Quantum Curvature as Key to the Quantum Universe

Curvature is a key notion in general relativity, characterizing the local physical properties of spacetime. By contrast, the concept of curvature has received scant attention in nonperturbative quantum gravity. One may even wonder whether in a Planckian regime, meaningful notions of (quantum) curvature exist at all. Remarkably, recent work in quantum gravity using causal dynamical triangulations (CDT) has demonstrated both the existence and usefulness of a new notion of quantum Ricci curvature (QRC), which relies neither on smooth structures nor on tensor calculus. This chapter recalls some classical notions related to curvature and parallel transport, as well as previous unsuccessful attempts to construct quantum curvature observables based on deficit angles and Wilson loops. It introduces the quasi-local QRC on piecewise flat triangulations and describes its behavior in a purely classical setting, its use in quantum observables, and its currently known results in (C)DT quantum gravity in two and four dimensions. The QRC opens the door to a range of interesting physical observables that were previously out of reach and will help to bridge the gap between the nonperturbative quantum theory and gravitational phenomena at lower energies.