Six-fold discrete symmetry of the QED Lagrangian

Using the Pauli algebra version of the Dirac equation six symmetric but non-equivalent copies of the QED Lagrangian are constructed. The group of symmetries is the group of outer automorphisms of the finite quaternion group, isomorphic to the group of permutations on three letters. Solutions to corresponding Dirac equations form six sectors of fermion fields. The intersections between different sectors can only contain massless fields, implying that the sectors are distinct. If the symmetry could be extended to the full Standard Model Lagrangian it would predict the ratio of dark matter to ordinary matter of 5 : 1 which is close to the ratio of 5.2 : 1 according to current observations.