Arcadian Functor

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Marni D. Sheppeard

Tuesday, December 02, 2008

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.


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