Quantum gravity in terms of topological observables

TL;DR

This work develops a covariant, background‑independent perturbation framework for quantum gravity by recasting four‑dimensional GR as a barely non‑linear BF theory based on SO(5), with gravity emerging from controlled symmetry breaking and topological invariants. The central result is that GR can be expressed as an expectation value of a topologically invariant observable, enabling a perturbative expansion around a topological theory using BF dynamics and, for nonzero Immirzi parameter, a natural regulator for torsion fluctuations. The formalism yields an exact generating functional for the BF sector and a nonlocal effective action that encodes gravitational degrees of freedom within a topological setting; a link to spin‑foam/state‑sum approaches and the Ashtekar‑Lewandowski measure is proposed to render the perturbation theory finite and triangulation independent. Collectively, the paper offers a pathway to a covariant, background‑independent quantum gravity framework that leverages topological quantum field theory and loop/spin‑foam techniques to address the nonrenormalizability concerns of quantum GR. The approach hinges on a tiny coupling $\alpha = G\Lambda/3 \approx 10^{-120}$ and a tunable Immirzi parameter $\gamma$, which also connects to quantum geometric spectra and potential CP‑violating effects in the quantum regime.

Abstract

We recast the action principle of four dimensional General Relativity so that it becomes amenable for perturbation theory which doesn't break general covariance. The coupling constant becomes dimensionless (G_{Newton} Λ) and extremely small 10^{-120}. We give an expression for the generating functional of perturbation theory. We show that the partition function of quantum General Relativity can be expressed as an expectation value of a certain topologically invariant observable. This sets up a framework in which quantum gravity can be studied perturbatively using the techniques of topological quantum field theory.