Glimmers of a post-geometric perspective

Quantum gravitational effects can become important at low energy if the wavefunction of the metric field fails to be peaked around a classical configuration. We try to understand such deviations from classicality within canonical quantum gravity by introducing a "fluid of observers" in the low energy theory and defining a distance operator "at equal time" among them. We find that, even in the presence of relevant fluctuations in the metric field, a locally flat limit is recovered in the neighbourhood of each observer. Deviations from classicality have no particular consequence, locally. However, at larger separations the expectation value of the distance operator behaves differently than a standard Riemannian distance. In particular, it is non-additive and thus cannot be obtained by the integral of a differential line element. This emerging "beyond Riemannian" geometry is a metric space similar to embedded Riemannian manifolds equipped with chord distances that cut through the ambient space. We study deviations from flat space by looking at triangles in the limit where their sizes go to zero. Beyond-Riemannian deviations with respect to flat space are of the same order as standard Riemannian ones, but qualitatively different. Possible connections with holography and with the black hole information paradox are briefly discussed.