Given my final success in writing the CKM matrix as the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix, the next step is to write down the parameterization that allows any 3×3 unitary matrix to be written in this elegant and natural form. Following the method used earlier to parameterize unitary 3×3 magic matrices, and correcting for a few typos (but the description of the method is correct).

We will use four real angles, , as parameters. First, define 6 real numbers I, J, K, R, G, and B as follows:

The the following matrix is unitary:

This is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. Note that the sum of the 3 elements of any row or column is equal to the complex phase .

The choice of the variable names I, J, K, R, G, B is to remind of the permutation group on three colors. I is the identity. J and K are the cyclic (even) permutations. And R, G, and B are the swaps that leave red, green, and blue unchanged, respectively. Earlier, we wrote the weak hypercharge and weak isospin quantum numbers as solutions to the idempotency problem for matrices of a similar form. See Electroweak Unification, Quarks for the rather difficult algebra involved in solving the resulting 6 quadratic equations in 6 unknowns.

The angles and show up three times each in the definitions of I, J, K, R, G, B. They are angles and have added to them constants of 0, 120, and 240 degrees. Equivalent angles show up in the Koide mass equations. As Marni Sheppeard says, this is the result of taking discrete Fourier transforms on three elements.

Of course one wonders what values of the four parameters define the CKM and MNS matrices.

CKM parameters, theta, alpha, beta, gamma:

+0.207642790, -0.006257792, -1.199063520, -0.273088590 (radians)

+11.89705551, -0.35854507, -68.701279, -15.64682364 (degrees)

The CKM parameter angle gamma = -15.65 degrees, defines the split between the average real part and average imaginary part. In the Koide formulas, this was the “valence” or “v” parameter. The angle alpha is close to zero. With gamma being relatively small, this makes I dominate over J and K. Finally, the angle theta defines the ratio of the “sea” (or “s” in the Koide mass formulas) sizes of I,J,K relative to R,G,B, that is, the part that is contributed with three angles alpha, alpha+120, and alpha+240 degrees. Since theta = 11.89 degrees, the CKM matrix is about 5x larger.

MNS parameters, theta, alpha, beta, gamma:

+1.030839502, +0.758336406, -0.424863234, +0.169918454 (radians)

+59.06275282, +43.44947551, -24.34287018, +9.735610275 (degrees)

As you can see, the MNS parameters seem to be pretty much random numbers. This is despite the fact that the MNS unitary matrix, when written in this parameterization, is remarkably simple:

So the arbitrariness of the CKM parameters (which are the result of approximate experimental measurement and consequently have error in them) does not give any sort of evidence that the CKM matrix cannot be written in a simple, elegant, and clean form as the sum of a real 1-circulant and an imaginary 2-circulant matrix.

What I’d like to do next is to rewrite the parameterization in such a way that the MNS parameterization becomes nice. That is, I’d like to see the elegance of the MNS matrix in its parameterization. This might give a clue as to how to analyze the CKM values, and this might allow us to derive a theoretical CKM matrix that is as successful as the tribimaximal theoretical MNS matrix.

I’m really irritated that 20 years ago I’d have understood where you were coming from.

Twenty years later, I find my brain has turned to mush.

That really bugs me…

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Hi

I thought that a general Unitary 3×3 matrix had 3 mixing angles and 6 phases. Can you write a general unitary matrix in this form?

Of course in the case of the CKM matrix you can remove 5 non-physical phases, but in general this is not possible. For example the PMNS matrix usually has 3 phases.

Jon,

The parameterization seems to work for unitary matrices with the freedom to multiply all rows and columns by arbitrary phases (such as the CKM matrix). To get a generic 3×3 unitary matrix you’d get back those five phases by multiplying rows and columns by complex phases.

By “seems to work” I mean that the “proof” is from counting degrees of freedom, and that it works for the fairly random matrices I’ve tried with it, (like the CKM). So I suspect it works for almost all 3×3 unitary matrices. If anyone has a proof, or knows counter examples I’d love to hear it. From the messing around I’ve done with it, I suspect it works for all unitary matrices or almost all.

The PMNS matrix will be different from the CKM matrix only if Majorana neutrinos are used, I think.

Hi Carl

Thanks for the reply. This is very interesting.

I am trying to represent a unitary matrix which is also symmetric.

I was reading an article which stated in passing that such a matrix has 6 real parameters. So I was thinking maybe your work would help me understand this.

In your representation, this type of matrix would require setting alpha=0 or pi. So that leaves 3 real parameters. That’s very good.

But then I need to add back 3 phases by multiplying row and columns as you mentioned. So there are 3 real parameters and 3 phases.

So I am wondering why there are 3 phases instead of 3 more real parameters as mentioned above.

So you use the same phase for a row as a column. Yes, that should be a total of six parameters. But I don’t see what the difference is between real parameters and phases. A phase is a multiplication by a particularly simple real parameter, .

What I find interesting about the parameters and or phases is that they’re all cyclic coordinates, cosines and sines instead of cosh or sinh. I guess that has to do with the Lie algebra being compact.

Hey, if you manage to prove that your observation gives a parameterization of all possible symmetric unitary 3×3 matrices could you leave me a note to that effect? Somehow I suspect that this is all in the mathematics literature of maybe the 1890s.

Hi Carl

Thanks again for the quick reply.

As I understand it many people in the literature talk of real parameter as those which lead to real quantities, and phases as those leading to complex quantities. So even though they may both be real numbers, the quantities they parameterize are different. If that makes sense.

Our conversation here has rekindled my interest in the subject so I’m going to set myself the task of proving that the parameterization is complete. I’ve got some new ideas for this, some stuff having to do with the unitary matrices being connected and compact, but I’m guessing that the problem will fall to some algebra drudge work. I’d also like to see what happens in 4×4 matrices but I don’t have any applications that need it.

To motivate myself I’ve written the equations into a paper which does related calculations. Now I have to finish the calculation or retract it… I’m guessing 6 hours of rock and roll should do it.

These things have to do with the discrete Fourier transform as was pointed out by Marni Sheppeard. My interest is in generalizations of pure density operators / matrices other than statistical mixtures.

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Did the question of completeness ever get resolved?