Relativistic weak quantum gravity, and it's significance for the standard model of particle physics

A question which has the attention of several experimentalists these days is: is gravity quantum? How to establish this through an experiment? A possibly doable experiment is to create a quantum superposition of two different position states of a massive object. And then to detect it's gravitational field and see if this field is non-classical, i.e. inconsistent with Newton's law of gravitation.

This is an example of non-relativistic weak quantum gravity. We might ask what could be an analogous relativistic scenario: relativistic matter fields in a quantum superposition of position states, and the consequent quantum gravitational fields they produce.

Nature already gives us such a setting - it is the standard model of particle physics! At low non-Planck energy scales. Say the electron satisfying the Dirac equation, which certainly obeys quantum linear superposition and we might be interested in the gravitational field it produces. How does it distort spacetime geometry? While carrying out an actual experiment could be next to impossible, we can try and make a theory for such relativistic weak quantum gravity, and see if we can predict something?

The electron is described by a spinorial wave function obeying the Dirac equation. It cannot be expected to produce a spacetime that is a superposition of many Minkowski spacetimes. Minkowski spacetime is vector based.

That is why we defined the electron state on a spinorial spacetime, described by quaternions, and more generally by the octonions. This spinorial spacetime is the square-root of Minkowski spacetime, same way that the Dirac equation is the square-root of the Klein-Gordon equation.

Remarkably, the non-commutative, non-associative nature of the square-root spacetime dictates the standard model of particle physics, predicting its observed symmetries, and predicting several of it's properties. For instance, quantisation of electric charge as observed [0, 1/3, 2/3, 1] and the value of the famous low-energy fine structure constant 1/137, and mass-ratios of the charged fermions. We interpret these findings as evidence for relativistic weak quantum gravity, and hence as evidence for the quantum nature of gravity. Further work is in progress.

From here, the extrapolation of the dynamics to higher energies can be carried out using conventional QFT on Minkowski spacetime. The significant new development is the realisation that the low energy free parameters of the standard model are being determined not by Planck energy scale physics, but at low energies itself. It is a quantum gravity effect in the infra-red. Relativistic weak quantum gravity comes into play whenever the matter field sources are relativistic, and quantum, i.e. having action of the order \hbar, and we want to know their spacetime geometry.

Reference: https://arxiv.org/abs/2110.02062