Higher-order curvature operators in causal set quantum gravity

Abstract

We construct higher-order curvature invariants in causal set quantum gravity. The motivation for this work is twofold: First, to characterize causal sets, discrete operators that encode geometric information on the emergent spacetime manifold, e.g., its curvature invariants, are indispensable. Second, to make contact with the asymptotic-safety approach to quantum gravity in Lorentzian signature and find a second-order phase transition in the phase diagram for causal sets, going beyond the discrete analog of the Einstein–Hilbert action may be critical. Therefore, we generalize the discrete d’Alembertian, which encodes the Ricci scalar, to higher orders. We prove that curvature invariants of the form R^{2} -2 Box R (and similar invariants at higher powers of derivatives) arise in the continuum limit.