We consider geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the Asymptotic Safety scenario for quantum gravity. We discuss the role of these operators from the Asymptotic Safety perspective, and compute their anomalous dimensions within the Einstein-Hilbert truncation. We also discuss certain subtleties arising in the definition of such geometric operators. Our results hint to an effective dimensional reduction of the considered geometric operators.Given the ansatz (2.16), one can show that the scaling operators of the theory have dimension, quantum corrections included, given by the eigenvalues of the matrix [66]Whenever a dot appears, as in J · χ, deWitt summation and integration convention is understood, i.e., J · χ = d d x J a (x) χ a (x).

We apply the new quantization scheme outlined in Phys. Rev. D102 (2020) 125001 to explore the influence which quantum vacuum fluctuations of the spacetime metric exert on the universes of Quantum Einstein Gravity, which is regarded an effective theory here. The scheme promotes the principle of Background Independence to the level of the regularized precursors of a quantum field theory ("approximants") and severely constrains admissible regularization schemes. Without any tuning of parameters, we find that the zero point oscillations of linear gravitons on maximally symmetric spacetimes do not create the commonly expected cosmological constant problem of a cutoff-size curvature. On the contrary, metric fluctuations are found to reduce positive curvatures to arbitrarily tiny and ultimately vanishing values when the cutoff is lifted. This suggests that flat space could be the distinguished groundstate of pure quantum gravity. Our results contradict traditional beliefs founded upon background-dependent calculations whose validity must be called into question therefore.

We determine the scaling properties of geometric operators such as lengths, areas, and volumes in models of higher derivative quantum gravity by renormalizing appropriate composite operators. We use these results to deduce the fractal dimensions of such hypersurfaces embedded in a quantum spacetime at very small distances.

We review the study of the scaling properties of geometric operators, such as the geodesic length and the volume of hypersurfaces, in the context of the Asymptotic Safety scenario for quantum gravity. We discuss the use of such operators and how they can be embedded in the effective average action formalism. We report the anomalous dimension of the geometric operators in the Einstein–Hilbert truncation via different approximations by considering simple extensions of previous studies.

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