Asymptotic safety in quantum gravity

Its key ingredient is a nontrivial fixed point of the theory's renormalization group flow which controls the behavior of the coupling constants in the ultraviolet (UV) regime and renders physical quantities safe from divergences.

The running of the coupling constants, i.e. their scale dependence described by the renormalization group (RG), is thus special in its UV limit in the sense that all their dimensionless combinations remain finite.

At the present time, there is accumulating evidence for a fixed point suitable for asymptotic safety, while a rigorous proof of its existence is still lacking.

According to the traditional point of view renormalization is implemented via the introduction of counterterms that should cancel divergent expressions appearing in loop integrals.

As this inevitably leads to an infinite number of free parameters to be measured in experiments, the program is unlikely to have predictive power beyond its use as a low energy effective theory.

It turns out that the first divergences in the quantization of general relativity which cannot be absorbed in counterterms consistently (i.e. without the necessity of introducing new parameters) appear already at one-loop level in the presence of matter fields.

[2] In order to overcome this conceptual difficulty the development of nonperturbative techniques was required, providing various candidate theories of quantum gravity.

For a long time the prevailing view has been that the very concept of quantum field theory – even though remarkably successful in the case of the other fundamental interactions – is doomed to failure for gravity.

By way of contrast, the idea of asymptotic safety retains quantum fields as the theoretical arena and instead abandons only the traditional program of perturbative renormalization.

After having realized the perturbative nonrenormalizability of gravity, physicists tried to employ alternative techniques to cure the divergence problem, for instance resummation or extended theories with suitable matter fields and symmetries, all of which come with their own drawbacks.

In 1976, Steven Weinberg proposed a generalized version of the condition of renormalizability, based on a nontrivial fixed point of the underlying renormalization group (RG) flow for gravity.

[4] [5] The idea of a UV completion by means of a nontrivial fixed point of the renormalization groups had been proposed earlier by Kenneth G. Wilson and Giorgio Parisi in scalar field theory[6][7] (see also Quantum triviality).

Introduced in 1993 by Christof Wetterich and Tim R Morris for scalar theories,[10][11] and by Martin Reuter and Christof Wetterich for general gauge theories (on flat Euclidean space),[12] it is similar to a Wilsonian action (coarse grained free energy)[6] and although it is argued to differ at a deeper level,[13] it is in fact related by a Legendre transform.

This work can be considered an essential breakthrough in asymptotic safety related studies on quantum gravity as it provides the possibility of nonperturbative computations for arbitrary spacetime dimensions.

Here the basic input data to be fixed at the beginning are, firstly, the kinds of quantum fields carrying the theory's degrees of freedom and, secondly, the underlying symmetries.

In this sense any action in theory space is a linear combination of field monomials, where the corresponding coefficients are the coupling constants,

The renormalization group (RG) describes the change of a physical system due to smoothing or averaging out microscopic details when going to a lower resolution.

Infinitesimal RG transformations map actions to nearby ones, thus giving rise to a vector field on theory space.

The construction of a quantum field theory amounts to finding an RG trajectory which is infinitely extended in the sense that the action functional described by

The key hypothesis underlying asymptotic safety is that only trajectories running entirely within the UV critical surface of an appropriate fixed point can be infinitely extended and thus define a fundamental quantum field theory.

Its critical exponents agree with the canonical mass dimensions of the corresponding operators which usually amounts to the trivial fixed point values

[14] It is the scale dependent version of the effective action where in the underlying functional integral field modes with covariant momenta below

As described in the previous section, the FRGE lends itself to a systematic construction of nonperturbative approximations to the gravitational beta-functions by projecting the exact RG flow onto subspaces spanned by a suitable ansatz for

In such BC formulation, the differential equation for the Ricci scalar R is overconstrained, but some of these constraints can be removed via the resolution of movable singularities.

[28] In combination these results constitute strong evidence that gravity in four dimensions is a nonperturbatively renormalizable quantum field theory, indeed with a UV critical surface of reduced dimensionality, coordinatized by only a few relevant couplings.

[16] Results of asymptotic safety related investigations indicate that the effective spacetimes of QEG have fractal-like properties on microscopic scales.

[29][30] In this context it might be possible to draw the connection to other approaches to quantum gravity, e.g. to causal dynamical triangulations, and compare the results.

As an example, asymptotic safety in combination with the Standard Model allows a statement about the mass of the Higgs boson and the value of the fine-structure constant.

Some researchers argued that the current implementations of the asymptotic safety program for gravity have unphysical features, such as the running of the Newton constant.

[33] Others argued that the very concept of asymptotic safety is a misnomer, as it suggests a novel feature compared to the Wilsonian RG paradigm, while there is none (at least in the quantum field theory context, where this term is also used).

Trajectories of the renormalization group flow in theory space, parametrized by infinitely many coupling constants. By convention, the arrows of the vector field (and the one on the green trajectory) point from UV to IR scales. The set of actions which lie inside the theory space and are pulled into the fixed point under the inverse RG flow (i.e., going in the direction opposite to the arrows) is referred to as UV critical surface. The asymptotic safety hypothesis is that a trajectory can only be realized in Nature if it is contained in the UV critical surface since only then it has a well-behaved high energy limit (orange, blue, and magenta trajectories, by way of example). Trajectories outside this surface escape theory space for since they develop unacceptable divergences in the UV, while going to lower scales they approach the UV critical surface. This situation is represented by the green trajectory which lies above the surface and runs away from it for increasing RG scale (opposite to the green arrow).
Flow diagram for the Einstein–Hilbert truncation of quantum gravity. Arrows point from UV to IR scales. Dark background color indicates a region of fast flow, in regions of light background the flow is slow or even zero. The latter case includes a vicinity of the Gaussian fixed point in the origin, and the non-Gaussian fixed point in the center of the spiralling arrows, respectively. The cross-over trajectory tangent to the green arrows connects the non-Gaussian to the Gaussian fixed point and plays the role of a separatrix .
Phase portrait for the Einstein–Hilbert truncation. Shown are the RG trajectories corresponding to the flow diagram on the left-hand side. (First obtained in Ref. [ 21 ] )