Spin(11,3), particles and octonions

Abstract

The fermionic fields of one generation of the Standard Model, 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=OxO', where O,O' are the usual and split octonions respectively. It is then well-known that choosing a unit imaginary octonion u in Im(O) equips O with a complex structure J. Similarly, choosing a unit imaginary split octonion u' in Im(O') equips O' with a complex structure J', except that there are now two inequivalent complex structures, one parametrised by a choice of a timelike and the other of a spacelike unit u'. In either case, the identification S=OxO' implies that there are two natural commuting complex structures J, J' on S. Our main new observation is that the subgroup of Spin(11,3) that commutes with both J, J' on S is the direct product Spin(6) x Spin(4) x Spin(1,3) of the Pati-Salam and Lorentz groups, when u' 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 J' corresponds to splitting of Lorentz Dirac spinors into two different chiralities. We also study the simplest possible symmetry breaking scenario with the "Higgs" field taking values in the representation that corresponds to 3-forms in R^{11,3}. 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 3-form Higgs field also produces the Dirac mass terms for all the particles.

arXiv comment27 pages, 2 tables

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• Unification of the four forces in the Spin(11,1) geometric algebra
• Algebraic realisation of three fermion generations with $S_3$ family and unbroken gauge symmetry from $\mathbb{C}\ell (8)$
• Warm Dark Matter from Higher-Dimensional Gauge Theories
• Physics with non-unital algebras? An invitation to the Okubo algebra
• Particle models from special Jordan backgrounds and spectral triples
• A New Division Algebra Representation of $E_6$
• Division Algebras, Triality, and Exceptional Magic
• Unification based on the mysterious cubic-structure grouping of quarks and leptons