Daily Archives: June 29, 2008

The MNS Matrix as Magic Square

Recently, Kea observed that the CKM matrix can be written as the sum of 1-circulant and 2-circulant matrices. The CKM matrix defines the relationship between flavor eigenstates and mass eigenstates for the quarks. Her observation naturally suggests one should look also at the MNS matrix, which defines the relationship between flavor and mass eigenstates for the leptons.

Current experimental measurements show that the MNS matrix is approximately tribimaximal:
Tribimaximal MNS matrix
The form of the matrix is chosen so that the rows and columns are orthonormal. This is the mathematical definition of unitary, it means that when the matrix is multiplied by its conjugate transpose, it will give the identity matrix. The physics definition of unitary is a little looser, it only requires that probabilities sum to 1.

For the case of a mixing matrix, probabilities are given by the squares of the absolute values of the entries. Conservation of probabilities means that the sum of the probabilities for any row or column is 1. Since in quantum mechanics, the phase of any quantum state is arbitrary, we can consider modifications to the tribimaximal matrix by multiplying any row or column by an arbitrary phase. The result will be a new matrix but its probabilities will be unchanged (and therefore consistent with experiment).
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