garrett said:
This is the main reason I think what you and Carl have been working on, with the CKM and MNS matrices, is interesting, and deserves more attention.
Wow, thanks for the compliment. I should read Physics Forums more often!
This seems like a good time to post the new results. On this particular subject, Marni's been providing the key insights and I've been doing the manual labor / number crunching. While CKM solution was found using a Java program, it's explainable without going through that.
The CKM and MNS are 3x3 matrices. They begin with experimental measurements, which are of the absolute values of the unitary matrix elements. To make such a matrix unitary, one multiplies the 9 components by complex phases in such a way that the rows and columns become orthonormal. Once you find such a unitary matrix, you can multiply a row or column by a complex phase and it will remain unitary. So there are a lot of choices to represent the same set of experimental numbers. Given a particular choice, or "parameterization", one needs 4 real variables to describe the possible sets of experimental numbers. A parameterization is supposed to associate each set of numbers with a specific unitary matrix.
The usual parameterizations are the K&M, Euler angles, and the Wolfenstein. They are given in the Wikipedia article on the CKM matrix. A week ago I found a pretty parameterization that chooses the phases such that a 3x3 unitary matrix splits into a real and imaginary part:
http://carlbrannen.wordpress.com/2008/10/09/a-new
The real part is 1-circulant (each row shifts one to the right), while the imaginary part is 2-circulant (each row shifts two to the right). Like the other parameterizations, this is (more or less) a unique way of describing a unitary matrix. I'm attracted to the form partly because I'm a fan of density matrices which also eliminate arbitrary complex phases in an elegant way. The best explanation is an example.
The MNS experimental data is approximately given by the "tribimaximal" values:
\begin{array}{c|ccc}<br />
&\nu_1&\nu_2&\nu_3\\ \hline<br />
e&2/3&1/3&0\\<br />
\mu&1/6&1/3&1/2\\<br />
\tau&1/6&1/3&1/2\end{array}
Rewriting this in real 1-circulant plus imaginary 2-circulant form we have a result dating to around June:
\left(\begin{array}{ccc}<br />
\sqrt{1/3}&\sqrt{1/6}&0\\<br />
0&\sqrt{1/3}&\sqrt{1/6}\\<br />
\sqrt{1/6}&0&\sqrt{1/3}<br />
\end{array}\right)<br />
\pm i\left(\begin{array}{ccc}<br />
\sqrt{1/3}&-\sqrt{1/6}&0\\<br />
-\sqrt{1/6}&0&\sqrt{1/3}\\<br />
0&\sqrt{1/3}&-\sqrt{1/6}<br />
\end{array}\right)
The CKM matrix experimental data (from
http://arxiv.org/abs/0706.3588 ) is:
\begin{array}{c|ccc}<br />
&d&s&b\\ \hline<br />
u&.974134&.225921&.0042982\\<br />
c&.225766&.973323&.0409179\\<br />
t&.009583&.040019&.9991530\end{array}
In real + imaginary form this is approximately given by:
\left(\begin{array}{ccc}<br />
+.97331&-.00858&+.00047\\<br />
+.00047&.+97331&-.00858\\<br />
-.00858&+.00047&.+97331<br />
\end{array}\right)<br />
\pm i\left(\begin{array}{ccc}<br />
+.04001&+.22576&-.00427\\<br />
+.22576&-.00427&+.04001\\<br />
-.00427&+.04001&+.22576<br />
\end{array}\right)
which dates to October 8:
http://carlbrannen.wordpress.com/2008/10/08/ckm
Note that the two matrices each have only 3 real degrees of freedom for a total of 6.
This is basically the latest stage of a project Marni Sheppeard started back in June. She had noticed that the CKM matrix was approximately the sum of a 1-circ and a 2-circ. From there, it was natural to work out the MNS matrix, since it had a simple form that was easy to deal with. And she's pointed out that all this is related to the discrete Fourier transform.
In the last week or so, a flurry of emails and blog comments has given a few more results. I should type them up and make them into a post, but things are busy.
Ignoring an overall factor of sqrt(1/3), the discrete Fourier transform (DFT) of a vector of 3 elements (a,b,c) is defined as the three results:
A = a + b + c,
B = a + wb + w*c,
C = a + w*b + wc,
where w = \exp(2i\pi/3). This can be accomplished by multiplying the vector on the right by a matrix with the nine values:
\left(\begin{array}{ccc}<br />
1&1&1\\<br />
1&w&w^*\\<br />
1&w^*&w<br />
\end{array}\right)
To take the DFT of a 3x3 matrix, you multiply on the right by the above matrix, and on the left by the inverse transform. In the case of these matrices, this amounts to taking a DFT over generations for both the electrons and the neutrinos, or for both quark charges.
The eigenvectors of an arbitrary 1-circulant matrix are always (1,1,1), (1,w,w*), and (1,w*,w). So it turns out that the discrete Fourier transform of a 3x3 1-circulant matrix is a 3x3 diagonal matrix. And similarly, the DFT of a 2-circulant matrix is an "anti-diagonal" matrix.
The usual DFT is linear, and so is the matrix version. This means that the real 1-circulant + imaginary 2-circulant forms we've written the CKM and MNS matrices in amount to a choice of phase such that the DFT of the unitary matrix is as simple as possible.
The DFT converts the CKM and MNS matrices, when this new parameterization is chosen, into matrices that are very reduced: the degrees of freedom of the unitary matrices are written so that as many values are zero as possible. They're not as reduced as a diagonal matrix but it's close. You could say that this is the least possible non commutative generalization of diagonal.
The implication is that the MNS matrix is peculiarly simple in this parameterization because the DFT has something to do with the generation structure of the leptons. And from that, one supposes that there may be a pattern in the CKM matrix as well. But no one has found that pattern so far. Nevertheless, the CKM numbers are suggestive in that they follow the usual generation pattern in that one gets a sort of scale between the three values of about 9. Since the Koide mass formulas are also related to the DFT, we suspect that the 6 CKM entries are also so related.
Where is this going? Like the Koide formula the neutrinos, I think that the lepton version of this is sufficiently elegant that it should start showing up on arXiv. Right now I'm busy writing up a paper extending Koide's mass formulas to the hadrons, which I think is more important.
