2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/44273Large neutrino mixing angles suggest that the Yukawa sector is invariant under permutations of the fermion families. This S_{3} permutation symmetry is broken at a large energy scale but much below the unification scale. Assuming that the lepton mass matrix is approximately diagonal, all neutrino mixing angles naturally come from the breaking of S_{3}-> S_{2}. In the neutrino sector, S_{2} remains (approximately) unbroken. As a consequence, we have a large atmospheric neutrino angle and U_{e3}=0. The S_{3} symmetry at the unification scale can also explain the large solar mixing angle. We give an explicit expression of the solar mixing angle in terms of the left-handed neutrino masses. We observe that this family permutation symmetry comes very naturally from a quantized theory of functionals [1], that is an extension of quantum field theory.6 pagesHigh Energy Physics - Phenomenologytext