Elementary particles as irreducible representations

Question

It is known that, in Wigner's view, free elementary particles are nothing but irreducible representations of Poincaré group. However, Arnold Neumaier recently suggested to me that this picture is not appropriate,  in fact it is misleading,  due to what  nowadays we know  about (physically) elementary particles: Both quarks and neutrinos are not described in terms of such representations because the mass is not a Casimir operator of the representation as the mass of these particles is not a constant observable, but is described by the Cabibbo-Kobayashi-Maskawa Quark Mass Matrix  and a Neutrino Mass Matrix. He also pointed out to me that these particles actually are irreducible representation of the global symmetry group.

Could you show me the irreducibility of these representations of the global symmetry group or provide me some technical reference where an explicit  proof appears?

Answer 1

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.

Comments

Very interesting Peter...Indeed, as a representative of Italian community, I am always confused when I have to decide between Galilean and Galileian.

There is a similar problem/confusion in Italian: "gruppo di Galileo" (name of the famous scientist)  or "gruppo di Galilei" (family name of the famous scientist).

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''neutrinos including the matrix of masses'' - this only breaks the generation symmetry, which is already broken because of the different masses of the electron and its heavier generation partners.

The 1-particle Hilbert space is the space of linear mappings from $R^3$ (coordinatizing the spatial momentum) - with time component of $p$ determined by the masses (in the mass eigenstates) - to a finite-dimensional complex representation space which is the direct sum of the representation spaces of the four irreps of the biggest symmetry group mentioned at the beginning of my answer. Thus it is not irreducible. The smaller one makes the group the more irreducible components one gets. (Various grand unification schemes therefore try to work with larger groups, where all particles fit into a single irrep, but there is no experimental evidence for that, as it invariably means that many additional particles must also exist, and none of them turned up so far.)

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It is not irreducible, you argued that it was irreducible. I quote from you comment to my answer in PSE:

"In today's theories, elementary particles are irreducible representations of the full symmetry group, which is not just the Poincare group. If the mass transforms nontrivially under the internal symmetry group, it is a matrix rather than a number."

What did you mean there so?

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My quoted remark referred to each class of elementary particles separately. 

The representation of all elementary particles is not irreducible. But this is because elementary particles with different spin cannot be in a common finite-dimensional irreducible representation of a group. But the discussion on PSE was not about all elementary particles but about neutrinos (or quarks), and these are in a single representation of Poincare x SU(2)_iso x U(3)_gen. All components of this representation space are mixed since there is no corresponding superselection rule. The representation splits into 3 (neutrinos) resp. 6 (quarks)  irreducible representations of the Poincare group. These representation are mixed by the interactions, hence neutrino and quark masses are mixed as well. 

In fact, this implies that there is even a simpler example of mass mixing (which I didn't think of before) - the nucleon. It is the basic particle in nuclear physics and appears in their models as superpositions of proton, neutron and their antiparticles. The charge superselection rule (for the U(1) charge of electromagnetism) doesn't apply on the nuclear level as nuclear physics is blind to electromagnetic interactions. 

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Thank you for posting this discussion.  (1)  Can you be more explicit about how U(3)_gen acts?  (2)  In the high energy limit where the SM gauge group is unbroken, shouldn't fermions be massless, and hence LH and and RH fermions in distinct irreps?

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Note that the generation U(3) is usually not considered to be a symmetry, as it is explicitly broken in the standard model Lagrangian to U(1) x U(1) x U(1). But it is naturally associated with the mixing matrices. Dropping it from the description doesn't change the main conclusion that even for single particle systems, mass may be a non-numerical operator.

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It's interesting to compare "Galileian,Galilean" using the Google Ngram Viewer for the English corpus and then for the Italian corpus, which shows something of why an Italian reader might have a different take on this (or both compared together). For this kind of comparison I'd generally choose to use the Google Ngram Viewer, which is based on published books and gives some insight into how usage has changed over the last 200 years, instead of using Google on the web, as it is today only.