Spin(11,3), particles and octonions

          @misc{ pirsa_PIRSA:21040006,
            doi = {10.48660/21040006},
            url = {https://pirsa.org/21040006},
            author = {Krasnov, Kirill},
            keywords = {Mathematical physics, Particle Physics, Quantum Fields and Strings},
            language = {en},
            title = {Spin(11,3), particles and octonions},
            publisher = {Perimeter Institute},
            year = {2021},
            month = {apr},
            note = {PIRSA:21040006 see, \url{https://pirsa.org}}
          }
          

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). I will 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 there is a choice of J,J' so that the subgroup of Spin(11,3) that commutes with both is the direct product SU(3)xU(1)xSU(2)_LxSU(2)_R x Spin(1,3) of the group of the left/right symmetric extension of the SM and Lorentz group. 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.