Relational observables in Asymptotically safe gravity

TL;DR

This paper develops a method to compute the renormalisation group flow of relational observables in quantum gravity, demonstrating their evolution from microscopic expressions to quantum expectation values within the asymptotic safety framework.

Contribution

It introduces a novel approach using the composite operator formalism to analyze the scale dependence of relational observables in quantum gravity.

Findings

Computed scaling dimensions of relational metric and scalar curvature at the fixed point.

Applied the method to scalar fields coupled to gravity as a physical coordinate system.

Established a foundation for analyzing running observables in asymptotic safety.

Abstract

We introduce an approach to compute the renormalisation group flow of relational observables in quantum gravity which evolve from their microscopic expressions towards the full quantum expectation value. This is achieved by using the composite operator formalism of the functional renormalisation group. These methods can be applied to a large class of relational observables within a derivative expansion for different physical coordinate systems. As a first application we consider four scalar fields coupled to gravity to represent the physical coordinate frame from which relational observables can be constructed. At leading order of the derivative expansion the observables are the inverse relational metric and the relational scalar curvature. We evaluate their scaling dimensions at the fixed point, both in the standard renormalisation group scheme and in the essential scheme. This represents the first steps to describe running observables within asymptotic safety; this treatment can be generalised to other observables constructed from different tensors and in different physical coordinate systems.