On the scaling of composite operators in asymptotic safety

Abstract

The Asymptotic Safety hypothesis states that the high-energy completion of gravity is provided by an interacting renormalization group fixed point. This implies nontrivial quantum corrections to the scaling dimensions of operators and correlation functions which are characteristic for the corresponding universality class. We use the composite operator formalism for the effective average action to derive an analytic expression for the scaling dimension of an infinite family of geometric operators d d x √ gR n . We demonstrate that the anomalous dimensions interpolate continuously between their fixed point value and zero when evaluated along renormalization group trajectories approximating classical general relativity at low energy. Thus classical geometry emerges when quantum fluctuations are integrated out. We also compare our results to the stability properties of the interacting renormalization group fixed point projected to f (R)-gravity, showing that the composite operator formalism in the single-operator approximation cannot be used to reliably determine the number of relevant parameters of the theory.

The phase diagram of the multi-matrix model with ABAB interaction from functional renormalization

“…For instance, the scaling behavior of the loop average serves as an order parameter which allows to relate the length L of a loop of A/B-type of links to the enclosed area A. To make similar arguments solely using FRG technology, the composite-operator renormalization construction [136,137] developed in the context of the Asymptotic Safety program in [138][139][140], would have to be carried over to the application of the FRG in the discrete quantum-gravity context. For the symmetric model (α 1 = α 2 ) with ABAB interaction term, it is possible to identify two distinct continuum phases.…”

“…Beyond the characterization of the universality class in terms of the critical exponents, an understanding of the emergent geometries is desirable. Within the FRG, the compositeoperator formalism [136][137][138][139][140] appears well-suited to provide access to, e.g., the scaling of geometric quantities. At the purely formal level, the evaluation of the corresponding flow equations for matrix/tensor models is not expected to be more challenging than for the flow equation for the effective action itself.…”

Why the Cosmological Constant Seems to Hardly Care About Quantum Vacuum Fluctuations: Surprises From Background Independent Coarse Graining

“…More general composite operators O ĝ µν can be included in the gravitational EAA[34][35][36][37][38] by coupling them to independent sources[39][40][41][42][43][44][45].Frontiers in Physics | www.frontiersin.org…”

Critical Reflections on Asymptotically Safe Gravity

“…Finally, let us recall that in other approaches to quantum gravity such as LQG, "geometrical" observables such as lengths, areas, volumes, and curvatures have played an important role. These have also been discussed to some extent in Asymptotic Safety, [323], and can be computed with a flow equation for composite operators [90,[324][325][326][327][328][329]. While presently it is not clear what type of measurement is required to access such observables, they can be used to explore whether different approaches to quantum gravity give rise to universal physical results.…”