Renormalisation of postquantum-classical gravity

One of the obstacles to reconciling quantum theory with general relativity, is constructing a theory which is both consistent with observation, and and gives finite answers at high energy, so that the theory holds at arbitrarily short distances. Quantum field theory achieves this through the process of renormalisation, but famously, perturbative quantum gravity fails to be renormalisable, even without coupling to matter. Recently, an alternative to quantum gravity has been proposed, in which the geometry of spacetime is taken to be classical rather than quantum, while still being coupled to quantum matter fields [1, 2]. This can be done consistently, provided the dynamics is fundamentally stochastic. Here, we find that the pure gravity theory is formally renormalisable. We do so via the path integral formulation by relating the classical-quantum action to that of quadratic gravity which is renormalisable. Because the action induces stochastic dynamics of space-time, rather than deterministic evolution of a quantum field, the classical-quantum theory is free of tachyons and negative norm ghosts. The key remaining question is whether the renormalisation prescription retains completely positive (CP) dynamics. This consideration appears to single out the scale invariant and asymptotically free theory. We give further evidence that the theory is CP, by showing that the two-point function of the scalar mode is positive. To support the use of precision accelerometers in testing the quantum nature of spacetime, we also compute the power spectral density of the acceleration. The results presented here have a number of implications for inflation, CMB data, and experiments to test the quantum nature of spacetime. They may also provide a way to compute probabilities in the regime of quantum gravity where spacetime can be treated as effectively classical.