The particles of the standard model fall into irreducible representation of the direct product of the Poincare group ISO(1,3), the gauge group S(U(3) x U(2)) - often written as SU(3) x SU(2) x U(1) -, and a group U(3) interchanging the particle generations participating in the weak interaction. This defines the symmetry group of the standard model. The unbroken symmetry group is only SU(3) x U(1) x U(1)^3; the remaining part of the symmetry group is broken in Nature: The Poincare symmetry through the nonuniform small-scale distribution of matter, the SU(2) x U(1) symmetry through the different masses of proton and neutron, and the generation SU(3) through the different masses of the electron generations.

There are two irreducible fermionic representation: the quarks form one, the leptons form the other. There are three irreducible bosonic representation: the gluons form one, photon, Z-boson and W-boson another, and the Higgs bosons the third.

Each of these representations splits into multiple irreducible representations under the Poincare group. Since it is customary to give names to the particles in each of these representations, we have 6 quarks, 6 leptons (3 electrons and 3 neutrinos), 8 gluons, 1 photon, 1 Z-boson, 2 W-boson, and several Higgs bosons, depending on the precise structure of the Higgs sector. Each of these particles has a fixed mass, the mass being a Casimir operator of the Poincare Lie algebra. The different particles in this sense may be viewed as the different components of the various fields appearing in the standard model Lagrangian if the (additional) spin index is suppressed.

If one splits instead into irreducible representations of Poincare x Isospin SU(2), one gets 3 quark generations, 3 lepton generations (each consisting of one electron and one neutrino), 8 gluons, 1 photon, 1 Z-boson, 1 W-boson generation, and in the minimal case one Higgs generation. The generation U(3) interchanges the three generations of quarks, and the three generations of leptons.

Quark can appear in a superposition of the mass states; this is due to the existence of a quark mass mixing matrix: http://en.wikipedia.org/wiki/Cabibbo–Kobayashi–Maskawa_matrix. Similarly, neutrinos can appear in a superposition of the mass states; this is due to the existence of a neutrino mass mixing matrix: http://en.wikipedia.org/wiki/Pontecorvo–Maki–Nakagawa–Sakata_matrix. Though the latter define by convention the standard quarks and neutrinos (as irreducible representations of the Poincare group), they appear in the form of superpositions in the weak interaction. For neutrinos, this explains the neutrino oscillations. There is no corresponding mixing matrix for the three generations of electrons since these are subject to the charge superselection rule, which forbids superpositions of different eigenstates of charges.

However, because of the mass mixing, the irreducible representations of the Poincare group do not form superselection sectors. As a consequence, there is no superselection rule for the mass, and the mass is a nontrivial operator (3 times 3) on the single particle sector of quarks (and of neutrinos).This contrasts with nonrelativistic mechanics, where the structure of the Galilean group forces such a (Bargmann) superselection rule, and hence every single-particle sector is characterized by a numerical mass.