Real-Space Renormalization Group for Quantum Gravity I: Significance of terms linear and quadratic in curvature

Abstract: Real-Space renormalization group techniques are developed for tackling large curvature fluctuations in quantum gravity. Within cells of invariant volume $a^4$, only certain types of fluctuations are allowed. Normal coordinates are used to avoid redundancy of the degrees of freedom. The relevant integration measure is read off from the metric on metrics. All fluctuations in a group of cells are averaged over to get an effective action for the larger cell. In this paper the simplest type of fluctuations are kept. The measure is simply an integration over independent components of the curvature tensor at the center of each cell. Terms of higher order in $a$ are required for convergence in case of Einstein-Hilbert action. With only next order (in $a$) contribution to the action, there is no renormalization of Newton's or cosmological constants. The `massless Gaussian surface' in the renormalization group space is given by actions that have linear and quadratic terms in curvature and determines the evolution of coupling constants away from it. Our techniques allow for systematic improvements.