Neutrinos Again V
The $2 \times 2$ component of the second factor diagonalises a $2 \times 2$ circulant 
which is why Harrison et al selected such an operator for their neutrino mass matrix. But mixing is about sending mass states to weak states, so it makes more sense to consider a factorisation $U_{m}^{\dagger}V_{w}$ where $U_{m}$ is the universal $3 \times 3$ circulant diagonalisation operator. One can have fun switching rows or columns. For example, a codiagonalisation of $3 \times 3$ 2-circulants is given by 
One can combine a row switch in $U_{m}^{\dagger}$ with a column switch in the second operator to obtain 
which is just the tribimaximal mixing matrix again, up to some phase factors. Let us imagine adding an identity matrix factor as a one dimensional operator on the right, thus forming a triple product of Fourier operators, one for each dimension up to three.
Aside: Check out Carl's post on Koide fits for mesons.
which is why Harrison et al selected such an operator for their neutrino mass matrix. But mixing is about sending mass states to weak states, so it makes more sense to consider a factorisation $U_{m}^{\dagger}V_{w}$ where $U_{m}$ is the universal $3 \times 3$ circulant diagonalisation operator. One can have fun switching rows or columns. For example, a codiagonalisation of $3 \times 3$ 2-circulants is given by 
One can combine a row switch in $U_{m}^{\dagger}$ with a column switch in the second operator to obtain 
which is just the tribimaximal mixing matrix again, up to some phase factors. Let us imagine adding an identity matrix factor as a one dimensional operator on the right, thus forming a triple product of Fourier operators, one for each dimension up to three.Aside: Check out Carl's post on Koide fits for mesons.
posted by Kea | 5:11 PM
3 Comments:
Kea, I'm so glad you're dealing with this. The MNS matrix is just as important as the masses and just the mass problem is more than enough for me. I think you understand this better than anyone else. The corresponding problem for the quarks, the MNS matrix, beckons.
Also, for an interesting paper on qubits and error correcting codes, see Dave Bacon's latest post and paper.
Heh, thanks for the link! Quarks? Gee, I could happily spend the rest of my waitressing career thinking about neutrinos, but I suppose I will start thinking about quarks again some time.
I'm beginning to suspect that the key to the Koide splitting of the b-bbar and c-cbar mesons is tied up with the MNS matrix for quark mixing.
In both cases, what's happening is due to the quark being composed of a mixture of stuff that wants to resonate at different frequencies.
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