We analyze the conceptual role of background independence in the application of the effective average action to quantum gravity. Insisting on a background independent renormalization group (RG) flow the coarse graining operation must be defined in terms of an unspecified variable metric since no rigid metric of a fixed background spacetime is available. This leads to an extra field dependence in the functional RG equation and a significantly different RG flow in comparison to the standard flow equation with a rigid metric in the mode cutoff. The background independent RG flow can possess a non-Gaussian fixed point, for instance, even though the corresponding standard one does not. We demonstrate the importance of this universal, essentially kinematical effect by computing the RG flow of Quantum Einstein Gravity in the ``conformally reduced'' Einstein--Hilbert approximation which discards all degrees of freedom contained in the metric except the conformal one. Without the extra field dependence the resulting RG flow is that of a simple $ϕ^4$-theory. Including it one obtains a flow with exactly the same qualitative properties as in the full Einstein--Hilbert truncation. In particular it possesses the non-Gaussian fixed point which is necessary for asymptotic safety.
We discuss various basic conceptual issues related to coarse graining flows in quantum gravity. In particular the requirement of background independence is shown to lead to renormalization group (RG) flows which are significantly different from their analogs on a rigid background spacetime. The importance of these findings for the asymptotic safety approach to Quantum Einstein Gravity (QEG) is demonstrated in a simplified setting where only the conformal factor is quantized. We identify background independence as a (the ?) key prerequisite for the existence of a non-Gaussian RG fixed point and…
In this work we study a significantly enlarged truncation of conformally reduced quantum gravity in the context of Asymptotic Safety, including all operators that can be resolved in such a truncation including up to the sixth order in derivatives. A fixed point analysis suggests that there is no asymptotically safe fixed point in this system once one goes beyond an Einstein-Hilbert approximation. We will put these findings into context and discuss some lessons that can be learned from these results for general non-perturbative renormalisation group flows.
Within the derivative expansion of conformally reduced gravity, the modified split Ward identities are shown to be compatible with the flow equations if and only if either the anomalous dimension vanishes or the cutoff profile is chosen to be power law. No solutions exist if the Ward identities are incompatible. In the compatible case, a clear reason is found for why Ward identities can still forbid the existence of fixed points; however, for any cutoff profile, a background independent (and parametrisation independent) flow equation is uncovered. Finally, expanding in vertices, the combined e…
In single-metric approximations to the exact renormalization group (RG) for quantum gravity, it has been not been clear how to treat the large curvature domain beyond the point where the effective cutoff scale $k$ is less than the lowest eigenvalue of the appropriate modified Laplacian. We explain why this puzzle arises from background dependence, resulting in Wilsonian RG concepts being inapplicable. We show that when properly formulated over an ensemble of backgrounds, the Wilsonian RG can be restored. This in turn implies that solutions should be smooth and well defined no matter how large …
We study the thermodynamics and dynamics of high-dimensional Einstein-power-Yang-Mills black holes in conformal gravity. Specifically, we investigate a class of conformally related black holes whose metrics differ by a scale factor. We show that a suitable scale factor cures the geodesic incompleteness and the divergence of Kretschmann scalars at the center of black holes. In the aspect of thermodynamics, we analyse the Hawking temperature, the entropy, and the specific heat, and verify the existence of second-order phase transitions. We find that the thermodynamics of this class of conformall…
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, i…
We propose a method for the (re)-construction of a regularized functional integral, well defined in the ultraviolet limit, from a solution of the functional renormalization group equation of the effective average action. The functional integral is required to reproduce this solution. The method is of particular interest for asymptotically safe theories. The bare action for the Einstein-Hilbert truncation of Quantum Einstein Gravity (QEG) is computed and its flow is analyzed. As a second example conformally reduced gravity is explored. Various conceptual issues related to the reconstruction pro…
We show that the standard Hamiltonian of isotropic loop quantum cosmology is selected by physical criteria plus one choice: that it have a `minimal' number of terms. We also show the freedom, and boundedness of energy density, even when this choice is relaxed. A criterion used is covariance under dilations, the continuous diffeomorphisms remaining in this context, which are not canonical but conformally canonical transformations. We propose how to implement conformally canonical transformations in quantum theory. Removal of the infrared regulator yields independence of ordering ambiguities.
In the average action approach to the quantization of gravity the fundamental requirement of "background independence" is met by actually introducing a background metric but leaving it completely arbitrary. The associated Wilsonian renormalization group defines a coarse graining flow on a theory space of functionals which, besides the dynamical metric, depend explicitly on the background metric. All solutions to the truncated flow equations known to date have a trivial background field dependence only, namely via the classical gauge fixing term. In this paper we analyze a number of conceptual …
The most momentous requirement a quantum theory of gravity must satisfy is Background Independence, necessitating in particular an ab initio derivation of the arena all non-gravitational physics takes place in, namely spacetime. Using the background field technique, this requirement translates into the condition of an unbroken split-symmetry connecting the (quantized) metric fluctuations to the (classical) background metric. If the regularization scheme used violates split-symmetry during the quantization process it is mandatory to restore it in the end at the level of observable physics. In t…
A perturbative quantum theory of the 2-Killing vector reduction of general relativity is constructed. Although non-renormalizable in the standard sense, we show that to all orders of the loop expansion strict cut-off independence can be achieved in a space of Lagrangians differing only by a field dependent conformal factor. In particular the Noether currents and the quantum constraints can be defined as finite composite operators. The form of the field dependence in the conformal factor changes with the renormalization scale and a closed formula is obtained for the beta functional governing it…
In this thesis we investigate various fundamental aspects of asymptotically safe quantum gravity, in particular the compatibility of Asymptotic Safety with the requirements for background independence and unitarity. The first part contains a detailed analysis of the space of metrics. We unveil a novel, specifically designed connection on this space and use the corresponding geodesics to connect all metrics having the same signature. In the second part we study the consequences of different metric parametrizations, i.e. of different connections on the space of metrics, for the Asymptotic Safety…
We explore the nonperturbative renormalization group flow of Quantum Einstein Gravity (QEG) on an infinite dimensional theory space. We consider "conformally reduced" gravity where only fluctuations of the conformal factor are quantized and employ the Local Potential Approximation for its effective average action. The requirement of "background independence" in quantum gravity entails a partial differential equation governing the scale dependence of the potential for the conformal factor which differs significantly from that of a scalar matter field. In the infinite dimensional space of potent…
This thesis is devoted to exploring various fundamental issues within asymptotic safety. Firstly, we study the reconstruction problem and present two ways in which to solve it within the context of scalar field theory, by utilising a duality relation between an effective average action and a Wilsonian effective action. Along the way we also prove a duality relation between two effective average actions computed with different UV cutoff profiles. Next we investigate the requirement of background independence within the derivative expansion of conformally reduced gravity. We show that modified W…
The methods of the renormalization group and the $\varepsilon$-expansion are applied to quantum gravity revealing the existence of an asymptotically safe fixed point in spacetime dimensions higher than two. To facilitate this, physical renormalization schemes are exploited where the renormalization group flow equations take a form which is independent of the parameterisation of the physical degrees of freedom (i.e. the gauge fixing condition and the choice of field variables). Instead the flow equation depends on the anomalous dimensions of reference observables. In the presence of spacetime b…
We use the Wetterich-equation to study the renormalization group flow of $f(R)$-gravity in a three-dimensional, conformally reduced setting. Building on the exact heat kernel for maximally symmetric spaces, we obtain a partial differential equation which captures the scale-dependence of $f(R)$ for positive and, for the first time, negative scalar curvature. The effects of different background topologies are studied in detail and it is shown that they affect the gravitational RG flow in a way that is not visible in finite-dimensional truncations. Thus, while featuring local background independe…
We study the non-perturbative renormalization group flow of f(R)-gravity in three-dimensional Asymptotically Safe Quantum Einstein Gravity. Within the conformally reduced approximation, we derive an exact partial differential equation governing the RG-scale dependence of the function f(R). This equation is shown to possess two isolated and one continuous one-parameter family of scale-independent, regular solutions which constitute the natural generalization of RG fixed points to the realm of infinite-dimensional theory spaces. All solutions are bounded from below and give rise to positive defi…
Motivated by conformal field theory studies we investigate Quantum Einstein Gravity with a new field parametrization where the dynamical metric is basically given by the exponential of a matrix-valued fluctuating field, $g_{μν}=\bar{g}_{μρ}(e^h)^ρ_ν$. In this way, we aim to reproduce the critical value of the central charge when considering $2+ε$ dimensional spacetimes. With regard to the Asymptotic Safety program, we take special care of possible fixed points and new structures of the corresponding RG flow in $d=4$ for both single- and bi-metric truncations. Finally, we discuss the issue of r…
The $\mathcal{O}(\partial^2)$ background independent flow equations for conformally reduced gravity are shown to be equivalent to flow equations naturally adapted to scalar field theory with a wrong sign kinetic term. This sign change is shown to have a profound effect on the renormalization group properties, broadly resulting in a continuum of fixed points supporting both a discrete and a continuous eigenoperator spectrum, the latter always including relevant directions. The properties at the Gaussian fixed point are understood in particular depth, but also detailed studies of the local poten…
A perturbative quantum theory of the two Killing vector reduction of Einstein gravity is constructed. Although the reduced theory inherits from the full one the lack of standard perturbative renormalizability, we show that strict cutoff independence can be regained to all loop orders in a space of Lagrangians differing only by a field dependent conformal factor. A closed formula is obtained for the beta functional governing the flow of this conformal factor. The flow possesses a unique fixed point at which the trace anomaly is shown to vanish. The approach to the fixed point is compatible with…
In this contribution, we discuss the asymptotic safety scenario for quantum gravity with a functional renormalisation group approach that disentangles dynamical metric fluctuations from the background metric. We review the state of the art in pure gravity and general gravity-matter systems. This includes the discussion of results on the existence and properties of the asymptotically safe ultraviolet fixed point, full ultraviolet-infrared trajectories with classical gravity in the infrared, and the curvature dependence of couplings also in gravity-matter systems. The results in gravity-matter s…
The formulation of an exact functional renormalization group equation for Quantum Einstein Gravity necessitates that the underlying effective average action depends on two metrics, a dynamical metric giving the vacuum expectation value of the quantum field, and a background metric supplying the coarse graining scale. The central requirement of "background independence" is met by leaving the background metric completely arbitrary. This bimetric structure entails that the effective average action may contain three classes of interactions: those built from the dynamical metric only, terms which a…
In this talk, I present a theory of quantum gravity beyond Einstein. The theory is established based on spinnic and scaling gauge symmetries by treating the gravitational force on the same footing as the electroweak and strong forces. A bi-frame space-time is initiated to describe the laws of nature. One frame space-time is a globally flat coordinate Minkowski space-time that acts as an inertial reference frame for the motions of fields, the other is a locally flat non-coordinate Gravifield space-time that functions as an interaction representation frame for the degrees of freedom of fields. T…
