possible that there is some hypoxia going on here. Let me write the MNS matrix in its simplest form and I'll put it up as a blog post. In the meantime, I'd like to see more about the CKM matrix and sums of circulants.
When you multiply rows and columns by arbitrary phases you lose orthogonality (which ensures conservation of probability), but phases are arbitrary and so this can always be done.
When you split the result into r,g,b,i,j,k, you have the freedom of moving around a constant. The various terms are small integer multiples of 1, i, sqrt(2), and sqrt(2)i, with the whole matrix then divided by 6. You can arrange for all the "small integers" to be in (-2,-1,0,1,2), and you can get cool patterns. Too late, time to go to bed.
Hypoxia at a mere 8000 feet? I think not. I had a little hypoxia once on a very windy day, but that was at 23000 ft. And yes, I have mentioned the magic squares.
Kea, the derivation was too long to realistically fit in a blog post, but the results are easy to verify and are now up. I can hardly wait to see what you do with the CKM matrix.
Meanwhile, I thought I would spend a few cycles thinking about what matrices are doubly magic (in the sense that applies here). My suspicion is that the MNS matrix is the only example doubly magic with sums 1 in both regular and squared magnitude form, but that the CKM matrix will have different sums.
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