Large curvature and background scale independence in single-metric approximations to asymptotic safety

TL;DR

This paper addresses background dependence issues in single-metric approximations of quantum gravity's RG flow, proposing a formulation over background ensembles that ensures smooth solutions at large curvature, with a novel scale-invariant variable construction.

Contribution

It introduces a background ensemble formulation and a modified Ward identity to achieve background scale independence in single-metric RG approximations for quantum gravity.

Findings

Solutions are smooth and well-defined at large curvature.

Background scale invariance is achieved through a modified Ward identity.

Compatibility requires six-dimensional spacetime in the approximation.

Abstract

In single-metric approximations to the exact renormalization group (RG) for quantum gravity, it has been not been clear how to treat the large curvature domain beyond the point where the effective cutoff scale $k$ is less than the lowest eigenvalue of the appropriate modified Laplacian. We explain why this puzzle arises from background dependence, resulting in Wilsonian RG concepts being inapplicable. We show that when properly formulated over an ensemble of backgrounds, the Wilsonian RG can be restored. This in turn implies that solutions should be smooth and well defined no matter how large the curvature is taken. Even for the standard single-metric type approximation schemes, this construction can be rigorously derived by imposing a modified Ward identity (mWI) corresponding to rescaling the background metric by a constant factor. However compatibility in this approximation requires the space-time dimension to be six. Solving the mWI and flow equation simultaneously, new variables are then derived that are independent of overall background scale.