Let us recall once again that there are four distinct Jordan braids for the neutrino sector, expressed in $B_3$ by two crossing diagrams. Now since neutrinos mix, the electroweak states are not equivalent to the mass states. The MINOS experiment has shown that the so called neutrinos and antineutrinos (which reflect different electroweak sectors and not mass states) have distinct masses. Thus one pairs annihilating $B_3$ diagrams to form one neutrino type, so that the braid composition of the two distinct pairs (forming mass states) does not annihilate. For example, the pair
sums to give a mass state that is characterised by a diagonal identity matrix. When correctly normalised, the Jordan product of these two diagrams gives another (photon) identity matrix, up to the factor of $1/2$ that reflects the annihilation of two particles to create one photon. So for the neutrino sector, even $2 \times 2$ matrices can carry interesting information about fairyless electroweak symmetry breaking!
6 years ago