M Theory Lesson 243

Recall that the two dimensional Fourier type operator, which diagonalises a $2\times 2$ circulant, is combined with a three dimensional Fourier operator to obtain the tribimaximal neutrino mixing matrix. Using the more conventional Fourier operators, with entries $F_{ij}={\omega }^{ij}$ and $0\le i,j\le d-1$, the mixing matrix is expressed Note that the zero sum of the last column of the $2\times 2$ Fourier operator, in combination with the top row of ones on the $3\times 3$ operator, is entirely responsible for the zero entry of the MNS matrix. A product of standard Fourier operators always has this property, since the final column cycles through all the $d$-th roots of unity, which sum to zero.

In two dimensions, instead of cycling all three Pauli operators, like our usual choice of operator, the standard Fourier operator generates a 2-cycle of the form The standard basis for ${ℂ}^2$ forms the eigenvector set for ${\sigma }_Z$, and ${\sigma }_Z$ along with ${\sigma }_X$ form the generators for the noncommutative Weyl algebra, out of which further MUBs are constructed, in this case simply ${\sigma }_Y=-i{\sigma }_Z{\sigma }_X$.

The wikipedia article links to the original MUB paper by Julian Schwinger, who already knew about the quantum Fourier transform.