Wednesday, May 26, 2010

M Theory Lesson 333

Recall that the $2 \times 2$ $R_2$ matrix is only one of three MUB matrices, namely the one corresponding to the Pauli matrix $\sigma_{Y}$. We could consider $3 \times 3$ product forms using the MUB matrices for $X$ or $Z$ directions. Recall that these are, respectively, a Fourier matrix and the identity matrix. But a mixed $XYZ$ product results in the same two factor (tribimaximal type) form, since a scaled identity does not alter the mixing matrix, and the use of $i F_2$ only affects the phases.


  1. Is that the same $b$ as in lesson 321's CKM parametrization?

    $b$=24 there is an interesting result. And I'd like to be able to judge whether it might mean something. But I'm stymied by still not understanding what motivated you to look at those particular matrices. So here is a list of things I don't understand.

    The CKM matrix was decomposed into a set of three "scaled $R_2$ factors". I can see that the notation $R_i$ comes from Combescure, for a type of MUB-generating circulant that can be constructed in any prime dimension. $R_2$ is the Combescure R-matrix in 2 dimensions, and it happens to coincide with one representation of one of the Pauli matrices.

    1) Do Combescure R-matrices in other dimensions correspond to generalized Pauli matrices?

    2) I assume that "scaling" refers to the introduction of the parameters $a$, $b$, $c$. And then you use *three*-dimensional matrices. What do these changes do to the interpretation of the matrix as a Pauli matrix? What do they do to its interpretation as generator of a MUB?

    3) All that I really understand of Carl's work is that he gets masses as eigenvalues of circulant matrices. I can understand that, having done that, you would want to look at other mass-related parameters in terms of circulants somehow. But I can't tell if the relationship here is comparable. Are the CKM matrix elements the eigenvalues of some of these matrices you have constructed?

    4) I also really don't understand the relationship between being a sum of a 1-circulant and an imaginary 2-circulant, and being a product of three of these generalized $R_2$ matrices. Is there a theorem that one implies the other? Does the theorem have a physical or "quantum information" interpretation?

    I also have no intuition yet for how different $abc$ values would affect phenomena (as you were beginning to explore in your preceding post), but I haven't even formulated the questions there yet, so I'll stick to issues of algebra and quasi-physical motivation for now.


  2. P.S. in question 2 I mean 3x3 matrices.

  3. Hi Mitchell. Yes, the same b may be used. Some brief initial answers:
    1) Yes
    2) I'm still not sure how to think of this, but the scaling only on the diagonal is allowed for the mass matrices. Note that it is essential for the complex parts to be pure imaginary, so that the two circulant components don't interfere. One is then free to normalise the scale onto the diagonal.
    3) Mixing elements are like MUB inner products, not eigenvalues. Recall that Carl derived circulant generation operators from spin path integrals, so that is one motivation for the cyclic ansatz on generations.
    4) There is a nice theorem here. We are working on it.


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