A glimpse into unification with E8

In a recent talk that Tevian Dray gave, he identified E8 with SU(3, OxO') where O is an octonion, and O' is a split octonion. This is very helpful, suggesting that this is the right way to construct particle states prior to the L-R symmetry breaking.

Because subsequent to this symmetry breaking we have an SU(3,O) for electro-color, with particle states describing electro-color and made from a Cl(6) and a LH Majorana neutrino. And we have an SU(3,O') for pre-gravity, made from a Cl(6) and a RH Majorana neutrino, with the source charge being square-root of mass.

It remains to be understood what is the Clifford algebra to be associated with SU(3, OxO') prior to the symmetry breaking. However, one can guess that the particle states are `scalar lepto-quarks' such as neutrino-antineutrino, anti-down-quark-electron, upquark-upquark, positron-downquark. These states have no electric charge defined for them. Instead the quantum number for the source charge is electric-charge-square-root mass. This is the source for the unified force which has the symmetry E8 and whose Lagrangian has E8 x E8 symmetry.

Fortunately, the Lagrangian is easy to construct, and in our view, very pretty. The action is

S ~ (1/L^2) \int d\tau \dot{q_1} \dot{q_2}

We are doing special relativity in 4, 6 or 10 dim spacetime, but always defining space first through a division algebra: a quaternion, or a split biquaternion, or an octonion, or a split bioctonion. Depending on which space the above action is defined, four different cases emerge:

quaternion: special relativity and GR in 4D

split biquaternion: special relativity in 6D, gravi-weak unification

octonion: special relativity in 10D and elecro-color interaction and electroweak

split bioctonion: special relativity in (?) D and full unification; electro-color unifies with pre-gravitation

To see symmetry breaking, we expand the dynamical variables as

\dot q_1 = \dot Q_1 + \alpha Q_1

\dot q_2 = \dot Q_2 + \alpha Q_2

where \alpha will be the Yang-Mills coupling constant, which comes into play only after symmetry breaking. The dotted part is over O' and describes gravity; the undotted part is over O and describes electro-color-weak.

Bosons and fermions arise from the expansion

Q_1 = Q_B + \beta_1 Q_F

Q_2 = Q_B + \beta_2 Q_F

\beta_1 and \beta_2 are two unequal Grassmann elements, making the action as one for a 2-brane evolving in Connes time \tau on OxO'. The universe is made of enormously many such 2-branes.

This in essence is all that is there to the theory...details remain to be worked out. The coupling constants are fixed by the algebra of the octonions, or equivalently by the octonionic geometry in which the 2-brane lives.

The interested reader can find a little more related detail in the Appendix D p.47-51 of

https://www.preprints.org/manuscript/202101.0474/v5

The figure below attempts to display the role of division algebras in unification.