Composite operators in asymptotic safety

Abstract

We study the role of composite operators in the Asymptotic Safety program for quantum gravity. By including in the effective average action an explicit dependence on new sources we are able to keep track of operators which do not belong to the exact theory space and/or are normally discarded in a truncation. Typical examples are geometric operators such as volumes, lengths, or geodesic distances. We show that this set-up allows to investigate the scaling properties of various interesting operators via a suitable exact renormalization group equation. We test our framework in several settings, including Quantum Einstein Gravity, the conformally reduced Einstein-Hilbert truncation, and two dimensional quantum gravity. Finally, we briefly argue that our construction paves the way to approach observables in the Asymptotic Safety program.

Composite operators in functional renormalization group approach

“…In the present paper, we analyze this problem as for the general gauge theories formulated in the field-antifield quantization formalism, in arbitrary admissible gauges [7,8]. We find that the modified FRG approach [6] suffers with the same problem as the standard one [1,2].…”

“…This means that there is no consistent physical description of the results obtained within the framework of the standard FRG approach [1,2], or of the modified one [6], in the case of gauge theories.…”

“…The standard FRG approach [1,2] corresponds to the case when K = 0 while the modified one [6] works with K = 0. For both these approaches the gauge dependence problem is studied in a similar way.…”

“…Just recently, a modification of the standard FRG approach has been proposed [6]. The modification consists of a single insertion of composite operators into the generating functional of Green functions.…”

On the scaling of composite operators in asymptotic safety

“…By now, the spectral dimension d s associated with the short-distance properties of spacetime has been computed in many quantum gravity programs [101] with the rather spectacular outcome that even vastly different approaches coincide in the prediction that d s = 2 on microscopic scales. A more refined characterization could then be based on the anomalous scaling dimension of geometric operators comprising for instance, the volumes of spacetime, volumes of surfaces embedded into spacetime, the geodesic length, or correlation functions of fields separated by a fixed geodesic distance [25,[102][103][104]. For instance [25] computed the anomalous dimension γ 0 associated with the d-dimensional volume operator O 0 = d d x √ g. At the NGFP in four dimensions this anomalous dimension turned out to be γ * 0 | d=4 = 3.986 which was taken as a pointer that "spacetime could me much more empty than one would naively expect" [25].…”