Critical reflections on asymptotically safe gravity
Alfio Bonanno, Astrid Eichhorn, Holger Gies, Jan M. Pawlowski, Roberto Percacci, Martin Reuter, Frank Saueressig, Gian Paolo Vacca
CCritical reflections on asymptotically safe gravity
Alfio Bonanno, Astrid Eichhorn,
2, 3
Holger Gies, Jan M. Pawlowski, Roberto Percacci, Martin Reuter, Frank Saueressig, and Gian Paolo Vacca INAF, Osservatorio Astrofisico di Catania, via S. Sofia 78 and INFN,Sezione di Catania, via S. Sofia 64, I-95123,Catania, Italy CP3-Origins, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark Institute for Theoretical Physics, Heidelberg University,Philosophenweg 16, D-69120 Heidelberg, Germany Theoretisch-Physikalisches Institut, Abbe Center of Photonics, Friedrich-Schiller-Universit¨at Jena,and Helmholtz Institute Jena, Max-Wien-Platz 1, D-07743 Jena, Germany SISSA, via Bonomea 265, I-34136 Trieste and INFN, Sezione di Trieste, Italy Institute of Physics (THEP), University of Mainz, Staudingerweg 7, D-55099 Mainz, Germany Institute for Mathematics, Astrophysics and Particle Physics (IMAPP),Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands INFN, Sezione di Bologna, Via Irnerio 46, I-40126 Bologna
Asymptotic safety is a theoretical proposal for the ultraviolet completion of quantum field the-ories, in particular for quantum gravity. Significant progress on this program has led to a firstcharacterization of the Reuter fixed point. Further advancement in our understanding of the natureof quantum spacetime requires addressing a number of open questions and challenges. Here, we aimat providing a critical reflection on the state of the art in the asymptotic safety program, specifyingand elaborating on open questions of both technical and conceptual nature. We also point out sys-tematic pathways, in various stages of practical implementation, towards answering them. Finally,we also take the opportunity to clarify some common misunderstandings regarding the program.
CONTENTS
I. Introduction and conclusions 2II. Asymptotic Safety 3A. The main idea 3B. Non-gravitational examples 3III. Functional Renormalization Group 4A. Brief introduction to the FRG 5B. FRG approach to quantum gravity 5C. Results for asymptotically safe gravity 6D. The convergence question 8E. Do backgrounds matter? 8IV. Additional methods for asymptotic safety 9V. Running couplings 11A. A clarification of semantics 11B. Remarks on dimensional regularization 12C. Correlation functions and form factors 12VI. Observables 13A. Particle physics at the Planck scale 14B. Low-energy imprints 14C. Asymptotically safe cosmology 15D. Remarks 15VII. Relation of asymptotic safety to theeffective-field theory approach 15A. Asymptotic safety and Effective FieldTheory 15B. Effective vs. fundamental AsymptoticSafety 16 C. The structure of the vacuum 17D. RG improvement 17VIII. Scale symmetry and conformal symmetry 18A. The RG as scale anomaly 18B. Black hole entropy 19IX. Unitarity 20A. General remarks 20B. Flat-space propagators 21C. Spectral function of the graviton 21D. Interpretation of potential ghost modes 22E. Remarks 22X. Lorentzian nature of quantum gravity 22Acknowledgments 23References 23 a r X i v : . [ g r- q c ] A p r I. INTRODUCTION AND CONCLUSIONS
Asymptotic Safety [1–3] is a candidate for a quantumtheory of the gravitational interactions. It does not re-quire physics beyond the framework of relativistic Quan-tum Field Theory (QFT) nor does it require fields be-yond the metric to describe the quantum geometry ofspacetime. Moreover, the inclusion of matter degrees offreedom, like the standard model or its extensions, is con-ceptually straightforward. Thus, ultimately, AsymptoticSafety may develop into a quantum theory comprising allfundamental fields and their interactions.The core idea of Asymptotic Safety was formulated byWeinberg [4, 5] in the late seventies. It builds on the in-sight of Wilson [6], linking the renormalizability and pre-dictive power of a quantum field theory to fixed pointsof its Renormalization Group (RG) flow: a theory whoseultraviolet (UV) behavior is controlled by an RG fixedpoint does not suffer from unphysical UV divergences inphysical processes like scattering events. The prototypi-cal example for such a behavior is Quantum Chromody-namics (QCD) where the UV completion is provided bythe free theory. In technical terms QCD is asymptoti-cally free with the UV completion provided by a Gaus-sian fixed point. It was then stressed in [5] that a validUV completion could also be obtained from fixed pointscorresponding to actions with non-vanishing interactions,so-called non-Gaussian fixed points. In order to contrastthis situation to asymptotic freedom, this non-trivial gen-eralization has been termed “asymptotic safety”. Re-markably, the space of diffeomorphism invariant actionsconstructed from a four-dimensional (Euclidean) space-time metric indeed seems to contain a non-Gaussian fixedpoint suitable for Asymptotic Safety, the so-called Reuterfixed point [7, 8].As in other approaches to quantum gravity, substantialprogress has brought the program to a point where a fair-minded assessment of its achievements and shortcomingswill be useful. Therefore, the purpose of this article isto provide a critical review of the current status of thefield, of the key open questions and challenges, and topoint out directions for future research. By necessity,the discussion also covers questions of a more technicalnature which is reflected in the character of some of thesections. This also entails that the article does not serveas an introduction to the asymptotic safety program, forwhich we refer the reader to the textbooks [2, 3] andreviews [9–15]. A list of key references related to the openquestions is provided within each section, pointing thereader towards the broader discussion in the literature.The rest of the paper is organized as follows. In Sec. IIwe start with a concise introduction to asymptotic safety, The terminology Gaussian fixed point reflects that the actionassociated with the fixed point does not contain interactions andis thus quadratic in the fields. also giving examples of non-Gaussian fixed points provid-ing a UV completion in non-gravitational settings. Thesubsequent sections critically review open questions alongthe following lines:1. Issues related to the use of the functional RG (FRG)(“uncontrollable approximations”, use of the back-ground field method) are discussed in Sec. III.2. Because of these theoretical uncertainties, it is impor-tant to cross-check the results with different methods.This is discussed in Sec. IV.3. The difficulty of computing observables, and compar-ing with observations, is discussed in Sec. VI.4. Closely related to this is the, partly semantic, issueof the physical meaning of running couplings (can Λand G run? If so, what are the physical implicationsof this running?) and other aspects where the lit-erature on asymptotic safety deviates from standardparticle physics procedures (power vs. log running,use of dimensional regularization). These points arediscussed in Sec. V.5. In Sec. VII we discuss whether and in what wayasymptotic safety could be matched to effective fieldtheory (EFT) at low energy. Here we also discuss thelimitations of the procedure of “RG improvement”.6. In Sec. VIII we address the relation between scalesymmetry and conformal symmetry and the FRG.(How can one have scale invariance in the presenceof G ?) We also critically review the argument thatthe entropy of black holes is incompatible with grav-ity being described by Asymptotic Safety (“Gravitycannot be Wilsonian” or “Gravity cannot be a con-formal field theory”).7. The unsolved issue of unitarity is discussed in Sec. IX(in particular: do higher derivatives imply ghosts?).8. Finally, we stress the need of calculations inLorentzian signature in Sec. X.The goal of this paper is threefold:i) reinforcing progress in the research field by clearlyspelling out key open questions,ii) strengthening a critical and constructive dialogue onasymptotically safe gravity within a larger commu-nity,iii) contributing to a broad and critical assessment of thecurrent status and future prospects of research av-enues in quantum gravity. II. ASYMPTOTIC SAFETYA. The main idea ...where we recall the notion of quantum scale invari-ance and the predictive power of RG fixed points.
Asymptotic Safety [2, 3] builds on Wilson’s modernview of renormalization, which links the renormalizabil-ity and predictive power of a quantum field theory tofixed points of its RG flow. It is equivalent to the no-tions of “quantum scale invariance in the UV” and also to“non-perturbative renormalizability”, resulting in a the-ory that is fully specified by only a finite number of freeparameters.In practice, asymptotic safety is studied in the follow-ing way. One has a functional of the fields, that couldbe either a Wilsonian action S Λ depending on a UV cut-off Λ or a generating functional Γ k for the one-particleirreducible (1PI) correlation functions depending on anIR cutoff k . We shall focus on the latter for definiteness,but at this stage the discussion is more general. For thepresent purposes, let us assume that this functional canbe expanded in a suitable basis of operators {O i } , inte-grals of monomials in the field and its derivativesΓ k = (cid:88) i ¯ u i ( k ) O i . (1)The beta functions of the, generally dimensionful, cou-plings ¯ u i ( k ) are given by the derivatives of ¯ u i ( k ) withrespect to t = log k . Then, one converts the dimension-ful couplings ¯ u i ( k ) into dimensionless ones by a suitablerescaling with the coarse-graining scale k , u i ≡ ¯ u i k − d i , (2)where d i is the canonical mass dimension of ¯ u i ( k ). In thisway one obtains a coupled set of autonomous differentialequations k∂ k u i ( k ) = β u i ( { u j } ) . (3)The solutions of this system are the RG trajectories andeach trajectory corresponds to a single physical theory.In general, it may happen that physical observables di-verge along a trajectory as k → ∞ (e.g., at a Landaupole). One simple way to avoid this is to require that the Generically, a fixed point will be neither UV nor IR, since it typ-ically has both IR attractive (irrelevant) and IR repulsive (rele-vant) directions. Depending on the choice of RG trajectory, thefixed point can therefore induce a UV or an IR scaling regime.Given two fixed points connected by an RG trajectory, the di-rection of the flow between them is fixed and the designation ofUV and IR fixed point becomes unambiguous. The notation u i for dimensionful quantities and ˜ u i for dimen-sionless quantities can also sometimes be found in the literature. trajectory describing the physical world emanates froma fixed point as k is lowered from the UV to the IR. Ata fixed point { u j ∗ } all beta functions vanish simultane-ously, β u i ( { u j ∗ } ) = 0 , ∀ i and, as we shall discuss in moredetail in Sec. VIII, scale invariance is realized . Such RGtrajectories are said to be either asymptotically free orasymptotically safe theories. This should be contrastedto the case where physical observables blow up at a finitevalue of k which indicates that one deals with an effectivefield theory.The predictive power of asymptotic safety originatesfrom the properties of the fixed point. Linearizing thebeta functions (3) about the fixed point, and diagonaliz-ing the stability matrix B ij ≡ ∂ u j β u i | u = u ∗ , one can de-termine which directions are attractive and which onesare repulsive. Eigenvalues with positive (negative) realparts correspond to eigenvectors along which the flow(from UV to IR) is dragged towards (repelled by) thefixed point. One typically works with the scaling expo-nents θ I = − eig B . Every irrelevant (IR attractive/UVrepulsive/ θ I <
0) direction fixes one parameter in theinitial conditions for Γ k , whereas relevant (IR repul-sive/UV attractive/ θ I >
0) directions correspond to freeparameters. Marginal directions ( θ I = 0) typically onlyoccur at Gaussian fixed points. Thus, the number ofindependent free parameters of an asymptotically safetheory is equal to the number of relevant directions ofthe fixed point that it originates from in the UV. At afree (Gaussian) fixed point, the relevant directions cor-respond to couplings with positive mass dimension. In alocal theory, there is only a finite number of such param-eters. In principle, an interacting fixed point could haveeven fewer relevant directions, and hence greater predic-tive power. If one could integrate the RG flow to theIR, one could test if the low-energy relations implied bythese properties of the UV fixed point are verified or not,cf. Sec. VI for further discussion. B. Non-gravitational examples ... where we provide a list of non-gravitational, asymp-totically safe theories together with the correspondingmechanism for asymptotic safety and we discuss howseveral techniques are used to study these examples.
Whereas the existence of UV-complete quantum fieldtheories based on the mechanism of asymptotic safetyhas been anticipated already in the early days of the In most cases this also implies conformal invariance. Note that the opposite sign convention, where the θ are definedwithout the additional negative sign, is also sometimes used inthe literature. More precisely, the “memory” of the initial condition for an ir-relevant direction is washed out by the RG flow and plays no rolefor the physics at k = 0. RG [16, 17], concrete examples have been identified onlymuch later, as a parametric control beyond perturba-tion theory is typically required. A paradigmatic classof examples is given by fermionic models in d = 3 dimen-sional spacetime including, for instance, the Gross-Neveumodel: though interactions of the type ∼ ( ¯ ψ m ψ ) (with m carrying some internal spin and/or flavor structure)belong to the class of perturbatively non-renormalizablemodels, there is by now convincing evidence that a largeclass of such models are in fact asymptotically safe in2 < d < /N expansions [18, 19]; in-deed, non-perturbative renormalizability has been provedfor specific models to all orders in the 1 /N expansion[20] with explicit results for higher orders being workedout, e.g., in [21–24]. Further quantitative evidence sub-sequently came from 2 + (cid:15) or 4 − (cid:15) expansions [25–30];the FRG for the first time facilitated analytic computa-tions directly in d = 3 [31–38]. For the asymptotic safetyprogram, these models are instructive for several reasons: (i) The fermionic non-Gaussian fixed point is typicallyconnected to a quantum phase transition. The latteris characterized by universal critical exponents whichcan also be studied using simulational methods [39–49] or the conformal bootstrap [22, 50]. In this way,the variety of available approaches have led to a con-firmation of asymptotic safety of these models to asubstantial degree of quantitative precision, summa-rized, e.g., in [51]. (ii)
While analytical as well as path integral MonteCarlo computations are typically performed in Eu-clidean spacetime, these models are relevant for lay-ered condensed-matter “Dirac materials” [52, 53], cor-responding to a d = 2 + 1 dimensional spacetime withLorentzian signature. The quantitative agreementalso with Quantum Monte Carlo methods (based on aHamiltonian formulation) [44–46], demonstrates thatasymptotic safety of these systems is visible in Eu-clidean as well as Lorentzian formulations. (iii) As a generic mechanism of asymptotic safety inthese models, an irrelevant (i.e., perturbatively non-renormalizable) operator such as the fermionic inter-action ∼ ( ¯ ψ m ψ ) becomes relevant as a consequenceof strong fluctuations. Correspondingly, the anoma-lous dimension of this and subsequent operators isshifted by an amount of O (1); see, e.g., [33, 54] fora determination of an infinite set of scaling dimen-sions for large N . As a consequence, strongly power-counting irrelevant operators remain irrelevant anddo not introduce an unlimited set of new physical pa-rameters. The same pattern is also observed in manystudies of asymptotically safe gravity [55–60]. (iv) The comparative simplicity of these models has en-abled a first study of the momentum dependenceof 4-point correlation functions at the non-Gaussian fixed point [61]. For instance, the Gross-Neveu model( m = ) in d = 2 + 1 at the non-Gaussian fixed pointcan be analyzed in terms of an s -channel-dependentGross-Neveu coupling g ∗ ( s ) which depends nontriv-ially and non-analytically on the dimensionless s vari-able at the UV fixed point. In fact, the s channeldependence can be shown to dominate over possible t and u channel dependences in a quantifiable man-ner at large N , resulting in a simpler form factor-likestructure of the 1PI 4-vertex at the UV fixed point.This illustrates that scattering properties in the scal-ing regime can develop nontrivial features beyond thescaling suggested by naive power-counting.Further examples for asymptotic safety include Yang-Mills theory in d = 4 + (cid:15) [62–65], and non-linear sigmamodels in d = 2 + (cid:15) [66–71]; for the latter, there is clearevidence for asymptotic safety even in d = 3 from latticesimulations [72]. The limit of large number of fermions N f in gauge theories has recently seen a resurgence ofinterest, e.g., [73–76], with early work in [77, 78], see also[79].Another recently discovered set of asymptoticallysafe models is given by gauged Yukawa models in theVeneziano limit of a suitably arranged large number ofvector fermions N f adjusted to the number of colors N c of the gauge group [80–86] in d = 3 + 1 dimensionalspacetime. Contrary to the lower-dimensional fermionicmodels, these gauged Yukawa models are power-countingrenormalizable to all orders in perturbation theory. Be-cause of the large number of fermions, fermionic screeningdominates the running of the gauge coupling, such thatasymptotic freedom is lost. The RG flow at high ener-gies nevertheless remains bounded, as it is controlled bya UV fixed point appearing in all RG marginal couplings.Whereas perturbative renormalizability of these modelssupports the use of perturbative RG beta functions inthe first place, the existence of non-Gaussian UV fixedpoints is parametrically controlled by a suitably smallVeneziano parameter, e.g., (cid:15) = N f N c − as in [80]. Despitethis technical vicinity to perturbative computations, thebehavior of the theory near the fixed point is very dif-ferent from the perturbative behavior near the Gaussianfixed point. For instance, the perturbatively marginaloperators turn into (ir-)relevant operators with anoma-lous dimensions reaching up to O (1) for (cid:15) (cid:46) O (0 . III. FUNCTIONAL RENORMALIZATIONGROUP
In Section II we have discussed the asymptotic-safetymechanism without referring to any specific calculationmethod. Now we introduce the Functional Renormal-ization Group (FRG), which has been the main toolenabling progress in Asymptotic Safety in the last 20years. It has been successfully applied to a large num-ber of other theories and physical phenomena, in par-ticular non-perturbative ones. Applications range fromthe phase structure of condensed matter systems, to con-finement and chiral symmetry breaking in QCD, to theelectroweak phase transition in the early universe andbeyond Standard Model physics. In cases, where re-sults from other non-perturbative methods (lattice sim-ulations, Dyson-Schwinger equations, Resurgence etc.)exist, the FRG results compare well to those obtainedby other methods. It is also worth emphasizing that,while the combination of conceptual and technical chal-lenges in quantum gravity is certainly unique, many ofthe technical challenges and physical effects encounteredhere have counterparts in other theories, most notablyin non-Abelian gauge theories, where they can also betested against other non-perturbative methods.
A. Brief introduction to the FRG ...where we briefly introduce the FRG as a tool tocalculate the effective action.
Currently, the primary tool to investigate AsymptoticSafety is the Functional Renormalization Group (FRG)equation for the effective average action Γ k introduced in[87–89] (Wetterich equation), and in [7] for gravity. Γ k depends on the content of the theory at hand, in quan-tum gravity it contains the metric degrees of freedom,Faddeev-Popov ghosts and possibly also matter fields.In the FRG approach the scale k is an infrared cutoffscale below which quantum fluctuations are suppressed.Thus, Γ k encodes the physics of quantum fluctuationsabove the cutoff scale. For k →
0, all quantum fluctua-tions have been taken into account and Γ k =0 is the fullquantum effective action,Γ = lim k → Γ k (4)whose minimum is the vacuum state of the QFT. Theflow equation for Γ k encodes the response of the effec-tive average action Γ k to the process of integrating outquantum fluctuations within a momentum shell, k∂ k Γ k [Φ; ¯Φ] = 12 Tr (cid:34) (2) k [Φ; ¯Φ] + R k k∂ k R k (cid:35) . (5)The term (Γ (2) k + R k ) − on the right hand side of (5) is thepropagator in the regularized theory. Here we have in-troduced Γ (2) k = Γ (Φ Φ) , the second derivative of Γ k w.r.t.the fields Φ. In (5) we have also introduced a genericbackground ¯Φ which typically is chosen as the solution tothe quantum equations of motion. Then, the fluctuationfield Φ encodes the fluctuations about this background, and the 1PI correlation functions of the fluctuation fields (cid:104) Φ i · · · Φ i n (cid:105) (proper vertices) in a given background¯Φ are given byΓ (Φ i ··· Φ in ) k [ ¯Φ] ≡ δδ Φ i · · · δδ Φ i n Γ k [Φ; ¯Φ] (cid:12)(cid:12)(cid:12)(cid:12) Φ=0 . (6)The term R k is a cutoff scale k - and momentum-dependent infrared regulator which suppresses fluctua-tions with momenta p (cid:46) k , decays rapidly for momenta p (cid:38) k , and vanishes at k = 0. The second propertyrenders the flow equation (5) finite due to the decay of k∂ k R k for large momenta. The regulator R k is indepen-dent of the fluctuation field, but may carry a dependenceon the background field. In a quantum field theory inflat space typically p is the plain momentum squared,while in gravity and gauge theories p may be associatedwith a background-covariant Laplacian. Finally, Tr com-prises a sum over all fluctuation fields and an integralover (covariant) loop momenta. The corresponding loopintegration is peaked about momenta p ≈ k , leading tothe momentum-shell integration. In summary, the flowequation (5) transforms the task of performing the pathintegral into the task of solving a functional differentialequation.Conceptually, the Wetterich equation implements theidea of the Wilsonian Renormalization Group: lowering k corresponds to integrating out quantum fluctuations shellby shell in momentum space. For k → ∞ , the theoryapproaches the bare or renormalized ultraviolet action,depending on the underlying renormalization procedure,for a detailed analysis see, e.g., [90–94]. The fact thateq. (5) does not require specifying a bare action a priorimakes it a powerful tool to scan for (interacting) RG fixedpoints and study their properties. The bare action canthen be reconstructed from the RG fixed point along thelines of [91, 93]. Essentially, the Wetterich equation canbe viewed as a tool to systematically test which choiceof bare action gives rise to a well-defined and predictivepath integral for quantum gravity.Notably, if one approximates Γ (2) k by the k -independentsecond functional derivative of a given bare action S (2) ,one obtains Γ k ≈ S + 12 Tr log (cid:16) S (2) + R k (cid:17) , (7)which reduces to the standard one-loop effective actionfor k = 0. Accordingly, approximations to the FRG al-ways contain one-loop results in a natural way. B. FRG approach to quantum gravity ...where we review the Functional RenormalizationGroup approach to quantum gravity, with a particularfocus on background-field techniques.
In the gravitational context, the construction of eq.(5) makes use of the background field method, decom-posing the physical metric g µν into a fixed, but arbitrarybackground metric ¯ g µν and fluctuations h µν , see [9] fortechnical details . The typical example is the linear split, g µν = ¯ g µν + h µν . (8)In the literature the fluctuation field h µν is commonlymultiplied with the square root of the Newton constantwhich makes it a standard dimension-one tensor field infour spacetime dimensions. The linear split (8) is thecommon choice not only in quantum gravity but also inapplications of the background field method to gauge the-ories or non-linear sigma models. In gravity it comes atthe price that the fluctuation field h µν is not a metricfield, indeed it has no geometrical meaning. While thisis not necessary, alternative parameterizations have beenused. These have the general form g µν = f ( h, ¯ g ) µκ ¯ g κν . (9)Of these alternative cases, the exponential split with f ( h, ¯ g ) = exp[¯g − h] has been explored, e.g., in [59, 102–105]. Further, the geometrical split in the Vilkovisky-deWitt approach with a diffeomorphism invariant flowhas been studied in [90, 106–109], for applications to non-linear sigma models see [110, 111].Different parameterizations (9) only constitute thesame quantization if they (i) cover the same configura-tion space and (ii) the Jacobian that arises in the pathintegral is taken into account (see [104] for a related dis-cussion). Condition (i) does not hold, e.g., for linearparameterization and exponential parameterization, see,e.g., [102, 104], while the linear split and the geometricalone with the Vilkovisky connection at least agree locally.However, it is well-known from two-dimensional gaugetheories, that quantizations on the algebra and on thegroup can differ, see, e.g., [112]. Moreover, studies of theparameterization dependence of results in truncations,e.g., [113–116], so far do not account for (ii).The presence of the background allows to discriminate“high-” and “low-momentum” modes by, e.g., comparingtheir eigenvalues with respect to the background Lapla-cian to the coarse-graining scale k . Moreover, it alsonecessarily enters gauge-fixing terms for the fluctuationfield. As a consequence, the effective action Γ k inheritstwo arguments, the set of fluctuation fields Φ and thecorresponding background fields ¯Φ for all cutoff scales k . We emphasize that this also holds true for vanishingcutoff scale, k = 0, due to the gauge fixing. Notably, the asymptotic-safety mechanism is not tied to thespacetime metric carrying the gravitational degrees of freedom.While explored in far less detail, the vielbein and the Palatiniformalisms may also lead to a theory which is asymptoticallysafe [95–101].
Conceptually, the Wetterich equation lives on the so-called theory space, the space containing all action func-tionals constructable from the field content of the theoryand compatible with its symmetry requirements. TheFRG then defines a vector field generating the RG flowon this space. We proceed by discussing two systematicexpansion schemes commonly used in quantum gravity(as well as other systems): the vertex expansion and the(covariant) derivative expansion.The proper vertices of the effective average action (6)can be used as coordinates in theory space as the setof (1PI) correlation functions { Γ (Φ i ··· Φ in ) k [ ¯Φ] } defines agiven action and hence a theory. The vertex expansion isthe expansion in the order of the fluctuation correlationfunctions and hence in powers of the fluctuation field,Γ k [Φ; ¯Φ] = ∞ (cid:88) n =1 n ! (cid:90) n (cid:89) i =1 (cid:2) d d x i Φ j i ( x i ) (cid:3) ×× Γ (Φ j ··· Φ jn ) k [ ¯Φ]( x , · · · , x n ) , (10)where Γ (Φ j ··· Φ jn ) k [ ¯Φ], for n >
2, are the proper vertices(6), that carry the measure factors (cid:81) i (cid:112) ¯ g ( x i ).The derivative expansion is best explained in the caseof the diffeomorphism invariant background effective ac-tion ¯Γ k [ ¯Φ] = Γ k [Φ = 0; ¯Φ]. This object can be expandedin diffeomorphism invariant operators such as powers ofthe curvature scalar and other invariants. Then, in thederivative expansion the sum in (1) contains all diffeo-morphism invariant terms with less than a certain num-ber of derivatives. The leading order of this expansionis ¯Γ k [ g µν ] (cid:39) πG k (cid:90) d d x √ g [2Λ k − R ] . (11)At the next order one has to add four-derivative terms in-cluding R , R µν R µν , and R µνρσ R µνρσ , and so on. In thislight, it should be understood that the Einstein-Hilbertaction just provides the leading terms in the derivativeexpansion of ¯Γ k [ g µν ] and does not constitute the bareaction underlying Asymptotic Safety. It has to be sup-plemented by gauge-fixing and ghost terms, and, if theapproximation is extended, additional terms (cid:98) Γ k [ h ; ¯ g ] de-pending on two arguments separately. “Bimetric” studiesdistinguishing g µν and ¯ g µν for the Einstein-Hilbert trun-cation can be found in [117–119]. C. Results for asymptotically safe gravity ...where we give a brief overview of the results obtainedwith the truncated FRG and provide a sketch of the fullflow from the UV fixed point down to the IR.
Most work has been done in the background-field ap-proximation, that is Γ k [Φ; ¯Φ] = ¯Γ k [Φ + ¯Φ]+ gauge fix-ing + ghosts . If one evaluates the FRG in a one-loop approximation, including terms quadratic in cur-vature, the known universal beta functions of the four-derivative couplings are reproduced, but additionallythe cosmological and Newton constant have a non-trivial fixed point [120–122]. Going beyond one loop,the following classes of operators have been studied inpure gravity: The Einstein-Hilbert truncation has beenexplored extensively [7, 8, 113, 123–131]. Einstein-Hilbert action plus R [132, 133]; Einstein-Hilbert ac-tion plus R and R µνρσ R µνρσ [56, 134–137]; Einstein-Hilbert action plus R , R µν R µν and R µνρσ R µνρσ [138];Einstein-Hilbert action plus the Goroff-Sagnotti coun-terterm R µνρσ R ρσαβ R αβµν [139]; polynomial functionsof the scalar curvature (polynomial “ f ( R ) truncation”)up to orders N = 6 [140, 141], N = 8 [55], N = 35[57, 58], and lately also N = 71 [142], or effective ac-tions of the form f ( R µν R µν ) + f ( R µν R µν ) R , where f and f are polynomials [143], effective actions of the form f ( R µνρσ R µνρσ ) + f ( R µνρσ R µνρσ ) R where f and f arepolynomials or finally effective actions consisting of a sin-gle trace of n Ricci tensors ( R µν R νρ . . . R αµ ) with n upto 35, [144]. The case of an “infinite number” of cou-plings has been addressed in the f ( R ) truncation by solv-ing [109, 145–159] a non-linear differential equation for f [58, 109, 116, 140–143, 145–161]. Global solutions forsuch “infinite” truncations can also be found for grav-ity coupled to a scalar field, see, e.g., [162]. For a moregeneral overview of the situation in gravity-matter sys-tems we refer to the review [14]. Notably, a fixed pointsuitable for Asymptotic Safety has been identified in allthese works.As is clear from this list, the terms included do notreflect the systematics of a derivative expansion. It hasalso to be said that in many of these calculations the betafunctions that one obtains are only unknown linear com-binations of the beta functions that would be obtained ifall curvature invariants of the same order were included.This is because the calculations are done on spheres,e.g., [55, 57, 58, 116, 132, 133, 140–143, 146–148, 150–153, 155, 157, 160, 163], a hyperbolic background [161] orsometimes on Einstein backgrounds, e.g., [56, 134], andthis does not permit to differentiate between functions ofRicci tensor and of the Ricci scalar, for example.In terms of the vertex expansion, most work has beenbuilt on an expansion around flat space while keepingpart of the full momentum dependence of propagatorsand vertices. For the vertices typically the symmetricpoint configuration is considered. For results in puregravity and gravity-matter systems see [164–174]. Theseworks have revealed the existence of a nontrivial fixedpoint in the two-, three-, and four-point functions com-patible with the findings in the background approxima-tions. Analogous calculations with compatible resultshave also been done for the two-and three-point func-tions on a spherical background [156, 159]. The resultsin [156, 159] for background curvature and backgroundmomentum-dependent two- and three-point function ofthe fluctuation field have then been used to compute the full f ( R )-potential in pure gravity and in the gravity-scalar system beyond the background approximation.Just like in the derivative expansion in asymptoticallysafe gravity, it has also not been possible to fully andsystematically implement the vertex expansion beyondthe lowest order: In particular, the three- and four-pointfunctions have only been calculated for a special kinemat-ical configuration and the symmetric background doesnot allow to fully disentangle different operators.To connect the UV fixed point to physics at k = 0,complete trajectories must be constructed. Currently,this part of the program is less advanced than the char-acterization of the fixed point itself; UV-IR flows havebeen computed, e.g., in [108, 164, 166, 169, 175]. It isexpected that complete solutions are most likely charac-terized by several regimes [125, 133, 176, 177], see also,e.g., [178, 179] for matter-gravity systems:- The first part of the flow from the Reuter fixed pointin the UV down to some scale M is in a linear regimeclose to the fixed point. At these extreme UV scales, thesystem could a priori either be in a strongly interactingnon-perturbative regime or be characterized by weak in-teractions. There are some tentative hints for the latter(see Sec. III D), but a conclusive statement regarding thenature of the fixed point cannot yet be made.- Close to the Planck scale, the flow has potentially al-ready left the linear regime around the fixed point. Insimple approximations, M = M Pl , i.e., the transitionscale at which fixed-point scaling stops, actually comesout equal to the Planck scale. The regime around thePlanck scale could again be characterized by either non-perturbative or near-perturbative physics – irrespectiveof the nature of the fixed point. Once one leaves the fixed-point regime, non-localities of order 1 /k , or dynamicallygenerated scales are expected to play a role. - Below the Planck scale, the description of the purelygravitational sector is expected to become much simpler.Once near the Gaussian fixed point, the flow is dominatedby the canonical scaling terms. For instance, the dimen-sionful Newton constant becomes scale independent. One It is important to realize that non-local operators, i.e., operatorswith inverse powers of derivatives, proliferate under the flow andare canonically increasingly relevant. They are therefore likelyto destroy the predictive nature of the fixed point, if includedin theory space explicitly. On the other hand, the flow nevergenerates an operator with inverse powers of derivatives within aquasi-local theory space, i.e., the requirement of quasilocality canbe imposed consistently on the theory space. Of course it is well-known that the full effective action contains physically importantnon-localities. These arise in the limit k → /k , and also encode the presence of dynamically generatedscales. expects that corrections obtained within the effective-field theory approach to quantum gravity are recoveredin this regime.In many cases these works on asymptotic safety basedon the FRG can be compared to, or substantiated by,other approximation methods or techniques. We defer adiscussion of such relations to Sects. IV and VII. D. The convergence question ...where we discuss the convergence (or lack thereof )of systematic expansion schemes in the FRG.
In practical applications, one has to work in trunca-tions of the theory space. These can also be infinite di-mensional, if a closed form for the flow of an appropriatefunctional can be found. In the gravitational case, closedflow equations for f ( R ) truncations constitute an exam-ple [58, 109, 116, 140–143, 145–161]. Further examplesare the scalar potential and a nonminimal functional inscalar-tensor theories, see, e.g., [104, 105, 181–183].A reasonable expansion scheme should capture the rel-evant physics already at low orders of the expansion. Fora fixed point, this includes the relevant operators. At thefree fixed point one simply expands according to canon-ical power counting. At a truly non-perturbative fixedpoint, the relevant operators are not known. Therefore,simple truncations that correctly model non-perturbativephysics can be difficult to devise. It is in such setupsthat the concerted use of several techniques can be mostuseful; the IR regime of QCD constitutes an excellent ex-ample. Finally, at an interacting, but near-perturbativefixed point, canonical power counting constitutes a vi-able guiding principle to set up truncations. Here, near-perturbative refers to the fact that the spectrum of criti-cal exponents exhibits deviations of O (1) from the canon-ical spectrum of scaling dimensions, but not significantlylarger, in other words, the anomalous contribution to thescaling of operators is η O (cid:46) O (1).The strategy that has (implicitly or explicitly) beenfollowed for the choice of truncations for the Reuterfixed point has been based on the assumption of near-perturbativity. This motivates a choice of truncationbased on canonical power counting. The self-consistencyof this assumption has to be checked by the resultswithin explicit truncations. Indeed, [57, 58, 142, 143]find a near-canonical scaling spectrum in the f ( R ) trun-cation. Moreover, [172–174] find close agreement of vari-ous “avatars” of the Newton coupling, something that isnot expected in a truly non-perturbative regime.As a self-consistent truncation scheme appears to beavailable for quantum gravity, the apparent convergenceof fixed-point results is a key goal. It is fair to say thatthe status of results is rather encouraging with regard tothis question, see [91, 118, 119, 124, 125, 132, 175, 184–199]. This has given rise to the general expectationthat the Reuter fixed point indeed exists in full the- ory space, and provides a universality class for quan-tum gravity. Nevertheless, it should be pointed outthat due to the technically very challenging nature ofthese calculations, the inclusion of a complete set ofcurvature-cube operators remains an outstanding task.In the vertex expansion, higher order derivative termsare captured by momentum-dependent correlation func-tions, which exhibit robust evidence for the Reuter fixedpoint [156, 164, 166–170, 172–174, 198–200]. E. Do backgrounds matter? ...where we highlight the technical challenges one faceswhen attempting to reconcile the use of a local coarse-graining procedure with the background independenceexpected of a non-perturbative quantum gravity approach.
When setting up the Wetterich equation for gravity [7]the background field formalism plays an essential role.The background metric ¯ g µν serves the double purposeof i) introducing a gauge fixing which is invariant underbackground-transformations, and ii) introducing a regu-lator, as required to implement a local notion of coarsegraining. At the same time, the decomposition of thephysical metric into a fixed, but arbitrary backgroundand fluctuations introduces a new symmetry, so-calledsplit-symmetry transformations: the linear split (8) isinvariant under¯ g µν (cid:55)→ ¯ g µν + (cid:15) µν , h µν (cid:55)→ h µν − (cid:15) µν . (12)While actions of the form (11) are invariant under thesetransformations, the gauge-fixing and the regulator terms∆ S k = (cid:90) d x √ ¯ g h µν [ R k ( − ¯ D )] µνκλ h κλ , (13)with ¯ D µ denoting the covariant derivative constructedfrom ¯ g µν violate this symmetry. Thus Γ k [ h µν ; ¯ g µν ] ≡ Γ k [ g µν , ¯ g µν ] genuinely depends on two metric-type argu-ments.Nevertheless, the gravitational effective average action[7] provides a background-independent approach to quan-tum gravity. The background metric ¯ g µν is not an “ab-solute element” of the theory but rather a second, freelyvariable metric-type argument which is determined fromits own equations of motion. At the most conservativelevel this feature follows from standard properties of thebackground field method satisfied by the effective ac-tion Γ and their extension to the effective average ac-tion Γ k [107, 108, 155, 156, 201–205]. Alternatively, ithas been proposed to achieve background independencenot by quantization in the absence of a background, butrather by quantization on all background simultaneously [119]. We now review these arguments.The fact that Γ k and the resulting effective action Γdepend on two arguments allows to derive a backgroundas well as a quantum equation of motion δ Γ[ h ; ¯ g ] δ ¯ g µν (cid:12)(cid:12)(cid:12)(cid:12) ¯ g =¯ g eom ,h =0 = 0 , δ Γ[ h ; ¯ g ] δh µν (cid:12)(cid:12)(cid:12)(cid:12) ¯ g =¯ g eom ,h =0 = 0 . (14)The Ward identity following from the transformation(12) then relates these two equations implying that asolution of one is also a solution of the other. This allowsto fix ¯ g in a dynamical way. In particular, it shows thatat k = 0 the background metric does not have the statusof an absolute element. At finite values of k , the Wardidentity satisfied by Γ k receives additional contributionsfrom the regulator (13) which introduce a genuine depen-dence on the background field. From these arguments, itis then clear that “background independence” is restoredat k = 0 only.The “all backgrounds is no background” proposal pro-vides an extension of “background independence” to fi-nite values of k . The underlying idea is to describe (onesingle) background-independent quantum field theory ofthe metric through the (infinite) family of “all possible”background-dependent field theories that live on a non-dynamical classical spacetime. Each family member hasits own classical metric ¯ g µν rigidly attached to the space-time manifold. For each given background ¯ g µν , standardmethods can be used to quantize the fluctuation fieldsΦ. Repeating this procedure for all ¯ g µν yields expec-tation values (cid:104)O(cid:105) ¯ g which are manifestly ¯ g dependent ingeneral. Loosely speaking, the family of backgrounds,which is at the heart of background independence in theabstract sense of the word, should be regarded as the setof all possible ground states, one of which will be pickeddynamically.Ultimately, the physical background metric that ispresent in the geometric phase of quantum gravity, isdetermined by the dynamics of the system in a self-consistent fashion by solving the quantum equations ofmotion at finite kδ Γ k [ h ; ¯ g ] δh µν (cid:12)(cid:12)(cid:12)(cid:12) ¯ g =¯ g sc k ,h =0 = 0 , (15)where the self-consistent background metric (¯ g sc k ) µν isinserted. Hence, the expectation value of the metricis a prediction rather than an input. Notably, setting( h = 0 , ¯ g = ¯ g sc k ) is a particular way of going “on-shell”(but not the only one). We refer to [3, 156] for furtherdetails.Given these remarks, it is clear that future work mustaddress the following challenges: (1) The different functional dependence of Γ k on h µν and¯ g µν induces differences in the propagators for the fluc-tuation field and the background field. Thus, the func-tional dependence of Γ k on h µν and ¯ g µν separately shouldbe computed for a class of background metrics as broadas possible, as ultimately background independence canonly be achieved if the dependence on the two distinctarguments of Γ k is disentangled cleanly. For computational feasibility the existing calculationsmainly employ either highly symmetric background ge-ometries or the Seeley-DeWitt (early time) expansionof the heat-kernel which encapsulates only local (albeituniversal) information [206]. It is important to high-light that computations evaluating the left-hand side ofthe Wetterich equation at h = 0 (i.e., equating fluctua-tion propagator and background propagator) can deformand/or remove fixed points and introduce unphysical ze-ros of beta functions [163]. (2) The difference between the g µν dependence of Γ k andits ¯ g µν dependence, driven by the distinct dependence ofregulator and gauge fixing on the two metrics, is encodedin the modified split Ward or Nielsen identity resultingfrom (12). In principle, by solving the flow equation to-gether with this Ward identity, one would obtain a flowfor a functional of a single metric. In practice, the solu-tions of the Ward identity has only been possible for thesimplest approximations [108, 155, 202–205]. (3) When Γ k and with it (¯ g sc k ) µν show a strong k depen-dence, the effective spacetime is likely to possess multi-fractal properties which were argued to lead to a dimen-sional reduction in the ultraviolet [133, 207–209] andto a “fuzzy” spacetime structure at even lower scales[197, 210, 211]. In the existing analyses the fractal-likeproperties were characterized in terms of ordinary, i.e.,smooth classical metrics, the trick being that one andthe same spacetime manifold was equipped not with onebut rather the one-parameter family of classical metrics, { (¯ g sc k ) µν } . As these fractal-like properties relate to the k dependence of Γ k , it is at present unclear whether an“echo” of this behavior exists in the physical limit k → k → canprovide an answer to this question. If there is, it shouldbe a mostly negligible effect at scales relevant for currentexperiments.In conclusion, the issue of the background dependenceis a main obstacle to progress in the application of theFRG to quantum gravity, both at the conceptual andtechnical level. IV. ADDITIONAL METHODS FORASYMPTOTIC SAFETY ... where we review other techniques used to search forasymptotic safety in gravity, including the (cid:15) expansion,numerical simulations, tensor models, and stress thebenefits of using multiple methods.
Sections III D and III E have highlighted the techni-cal challenges one faces when employing the FRG tostudy asymptotically safe gravity. Therefore, there is astrong case for the use of complementary methods, es-pecially those where background independence can beimplemented, such as Regge calculus or random latticetechniques, as well as specific tensor models. Due to therather different nature of the systematic errors in these0approaches, this simultaneously addresses the challengelinked to the convergence of truncations. Furthermore,other techniques may be better suited to explore the com-plete phase diagram of quantum gravity potentially in-cluding pre-geometric phases.Historically, the starting point for studies of asymp-totically safe gravity has been the (cid:15) expansion around d = 2, [212–216], which has been pushed to two-loop or-der in [217]. It has been shown that the Reuter fixedpoint in d = 4 dimensions is continuously connected tothe perturbative fixed point seen in 2 + (cid:15) spacetime di-mensions [7, 125]. The connection between AsymptoticSafety and Liouville gravity in d = 2 dimensions has beenmade in [196]. An (off-shell) gauge and parameterizationdependence, as exhibited by truncated FRG studies, isalso present in the (cid:15) expansion. Higher-loop terms arerequired in order to resum the (cid:15) expansion for the criti-cal exponent to learn about the d = 4-dimensional case.This appears to be merely a technical challenge, to whichthe advanced techniques developed in the context of su-pergravity [218] might potentially be adapted.In line with the near-perturbative nature of the fixedpoint in d = 4, expected from FRG studies [57, 58, 142,143, 173], a Pad´e resummation might yield a fixed pointthat is continuously connected to the fixed point in thevicinity of two dimensions.Lattice approaches provide access to a statistical the-ory of random spatial geometries, thereby being in aposition to provide evidence for or against asymptoticsafety in the Euclidean regime. There are two main waysin which discrete random geometries are explored: Onecan hold a triangulation fixed and vary the edge lengths,as in Regge calculus, or hold the edge lengths fixed butvary the triangulation, as in dynamical triangulations.The latter have developed in two research branches: Eu-clidean Dynamical Triangulations [219], and Causal Dy-namical Triangulations [220, 221].Regge calculus (see [222] for a review) based on theEinstein-Hilbert action is subject to the well-known con-formal factor instability, which requires an extrapolationin order to extract information about a critical point, seethe discussion in [223]. With this caveat in mind, indi-cations for asymptotic safety are found in Monte Carlosimulations of Regge gravity [223] based on the Einstein-Hilbert action. Testing the effect of additional, e.g.,curvature-squared operators, which could correspond toadditional relevant directions and have an important im-pact on the phase structure, is an outstanding challengein Regge gravity. A first comparison of scaling exponentsobtained with the FRG to the leading-order exponent inRegge gravity can be found in [130, 224, 225].In the case of Causal Dynamical Triangulations (CDT),the configuration space includes only configurations thatadmit a Wick rotation, see [221] for a review. Therefore,an analytical continuation to a Lorentzian path integralis in principle possible. In two dimensions, one can solveCDT analytically. Owed to the fact that in this case thereare no local degrees of freedom, it has been shown in [226] and [227, 228] that the Hamiltonian appearing in thecontinuum limit agrees with the one for two-dimensionalcontinuum quantum gravity and Horava-Lifshitz gravity[229], respectively. Moreover, Liouville gravity can berecovered by allowing for topology change of the spatialslices [226]. It has been stressed in [221] though that theequivalence of CDT and Horava-Lifshitz gravity may notextend beyond the two-dimensional case.In higher dimensions, one searches for the contin-uum limit numerically. In practice, evidence for sev-eral [230, 231] second-order phase transition lines/pointsexists in numerical simulations, both in spherical andtoroidal spatial topology. The large-scale spatial topol-ogy does not appear to impact the phase structure [232],but can actually improve the numerical efficiency of thestudies, as observed in [233]. The higher-order transitioncan be approached from a phase in which several geo-metric indicators (spatial volume of the geometry as afunction of time [234]; Hausdorff dimension and spectraldimension [235]) signal the emergence of a spacetime withsemi-classical geometric properties. The properties of thecontinuum limit remain to be established, as the pro-cess of following RG trajectories along lines of constantphysics towards the phase transition has not yet led toconclusive results regarding asymptotic safety [236, 237].In Euclidean Dynamical Triangulations (EDTs), the con-figuration space differs from CDTs, as configurations donot in general admit a Wick rotation. This gives rise tospatial topology change and the proliferation of so-called“baby universes”. Early work [238–241] has not shown ahigher-order phase transition [242–244]. The inclusion ofa measure-term has led to the hypothesis that the first-order transition line could feature a second-order end-point, and some evidence exists that the volume profileof the “emergent universe” approaches that of Euclideande Sitter, i.e., a sphere, as one tunes towards the tenta-tive critical point [245, 246]. This measure term could bereinterpreted as a sum of higher-order curvature invari-ants [245, 246] contributing to the action. The investiga-tion [246] was unable to corroborate the appearance of asecond-order endpoint though.Finally, dynamical triangulations can be encoded in apurely combinatorial, “pre-geometric” class of models,so-called tensor models [247–254], that attempt to gen-eralize matrix models [255] for two-dimensional gravityto the higher-dimensional case. FRG tools which in-terpret the tensor size N as an appropriate notion of“pre-geometric” (i.e., background-independent) coarse-graining scale [256], allow to recover the well-knowncontinuum limit in two-dimensional quantum gravitywithin systematic uncertainties related to truncations[257]. First tentative hints for universal critical behav-ior in models with 3- and 4-dimensional building blockshave been found [258, 259]. This method could in thefuture provide further evidence for asymptotic safety, see[260] for a discussion, once the systematic uncertaintiesare reduced by suitable extensions of the truncation, andan understanding of the emergent geometries has been1developed.More broadly, the framework of the RenormalizationGroup and the notion of a universal continuum limitlinked to a fixed point have recently been gaining trac-tion in several approaches to quantum gravity, includ-ing group field theories [261–263] as well as spin foammodels [264]. Accordingly, the concept of asymptoticsafety might play an important role in several distinctapproaches to quantum gravity. In particular, in spinfoams, a search for interacting fixed points in numericalsimulations has started recently in reduced configurationspaces, see, e.g., [265–267].In summary, the further development and applicationof a broad range of tools to explore asymptotic safetycould be key to gain quantitative control over a poten-tial fixed point, establish its existence and to developrobust links to phenomenology, which rely on a goodunderstanding and control over systematic errors withinvarious techniques. V. RUNNING COUPLINGSA. A clarification of semantics ...where we clarify that the term “running coupling”is used with different meanings in different contexts.
Much of the current work on asymptotic safety of grav-ity uses techniques and jargon that are more common instatistical than in particle physics. This concerns evenbasic notions such as the RG. If one aims at detect-ing asymptotic safety by means of standard perturbativeparticle physics observables, there is thus much room formisunderstanding.The RG was used in particle physics largely as a toolto resum “large logarithms”, terms in the loop correc-tions to physical observables of the form log( p/µ ) =log( p/ Λ) + log(Λ /µ ), where p is a momentum, µ a refer-ence scale and Λ a UV cutoff. From the way they emerge,the beta functions that resum the large logs are justthe coefficients of the logarithmic divergences log(Λ /µ ).One important feature of these logarithmic terms is thattheir coefficients are “universal”, up to next-to-leadingnon trivial order (NLO) in the coupling expansion. Thisentails two things: on the one hand, it means that, upto NLO, they are independent of the way one computesthem . On the other hand, one can use them to “RG im-prove” any tree level observable, and one is guaranteed toobtain the correct result (not the full result, of course, butthe part that comes from calculating and then resummingthe logs). Here by “RG improvement” we mean the sub- They are almost always derived in dimensional regularization,which for technical reasons is the most convenient method, e.g.it respects gauge symmetries. stitution of the running coupling into a tree-level expres-sion, and the subsequent identification of the RG scalewith an appropriate physical scale of the system. Ifone demands these properties of a running coupling, thenone would say that only dimensionless couplings can run .Dimensionful couplings have power divergences that aresimply subtracted in perturbation theory. In line withthese arguments, it has been pointed out in [268, 269],that the one-loop corrections to gravity-mediated scat-tering amplitudes cannot be obtained from applying theRG improvement to Newton’s coupling.In Wilson’s non-perturbative approach to renormaliza-tion, all possible terms consistent with symmetries arepresent in the action. Quite often, the Wilsonian mo-mentum cutoff has a direct physics interpretation, e.g.,as lattice spacing in condensed-matter applications (witha relation to the Kadanoff block-spinning [270] underly-ing Wilson’s renormalization idea), and as the mass ofstates that are “integrated out” in effective field theo-ries. In lattice gauge theories the Wilsonian momentumcutoff is finally removed (in the continuum limit), butkeeps its physics interpretation similar to the condensed-matter applications at intermediate stages. Nevertheless,the momentum cutoff is treated mathematically as an in-dependent variable, and all couplings in the Wilsonian ac-tion depend on it. Apart from a few relevant parametersto be tuned to criticality, the remaining set of “runningcouplings” is not constrained by the demands of univer-sality; still, this notion of running couplings remains alsovalid at the non-perturbative level.The relation between the two definitions of the RGis this: At energy scales much higher than all themasses, the leading- and next to leading-order termsof the perturbative beta functions, that are indepen-dent of the renormalization scheme, can also be obtainedfrom the Wilsonian RG and are independent of detailsof the coarse-graining scheme. In particular, the one-loop terms can be easily found from (7). The recoveryof 2-loop terms from the FRG has been addressed, e.g.,in [271–277]. At energies comparable to the masses, thebeta functions extracted from the Wilsonian RG includethreshold effects which encode the automatic decouplingof massive modes from the flow at scales below the mass.This is an advantage over setups in which this decouplingis not accounted for automatically and must instead bedone by hand.If one accepts the more general Wilsonian definitionof running coupling, then the statement “dimensionfulcouplings cannot run” translates into the statement thatthe Wilsonian running of dimensionful couplings does notcarry the same direct physical meaning as the running For example in a process e + e − → e + e − at center of mass energy √ s >> m e at n -loop order the renormalized leading contributionwith subtraction scale µ is proportional to the tree level cross-section (at scale µ ) times (cid:80) l ≤ n [ α ( µ ) c log sµ ] l , which shows thatthe most convenient choice is µ = √ s . T c , which varies from one material to another.If one is interested in this physical information, the ac-curate scaling of the corresponding quadratic compositeoperator or the behavior of the two-point function shouldbe determined.Similar considerations may apply in quantum gravity,where the running Planck mass (the coefficient of the “ R ”operator in the effective Lagrangian) is a non-universalquantity which is just one of the parameters defining theposition of a possible UV fixed point and of the criti-cal surface containing it. Note that in an asymptoticallysafe theory of quantum gravity, the physics is related notjust to the UV fixed point, but to the particular renor-malized trajectory flowing away from it towards lowerenergy scales. Therefore it depends indirectly on all suchWilsonian (dimensionful) couplings. Observables, as al-ready discussed, are computed at k = 0 on the on-shellconfigurations and are mostly sensible to a number ofnon-universal parameters related to the finite number ofrelevant directions, including the (flowing) Planck mass.We shall discuss in Sect. V C how one could define theeffective couplings. B. Remarks on dimensional regularization ...where we explain in which cases some care isrequired for the correct interpretation of results achievedwithin dimensional regularization.
A seemingly technical point where the Wilsonian RGapproach differs from a perturbative particle-physics per-spective is the regularization of quantum modes. Whilethe FRG works with explicit momentum-space regulators(or spectral regulators of curved-spacetime Laplacians),conventional perturbation theory mostly uses dimen-sional regularization for reasons of convenience. Physicsmust not depend on the choice of the regularizationscheme, hence it is an obvious question as to whether dimensional regularization can also be brought to workin a FRG context and for the asymptotic-safety scenarioof gravity.In fact, one-loop results for power-counting marginaloperators quadratic in the curvature with dimensionlesscouplings exhibit the expected universality [120, 279,280]. However, this is no longer true for the RG run-ning of power-counting relevant and irrelevant operators,simply because they do not feature the same degree ofuniversality. Even worse, dimensional regularization isblind to power divergencies and hence acts as a projec-tion onto logarithmic divergences appearing as 1 /(cid:15) poles.For such reasons, Weinberg calls dimensional regulariza-tion “a bit misleading” in the context of asymptoticallysafe theories [5].Dimensional regularization relies on the virtues of ana-lytic continuation. Hence, its application requires to payattention to the analytic structure of a problem at hand.This is well known, for instance, from non-relativisticscattering problems where a naive application of dimen-sional regularization fails because of a different analyticstructure of the propagators and more care is needed toapply analytic continuation methods to regularize andcompute observables [281, 282]. The same is true forcomputations in large background fields where a naivestraightforward application of dimensional regularizationis not possible, but requires a careful definition in termsof a dimensionally continued propertime or ζ functionregularization [283, 284]. The latter techniques can belinked to heat-kernel methods and allow to access infor-mation related to power divergences [285].As most computations for asymptotically safe gravityare performed in “large backgrounds”, i.e., in a fiducialbackground spacetime, a proper use of dimensional regu-larization would similarly require a definition in terms of,e.g., a propertime or ζ function definition based on theheat kernel. In fact, approximations of the FRG havebeen mapped onto a propertime representation (proper-time RG). Applications to gravity do lend further sup-port to the existence of the Reuter fixed point and theasymptotic-safety scenario [187]. C. Correlation functions and form factors ...where we clarify the distinction between RG scaledependence and physical scale dependence within theFRG context. We further detail how the physics ofasymptotically safe theories is encoded in momentum-dependent correlation functions and form factors, discussthe definition of non-perturbative running couplings andthe construction of observables from these objects.
The idea of the Wilsonian renormalization group is tosolve the theory by integrating out quantum fluctuations,one (covariant) momentum shell at the time. It is crucialto distinguish the k -dependence from the dependence onphysical scales. In the FRG approach governed by the3Wetterich equation (5) with an infrared cutoff, generalcorrelation functions C k ( p , ..., p n ) = (cid:104) Φ i ( p ) · · · Φ i n ( p n ) (cid:105) k (16)are trivial for all (covariant) momentum scales p i /k (cid:28)
1, and carry the momentum dependence of the full theoryfor all momentum scales p i /k (cid:29) p i /k (cid:28) m (cid:38) k : the quantum dynamics dies off with powersof p i /k . In asymptotically safe theories, these correla-tion functions will exhibit indications of quantum scaleinvariance at large p i /k .Evidently, the correlation function C k ( p , ..., p n ) is ahighly non-trivial function of all p i , other physics scalesin the theory, such as mass scales, and the cutoff scale k . The latter is instrumental for the transition from thefull quantum dynamics of the theory to the trivial one inthe gapped regime. This non-trivial behavior is compli-cated by the fact that the n -point correlation functionscarry n momenta p i with i = 1 , ..., n . This results in amultiscale problem, unless we restrict ourselves to a sym-metric point with p i = p . We also remark that both UVand IR regimes may exhibit asymptotic power-law mo-mentum scaling or anomalous scaling and the momen-tum and cutoff dependence in the transition regime at p i /k ≈ S matrix elements viathe proper background vertices ¯Γ (¯Φ i ··· ¯Φ in ) k . To computesuch observables, in the FRG approach to asymptoticallysafe gravity we first have to compute the proper verticesof the fluctuation fields, Γ (Φ i ··· Φ in ) k . Their scale- and(covariant) momentum dependence indirectly encode thephysics of asymptotically safe gravity despite not beingobservables themselves. An important step towards ob-servables is made by considering running couplings, thatare renormalization group invariant combinations of the form factors or dressings of these vertices as defined instandard gauge theories and scalar and fermionic QFTs.These are defined from the k - and momentum-dependentvertices together with appropriate factors of the wave-function renormalizations. For instance, in the case ofscalar and fermionic QFTs, these are directly related to S matrix elements. In turn, in gauge theories such asQCD they lack gauge invariance but nonetheless carryimportant physics information: In QCD these runningcouplings derived from the proper vertices of the fluc-tuating or background fields give direct access to themomentum scaling in the perturbative regime as mea-sured by high-energy experiments, see, e.g., [286]. Fur-ther, non-perturbative physics, such as the emergence ofthe confinement mass gap, is also captured by these run-ning couplings, see, e.g., [287–289]. This implies that the momentum dependence of theserunning couplings at k = 0 provides rather non-trivialphysics information. In asymptotically safe gravity, itcan in particular be used to identify scaling regimes inthe UV and the IR as well as the transition scale: In[164, 166, 168, 169] non-perturbative generalizations ofthe Newton coupling G ( n ) k ( p ) with n = 3 ,
4, defined fromthe n -point functions, have been computed from combi-nations of the proper two-, three- and four-point func-tions of the fluctuation fields in a flat background andall cutoff scales. For the generalization to the case withmatter, see [170, 172–174]. In these calculations, the de-pendence on the n − n -point vertex hasbeen simplified by going to the momentum-symmetricpoint, allowing the definition of a running coupling thatdepends on a single momentum. A flat background, asused in the above studies is of course a first step towardsa comprehensive understanding of the physical scale de-pendence of quantum gravity. For first steps towards anextension to generic background see [156, 159].On a generic background, the dependence on physi-cal scales can also be captured in the language of formfactors. In the background effective action these formfactors appear naturally, see [290],¯Γ[ g µν ] = (cid:90) d x √ g (cid:2) f ( R ) + f ( R µν R µν )+ C µνρσ W T (∆) C µνρσ − R W R (∆) R + · · · (cid:3) , (17)Equation (17) also summarizes concisely the approxima-tion considered so far for the background effective action.The corresponding form factors W R and W T have beencomputed in [175]. Note that (17) can also be understoodas the dynamical effective action in the diffeomorphisminvariant single metric approach put forward in [278, 291–294]. There it has been argued that the physical gaugethere facilitates the direct physics interpretation of formfactor such as W T and W R .Both within the language of momentum-dependentcorrelation functions as well as with form factors, theasymptotically safe regime, the transition regime and along infrared regime with classical scaling have been iden-tified. The results are rather promising and open a pathtowards the computation of observables or their local in-tegral kernels. Still, the approximations used so far doin particular not sustain large curvatures and have to beupgraded significantly. VI. OBSERVABLES ...where we emphasize the necessity to investigateobservables in order to make quantum gravity testable,and discuss three possible classes of observables.
The physical behavior of a system is probed throughobservables. While their definition and construction is4not a problem in many interesting cases of quantum andstatistical field theories in flat, and possibly some specificclassical curved spacetimes, it is in general very difficultto define meaningful observables in quantum gravity. Tobegin with, already in classical gravity diffeomorphisminvariance makes the notion of a spacetime point unphys-ical and hence implies that there cannot exist any localobservable: any gauge invariant observable must be theintegral of a scalar density over all spacetime. The situ-ation is somewhat better in the presence of matter, forexample it makes sense to define the value of the scalarcurvature at the position of a particle, or at a point wherecertain matter fields have predetermined values [295].These observables are however difficult to work with inpractice. These problems persist in quantum gravity, see,e.g., [296]. Nevertheless the construction of observablesremains a crucial task.In the following, we will focus on observables in thesense of quantities that are of direct phenomenologicalrelevance. These often rely on introducing a (dynami-cally generated) background that provides a suitable no-tion of locality. The type of observables that one willconsider depends very strongly on the type of observa-tions that one has in mind. We will distinguish threepossible classes of observations that could be used to testasymptotic safety.
A. Particle physics at the Planck scale
The first is appropriate when we imagine living in amacroscopic classical spacetime and probing its short dis-tance structure by some “microscope” of the kind that isused in particle physics. For example, we could try to di-rectly measure scattering cross-sections and decay ratesat Planckian scale or beyond. In this case the issue ofdiffeomorphism invariance is circumvented by postulat-ing the existence of an asymptotically flat background,which is necessary in order to define the appropriate no-tions of particles and asymptotic states. The validity ofthis postulate remains to be investigated in a given quan-tum theory of gravity. In principle, the integral kernelsof these particle-physics observables can be constructedfrom the proper vertices of the background effective ac-tion ¯Γ (¯Φ i ··· ¯Φ in ) k for an asymptotically flat spacetime, see[297]. We provide some details on the calculation of thesequantities in Sec. V C. Indeed, the original formulation ofAsymptotic Safety by Weinberg was formulated in theseterms: as stated in [4, 5], ideally, the couplings whoserunning one wants to study should be defined directlyin terms of such observables. However, most of the ac-tual work on Asymptotic Safety is based on the runningof parameters in the Lagrangian, that are not directlyobservable or not even directly related to observables.Assuming that this notion makes sense, measurementof the S matrix at the Planck scale and beyond wouldgive the most direct and unambiguous test of Asymp-totic Safety. Unfortunately, neither the theoretical nor the experimental sides of the comparison are available.In settings with extra dimensions, scattering cross sec-tions have been calculated within the framework of RGimprovement [298–300], see Sec. VII D for a discussionof the potential pitfalls of this procedure. With currenttechnology, these observables are also unlikely to ever bemeasured. Furthermore, the postulate of an asymptoti-cally flat background leaves out many situations that areof interest in the context of quantum gravity. B. Low-energy imprints
A second possibility, still closely related to the worldof particle physics, but not requiring Planckian energy,is the observation of properties of the low-energy worldthat could carry an imprint of asymptotically safequantum gravity. One can distinguish two sub-cases,that we shall refer to as “higher-order observables” and“marginal observables”. Both sets of observables aremost directly calculable if one assumes a “great desert”between the Planck and the Fermi scale. Else, onerequires a specific model for the intervening physics. i) The high energy theory will leave traces in thelow energy effective field theory in the form of higherorder operators that are suppressed by inverse powersof the high scale. In particular, higher-order matterself-interactions are very likely both nonvanishing andirrelevant in the UV, if an asymptotically safe matter-gravity fixed point exists [301–306]. This results inpredictions for these higher-order couplings in the IR.The separation of scales between the Planck scale and IRscales is so large that, typically, these quantum-gravityeffects are unmeasurably tiny. Still, one may hopethat there exists a signature that is forbidden in anynon-gravitational process and that becomes detectableunder rather unexpectedly favorable circumstances. ii)
The other, significantly more promising, possibility isthat some gross features of the low energy world, probedat present or future colliders and linked to canonicallymarginal couplings, i.e., dimensionless operators, couldbe directly “explained” by properties of a UV-completequantum theory of gravity and matter. This is due tothe fact, explained in section III, that Asymptotic Safetymay yield more predictions than a perturbatively renor-malizable model. In the gravitational sector, this mech-anism may not lead to testable predictions: here only ahandful of parameters are experimentally accessible andthere are essentially no constraints on the value of thecurvature-squared couplings. In the matter sector, thispicture changes completely. In this case literally thou-sands of observables are available, depending on at leasttwo dozen free parameters. Some of these canonicallymarginal couplings could become irrelevant directions ofan asymptotically safe gravity-matter model. First ten-tative hints have been obtained in this direction, for ex-ample a proposed scenario for a prediction of the Higgsmass [307] and a calculation of the top mass [178], the5Abelian gauge coupling [193, 308] and the bottom mass[309]. These are obtained in comparatively small trunca-tions and are subject to the assumption that Euclideanresults carry over to Lorentzian gravity-matter systems.These are of course not “smoking guns” for AsymptoticSafety, but it is not unreasonable to expect that a micro-scopic description of quantum gravity constrains the fea-tures of a matter sector that can consistently be coupledto it. Indeed, the swampland program in string theory isbased upon the same assumption. Ultimately, one couldhope to arrive at an extended list of calculable propertiesof matter models from various quantum gravity theories,allowing to rule out some of the latter observationallywithout the need to probe Planck-scale physics directly.It therefore seems worthwhile to more systematically de-velop the predictions that an asymptotically safe theoryof gravity and matter can make for low-energy observ-ables. In particular, the dark-matter sector could al-low to make genuine predictions [310–312], in contrastto the consistency tests that the already measured prop-erties of the Standard Model provide. We shall discussin Sec. VII A how such effects could be calculated.
C. Asymptotically safe cosmology
The third class of observations is related to cosmol-ogy. As long as a Friedmann-Robertson-Walker (or someother) background is a good approximation, there is awell-developed machinery for the treatment of fluctua-tion correlators [313]. At the formal level, observablesin quantum gravity are given by integrated correlators,for example spacetime integrals of n -point correlations ofthe Ricci scalar O ( n ) R = (cid:104) (cid:89) i (cid:90) d x i (cid:112) g ( x i ) R ( x ) · · · R ( x n ) (cid:105) . (18)Naturally, while the spacetime dependent curvature fluc-tuations in (18) are not observables themselves, theycarry the physics information encoded in its spacetimeor momentum-scale dependence.Inflation is believed to occur at sub-Planckian ener-gies, but it may be close enough to a fixed-point regimeto be directly influenced by it. Further, in settings likeStarobinsky inflation, higher-order operators in the grav-itational theory actually drive inflation. Along this line,it was explored in [177] whether the freedom in the R coupling offered by Asymptotic Safety can be used torealize Starobinsky inflation giving power spectra com-patible with present observations. Moreover, there aresome tentative hints that quantum-gravity effects typi-cally drive scalar potentials towards flatness, see, e.g.,[181, 293, 310], and generally impose strong constraintson the inflationary potential that is usually introducedin a rather ad-hoc manner, see also [314]. In a more un-orthodox approach to early cosmology, the idea is beingexplored that quantum gravity directly solves the hori-zon, flatness and monopole problems and generates the appropriate spectrum of fluctuations without the need foradditional degrees of freedom together with an ad-hoc po-tential. In particular, in [315] it has been demonstratedthat an action including all gravitational four-derivativeinvariants leads to the suppression of spacetime configu-rations with an initial singularity as well as anisotropiesand inhomogeneities. In the early universe the usual flatspace QFT machinery is not available and one has touse different observables that are geared to high tem-perature/high curvature situations. Then one may hopethat features of the fixed point such as scaling exponentsand OPE coefficients - that in statistical physics are gen-erally considered measurable physical quantities - couldleave an imprint in these cosmological observables. D. Remarks
As with other situations where non-perturbativephysics is involved, one could try to cross-check re-sults obtained with continuum QFT methods with lat-tice studies. It is worth mentioning that also in latticeapproaches to quantum gravity, observables are very hardto define and especially to implement in the simulations,see, e.g., [316] for encouraging recent results. This is instark contrast to the large number of observables thatcan be defined in the presence of an asymptotically flatbackground.Finally, let us recall that in other approaches to quan-tum gravity such as LQG, “geometrical” observables suchas lengths, areas, volumes, and curvatures have played animportant role. These have also been discussed to someextent in Asymptotic Safety, [317], and can be computedwith a flow equation for composite operators [90, 318–323]. While presently it is not clear what type of mea-surement is required to access such observables, they canbe used to explore whether different approaches to quan-tum gravity give rise to universal physical results. Fur-ther, such geometrical observables have been used in [324]to set up a physical renormalization scheme.
VII. RELATION OF ASYMPTOTIC SAFETY TOTHE EFFECTIVE-FIELD THEORY APPROACHA. Asymptotic safety and Effective Field Theory ...where we discuss the relation of the EFT frameworkto Asymptotic Safety and also outline a strategy how todevise approximations in which the link between the twodescriptions can be established in practice.
The framework of EFT is pervasive in modern particlephysics. EFT is based on an expansion in
E/M , where E is the typical energy scale of the experiment, and M isthe scale above which the EFT description may no longerbe meaningful. In EFT one finds that higher loop correc-tions are suppressed by higher powers of E/M , so that6the tree level and one loop are usually enough to explainmost of the phenomenology, provided the system is in-deed perturbative in nature, as happens to be the case inmany particle-physics applications. Physical predictionsare possible even when the theory is not perturbativelyrenormalizable, as long as one considers only low-energyobservables and assumes that the dimensionless counter-parts of all couplings are roughly of O (1).Einstein gravity is a paradigmatic example of this pointof view. It is perturbatively non-renormalizable, but onecan still reliably compute observables in perturbationtheory, as long as they are not affected by the higher-derivative terms in the action, whose coefficients are notcalculable. This is the case for some non-analytic partsof scattering amplitudes. The calculation of the quan-tum corrections to the Newtonian potential is the mostreliable calculation ever performed in quantum gravity[325]. It is also the most accurate, since the separationof scales between the characteristic scale of the theory(the Planck scale) and the scale where one performs ex-periments (even at the LHC) is the largest of any EFT,so that loop corrections are suppressed by enormous fac-tors. In this way, every test of Einstein’s GR is also atest of this EFT of gravity.In view of this, the motivation for asymptotic safetyis twofold: first, to have predictions for what happensat and beyond the Planck scale and second, the promiseof increased predictivity, in particular also at lower en-ergies. This is especially desirable in the presence ofstandard-model or beyond-standard-model matter whichis or might be detected in present and future collidersor, e.g., in dark-matter detection experiments. We stressagain that the enhanced predictivity comes from the factthat asymptotic safety selects a class of RG trajectorieswhich are expected to be parametrized by only a few freeparameters. In principle, all the remaining coefficientsin the effective action are calculable, including the coeffi-cients of local higher dimensional operators that appearperturbatively divergent and are therefore not calculablein the EFT.When one follows a realistic RG trajectory from theUV fixed point, crossing the Planck scale and movingtowards the IR, one must eventually arrive in the im-mediate neighborhood of the free-theory fixed point ofEinstein theory, which is the domain where EFT is appli-cable. In this regime, all the predictions of EFT must stillhold true. Indeed, in the FRG formalism, the loop ex-pansion can be reconstructed systematically by expand-ing the right-hand side of the equation in powers of (cid:126) ,cf. Eq. (7). This is usually not done, because there arealready other methods that are perhaps better suited forthis task; but in principle, the FRG can reproduce all theresults of the EFT in this way.In practice, constructing a flow that links the descrip-tion of the fixed point, which might or might not be near-perturbative, to the perturbative low-energy regime af-ter potentially passing through a more strongly-coupledtransition regime, is a challenge. A possible strategy to deal with this complex problem is to figure out whichparts of the flow can be captured by perturbation the-ory, and then use different tools (perturbation theory,one’s favourite FRG approach) in the respective regimeso as to obtain maximally reliable predictions of the ob-servables. In order to link the description in terms of theFRG for the effective average action Γ k to the perturba-tive EFT setup, one needs to calculate Γ k = M , where M is the scale at which a perturbative description becomespossible. This procedure has been performed and care-fully checked in QCD, where we flow from an asymptoti-cally free theory of quarks and hadrons in the ultravioletto chiral perturbation theory and low energy effectivemodels in the infrared.First steps towards using the FRG to derive the effec-tive action of quantum gravity and matter systems havebeen taken in [326, 327]. Such calculations overlap sig-nificantly with EFT calculations.Investigations in low energy effective theories are typ-ically based on the Wilsonian action S eff , Λ (regularizedwith a UV cutoff) both in QCD and in standard pertur-bative low energy EFT approach which is used in colliderphysics. The Wilsonian action is the generalized Legen-dre transform of the effective average action [88, 89] andobeys the Polchinski equation [328]. So far, the Wilso-nian action has been less used in asymptotic-safety inves-tigations, but it could help to compare to results obtainedfrom collider measurements of scattering observables forits closeness to low energy effective theories. These workscan be based on recent proposals in [329, 330] for the flowof the Wilsonian action based on proper time regulatorschemes [330].The choice of truncation used for the effective averageaction down to the scale M might be crucial to correctlyencode the various consequences of the UV fixed point,both in the matter and gravity sector. Γ k = M or S eff , Λ= M provide the initial condition for a subsequent perturba-tive calculation at one or two loops; of course, also RGschemes would need to be matched for precision calcula-tions. The perturbative part of the RG evolution givesrise to the non-localities in Γ k → and all IR effects whichare necessary to include to correctly describe observables.In this way, the FRG and perturbation theory can beused concertedly in order to link the UV fixed point toobservable physics in the IR (see also Sec. V C), and theuse of different RG equations (Wetterich and Polchinski)would offer non-trivial consistency checks. B. Effective vs. fundamental Asymptotic Safety ...where we discuss why an asymptotically safe fixedpoint could matter even if the deep UV of quantumgravity is described by a completely different theory.
The RG fixed point underlying asymptotic safetyfeatures infinitely many infrared attractive directions.Therefore, a fixed point can serve various purposes in7different scenarios: 1) it can be the UV starting pointof an RG trajectory, 2) it can be the IR endpoint of anRG trajectory, 3) it can generate an intermediate scal-ing regime at finite scales. The latter option can playa role in settings where a more “fundamental” descrip-tion of quantum gravity holds at small distance scales,i.e., beyond a finite momentum cutoff k UV . Indeed, for k < k UV , an effective description (with the metric as theeffective gravitational field – not necessarily in the senseof perturbative EFT) holds, i.e., we are in the theoryspace of asymptotically safe gravity. The more funda-mental description provides the initial condition for theRG flow at k UV . If the initial condition satisfies a finitenumber of conditions related to the relevant directions ofthe fixed point, the flow will pass close by the fixed pointand exhibit an approximate scaling regime over a finiterange of scales. The flow towards the deep IR will thenclosely resemble that of an actual fixed-point trajectory,resulting in essentially the same predictivity [331], see[332] for a general discussion and [333] for a discussion inthe context of string theory.In this sense, an asymptotically safe fixed point canplay a role in an EFT setup for gravity, and serve asa way to extend the regime of validity of the standardperturbative EFT framework. C. The structure of the vacuum ...where we caution that the true ground state ofgravity might not be a flat background, making the bridgeto the EFT setting potentially more intricate. Thisquestion has so far only been addressed within a severeapproximation of the dynamics and degrees of freedom.
The EFT approach to quantum gravity typically quan-tizes (small) fluctuations about a flat background. Tolink asymptotic safety to the EFT regime, one musttherefore explore whether a flat background is a self-consistent choice, i.e., whether the flat background cor-responds to the ground state of the theory.To date the only explicit investigation of the vacuumstructure of asymptotically safe gravity based on the ef-fective action Γ k =0 has been performed within the con-formally reduced R + R -approximation and a layeredstructure of the effective spacetime has been found withinthis simple truncation (borrowing terminology from avacuum model of Yang Mills theory, it has been termed“lasagna vacuum”) [334]. Thereby the spatial modula-tion of the metric cures the notorious conformal factorinstability generating a phase similar to those present In this approximation, only fluctuations of the conformal fac-tor are taken into account. Quite surprisingly, this appears tosuffice to generate an asymptotically safe fixed point in simpletruncations [188], contrary to the expectation that the importantdegrees of freedom in gravity are the spin-2 ones. also in higher-derivative low-dimensional condensed mat-ter systems. While this proposed vacuum structure hasonly been found in a severe approximation of the dynam-ics and degrees of freedom, this can be read as a firmwarning regarding all backgrounds that are not shownto be solutions of the effective field equations. They areof no physical relevance and might convey an incorrectgeneral picture. In particular, one typically expects trun-cations to converge faster when the field configurationsare expanded about the true ground state of the theory –an expectation that can be tested within, e.g., the O( N )model. On the other hand, it is crucial to remark that aspatially modulated ground state appears to be difficultto reconcile with stringent tests of Lorentz symmetry inthe gravitational and the matter sector. Further, whilethe conformal approximation could suffice to capture thepresence of a fixed point, it is to be expected that theinclusion of spin-2 modes will have a strong impact onsuch studies.Moreover, the importance of properly accounting forthe ( k -dependent) ground state in studies of the flowis emphasized in a recent background-independent re-analysis of the cosmological constant problem allegedlycaused by quantum vacuum fluctuations. Paying carefulattention to identifying the correct ground state, the of-ten discussed naturalness problem disappears, see [197].Understanding the ground state of the theory at k = 0is important. It is expected that since Λ k = λ ∗ k → ∞ inthe quantum regime governed by the Reuter fixed point,the self-consistent metrics (cf. Sec. III E) ¯ g sc k →∞ ∝ k − will display increasing and ultimately diverging curva-ture. It is an open question how this manifests itselfat the level of Γ k → and its effective field equations.Whether this is an unphysical effect and only presentat large k or whether it translates into a physical scaledependence is presumably important for questions ofsingularity resolution in black-hole spacetimes and theearly universe. More generally, accounting for true vac-uum of the theory, with the help of the self-consistentbackground is important for a quantitatively precise ex-ploration of the phenomenological implications of thequantum-gravity effects. D. RG improvement ...where we critically review and discuss the proce-dure of RG improvement, discuss its interpretation as“quantum-gravity inspired” phenomenology, and cautionregarding the quantitative reliability of this tool.
Since the task of calculating the effective action Γ k → ,including its non-local contributions, is an extremelychallenging one, one may hope to extract qualitative in-formation on the effects of quantum fluctuations by ap-plying the procedure of “RG improvement” in gravity.In Sect.VI.A, we have already defined what is meant byRG improvement in a perturbative context. Proceeding8in a similar way in a gravitational context, it has been acommon strategy to retain the dependence of some of thecouplings, G k and Λ k say, on the RG scale k and iden-tify the latter with a geometrical quantity or momen-tum. Based on such RG improvement ideas there is asubstantial body of work investigating black-hole physics[335–349], gravitational collapse [350–356], and cosmo-logical scenarios [314, 354, 357–370] inspired by Asymp-totic Safety. One might expect that this procedure couldbe justified in some cases where the external scale in ques-tion acts as an IR cutoff for fluctuations.The “improvement” could be applied at differentstages, for instance, at the level of the action or the fieldequations, or of the solution of the field equation. Thisfreedom already implies that this procedure could lead toambiguous results. As an example, we may consider theRG-improvement procedure based on the effective aver-age action approximated by the Einstein-Hilbert action,Γ k = 116 πG k (cid:90) d x √ g [2Λ k − R ] . (19)Dimensional analysis implies that at the fixed point G k = g ∗ k − , Λ k = λ ∗ k . Identifying k with the Ricciscalar and substituting the result back into (19) leads toa higher-derivative R action. This is precisely what onewould expect for the fixed-point action for an f ( R ) theoryin the large R limit. Indeed RG improving any f ( R ) the-ory in the same way results in an R action, as expectedfrom classical scale invariance. Thus the scale identifica-tion generates interactions that have a natural place inthe effective action (17). However, this can lead at mostto qualitative insights, as is made clear, for example, bythe fact that even the simple identification k ∼ R canonly be made up to some arbitrary numerical factor.To understand better whether an RG improvement isjustified, let us consider some classic QFT examples, andcontrast them with their gravitational counterparts. TheUehling potential in QED is probably the paradigmaticexample: the correct form of the one-loop potential be-tween two point charges can be obtained by inserting theone-loop form of the running coupling in place of theclassical coupling and identifying the RG scale with theFourier momentum of the static potential between thepoint charges. Conversely, one can read off the screeningnature of the QED coupling from the one-loop effectiveaction. Similarly, the Coleman-Weinberg effective poten-tial is obtained, in a classically scale-invariant theory, byreplacing the classical quartic scalar coupling by its one-loop counterpart, evaluated at a renormalization scale k ∼ φ . This is justified, insofar as the classical VEV ofthe scalar is the only scale in the problem. Similar con-siderations have also been applied to non-Abelian gaugetheories [371–373].Coming closer to gravity, a recent example in curvedspace where RG improvement works, is the case of inter-acting conformally coupled fields in de Sitter spacetime.A correlator evaluated at the fixed point can be related toa CFT correlator in flat space by a Weyl transformation. Then, the late time power-like behavior of correlators canbe obtained as a resummation of secular terms controlledby the anomalous dimensions in flat space, with an RGimprovement at the renormalization scale µ = H [374],where the Hubble scale H of the de Sitter background isthe only non-trivial scale in the problem.Even more relevant for us, the running of G and thequantum corrections to the Newtonian potential due toa scalar field loop have been compared in [375]. Theyfind that in general the RG improvement gives the ex-pected qualitative behavior, and also reproduces the cor-rect numerical coefficients for minimal coupling ( ξ = 0)or conformal coupling ( ξ = 1 / G is logarithmic andproportional to the mass of the scalar field. This gives aresult that is in agreement with the quantum correctionto the potential. On the other hand, if one extracts the(quadratic) running of G from the FRG, and tries to de-rive the analog of the Uehling potential from there, onegets a term with the opposite sign of the quantum correc-tion calculated in EFT [325]. This is a clear failure of theRG improvement: the EFT calculation gives a screeningcontribution, whereas the FRG seems to give an antis-creening one, as required by asymptotic safety. The sit-uation has been clarified in part in [119]: due to the useof the background field method, there are different waysof defining Newton’s coupling that have different typesof behavior at low energy (where the EFT result holds)and at high energy, where one is assumed to approach afixed point. However, this leads us back to the issue ofthe shift Ward identities, cf. Sec. III E that, as discussedearlier, does not currently have a satisfactory solution.In conclusion, physical quantum effects in an asymp-totically safe theory have to be calculated, as in any otherQFT, from the effective action, where all fluctuationshave been integrated out. We stress that the results oneobtains from the RG improvement, e.g., for black holesor the early universe, cannot be viewed as actual deriva-tions from a fundamental theory of quantum gravity, butshould still be viewed as “quantum-gravity-inspired mod-els”, providing qualitatively sensible, though not neces-sarily precise, answers in some cases where there is aclearly identifiable single scale in the problem. VIII. SCALE SYMMETRY AND CONFORMALSYMMETRYA. The RG as scale anomaly ...where we clarify the meaning of scale symmetry inthe context of asymptotic safety.
A point that tends to generate confusion concerns the9interpretation of the RG flow as an anomalous breaking ofscale invariance. It may seem puzzling that the asymp-totic safety program claims (quantum) scale invarianceeven though Γ k contains dimensionful couplings. Thegoal of this subsection is to clarify this point. We follow[376], see also [278], section 6.9. Consider a perturbatively renormalizable QFT, withan interaction term u O ≡ u (cid:82) d x L , where L is adimension-four operator and u a dimensionless coupling.If there is no mass term, the theory is scale invariantunder the standard realization of scale transformationswhich act on the fields but not on the couplings. In thequantum theory, however, scale invariance is broken bythe beta function δ (cid:15) Γ = −A ( (cid:15) ) ∼ − (cid:15)β u O . (20)Here (cid:15) is the infinitesimal parameter generating the trans-formation, δ (cid:15) g µν = 2 (cid:15)g µν , etc. and A ( (cid:15) ) is the traceanomaly which can be formally seen as due to non-invariance of the functional integration measure. At afixed point β u = 0 and scale invariance is recovered.Eq. (20) can be generalized to the Wilsonian RG. Inthis case there is an additional term coming from thepresence of an explicit momentum cutoff which is givenby the “beta functional” defined in (5): δ (cid:15) Γ k = −A ( (cid:15) ) + (cid:15)k∂ k Γ k . (21)For the effective average action given in (1) one finds thatthe anomaly is given by [376] A ∼ (cid:15) (cid:88) i β u i k d i O i , (22)where d i is the canonical mass dimension of O i . Again A vanishes at a fixed point. Nevertheless, the standardrealization of scale invariance, acting on fields only, isbroken due to the extra term in (21) δ (cid:15) Γ k ∼ (cid:15) (cid:88) i d i ¯ u i O i . (23)There is however an alternative realization of scale invari-ance acting on both the fields and the cutoff. Here thetransformation of the fields remaining unaltered ˆ δ (cid:15) g µν =2 (cid:15)g µν , etc. while the cutoff transforms as ˆ δ (cid:15) k = − (cid:15)k . Un-der this alternative realization,ˆ δ (cid:15) Γ k ∼ −A ( (cid:15) ) , (24)which vanishes at a fixed point.In conclusion, we see that in a “Wilsonian” formulationof the RG, quantum scale invariance is realized at a fixedpoint, albeit with respect to a different implementation ofrescalings than the one generally used in particle physics. Here we discuss global scale invariance. It has been shown thatlocal scale invariance can be maintained in the RG flow provideda dilaton field is present [377–379].
B. Black hole entropy ...where we discuss an argument against a QFTfor gravity based on black-hole entropy and point outwhere assumptions are being made which require furtherinvestigation.
Refs. [380, 381] presented a chain of arguments indicat-ing that a quantum-field theoretic description of gravityin four dimensions cannot be UV complete. In short,this chain proceeds along the following lines. First, it isassumed that, at high energies, the density of states inquantum gravity is dominated by black holes, which alsogoes by the name of “asymptotic darkness”. Black holethermodynamics, building on quantum field theory on acurved background, implies that the leading term in theentropy S of the black hole is proportional to the area A of its horizon. For a d -dimensional Schwarzschild blackhole S ∝ A ∝ M d − d − (25)where M is the ADM mass of the black hole. Identifying M with a typical energy scale E , the asymptotic dark-ness hypothesis then suggests that the number of statesavailable at high energy should scale as S BH ∝ E d − d − (26)In four dimensions this implies that S BH ∝ E . On theother hand, the degrees of freedom of a conformal fieldtheory (CFT) living on a d -dimensional Minkowski spacefollow the scaling law S CFT ∝ E d − d (27)which in four dimensions becomes S CFT ∝ E . Themismatch between the density of available states (26)and (27) is then taken as an indication that the high-energy completion of four-dimensional gravity cannot begiven by a conformal field theory.We now critically review the assumptions entering intothis chain of arguments: ) Scales involved in the problem:Seeing quantum-gravity effects in scattering events re-quires going to large energies and small impact parame-ters relative to the Planck scale. This is not the sameas considering just trans-Planckian energies: the energyinvolved in the merger of two astrophysical black holesclearly exceeds the Planck mass m Pl ≈ − g by manyorders of magnitude. Nevertheless, classical general rela-tivity provides a very accurate description of these events,for which the impact parameter is large compared to thePlanck length. ) The asymptotic darkness hypothesis:The idea of asymptotic darkness relies on the hoop con-jecture [382] which states that scattering at sufficientlyhigh energy results in black-hole formation. While nu-merical simulations confirm this expectation in classical0gravity [383, 384], a corresponding study in the quantumcase is lacking, see also the discussion in [385]. Whenphrased in terms of the effective action (17), it is ex-pected that the form-factors W (∆) (or, more generally,the 1PI vertices) will play a central role in correctly de-scribing scattering processes at trans-Planckian scales.Currently, little is known about these effects though, andit is an open question whether or not Planckian scatter-ing in asymptotically safe gravity does or does not leadto black-hole formation. In [386], it has been proposedthat black-hole formation in Planckian scattering is a keyproperty of gravity that allows the theory to self-unitarize(classicalisation). Whether this has anything to do withasymptotic safety is an open question. See [387, 388]for related discussions in the context of non-linear sigmamodels. ) Corrections to the entropy formula:The semi-classical area law (25) is a good approximationfor large black holes. It receives further corrections fromquantum gravity though. Logarithmic corrections weredetermined in [389], indicating that S = A G −
32 log (cid:18) A G (cid:19) + · · · (28)Clearly, these corrections become increasingly importantfor small (i.e., near-Planckian) black holes, see, e.g., [389–392]. Thus, it is a priori unclear if the simple scaling law(25) is applicable in the quantum gravity regime. Dimensional reduction of the momentum space:A critical point in extending scaling arguments to quan-tum gravity is the identification of the correct notion ofdimensionality which actually controls the scaling laws.While in flat Minkowski space there is just the dimen-sion of spacetime d , fluctuating spacetimes are typicallycharacterized by a whole set of “generalized dimensions”(spectral dimension, Hausdorff dimension, etc.) which donot necessarily agree. In particular, a rather universal re-sult about quantum gravity [393, 394] indicates that thedimension of the theory’s momentum space (spectral di-mension) undergoes a dimensional reduction to d s = 2 atenergies above the Planck scale. In [395], it was arguedthat such a mechanism could constitute a potential wayto reconcile the semi-classical scaling in gravity with thescaling of states in the conformal field theory. In orderto make such proposals robust, it is important to iden-tify the proper notion of dimensionality which controlsthe scaling of the quantity of interest. In the context ofblack hole thermodynamics, it has been suggested thatthis could be achieved with the “Unruh dimension” [396]governing the scaling laws in the black-hole evaporationprocess. ) Entropy of asymptotically safe black holes:The entropy of black holes in asymptotic safety has beeninvestigated in [336, 339, 340, 345] based on RG improve-ment techniques (the cautionary remarks regarding RGimprovement from Sec. VII D apply in this case). Oneoutcome of this investigation was that the entropy ofPlanck-size black holes follows the Cardy-Verlinde for- mula [345] indicating compatibility with a conformal fieldtheory description. Concerning macroscopic black holes,the semi-classical result for the black-hole entropy canpresumably be understood entirely in terms of the entan-glement entropy of matter fields living on the black-holebackground geometry [397], see [398] for a comprehen-sive review, and [399, 400] for discussions in the contextof the FRG and asymptotic safety.In conclusion, combining semi-classical argumentsbased on the asymptotic darkness hypothesis and con-formal field theory in flat space gives rise to results intension with the asymptotic-safety conjecture. It is clearthat much more work is needed in order to actually showthat these arguments also apply in the framework ofquantum gravity. IX. UNITARITYA. General remarks ...where we point out that the concept of unitarity inquantum gravity is way more subtle than for a quantumfield theory on flat Minkowski space.
Conservation of probabilities is a cornerstone of quan-tum mechanics. For a QFT in a flat Lorentzian back-ground, this feature is reflected by the S matrix, con-necting the initial state and the final state of a physicalsystem, being unitary. Starting from a QFT defined on aEuclidean signature spacetime the Osterwalder-Schraderaxioms [401, 402], including the requirement of reflectionpositivity, guarantee that the theory has an analytic con-tinuation to a unitary QFT.Notably, it is highly non-trivial to generalize the con-cept of a unitary S matrix to more general backgrounds[403] or to the gravitational interactions [404, 405]. Ex-amples for such generalizations are the local S matrixin de Sitter space studied in [406] or the one recentlyconstructed in [407].Along a different line, the existence of unphysicalmodes such as tachyons, negative norm states, etc., ina given background ¯ g µν does not automatically signalthe inconsistency of the theory. It may just indicate theinstability of this particular background . As an ex-ample, [334] highlights how a non-standard backgroundremoves the conformal-mode problem in the Euclideanpath-integral. From a phenomenological point of view,a minimal requirement is to impose that cosmologicallyrelevant backgrounds of Friedman-Lemaitre-Robertson-Walker-type are stable on cosmic time-scales. This is a well-known situation, for instance, in scalar theories,where an expansion about a saddle point of the potential leads totachyonic instabilities, but does of course not signal an inconsis-tency of the theory. For instance, in inflationary scenarios theseinstabilities are key to the resulting physics.
B. Flat-space propagators ...where we review Ostrogradsky’s theorem and itsloopholes.
With the above cautionary remarks in mind, let us dis-cuss the gravitational propagator on a flat background.In the presence of a finite number of higher-derivativeterms, a partial-fraction decomposition of the propagatorreveals the presence of additional modes. For example, apropagator derived from a four-derivative theory yields1 p ( p + m ) = 1 m (cid:18) p − p + m (cid:19) . (29)The terms in the partial-fraction decomposition comewith alternating signs with the modes associated to thenegative residues corresponding to ghosts. In the case ofphysical fields related to asymptotic states, this violatesreflection positivity [410]. The latter signals the violationof unitarity in the Lorentzian theory and is related to aspectral function with negative parts. The violation ofunitarity is already present at the classical level where itcorresponds to an instability of the theory according toOstrogradsky’s theorem. Any non-degenerate Hamilto-nian with higher time derivatives of finite order unavoid-ably features such an instability, see, e.g, the pedagogicaldiscussion in [411]. This directly implies that truncationsto finite order in momenta generically feature truncation-induced instabilities and are not suitable to investigatethe unitarity of the theory.There are three prominent ways to avoid the Ostro-gradsky instability: Propagators consisting of an entire function with asingle zero at vanishing momentum may avoid the oc-currence of negative residues. This is the path takenby non-local ghost-free gravity [412–415]. At the classi-cal level, the well-posedness of the corresponding initial-value problem has been discussed in [416]. One can give up Lorentz invariance, introducing higher-order spatial derivative terms while keeping twotime-derivatives. This is the idea underlying Hoˇrava-Lifshitz gravity [229] which, by construction, is a power-counting renormalizable, unitary theory of gravity. Accept that Nature allows for the violation of causalityat microscopic levels [417–420]. In this case, the degreesof freedom associated with negative residues are inter-preted as “particles propagating backward in time”. Ifthese particles are sufficiently heavy this may not leavean experimentally detectable trace.We stress that in any case unitarity should be assessedbased on the propagators derived from the effective ac-tion Γ k =0 . Propagators derived from the effective averageaction Γ k at intermediate k may feature artificial poles.Under the flow in k , the mass of a ghost might diverge sothat the corresponding degrees of freedom decouple, see[421]. C. Spectral function of the graviton ...where we discuss the consequences of potentialnegative spectral weights of the graviton.
The ghost mode discussed in the last Sect. IX B is butone example for a spectral function that has negativespectral weights: Evidently, the second term in parenthe-sis in eq. (29) leads to a δ -function with negative prefactorin the spectral function of the graviton. In asymptoticallysafe gravity the graviton propagator is a general functionof momentum. Consequently the spectral function moregenerally may simply have negative parts.To begin with, negative spectral weights are a well-known feature of the gluon spectral function in Yang-Mills theory: upon the assumption of a spectral represen-tation of the gluon, it can be shown that its total spectralsum is vanishing due to the Oehme-Zimmermann super-convergence relation [422, 423]. This relation already im-plies that in the asymptotically free regime of Yang-Millstheory, the spectral function of the gluon is negative forlarge spectral values. Indeed, the analytic form for largespectral values can be computed within perturbation the-ory. More recently it could also be shown by similar ar-guments that the spectral function is also negative forsmall spectral values [424].These investigations highlight the fact that even theexistence of spectral representations for gauge fields withnon-linear gauge symmetries is an open issue. This istied to the fact that these fields are not directly relatedto asymptotic states even in regimes where they heuris-tically can be interpreted as particles. In QCD this ismanifest in gluon jets at colliders. In the context of grav-ity, this feature is intrinsic to the proposal made in point3) of the previous subsection: owing to their large mass,the states associated with the negative residue terms donot correspond to asymptotic states, see [425] for a recentdiscussion.In summary, even if the spectral representation of a2gauge fields exists, it very well can – and in the case ofthe gluon must – contain negative parts. Evidently, thisadds significantly to the already discussed intricacies ofdiscussing unitarity and the interpretation of positivityviolations in quantum gravity: negative spectral weightsmay be present without spoiling unitarity but clearlytheir presence casts doubts on unitarity. This situationasks for the investigation of the spectral representationof correlations of well-defined diffeomorphism-invariantobservables, see Sect. VI. D. Interpretation of potential ghost modes ...where we refer back to the concept of effectiveasymptotic safety discussed in Sec. VII B and discuss theinterpretation of the masses of unstable graviton modesin this context.
Future studies of unitarity may reveal that asymptoticsafety features physical ghost modes and hence is not aunitary fundamental QFT. Even in this case, an asymp-totically safe fixed point can still play a role within thesetting described in Sec. VII B, and serve as an extensionof the EFT regime for gravity. Then, the asymptoticallysafe description in this setting could inherit unitarityfrom the more “fundamental” description. In particular,the asymptotically safe setting can in this case exhibitunstable modes, with masses m > k UV – these signal theneed for a more “fundamental” description. Conversely,the masses of ghost modes can be used as an estimatefor the scale of new physics. E. Remarks
In summary, it is currently unclear whether or notasymptotically safe gravity is unitary, it shares with otherapproaches to quantum gravity. The question combinesboth conceptual and technical challenges in quantumgravity: there is the conceptual question of the complexstructure of correlation functions in the presence of adynamical metric field, as well as the necessity of non-perturbative numerical computations in Lorentzian sig-nature. As already emphasized in Sect. VI, cross-checksbetween quantum-gravity approaches and the concerteduse of more than one method are called for.
X. LORENTZIAN NATURE OF QUANTUMGRAVITY ...where we highlight the expected fundamental differ-ence between Lorentzian and Euclidean quantum gravityand explain why the flow equation is typically set up ina Euclidean setting.
Hitherto, the bulk of the Asymptotic Safety literatureemploys background spacetimes carrying Euclidean sig-nature metrics. This brings two technical advantages:Firstly, Euclidean signature entails that the squared mo-mentum of the fluctuation fields is positive semi-definite.Thus it is straightforward to define the “direction of theRG flow”, first integrating out fluctuations with a largesquared momentum before successively moving towardslower values. Secondly, the regulated propagators do notexhibit poles, as the particle cannot go on shell.For a QFT defined on flat Euclidean space R d , one cancarry out the computation and analytically continue theresults to Lorentzian signature by a Wick rotation. In thecontext of quantum gravity, including Asymptotic Safety,this strategy is very challenging for several reasons listedbelow, part of which has been already discussed in detailin Sect. IX.1. A generic background metric may not admit a (global)Killing vector which lends itself to an analytic con-tinuation to a well-defined Lorentzian time direction[426].2. The complex structure of the full graviton propaga-tors may obstruct the simple analytic continuation ofthe Euclidean propagator, for example there may verywell be cuts touching the Euclidean axis. Within theFRG this is complicated further as a momentum regu-larization either breaks the underlying (global) space-time symmetry or leads to additional poles and/orcuts [427]. There has been much progress in the pastyears on this in standard QFTs, see e.g., [427–432],but the extension to asymptotically safe gravity hasnot been put forward yet.3. At the structural level, there are solid arguments toexpect that the effective actions obtained from inte-grating out fluctuations in a Lorentzian and Euclideansignature setting will be different. Firstly, the space ofmetrics of the two settings comes with different topo-logical properties: while all Euclidean metrics can beconnected by geodesics (defined with respect to a suit-able connection) this property does not hold in theLorentzian case [433]. Secondly, the heat kernels fordifferential operators constructed from a Euclideanand Lorentzian signature metric differ by non-localterms [434]. While the latter do not affect the singu-larity structure of the heat kernel underlying pertur-bative renormalization, they may lead to differencesin Γ.A way to address the first point comes from studyingAsymptotic Safety in the Arnowitt-Deser-Misner (ADM)formalism. In this case, spacetime has a built-in foliationstructure which defines a natural time direction. A firstinvestigation of Asymptotic Safety in this framework hasbeen performed in [435] and further developed in a seriesof works [225, 436–442]. This provided first-hand indica-tions that the asymptotic-safety mechanism remains op-erative for Lorentzian signature metrics as well, at least3within very small truncations. At this stage the compu-tations in the Lorentzian signature framework have notreached a level of sophistication where the structural dif-ferences outlined in point 3) can be resolved. In gen-eral, the systematic development of the FRG applicableto Lorentzian signature spacetimes is a research area tobe developed in the future.This point could in the future become another exam-ple for the progress that can become possible if toolsand concepts from various quantum-gravity approachesare brought together. Specifically, causal set theory (see[443] for a review), at least when restricted to so-called“sprinklings”, can be viewed as a discretization of theLorentzian path integral over geometries. See also [444].This motivates the search for a universal continuum limit,linked to a second-order phase transition in the phase di-agram for causal sets. Monte Carlo studies of the phasediagram for restricted configuration spaces in low dimen-sionalities can be found in [445–448]. ACKNOWLEDGMENTS
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