On Characterizing the Quantum Geometry Underlying Asymptotic Safety

Abstract

The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a "perturbative regime" where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R 2 -operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.

Scalar-tensor theories within Asymptotic Safety

“…Investigating large-scale truncations within the composite operator formalism suggests that there can also gravity-matter fixed points beyond near-canonical scaling[195,196]. These are difficult to track based on FRG computations though.…”

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.…”

Editorial: Coarse graining in quantum gravity -- Bridging the gap between microscopic models and spacetime physics

“…In the absence of experimentally measurable observables, the comparison of characteristic properties of the quantum geometry between approaches to quantum gravity is a potential pathway to find commonalities or differences. In this spirit, A. Kurov and F. Saueressig link functional RG techniques to the analysis of geometric operators and observables in quantum gravity in [46].…”