Quantum curvature fluctuations and the cosmological constant in a single plaquette quantum gravity model

Understanding the microscopic behavior of spacetime, in particular quantum uncertainty in the Ricci scalar, is critical for developing a theory of quantum gravity and perhaps solving the cosmological constant problem. To test this, we compute this quantity for a simple but exact discrete quantum gravity model based on a single plaquette of spacetime. Our results confirm initial speculations of Wheeler from 1955 in finding a UV divergence in the quantum uncertainty. We further show that this behavior is stable under renormalization, but potentially unstable with the introduction of a cosmological constant, suggesting that a bare cosmological constant is ruled out.