On the reconstruction problem in Quantum Gravity

TL;DR

This paper explores the relationship between path integrals and the Wilsonian renormalization group in quantum gravity, showing that at quadratic order in curvature, their connection involves only local terms, thus addressing the reconstruction problem.

Contribution

It demonstrates that the map between the path integral and fixed-point actions in quantum gravity remains local at quadratic order in curvature, clarifying the reconstruction problem.

Findings

The map does not generate non-localities at quadratic order.

The bare and fixed-point actions differ only by local terms.

The results do not hold if a physical UV cutoff or non-locality scale is present.

Abstract

Path integrals and the Wilsonian renormalization group provide two complementary computational tools for investigating continuum approaches to quantum gravity. The starting points of these constructions utilize a bare action and a fixed point of the renormalization group flow, respectively. While it is clear that there should be a connection between these ingredients, their relation is far from trivial. This results in the so-called reconstruction problem. In this work, we demonstrate that the map between these two formulations does not generate non-localities at quadratic order in the background curvature. At this level, the bare action in the path integral and the fixed-point action obtained from the Wilsonian renormalization group differ by local terms only. This conclusion does not apply to theories coming with a physical ultraviolet cutoff or a fundamental non-locality scale.