Higher-order curvature operators in causal set quantum gravity

TL;DR

This paper develops higher-order curvature invariants within causal set quantum gravity, aiming to better characterize spacetime geometry and explore phase transitions relevant to quantum gravity approaches.

Contribution

It introduces generalized discrete operators encoding higher-order curvature invariants, connecting causal set theory with continuum limits and asymptotic safety scenarios.

Findings

Higher-order curvature invariants derived from generalized discrete operators.

Continuum limits recover invariants like R^2 - 2 □ R.

Potential implications for phase transitions in 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 analogue 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.