A little bit of historical background about the octonionic theory

Over the last two and a half decades or so, the following developments have taken place:

Stream I

Stephen Adler and collaborators proposed the theory of Trace Dynamics, i.e. quantum theory is an emergent phenomenon, arising in the thermodynamic limit of an underlying pre-quantum dynamics on a flat spacetime. Adler expressed the view that classical gravitation is also an emergent phenomenon, that gravity should not be quantised, and that the emergence of gravity might be shown from a generalisation of trace dynamics.

Stream II

Completely independent of Stream I, Chamseddine and Connes developed the spectral action principle: the Einstein-Hilbert action can be expressed as the spectrum of the Dirac operator.

Stream III

Independent of Stream I and II, several researchers showed that some properties of the standard model could be explained in an algebraic way, using Clifford algebras made from the algebra of the octonions. This is an ongoing research programme. Space-time here is flat four dimensional Minkowski spacetime, in the spirit of GUTs.

Stream IV

Independent of Streams I, II and III, I had been making naive attempts at developing a formalism from which quantum theory and gravitation both are emergent.

It appears useful to employ the spectral action principle to generalise trace dynamics to include gravitation. Each of the eigenvalues in the spectrum of the Dirac operator is raised to the status of a matrix [in the spirit of trace dynamics], permitting one to construct an action principle for a pre-spacetime, pre-quantum theory which generalises trace dynamics so as to include gravitation. Because of the phenomenon of spontaneous localisation, it seems possible that both quantum theory and classical gravitation are emergent in this theory, in a thermodynamic sense.

For various reasons, it appears useful to assume that the pre-quantum pre-spacetime degrees of freedom live on an 8D space labelled by the octonions, and evolve in this space in an absolute time known as the Connes time.

This suggests a connection with stream III namely that the elementary particles of the standard model, described by the algebra of the octonions, live on this physical octonionic space, in the spirit of a Kaluza-Klein theory. This suggests an approach to unification, in which the non-commutative space determines the dimensionless parameters of the standard model, without appealing to physics at very high energy scales which are not currently accessible by present day accelerators. This is relativistic weak quantum gravity impacting on the standard model: unification of gravity with the other forces is a process independent of the energy scale: a collection of physical systems each having an action of order \hbar, between themselves give rise to (weak) quantum gravitation [even at low energies] and this gravitation must be described in a unified way with the physical systems which source it. The non-commutative nature of the space transforms elementary particles from points to extended objects with extent of the order of Planck length, making it a quantum gravitational situation [also because action is order \hbar so that quantum geometries are superposed]. However, the time scales of interest (in Connes time) are much larger than Planck time, so that Planck energy scales are not probed, nor are Planck length scales actually probed. This makes it a low-energy quantum gravitational situation.

It might be possible to show from this framework, following the principles of trace dynamics and spontaneous localisation, that 4D classical spacetime and classical gravitation, quantum field theory, and the standard model as we know it, are emergent from this framework. This is of course extremely challenging and ambitious. A modest effort in this direction is currently in progress.