A spinor spacetime, and the standard model

Just as the Dirac equation is the square-root of the Klein-Gordon equation, a spinor spacetime is the square-root of Minkowski spacetime.

If we are not interested in the gravitational field of an electron, it is perfectly fine to work with the Dirac equation written on a Minkowski spacetime.

However, if we want to know the gravitational field produced by the electron, we must first define the electron states on a spinor spacetime.

Octonions define a spinor spacetime, which has eight octonionic dimensions. It's square is a ten-dimensional Minkowski spacetime.

Using Clifford algebras, the spinorial states for fermions can be defined on 8D octonionic spacetime. The symmetries of the octonionic space restrict what properties the fermions can have. Charge and (square-root) mass are both defined as eigenvalues of certain symmetry operators of the 8D space, and take discrete values consistent with what is observed experimentally in the standard model. The allowed properties show that the only fermions possible are left-handed and right-handed quarks and leptons of the three generations. This includes three right handed sterile neutrinos, one per generation.

The gravitational effect of an electron is equivalent to curving of this 8D octonionic space-time, and is described by the equations of trace dynamics.

The description of the standard model using the laws of QFT on Minkowski spacetime, while extremely successful, is an approximate description. It does not tell us why the standard model is what it is, and why the dimensionless constants take those particular values which we see in experiments.

On the other hand, when we describe the dynamics of elementary particles using trace dynamics on a spinor spacetime, the symmetries of the standard model and its dimensionless constants are determined by the algebraic properties of the 8D octonionic spacetime. There is no freedom.

This description in terms of a spinor spacetime is available at all energy scales, low as well as high. That is the reason why the low energy fine structure constant gets determined in this theory. By a pre-spacetime pre-quantum theory we do not just mean a pre-theory at Planck scale energies. We also employ this pre-theory at low energies to understand the standard model at low energies. We can call this relativistic weak quantum gravity coupled to the standard model.

QFT on Minkowski spacetime can be recovered from trace dynamics on a spinor spacetime.