Mass spectral condition for leptons and quarks

Consideration is given to the mass spectral conditiona(a,_m)a=a_m, wherea is the annihilation operator forQ=–1 states in the Hilbert space of leptons and quarks anda_m is the commutator resolvent ofa with respect tom. It is observed that this spectral condition, which simply requires a to be a congruent automorph ofa_m, implies that the third-, fourth-, and fifth-generationQ=-1 leptons have the masses 1788.03 MeV, 42.1649 GeV, and 1.33422 TeV, respectively. With the assumption that the mass spectral condition also holds forQ=0 states in the Hilbert space, one obtains new theoretical upper and lower bounds on the neutrino masses.