Classification of Elementary Particles Based on the Representation Types of the Poincaré Group Including Space, Time, and Charge Reflections
It is argued that a type analysis of the Poincaré group including space, time, and charge reflections suffices to determine all the known quantum numbers of elementary particles in a way consistent with the superselection rules. The charged Poincaré group, defined as the direct product of the full Poincaré group with the involution group of charge conjugations, is studied, and its representation types are classified. It is shown that these representations can be grouped into multiplets. While there exist superselection rules separating the different multiplets, there is no superselection between the states of each multiplet. The theory defines exactly three Dirac fields for massive spin- 1/2 representations. One of these fields transforms like an ordinary Dirac field under C, P, T. The other two fields form a doublet where the charge conjugation has the significance of G parity. They are identified with the λ, p, and n quarks of Gell-Mann. The meaning of the SU3 symmetries becomes self-evident after this identification. The leptons are assumed to have zero masses in the unitary-symmetry limit. The γ5 invariance is exploited to choose a "neutral" type with trivial charge conjugations. The theory defines six neutral two-component fields, which are identified with the neutrinos, the left- and right-handed electron, and muon fields. The discrete symmetries lead in a natural way to the V-A coupling for the lepton-lepton interactions. Thus it is concluded that the lepton-lepton weak interactions preserve the space, time, and charge reflection symmetries of an unusual type of representation.