### 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 $0 \leq i,j \leq d - 1$, 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 $\mathbb{C}^{2}$ 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.

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 $\mathbb{C}^{2}$ 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.

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