REFLECTION GROUPS AND INTERNAL-SYMMETRY ALGEBRAS

It is shown that to discrete Abelian reflection groups in pseudo-Euclidean space there correspond, in spinor index space, non-Abelian groups whose anticommuting elements build up Lie algebras; in particular, two reflections, with respect to orthogonal planes, and their product build up a su 2, c algebra. Conditions are given for these «reflection algebras» to be internal-symmetry algebras (commuting with the Poincaré algebras). When covariance with respect to the «extended» conformal group (conformal transformations and reflections) is postulated, the full use of 8-component conformal spinors is necessary. The conformal spinor is either a doublet of canonical Dirac spinors (Dirac basis) or a doublet of conformal Cartan semi-spinors (eigenstates of Γ 7) (semi-spinor basis). The two bases correspond to equivalent representations in index space, but they give rise to physically nonequivalent spinor equations in Minkowski space: Dirac spinors of the doublet may give rise separately to free particles; semi-spinors do not: they obey coupled spinor equations and they transform into each other for any reversal (including space reversal). The conformal reflection group may generate «internal symmetry algebras», u 2, c for massive spinors, u 2, c,L⊕u 2, c,R for massless ones. Possible conformal covariant Lagrangians with «internal-symmetry algebras» are discussed. It appears that the simplest conformally covariant interaction Lagrangians may naturally represent weak interactions. Next, the group O 4,4 (or O 5,3 or O 6,2) containing conformal and Poincaré groups as subgroups is studied. Reflection groups in V 9 give rise to u 4 «internal symmetry algebra», which becomes u 4,L⊕u 4,R for massless spinors. The u 4 internal symmetry algebra acts on V 8 spinors, which may either be quadruplets of Dirac spinors (Dirac basis) or three independent quadruplets of conformal semi-spinors (semi-spinor basis); on these acts a u 3 algebra orthogonal to the u 4 above. It is shown that conformal semi-spinors may not exist as free fields individually in Minkowski space, since they obey coupled equations, but only in systems and precisely an even number (≥2) for bosons and odd (≥3) for fermions. Furthermore, these systems must be singlets of the u 3 algebra. The analogy of V 8 spinor properties with leptons, when in the Dirac basis, and with coloured quarks, when in the Cartan semi-spinor basis, is emphasized and discussed. © 1979 Società Italiana di Fisica.