Spin(11, 3), particles, and octonions

The fermionic fields of one generation of the Standard Model (SM), including the Lorentz spinor degrees of freedom, can be identified with components of a single real 64-dimensional semi-spinor representation S+ of the group Spin(11, 3). We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with S+=O⊗Õ, where O,Õ are the usual and split octonions, respectively. It is then well known that choosing a unit imaginary octonion u∈Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion ũ∈Im(Õ) equips Õ with a complex structure J̃, except that there are now two inequivalent complex structures, one parameterized by a choice of a timelike and the other of a spacelike unit ũ. In either case, the identification S+=O⊗Õ implies that there are two natural commuting complex structures J,J̃ on S+. Our main new observation is that the subgroup of Spin(11, 3) that commutes with both J,J̃ on S+ is the direct product Spin(6) × Spin(4) × Spin(1, 3) of the Pati–Salam and Lorentz groups, when ũ is chosen to be timelike. The splitting of S+ into eigenspaces of J corresponds to splitting into particles and anti-particles. The splitting of S+ into eigenspaces of J̃ corresponds to splitting of Lorentz Dirac spinors into two different chiralities. This provides an efficient bookkeeping in which particles are identified with components of such an elegant structure as O⊗Õ. We also study the simplest possible symmetry breaking scenario with the “Higgs” field taking values in the representation that corresponds to three-forms in R11,3. We show that this Higgs can be designed to transform as the bi-doublet of the left/right symmetric extension of the SM and thus breaks Spin(11, 3) down to the product of the SM, Lorentz, and U(1)B−L groups, with the last one remaining unbroken. This three-form Higgs field also produces the Dirac mass terms for all the particles.