M Theory Lesson 328

Even when reducing to $2 \times 2$ submatrices, the scaled $R_2$ factors define a basis for a three dimensional space via their column vectors. Let us consider how the two parameters $(a,b)$ give rise to a tribimaximal type probability matrix.

When two bases are mutually unbiased, the inner product of the column vectors between bases always gives the same result. That is, the probabilities are all equal. For two distinct $R_2$ factors, on the other hand, the nine possible column products are all distinct! With the denominator $(a^2 + 1)(b^2 + 1)$ (for the squares) we have the nine numerators: For the neutrino mixing choice of $(a,b) = (\sqrt{2}, 1)$ we observe that these nine numbers become the entries of the tribimaximal matrix. In order to generalise to the $27$ possibilities for three variables, we will place these $9$ numbers in a square, which is basically the mixing matrix. Interpreting the zero as $(a^2 + 1)(b^2 + 1)$, the three points on the edges all obey the sum rule $P + Q = R$. This rule extends naturally to the cube in three dimensions, where the three parameters $(a,b,c)$ are symmetrically represented.