### M Theory Lesson 230

The four sets of eigenvectors associated to the matrices $M_{i}$ are

$B_{1}: (\omega^{2}, \omega^{2}, 1),(1, \omega^{2}, \omega^{2}),(1, \omega, 1)$

$B_{2}: (\omega, \omega, 1),(1, \omega, \omega),(1, \omega^{2}, 1)$

$B_{3}: (1,1,\omega),(\omega,1,1),(\omega^{2}, 1, \omega^{2})$

$B_{4}: (1,1,\omega^{2}),(\omega^{2},1,1),(\omega,1,\omega)$

Observe that the duality maps $B_{1} \rightarrow B_{2}$ and $B_{3} \rightarrow B_{4}$ are simply given by complex conjugation on the eigenvector entries. The 12 vector points lie in a projective space in three (complex) dimensions, just as the 6 points of the two dimensional MUBs lie on a Bloch sphere. Points in twistor space $\mathbb{CP}^{3}$ naturally appear with MUBs in dimension four.

The general connection between the Fourier transform for finite fields and MUBs is explained in papers by Planat et al. In M theory, we will view the character maps from Galois groups into the complex numbers as pieces of functors into the infinite dimensional categories where the complex numbers rightfully belong.

$B_{1}: (\omega^{2}, \omega^{2}, 1),(1, \omega^{2}, \omega^{2}),(1, \omega, 1)$

$B_{2}: (\omega, \omega, 1),(1, \omega, \omega),(1, \omega^{2}, 1)$

$B_{3}: (1,1,\omega),(\omega,1,1),(\omega^{2}, 1, \omega^{2})$

$B_{4}: (1,1,\omega^{2}),(\omega^{2},1,1),(\omega,1,\omega)$

Observe that the duality maps $B_{1} \rightarrow B_{2}$ and $B_{3} \rightarrow B_{4}$ are simply given by complex conjugation on the eigenvector entries. The 12 vector points lie in a projective space in three (complex) dimensions, just as the 6 points of the two dimensional MUBs lie on a Bloch sphere. Points in twistor space $\mathbb{CP}^{3}$ naturally appear with MUBs in dimension four.

The general connection between the Fourier transform for finite fields and MUBs is explained in papers by Planat et al. In M theory, we will view the character maps from Galois groups into the complex numbers as pieces of functors into the infinite dimensional categories where the complex numbers rightfully belong.

## 5 Comments:

I found it here.

Ah, OK, thanks Carl.

Actually it's better than the original because it includes the inane comments by the editor.

Every now and then I figure out where you're going with this. The stuff you're talking about has to do with how one stuffs the quarks and leptons into 3x3 matrices. I think that what you're discussing here is related to the generation structure that you need when you derive the standard model fermions weak hypercharge and weak isospin quantum numbers from 3x3 idempotent matrices.

That is, what I linked to are the density matrix versions of the same thing. This sort of explains why we need MUBs in the mass equations.

And it makes me think that the clue understanding the quark mixing matrix is to look at qutrit MUBs rather than Pauli MUBs.

OK, I added a link to your qutrit MUB post from February. I'm sure we'll get back to quark mixing soon enough.

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