Abstract
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.
Highlights
We describe an octonionic model for Spin(11, 3) in which the semi-spinor representation gets identified with S+ = O ⊗ O, where O, Oare the usual and split octonions, respectively
The suggestion that octonions may provide a useful language for describing elementary particles is almost as old as the Standard Model (SM) itself; see Ref. 1 and references therein. (Note, that in this reference, somewhat confusingly, the usual octonion division algebra written in a complex basis is referred to as the split octonion algebra.) it was observed early on that the eight-dimensional space of octonions, after a complex structure is chosen, splits as O = C ⊕ C3
The observations of this paper provide an embedding of all known particles into such an elegant mathematical structure as O ⊗ Oand characterize which choices need to be made to fix the embedding of the groups relevant for physics into Spin(11, 3)
Summary
A choice of a unit imaginary octonion u ∈ Im(O) gives yet another special direction From these data, one can construct a complex structure on semi-spinors whose commutant is the Pati–Salam group Spin(6) × Spin(4). There exist two complex structures J, ̃J on the space of semi-spinors of Spin(11, 3), one parameterized by a unit imaginary octonion u ∈ Im(O) and the other parameterized by a unit imaginary octonion u ∈ Im(O ) that is timelike Their common commutant in Spin(11, 3) is the product of the Pati–Salam Spin(6) × Spin(4) and Lorentz Spin(1, 3) groups. The novelty here is that a new type of complex structure arises, one related to a timelike unit octonion The commutant of this complex structureJ is the product of the Lorentz Spin(1, 3) and “weak” SU(2) gauge groups. The last property is equivalent to saying that any two imaginary octonions (as well as the identity) generate a subalgebra that is associative and is a copy of the quaternion algebra H
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