The Asymptotic Safety Scenario in Quantum Gravity

1.1 Survey of the scenario

First, the gravitational field itself is taken seriously as the prime carrier of the relevant classical and quantum degrees of freedom. Second, a physics premise (“antiscreening”) is made about the self-interaction of these quantum degrees of freedom in the ultraviolet. Third, the effective diminution of the relevant degrees of freedom in the ultraviolet (on which morally speaking all approaches agree) is interpreted as universality in the statistical physics sense in the vicinity of an ultraviolet renormalization group fixed point. The resulting picture of microscopic geometry is fractal-like with a local dimensionality of two.

This amounts to a strategy centered around a functional integral picture, which was indeed the strategy adopted early on [144, 78], but which is now mostly abandoned. A functional integral over geometries of course has to differ in several crucial ways from one for fields on a fixed geometry. This led to the development of several formulations (canonical, covariant [64, 65, 66], proper time [212, 213], and covariant Euclidean [104, 92]). As is well-known the functional integral picture is also beset by severe technical problems [210, 63]. Nevertheless this should not distract attention from the fact that a functional integral picture has a physics content which differs from the physics content of other approaches. For want of a better formulation we shall refer to this fact by saying that a functional integral picture “takes the degrees of freedom of the gravitational field seriously also in the quantum regime”.

Let us briefly elaborate on that. Arguably the cleanest intuition to ‘what quantizing gravity might mean’ comes from the functional integral picture. Transition or scattering amplitudes for nongravitational processes should be affected not only by one geometry solving the gravitational field equations, but by a ‘weighted superposition’ of ‘nearby possible’ off-shell geometries. The rationale behind this intuition is that all known (microscopic) matter is quantized that way, and using an off-shell matter configuration as the source of the Einstein field equations is in general inconsistent, unless the geometry is likewise off-shell. Moreover, relativistic quantum field theory suggests that the matter-geometry coupling is effected not only through averaged or large scale properties of matter. For example nonvanishing connected correlators of a matter energy momentum tensor should be a legitimate source of gravitational radiation as well (see [81]). Of course this does not tell in which sense the geometry is off-shell, nor which class of possible geometries ought to be considered and be weighed with respect to which measure. Rapid decoherence, a counterpart of spontaneous symmetry breaking, and other unknown mechanisms may in addition mask the effects of the superposition principle. Nevertheless the argument suggests that the degrees of freedom of the gravitational field should be taken seriously also in the quantum regime, roughly along the lines of a functional integral.

Doing so one has to face the before mentioned enormous difficulties. Nevertheless facing these problems and maintaining the credible physics premise of a functional integral picture is, in our view, more appropriate than evading the problems in exchange for a less credible physics premise. Of course in the absence of empirical guidance the ‘true’ physics of quantum gravity is unknown; so for the time being it will be important to try to isolate differences in the physics content of the various approaches. By physics content we mean here qualitative or quantitative results for the values of “quantum gravity corrections” to generic physical quantities in the approach considered. Generic physical quantities should be such that they in principle capture the entire invariant content of a theory. In a conventional field theory S-matrix elements by and large have this property, in canonical general relativity Dirac observables play this role [9, 219, 70]. In quantum gravity, in contrast, no agreement has been reached on the nature of such generic physical quantities.

Quantum gravity research strongly draws on concepts and techniques from other areas of theoretical physics. As these concepts and techniques evolve they are routinely applied to quantum gravity. In the case of the functional integral picture the transferral was in the past often dismissed as eventually inappropriate. As the concepts and techniques evolved further, the reasons for the original dismissal may have become obsolete but the negative opinion remained. We share the viewpoint expressed by Wilczek in [230]: “Whether the next big step will require a sharp break from the principles of quantum field theory, or, like the previous ones, a better appreciation of its potentialities, remains to be seen”. As a first (small) step one can try to reassess the prospects of a functional integral picture for the description of the quantized gravitational field, which is what we set out to do here. We try to center the discussion around the above main ideas, and, for short, call a quantum theory of gravity based on them Quantum Gravidynamics. For the remainder of Section 1.1 we now discuss a number of key issues that arise.

In any functional integral picture one has to face the crucial renormalizability problem. Throughout we shall be concerned exclusively with (non-)renormalizability in the ultraviolet. The perspective on the nature of the impasse entailed by the perturbative non-renormalizability of the Einstein-Hilbert action (see Bern [30] for a recent review), however, has changed significantly since the time it was discovered by ’t Hooft and Veltmann [210]. First, the effective field theory framework applied to quantum gravity (see [50] for a recent review) provides unambiguous answers for ‘low energy’ quantities despite the perturbative non-renormalizability of the ‘fundamental’ action. The role of an a-priori microscopic action is moreover strongly deemphasized when a Kadanoff-Wilson view on renormalization is adopted. We shall give a quick reminder on this framework in Appendix A. Applied to gravity it means that the Einstein-Hilbert action should not be considered as the microscopic (high energy) action, rather the (nonperturbatively defined) renormalization flow itself will dictate, to a certain extent, which microscopic action to use and whether or not there is a useful description of the extreme ultraviolet regime in terms of ‘fundamental’ (perhaps non-metric) degrees of freedom. The extent to which this is true hinges on the existence of a fixed point with a renormalized trajectory emanating from it. The fixed point guarantees universality in the statistical physics sense. If there is a fixed point, any action on a renormalized trajectory describes identically the same physics on all energy scales lower than the one where it is defined. Following the trajectory back (almost) into the fixed point one can in principle extract unambiguous answers for physical quantities on all energy scales.

Compared to the effective field theory framework the main advantage lies not primarily in the gained energy range in which reliable computations can be made, but rather that one has a chance to properly identify ‘large’ quantum gravity effects at low energies. Indeed the (presently known) low energy effects that arise in the effective field theory framework, although unambiguously defined, are suppressed by the powers of energy scale/Planck mass one would expect on dimensional grounds. Conversely, if there are detectable low energy imprints of quantum gravity they presumably arise from high energy (Planck scale) processes, in which case one has to computationally propagate their effect through many orders of magnitudes down to accessible energies.

This may be seen as the the challenge a physically viable theory of quantum gravity has to meet, while the nature of the ‘fundamental’ degrees of freedom is of secondary importance. Indeed, from the viewpoint of renormalization theory it is the universality class that matters, not the particular choice of dynamical variables. Once a functional integral picture has been adopted, even nonlocally and nonlinearly related sets of fields or other variables may describe the same universality class — and hence the same physics.

The arena on which the renormalization group acts is a space of actions or, equivalently, a space of measures. A typical action has the form ∑ _α u _α P _α , where P _α are interaction monomials (including kinetic terms) and the u _α are scale dependent coefficients. The subset u _i which cannot be removed by field redefinitions are called essential parameters, or couplings. Usually one makes them dimensionless by taking out a suitable power of the scale parameter μ, \({g_i}(\mu) = {\mu ^{- {d_i}}}{u_i}(\mu)\). In the following the term “essential coupling” will always refer to these dimensionless variants. We also presuppose the principles according to which a (Wilson-Kadanoff) renormalization flow is defined on this area. For the convenience of the reader a brief reminder is included in Appendix A. In the context of Quantum Gravidynamics some key notions (unstable manifold and continuum limit) have a somewhat different status which we outline below.

Initially all concepts in a Wilson-Kadanoff renormalization procedure refer to a choice of coarse graining operation. It is part of the physics premise of a functional integral type approach that there is a description independent and physically relevant distinction between coarse grained and fine grained geometries. On a classical level this amounts to the distinction, for example, between a perfect fluid solution of the field equations and one generated by its 10³⁰ or so molecular constituents. A sufficiently large set of Dirac observables would be able to discriminate two such spacetimes. Whenever we shall refer later on to “coarse grained” versus “fine grained” geometries we have a similar picture in mind for the ensembles of off-shell geometries entering a functional integral.

With respect to a given coarse graining operation one can ask whether the flow of actions or couplings has a fixed point. The existence of a fixed point is the raison d’être for the universality properties (in the statistical field theory sense) which eventually are ‘handed down’ to the physics in the low energy regime. By analogy with other field theoretical systems one should probably not expect that the existence (or nonexistence) of a (non-Gaussian) fixed point will be proven with mathematical rigor in the near future. From a physics viewpoint, however, it is the high degree of universality ensued by a fixed point that matters, rather than the existence in the mathematical sense. For example non-Abelian gauge theories appear to have a (Gaussian) fixed point ‘for all practical purposes’, while their rigorous construction as the continuum limit of a lattice theory is still deemed a ‘millennium problem’. In the case of quantum gravity we shall present in Sections 3 and 4 in detail two new pieces of evidence for the existence of a (non-Gaussian) fixed point.

Accepting the existence of a (non-Gaussian) fixed point as a working hypothesis one is led to determine the structure of its unstable manifold. Given a coarse graining operation and a fixed point of it, the stable (unstable) manifold is the set of all points connected to the fixed point by a coarse graining trajectory terminating at it (emanating from it). It is not guaranteed though that the space of actions can in the vicinity of the fixed point be divided into a stable and an unstable manifold; there may be trajectories which develop singularities or enter a region of coupling space deemed unphysical for other reasons and thus remain unconnected to the fixed point. The stable manifold is the innocuous part of the problem; it is the unstable manifold which is crucial for the construction of a continuum limit. By definition it is swept out by flow lines emanating from the fixed point, the so-called renormalized trajectories. Points on such a flow line correspond to actions or measures which are called perfect in that they can be used to compute continuum answers for physical quantities even in the presence of an ultraviolet (UV) cutoff, like one which discretizes the base manifold. In practice the unstable manifold is not known and renormalized trajectories have to be identified approximately by a tuning process. What is easy to determine is whether in a given expansion “sum over coupling times interaction monomial” a coupling will be driven away from the value the corresponding coordinate has at the fixed point after a sufficient number of coarse graining steps (in which case it is called relevant) or will move towards this fixed point value (in which case it is called irrelevant). Note that this question can be asked even for trajectories which are not connected to the fixed point. The dimension of the unstable manifold equals the number of independent relevant interaction monomials that are ‘connected’ to the fixed point by a (renormalized) trajectory.

Typically the unstable manifold is indeed locally a manifold, though it may have cusps. Although ultimately it is only the unstable manifold that matters for the construction of a continuum limit, relevant couplings which blow up somewhere in between may make it very difficult to successfully identify the unstable manifold. In practice, if the basis of interaction monomials in which this happens is deemed natural and a change of basis in which the pathological directions could simply be omitted from the space of actions is very complicated, the problems caused by such a blow up may be severe. An important issue in practice is therefore whether in a natural basis of interaction monomials the couplings are ‘safe’ from such pathologies and the space of actions decomposes in the vicinity of the fixed point neatly into a stable and an unstable manifold. This regularity property is one aspect of “asymptotic safety”, as we shall see below.

A second caveat appears in infinite-dimensional situations. Whenever the coarse graining operates on an infinite set of potentially relevant interaction monomials, convergence issues in the infinite sums formed from them may render formally equivalent bases inequivalent. In this case the geometric picture of a (coordinate independent) manifold breaks down or has to be replaced by a more refined functional analytic framework. An example of a field theory with an infinite set of relevant interaction monomials is QCD in a lightfront formulation [174] where manifest Lorentz and gauge invariance is given up in exchange of other advantages. In this case it is thought that there are hidden dependencies among the associated couplings so that the number of independent relevant couplings is finite and the theory is eventually equivalent to conventional QCD. Such a reduction of couplings is nontrivial because a relation among couplings has to be preserved under the renormalization flow. In quantum gravity related issues arise to which we turn later.

The unstable manifold of a fixed point is crucial for the construction of a continuum limit. The fixed point itself describes a strictly scale invariant situation. More precisely the situation at the fixed point is by definition invariant under the chosen coarse graining (i.e. scale changing) operation. In particular any dependence on an ultraviolet cutoff must drop out at the fixed point, which is why fixed points are believed to be indispensable for the construction of a scaling limit. If one now uses a different coarse graining operation the location of the fixed point will change in the given coordinate system provided by the essential couplings. One aspect of universality is that all field theories based on the fixed points referring to different coarse graining operations have the same long distance behavior.

This suggests to introduce the notion of a continuum limit as an ‘equivalence class’ of scaling limits in which the physical quantities become independent of the UV cutoff, largely independent of the choice of the coarse graining operation, and, ideally, invariant under local reparameterizations of the fields.

In the framework of statistical field theories one distinguishes between two construction principles, a massless scaling limit and a massive scaling limit. In the first case all the actions/measures on a trajectory emanating from the fixed point describe a scale invariant system, in the second case this is true only for the action/measure at the fixed point. In either case the unstable manifold of the given fixed point has to be at least one-dimensional. Here we shall exclusively be interested in the second construction principle. Given a coarse graining operation and a fixed point of it with a nontrivial unstable manifold a scaling limit is then constructed by ‘backtracing’ a renormalized trajectory emanating from the fixed point. The number of parameters needed to specify a point on the unstable manifold gives the number of possible scaling limits — not all of which must be physically distinct, however.

In this context it should be emphasized that the number of relevant directions in a chosen basis is not directly related to the predictive power of the theory. A number of authors have argued in the effective field theory framework that even theories with an infinite number of relevant parameters can be predictive [126, 16, 32]. This applies all the more if the theory under consideration is based on a fixed point, and thus not merely effective. One reason lies in the fact that the number of independent relevant directions connected to the fixed point might not be known. Hidden dependencies would then allow for a (genuine or effective) reduction of couplings [236, 160, 174, 11, 16]. For quantum gravity the situation is further complicated by the fact that generic physical quantities are likely to be related only nonlocally and nonlinearly to the metric. What matters for the predictive power is not the total number of relevant parameters but how the observables depend on them. To illustrate the point imagine a (hypothetical) case where n² observables are injective functions of n relevant couplings each; then n measurements will determine the couplings, leaving n² − n predictions. This gives plenty of predictions, for any n, and it remains true in the limit n → ∞, despite the fact that one then has infinitely many relevant couplings.

Infinitely many essential couplings naturally arise when a perturbative treatment of Quantum Gravidynamics is based on a 1/p² type propagator. As first advocated by Gomis and Weinberg [94] the use of a 1/p² type graviton propagator in combination with higher derivative terms avoids the problems with unitarity that occur in other treatments of higher derivative theories. Consistency requires that quadratic counterterms (those which contribute to the propagator) can be absorbed by field redefinitions. This can be seen to be the case [10] either in the absence of a cosmological constant term or when the background spacetime admits a metric with constant curvature. The price to pay for the 1/p² type propagator is that all nonquadratic counterterms have to be included in the bare action, so that independence of the UV cutoff can only be achieved with infinitely many essential couplings, but it can be [94]. In order to distinguish this from the familiar notion of perturbative renormalizability with finitely many couplings we shall call such theories (perturbatively) weakly renormalizable. Translated into Wilsonian terminology the above results then show the existence of a “weakly renormalizable” but “propagator unitary” Quantum Gravidynamics based on a perturbative Gaussian fixed point.

The beta functions for this infinite set of couplings are presently unknown. If they were known, expectations are that at least a subset of the couplings would blow up at some finite momentum scale μ = μ_term and would be unphysical for μ > μ_term. In this case the computed results for physical quantities (“reaction rates”) are likely to blow up likewise at some (high) energy scale μ = μ_term.

A similar remark applies to the signs of coupling constants. When defined through physical quantities certain couplings or coupling combinations will be constrained to be positive. For example in a (nongravitational) effective field theory this constrains the couplings of a set of leading power counting irrelevant operators to be positive [2]. In an asymptotically safe theory similar constraints are expected to arise and are crucial for its physics viability.

Note that whenever the criterion for asymptotic safety is met, all the relevant couplings lie in the unstable manifold of the fixed point (which is called the “UV critical surface” in [227], Page 802, a term now usually reserved for the surface of infinite correlation length). The regularity property described earlier is then satisfied, and the space of actions decomposes in the vicinity of the fixed point into a stable and an unstable manifold.

Comparing the two perturbative treatments of Quantum Gravidynamics described earlier, one sees that they have complementary advantages and disadvantages: Higher derivative theories based on a 1/p⁴ propagator are strictly renormalizable with couplings that are presumed to be asymptotically safe; however unphysical propagating modes are present. Defining higher derivative gravity perturbatively with respect to a 1/p² propagator has the advantage that all propagating modes are physical, but infinitely many essential couplings are needed, a subset of which is presumed to be not asymptotically safe. From a technical viewpoint the challenge of Quantum Gravidynamics lies therefore not so much in achieving renormalizability but to reconcile asymptotically safe couplings with the absence of unphysical propagating modes.

The solution of this ‘technical’ problem is likely also to give rise to enhanced preditability properties, which should be vital to make the theory phenomenologically interesting. Adopting the second of the above perturbative constructions one sees that situation is similar to, for example, perturbative QED. So, apart from esthetic reasons, why not be content with physically motivated couplings that display a ‘Landau’ pole, and hence with an effective field theory description? Predictability in principle need not a be problem. The previous remarks about the predictability of theories with infinitely many essential couplings apply here. Even in Quantum Gravidynamics based on the perturbative Gaussian fixed point, some lowest order corrections are unambiguously defined (independent of the scale μ_term), as stressed by Donoghue (see [32] and references therein). In our view [82], as mentioned earlier, the main rationale for trying to go beyond Quantum Gravidynamics based on the perturbative Gaussian fixed point is not the infinite number of essential couplings, but the fact that the size of the corrections is invariably governed by power-counting dimensions. As a consequence, in the energy range where the computations are reliable the corrections are way too small to be phenomenologically interesting. Conversely, if there is a physics of quantum gravity, which is experimentally accessible and adequately described by some Quantum Gravidynamics, the above two features need to be reconciled — perturbatively or nonperturbatively.

Assuming that this can be achieved certain qualitative features such a gravitational functional integral must have can be inferred without actually evaluating it. One is the presence of anti-screening configurations, the other is a dimensional reduction phenomenon in the ultraviolet.

In non-Abelian gauge theories the anti-screening phenomenon can be viewed as the physics mechanism underlying their benign high energy behavior (as opposed to Abelian gauge theories, say); see e.g. [175] for an intuitive discussion. It is important not to identify “anti-screening” with its most widely known manifestation, the sign of the dominant contribution to the one-loop beta function. In an exact continuum formulation of a pure Yang-Mills theory, say, the correlation functions do not even depend on the gauge coupling. Nevertheless they indirectly do know about “asymptotic freedom” through their characteristic high energy behavior. In the functional integral measure this comes about through the dominance of certain configurations/histories which one might also call “anti-screening”.

By analogy one would expect that in a gravitational functional integral which allows for a continuum limit, a similar mechanism is responsible for its benign ultraviolet behavior (as opposed to the one expected by power counting considerations with respect to a 1/p² propagator, say). Some insight into the nature of this mechanism can be gained from a Hamiltonian formulation of the functional integral (authors, unpublished) but a concise characterization of the “anti-screening” geometries/histories, ideally in a discretized setting, remains to be found. By definition the dominance of these configurations/histories would be responsable for the benign ultraviolet properties of the discretized functional integral based on a non-Gaussian fixed point. Conversely understanding the nature of these antiscreening geometries/histories might help to design good discretizations. A discretization of the gravitational functional integral which allows for a continuum limit might also turn out to exclude or dynamically disfavor configurations that are taken into account in other, off-hand equally plausible, discretizations. Compared to such a naive discretization it will look as if a constraint on the allowed configurations/histories has been imposed. For want of a better term we call this an “anti-screening constraint”. A useful analogy is the inclusion of a causality constraint in the definition of the (formal Euclidean) functional integral originally proposed by Teitelboim [212, 213], and recently put to good use in the framework of dynamical triangulations [5]. Just as the inclusion of a good causality constraint is justified retroactively, so would be the inclusion of a suitable “antiscreening” constraint.

In accordance with this argument a 1/p⁴ type propagator goes hand in hand with a non-Gaussian fixed point for g_N in two other computational settings: in strictly renormalizable higher derivative theories (see Section 2.3.2 and in the 1/N expansion [216, 217, 203]. In the latter case a nontrivial fixed point goes hand in hand with a graviton propagator whose high momentum behavior is of the form 1/(p⁴ ln p²), in four dimensions, and formally 1/p ^d in d dimensions.

The fact that a large anomalous dimension occurs at a non-Gaussian fixed point was first observed in the context of the 2 + ϵ expansion [116, 117] and then noticed in computations based on truncated flow equations [133]. The above variant of the argument [157] shows that no specific computational information enters. It highlights what is special about the Einstein-Hilbert term (within the class of local gravitational actions): it is the kinetic (second derivative) term itself which carries a dimensionful coupling. Of course one could assign to the metric a mass dimension 2, in which case Newton’s constant would be dimensionless. However one readily checks that then the wave function renormalization constant of a standard matter kinetic term acquires a mass dimension d − 2 for bosons and d − 1 for fermions, respectively. Assuming that the dimensionless parameter associated with them remains nonzero as μ → ∞, one can repeat the above argument and finds now that all matter propagators have a 1/p ^d high momentum behavior, or a ln x² short distance behavior. It is this universality which justifies to attribute the modification in the short distance behavior of the fields to a modification of the underlying (random) geometry. This may be viewed as a specific variant of the old expectation that gravity acts as a short distance regulator.

Let us stress that while the anomalous dimension always governs the UV behavior in the vicinity of a (UV) fixed point, it is in general not related to the geometry of field propagation (see [125] for a discussion in QCD). What is special about gravity is ultimately that the propagating field itself determines distances. In the context of the above argument this is used in the reshuffling of the soft UV behavior to matter propagators. The propagators used here should be viewed as “test propagators”, not as physical ones. One transplants the information in η_N derived from the gravitational functional integral into a conventional propagator on a (flat or curved) background spacetime. The reduced dimension two should be viewed as an “interaction dimension” specifying roughly the (normalized) number of independent degrees of freedom a randomly picked one interacts with.

The same conclusion (1/p ^d propagators or interaction dimension 2) can be reached in a number of other ways as well, which are described in Section 2.4. A more detailed understanding of the microstructure of the random geometries occuring in an asymptotically safe functional integral remains to be found (see however [135, 134]).

Accepting this dimensional reduction as a working hypothesis it is natural to ask whether there exists a two-dimensional field theory which provides an quantitatively accurate (‘effective’) description of this extreme UV regime. Indeed, one can identify a number of characteristics such a field theory should have, using only the main ideas of the scenario (see the end of Section 2.4). The asymptotic safety of such a field theory would then strongly support the corresponding property of the full theory and the self-consistency of the scenario. In summary, we have argued that the qualitative properties of the gravitational functional integral in the extreme ultraviolet follow directly from the previously highlighted principles: the existence of a nontrivial UV fixed point, asymptotic safety of the couplings, and antiscreening. Moreover these UV properties can be probed for self-consistency.

1.2 Evidence for asymptotic safety

Presently the evidence for asymptotic safety in quantum gravity comes from the following very different computational settings: the 2 + ϵ expansion, perturbation theory of higher derivative theories, a large N expansion in the number of matter fields, the study of symmetry truncations, and that of truncated functional flow equations. Arguably none of the pieces of evidence is individually compelling but taken together they make a strong case for asymptotic safety.

The results from the 2 + ϵ expansion were part of Weinberg’s original motivation to propose the scenario. Since gravity in two and three dimensions is non-dynamical, however, the lessons for a genuine quantum gravitational dynamics are somewhat limited. Higher derivative theories were known to be strictly renormalizable with a finite number of couplings, at the expense of having unphysical propagating modes (see [207, 206, 83, 19, 59]). In hindsight one can identify a non-Gaussian fixed point for Newton’s constant already in this setting (see [54] and Section 2.3). The occurance of this non-Gaussian fixed point is closely related to the 1/p⁴-type propagator that is used. The same happens when (Einstein or a higher derivative) gravity is coupled to a large number N of matter fields and a 1/N expansion is performed. A nontrivial fixed point is found that goes hand in hand with a 1/p⁴-type progagator (modulo logs), which here arises from a resummation of matter self-energy bubbles, however.

As emphasized before the challenge of Quantum Gravidynamics is not so much to achieve (per-turbative or nonperturbative) renormalizability but to reconcile asymptotically safe couplings with the absence of unphysical propagating modes. Two recent developments provide complementary evidence that this might indeed be feasible. Both of these developments take into account the dynamics of infinitely many physical degrees of freedom of the four-dimensional gravitational field. In order to be computationally feasible the ‘coarse graining’ has to be constrained somehow. To do this the following two strategies have been pursued (which we label Strategies (c) and (d) according to the discussion below):

(c) The metric fluctuations are constrained by a symmetry requirement, but the full (infinite-dimensional) renormalization group dynamics is considered. We shall refer to this as the strategy via symmetry reductions.

(d) All metric fluctuations are taken into account but the renormalization group dynamics is projected onto a low-dimensional submanifold. Since this is done using truncations of functional renormalization group equations, we shall refer to this as the strategy via truncated functional flow equations.

Both strategies (truncation in the fluctuations but unconstrained flow and unconstrained quantum fluctuations but constrained flow) are complementary. Tentatively both results are related by the dimensional reduction phenomenon described before (see Section 2.4). The techniques used are centered around the background effective action, but are otherwise fairly different. For the reader’s convenience we included summaries of the relevant aspects in Appendices A and B. The main results obtained from Strategies (c) and (d) are reviewed in Sections 3 and 4, respectively.

For the remainder of this section we now first survey the pieces of evidence from all the computational settings (a-d):

(a) Evidence from 2 + ϵ expansions: In the non-gravitational examples of perturbatively non-renormalizable field theories with a non-Gaussian fixed point the non-Gaussian fixed point can be viewed as a ‘remnant’ of an asymptotically free fixed point in a lower-dimensional version of the theory. It is thus natural to ask how gravity behaves in this respect. In d = 2 spacetime dimensions Newton’s constant g_N is dimensionless, and formally the theory with the bare action \(g_{\rm{N}}^{- 1}\int {{d^2}x\sqrt g R(g)}\) is power counting renormalizable in perturbation theory. However, as the Einstein-Hilbert term is purely topological in two dimensions, the inclusion of local dynamical degrees of freedom requires, at the very least, starting from 2 + ϵ dimensions and then studying the behavior near ϵ → 0⁺. The resulting “ϵ-expansion” amounts to a double expansion in the number of ‘graviton’ loops and in the dimensionality parameter ϵ. Typically dimensional regularization is used, in which case the UV divergencies give rise to the usual poles in 1/ϵ. Specific for gravity are however two types of complications. The first one is due to the fact that \(\int {{d^{2 + \epsilon}}x\sqrt g R(g)}\) is topological at ϵ = 0, which gives rise to additional “kinematical” poles of order 1/ϵ in the graviton propagator. The goal of the renormalization process is to remove both the ultraviolet and the kinematical poles in physical quantities. The second problem is that in pure gravity Newton’s constant is an inessential parameter, i.e. it can be changed at will by a field redefinition. Newton’s constant g_N can be promoted to a coupling proper by comparing its flow with that of the coefficient of some reference operator, which is fixed to be constant.

Technically the non-universality of γ arises from the before-mentioned kinematical poles. In the early papers [86, 53, 227] the Choice i was adopted giving γ = 19/(24π), or γ = (19 − c)/(24π) if free matter of central charge c is minimally coupled. A typical choice for Choice ii is a mass term of a Dirac fermion, a typical choice for Choice iii is the coupling of a four-fermion (Thirring) interaction. Then γ comes out as γ = (19 + 6Δ₀ − c)/(24π), where Δ₀ = 1/2, 1, respectively. Here Δ₀ is the scaling dimension of the reference operator, and again free matter of central charge c has been minimally coupled. It has been argued in [118] that the loop expansion in this context should be viewed as double expansion in powers of ϵ and 1/c, and that reference operators with Δ₀ = 1 are optimal. The Choice iv has been pursued systematically in a series of papers by Kawai et al. [116, 117, 3]. It is based on a parameterization of the metric in terms of a background metric ḡ _μν , the conformal factor e ^σ , and a part f _μν which is traceless, ḡ ^μν f _μν = 0. Specifically \({g_{\mu v}} = {{\bar g}_{\mu \rho}}{({e^f})^\rho}_v{e^\sigma}\) is inserted into the Einstein-Hilbert action; propagators are defined (after gauge fixing) by the terms quadratic in σ and f _μν , and vertices correspond to the higher order terms. This procedure turns out to have a number of advantages. First the conformal mode σ is renormalized differently from the f _μν modes and can be viewed as defining a reference operator in itself; in particular the coefficient γ comes out as ν = (25 − c)/(24π). Second, and related to the first point, the system has a well-defined ϵ-expansion (absence of poles) to all loop orders. Finally this setting allows one to make contact to the exact (KPZ [122]) solution of two-dimensional quantum gravity in the limit ϵ → 0.

The action (1.7) can be supplemented by a matter action, containing a large number, O(N), of free matter fields. One can then keep the product N · c _d G_N fixed, retain the usual normalization of the matter kinetic terms, and expand in powers of 1/N. Renormalizability of the resulting ‘large N expansion’ then amounts to being able to remove the UV cutoff order by order in the formal series in 1/N. This type of studies was initiated by Tomboulis where the gravity action was taken either the pure Ricci scalar [216], Ricci plus cosmological term [203], or a higher derivative action [217], with free fermionic matter in all cases. More recently the technique was reconsidered [169] with Equation (1.7) as the gravity action and free matter consisting of Nn_S scalar fields, Nn_D Dirac fields, and Nn_M Maxwell fields.

Starting from the Einstein-Hilbert action the high energy behavior of the usual 1/p²-type propagator gets modified. To leading order in 1/N the modified propagator can be viewed as the graviton propagator with an infinite number of fermionic self-energy bubbles inserted and resummed. The resummation changes the high momentum behavior from 1/p² to 1/(p⁴ ln p²), in four dimensions. In 2 < d < 4 dimensions the resulting 1/N expansion is believed to be renormalizable in the sense that the UV cutoff Λ can strictly be removed order by order in 1/N without additional (counter) terms in the Lagrangian. In d = 4 the same is presumed to hold provided an extra C² term is included in the bare Lagrangian, as in Equation (1.7). After removal of the cutoff the beta functions of the dimensionless couplings can be analyzed in the usual way and already their leading 1/N term will decide about the flow pattern.

The qualitative result (due to Tomboulis [216] and Smolin [203]) is that there exists a nontrivial fixed point for the dimensionless couplings g_N, λ, and s. Its unstable manifold is three dimensional, i.e. all couplings are asymptotically safe. Repeating the computation in 2 + ϵ dimensions the fixed point still exists and (taking into account the different UV regularization) corresponds to the large c (central charge) limit of the fixed point found the 2 + ϵ expansion.

As a caveat one should add that the 1/p⁴-type propagators occuring both in the perturbative and in the large N framework are bound to have an unphysical pole at some intermediate momentum scale. This pole corresponds to unphysical propagating modes and it is the price to pay for (strict) perturbative renormalizability combined with asymptotically safe couplings. From this point of view, the main challenge of Quantum Gravidynamics lies in reconciling asymptotically safe couplings with the absence of unphysical propagating modes. Precisely this can be achieved in the context of the 2 + 2 reduction.

As mentioned the effective dynamics looks two-dimensional. Concretely the classical action describing the dynamics of the 2-Killing vector subsector is that of a non-compact symmetric space sigma-model non-minimally coupled to 2D gravity via the “area radius” \(\rho : = \sqrt {\det {{({M_{ab}})}_{1 \leq a,b \leq 2}}}\), of the two Killing vectors. To avoid a possible confusion let us stress, however, that the system is very different from most other models of quantum gravity (mini-superspace, 2D quantum gravity or dilaton gravity, Liouville theory, topological theories) in that it has infinitely many local and self-interacting dynamical degrees of freedom. Moreover these are literally (an infinite subset of) the degrees of freedom of the four-dimensional gravitational field, not just analogues thereof. The corresponding classical solutions (for both signatures of the Killing vectors) have been widely studied in the general relativity literature, c.f. [98, 26, 121]. We refer to [45, 46, 56] for details on the reduction procedure and [197] for a canonical formulation.

In summary, in the context of the 2 + 2 reduction an asymptotically safe coupling flow can be reconciled with the absence of unphysical propagating modes. In contrast to the technique on which Evidence (d) below is based the existence of an infinite cutoff limit here can be shown and does not have to be stipulated as a hypothesis subsequently probed for self-consistency. Since the properties of the 2 + 2 truncation qualitatively are the ones one would expect from an ‘effective’ field theory describing the extreme UV aspects of Quantum Gravidynamics (see the end of Section 2.4), its asymptotic safety is a strong argument for the self-consistency of the scenario.

The effective average action has been generalized to gravity [179] and we shall describe it and its properties in more detail in Sections 4.1 and 4.2. As before the metric is taken as the dynamical variable but the bare action Γ_λ,λ is not specified from the outset. In fact, conceptually it is largely determined by the requirement that a continuum limit exists (see the criterion in Section 2.2). Γ_λ,λ can be expected to have a well-defined derivative expansion with the leading terms roughly of the form (1.7). Also the gravitational effective average action Γ_λ,k obeys an ‘exact’ FRGE, which is a new computational tool in quantum gravity not limited to perturbation theory. In practice Γ_λ,k is replaced in this equation with a Λ independent functional interpreted as Γ_∞,k. The assumption that the ‘continuum limit’ Γ_∞,k for the gravitational effective average action exists is of course what is at stake here. The strategy in the FRGE approach is to show that this assumption, although without a-priori justification, is consistent with the solutions of the flow equation \(k{d \over {dk}}{\Gamma _{\infty, k}} = {\rm{rhs}}\) (where right-hand-side now also refers to the Hessian of Γ_∞,k). The structure of the solutions Γ _k of this cut-off independent FRGE should be such that they can plausibly be identified with Γ_∞,k. Presupposing the ‘infrared safety’ in the above sense, a necessary condition for this is that the limits lim_k→∞ Γ _k and lim_k→0 Γ _k exist. Since k ≤ Λ the first limit probes whether Λ can be made large; the second condition is needed to have all modes integrated out. In other words one asks for global existence of the Γ _k flow obtained by solving the cut-off independent FRGE. Being a functional differential equation the cutoff independent FRGE requires an initial condition, i.e. the specification of a functional Γ_initial which coincides with Γ _k at some scale k = k_initial. The point is that only for very special ‘fine tuned’ initial functionals Γ_initial will the associated solution of the cutoff independent FRGE exist globally [157]. The existence of the k → ∞ limit in this sense can be viewed as the counterpart of the UV renormalization problem, namely the determination of the unstable manifold associated with the fixed point lim_k→∞ Γ _k . We refer to Section 2.2 for a more detailed discussion of this issue.

The impact of matter has been studied by Percacci et al. [72, 171, 170]. Minimally coupling free fields (bosons, fermions, or Abelian gauge fields) one finds that the non-Gaussian fixed point is robust, but the positivity of the fixed point couplings \(g_0^{\ast} > 0\), \(g_1^{\ast} > 0\) puts certain constraints on the allowed number of copies. When a self-interacting scalar χ is coupled non-minmally via \(\sqrt g \left[ {\left({{k_0} + {k_2}{{\rm{\chi}}^2} + {k_4}{{\rm{\chi}}^4} + \ldots} \right)R\left(g \right) + {\lambda _0} + {\lambda _2}{{\rm{\chi}}^2} + \cdots + \partial {\rm{\chi}}\partial {\rm{\chi}}} \right]\), one finds a fixed point \(k_0^{\ast} > 0\), \(\lambda _0^{\ast} > 0\) (whose values are with matched normalizations the same as \(g_1^{\ast}\), \(g_0^{\ast}\) in the pure gravity computation) while all self-couplings vanish, \(k_2^{\ast} = k_4^{\ast} = \cdots = 0\), \(\lambda _2^{\ast} = \lambda _4^{\ast} = \cdots = 0\). In the vicinity of the fixed point a linearized stability analysis can be performed; the admixture with λ₀ and κ₀ then lifts the marginality of λ₄, which becomes marginally irrelevant [171, 170]. The running of κ₀ and λ₀ is qualitatively unchanged as compared to pure gravity, indicating that the asymptotic safety property is robust also with respect to the inclusion of self-interacting scalars.

Both Strategies (c) and (d) involve truncations and one may ask to what extent the results are significant for the (intractable) full renormalization group dynamics. In our view they are significant. This is because even for the truncated problems there is no a-priori reason for the asymptotic safety property. In the Strategy (c) one would in the coupling space considered naively expect a zero-dimensional unstable manifold rather than the co-dimension zero one that is actually found! In Case (d) the ansatz (1.13, 1.14) implicitly replaces the full gravitational dynamics by one whose functional renormalization flow is confined to the subspace (1.13, 1.14) (similar to what happens in a hierarchical approximation). However there is again no a-priori reason why this approximate dynamics should have a non-Gaussian fixed point with positive fixed point couplings and with an unstable manifold of co-dimension zero. Both findings are genuinely surprising.

Nevertheless even surprises should have explanations in hindsight. For the asymptotic safety property of the truncated Quantum Gravidynamics in Strategies (c) and (d) the most natural explanation seems to be that it reflects the asymptotic safety of the full dynamics with respect to a nontrivial fixed point.

Tentatively both results are related by the dimensional reduction of the residual interactions in the ultraviolet. Alternatively one could try to merge both strategies as follows. One could take the background metrics in the background effective action generic and only impose the 2-Killing vector condition on the integration variables in the functional integral. Computationally this is much more difficult; however it would allow one to compare the lifted 4D flow with the one obtained from the truncated flows of the effective average action, presumably in truncations far more general than the ones used so far. A better way to relate both strategies would be by trying to construct a two-dimensional UV field theory with the characteristics to be described at the end of Section 2.4 and show its asymptotic safety.

1.3 Some working definitions

Here we attempt to give working definitions for some of the key terms used before.

Quantum Gravidynamics: We shall use the term “Quantum Gravidynamics” to highlight a number of points in the present circle of ideas. First, that one aims at relating the micro- and the macro-physics of the gravitational field through a renormalization flow defined conceptually in terms of a functional integral. In contrast to “Quantum General Relativity” the microscopic action is allowed to be very different from the Einstein-Hilbert action or a discretization thereof. Plausibly it should be still quasilocal, i.e. have a well-defined derivative expansion, and based on perturbatively renormalizable higher derivative theories one would expect it to contain at least quartic derivative terms. This means that also the number of physical propagating degrees of freedom (with respect to a background) may be different from the number entailed by the Einstein-Hilbert action. The second motivation for the term comes from the analogy with Quantum Chromodynamics. Indeed, the premise is that the self-interaction for the quantized gravitational field is predominantly “anti-screening” in the ultraviolet in a similar sense as in Quantum Chromodynamics, where it is responsible for the characteristic high energy behavior of physical quantities. As in Quantum Chromodynamics the proper identification of the antagonistic degrees of freedom (screening versus anti-screening) may well depend on the choice of field variables.

As with “Quantum General Relativity” we take the term “Gravidynamics” in a broad sense, allowing for any set of field variables (e.g. vielbein and spin connection, Ashtekar’s variables, Plebanski and BF type formulations, teleparallel, etc.) that can be used to recast general relativity (see e.g. the review [167]). For example the coupling of fermions might be a good reason to use a vielbein formulation. If the metric is taken as dynamical variable in four dimensions we shall also use the term “Quantum Einstein Gravity” as in [154, 133, 131]. It is of course not assumed from the outset that the quantum gravidynamics based on the various set of field variables are necessarily equivalent.

Gaussian fixed point: A fixed point is called Gaussian if there exists a choice of field variables for which the fixed point action is quadratic in the fields and the functional measure is Gaussian. This includes the local case but also allows for nonlocal quadratic actions. The drawback of this definition is that the proper choice of field variables in which the measure reveals its Gaussian nature may be hard to find. (For example in the correlation functions of the spin field in the two-dimensional Ising model the underlying free fermionic theory is not visible.)

A non-Gaussian fixed point is simply one where no choice of fields can be found in which the measure becomes Gaussian. Unfortunately this, too, is not a very operational criterion.

Unstable manifold: The unstable manifold of a fixed point with respect to a coarse graining operation is the set of all points that can be reached along flow lines emanating from the fixed point, the so-called renormalized trajectories. Points on such a flow line correspond to perfect actions. The stable manifold is the set of points attracted to the fixed point in the direction of coarse graining.

Strict (weak) renormalizability: We call a field theory strictly (weakly) renormalizable with respect to a fixed point and a coarse graining operation if the dimension of its unstable manifold is finite (infinite). It is implied that if a field theory has this property with respect to one coarse graining operation it will have it with respect to many others (“universality”). Strict or weak renormalizability is believed to be a sufficient condition for the existence of a genuine continuum limit for observables.

Relevant coupling: Given an expansion “sum over couplings times interaction monomials”, a coarse graining operation, and a fixed point of it, a coupling is called relevant (irrelevant) if it is driven away from (towards) the value the corresponding coordinate has at the fixed point, under a sufficient number of coarse graining steps. Note that this distinction makes sense even for trajectories not connected to the fixed point (because they terminate). It is however an explicitly ‘coordinate dependent’ notion. The same terms are used for the interaction monomials associated with the couplings. The dimension of the unstable manifold equals the maximal number of independent relevant interaction monomials ‘connected’ to the fixed point. All points on the unstable manifold are thus parameterized by relevant couplings but not vice versa.

Couplings which are relevant or irrelevant in a linearized analysis are called linearly relevant or linearly irrelevant, respectively. A coupling which is neither linearly relevant nor linearly irrelevant is called (linearly) marginal.

Usually one stipulates Properties (C1) and (C2) for the functional measure after which Property (C3) should be a provable property of physical quantities like the S-matrix. The requirement of having also Properties (C1) and (C2) only for observables is somewhat weaker and in the spirit of the asymptotic safety scenario.

Typically the Properties (C1, C2, C3) cannot be rigorously established, but there are useful criteria which render the existence of a genuine continuum limit plausible in different computational frameworks. In Sections 2.1 and 2.2 we discuss in some detail such criteria for the perturbative and for the FRGE approach, respectively. For convenience we summarize the main points here.

In Criterion (FRGC1) the existence of the k → 0 limit in theories with massless degrees of freedom is nontrivial and the problem of gaining computational control over the infrared physics should be separated from the UV aspects of the continuum limit as much as possible. However the k → 0 limit is essential to probe stability/positivity/unitarity. For example, to obtain a (massive) Euclidean quantum field theory the Schwinger functions constructed from the vertex functions have to obey nonlinear relations which ensure that the Hilbert space reconstructed via the Osterwalder-Schrader procedure has a positive definite inner product.

Perturbative (weak) renormalizability: We call a theory perturbatively (weakly) renormalizable if Criterion (PTC1) can be achieved with finitely (infinitely) many essential couplings. A theory were neither can be achieved is called perturbatively non-renormalizable. Perturbative (weak) renormalizability is neither necessary nor sufficient for (weak or strict) renormalizability in the above nonperturbative sense. It is only in combination with Criterion (PTC2) that perturbative results are indicative for the existence of a genuine continuum limit.

Asymptotically free coupling: A non-constant coupling in the unstable manifold of a Gaussian fixed point.

The “non-constant” proviso is needed to exclude cases like a trivial \(\phi _4^4\) coupling. In a nonperturbative lattice construction of \(\phi _4^4\) theory only a Gaussian fixed point with a one-dimensional unstable manifold (parameterized by the renormalized mass) is thought to exist, along which the renormalized \(\phi _4^4\) coupling is constant and identically zero. The Gaussian nature of the fixed-point, on the other hand, is not crucial and we define:

Asymptotically safe coupling: A non-constant coupling in the unstable manifold of a fixed point.

Asymptoticaly safe functional measure: The functional measure of a statistical field theory is said to be asymptotically safe if is perturbatively weakly renormalizable or non-renormalizable, but possesses a fixed point with respect to which it is strictly renormalizable. Subject to the regularity assumption that the space of actions can in the vicinity of the fixed point be decomposed into a stable and an unstable manifold, this is equivalent to the following requirement: All relevant couplings are asymptotically safe and there is only a finite number of them. Note that unitarity or other desirable properties that would manifest itself on the level of observables are not part of this definition.

In a non-gravitational context the functional measure of the 3D Gross-Neveu model is presently the best candidate to be asymptotically safe in the above sense (see [101, 60, 198, 105] and references therein). Also 5D Yang-Mills theories (see [93, 148] and references therein) are believed to provide examples. In a gravitational context, however, there are good reasons to modify this definition.

First the choice of couplings has to be physically motivated, which requires to make contact to observables. In the above nongravitational examples with a single coupling the ‘meaning’ of the coupling is obvious; in particular it is clear that it must be finite and positive at the non-Gaussian fixed point. In general however one does not know whether ill behaved couplings are perverse redefinitions of better behaved ones. To avoid this problem the couplings should be defined as coefficients in a power series expansion of the observables themselves (Weinberg’s “reaction rates”; see the discussion in Section 1.1). Of course painfully little is known about (generic) quantum gravity observables, but as a matter of principle this is how couplings should be defined. In particular this will pin down the physically adequate notion of positivity or unitarity.

Second, there may be good reasons to work initially with infinitely many essential or potentially relevant couplings. Recall that the number of essential couplings entering the initial construction of the functional measure is not necessarily equal to the number eventually indispensable. In a secondary step a reduction of couplings might be feasible. That is, relations among the couplings might exist which are compatible with the renormalization flow. If these relations are sufficiently complicated, it might be better to impose them retroactively than to try to switch to a more adapted basis of interaction monomials from the beginning.

Specifically in the context of quantum gravity microscopic actions with infinitely many essential couplings occur naturally in several ways. First, when starting from the Gomis-Weinberg picture [94] of perturbative quantum gravity (which is implemented in a non-graviton expansion in Section 3 for the 2 + 2 reduction). Second, when power counting considerations are taken as a guideline one can use Newton’s constant (frozen in Planck units) to build dimensionless scalars (dilaton, conformal factor) and change the conformal frame arbitrarily. The way how these dimensionless scalars enter the (bare versus renormalized) action is not constrained by power counting considerations. This opens the door to an infinite number of essential couplings. The effective action for the conformal factor [149] and the dilaton field in the 2 + 2 reduction [154] provide examples of this phenomenon.

Third, the dimension of the unstable manifold is of secondary importance in this context. Recall that the dimension of the unstable manifold is the maximal number of independent relevant interaction monomials ‘connected’ to the fixed point. This maximal number may be very difficult to determine in Quantum Gravidynamics. It would require identification of all renormalized trajectories emanating from the fixed point — which may be more than what is needed physicswise: The successful construction of a subset of renormalized trajectories for physically motivated couplings may already be enough to obtain predictions/explanations for some observables. Moreover, what matters is not the total number of relevant couplings but the way how observables depend on them. Since generic observables (in the sense used in Section 1.1) are likely to be nonlinearly and nonlocally related to the metric or to the usual basis of interaction monomials (scalars built from polynomials in the curvature tensors, for instance) the condition that the theory should allow for predictions in terms of observables is only indirectly related to the total number of relevant couplings.

In summary, the interplay between the microscopic action, its parameterization through essential or relevant couplings, and observables is considerably more subtle than in the presumed non-gravitational examples of asymptotically safe theories with a single coupling. The existence of an asymptotically safe functional measure in the above sense seems to be neither necessary nor sufficient for a physically viable theory of Quantum Gravidynamics. This leads to our final working definition.

1.4 Relation to other approaches

For orientation we offer here some sketchy remarks on how Quantum Gravidynamics relates to some other approaches to Quantum Gravity, notably the Dynamical Triangulations approach, Loop Quantum Gravity, and String Theory. These remarks are of course not intended to provide a comprehensive discussion of the relative merits but merely to highlight points of contact and stark differences to Quantum Gravidynamics.

1.5 Discussion of possible objections

Here we discuss some of the possible objections to a physically viable asymptotically safe theory of quantum gravidynamics.

Q1 Since the microscopic action is likely to contain higher derivative terms, don’t the problems with non-unitarity notorious in higher derivative gravity theories reappear?

A1 In brief, the unitarity issue has not much been investigated so far, but the presumed answer is No.

First, the problems with perturbatively strictly renormalizable higher derivative theories stem mostly from the 1/p⁴-type propagator used. The alternative perturbative framework already mentioned, namely to use a 1/p²-type propagator at the expense of infinitely many essential (potentially ‘unsafe’) couplings, avoids this problem [94, 10]. The example of the 2 + 2 reduction shows that the reconcilation of safe couplings with the absence of unphysical propagating modes can be achieved in principle. Also the superrenormalizable gravity theories with unitary propagators proposed in [218] are intriguing in this respect.

Second, when the background effective action is used as the central object to define the quantum theory, the ‘background’ is not a solution of the classical field equations. Rather it is adjusted self-consistenly by a condition involving the full quantum effective action (see Appendix B). If the background effective action is computed nonperturbatively (by whatever technique) the intrinsic notion of unitarity will not be related to the ‘propagator unitarity’ around a solution of the classical field equations in any simple way.

One aspect of this intrinsic positivity is the convexity of the background effective action. In the flow equation for the effective average action one can see, for example, that the wrong-sign of the propagator is not an issue: If Γ _k is of the R + R² type, the running inverse propagator \(\Gamma _k^{(2)}\) when expanded around flat space has ghosts similar to those in perturbation theory. For the FRG flow, however, this is irrelevant since in the derivation of the beta functions no background needs to be specified explicitly. All one needs is that the RG trajectories are well defined down to k = 0. This requires that \(\Gamma _k^{(2)} + {{\mathcal R}_k}\) is a positive operator for all k. In the untruncated functional flow this is believed to be the case. A rather encouraging first result in this direction comes from the R² truncation [131].

More generally, the reservations towards higher derivative theories came from a loop expansion near the perturbative Gaussian fixed point. In contrast in Quantum Gravidynamics one aims at constructing the continuum limit nonperturbatively at a different fixed point. In the conventional setting one quantizes R + R² as the bare action, while in Quantum Gravidynamics the bare action, defined by backtracing the renormalized trajectory to the non-Gaussian fixed point, may in principle contain all sorts of curvature invariants whose impact on the positivity and causality of the theory is not even known in perturbation theory.

In the previous discussion we implicitly assumed that generic physical quantities are related in a rather simple way to the interaction monomials entering the microscopic action. For Dirac observables however this is clearly not the case. Assuming that the physically correct notion of unitarity concerns such observables it is clear that the final word on unitarity issues can only be spoken once actual observables are understood.

Q2 Doesn’t the very notion of renormalizability presuppose a length or momentum scale? Since the latter is absent in a background independent formulation, the renormalizability issue is really an artifact of the perturbative expansion around a background.

A2 No. Background independence is a subtle property of classical general relativity (see e.g. [199] for a discussion) for which it is unclear whether or not it has a compelling quantum counterpart. As far as the renormalization problem is concerned it is part of the physics premise of a functional integral type approach that there is a description independent and physically relevant distinction between coarse grained and fine grained geometries. On a classical level this amounts to the distinction, for example, between a perfect fluid solution of the field equations and one generated by its 10³⁰ or so molecular constituents. A sufficiently large set of Dirac observables would be able to discriminate two such spacetimes. Whenever we shall refer later on to “coarse grained” versus “fine grained” geometries we have a similar picture in mind for the ensembles of off-shell geometries entering a functional integral.

Once such a physics premise is made, the renormalization in the Kadanoff-Wilson sense is clearly relevant for the computation of observable quantities and does not just amount to a reshuffling of artifacts. Renormalization in this sense is, for example, very likely not related to the regularization ambiguities [151, 173] appearing in loop quantum gravity. A minimal requirement for such an interpretation of the regularization ambiguites would be that reasonable coarse graining operations exist which have a preferred discretization of the Einstein-Hilbert action as its fixed point. This preferred discretization would have to be such that the observables weakly commuting with the associated Hamiltonian constraint reproduce those of loop quantum gravity.

For clarity’s sake let us add that the geometries entering a functional integral are expected to be very rough on the cutoff scale (or of a distributional type without a cutoff) but superimposed to this ‘short wavelength zigzag’ should be ‘long wavelength’ modulations (defined in terms of dimensionless ratios) to which different observables are sensitive in different degrees. In general it will be impractical to base the distinction between ensembles of fine grained and coarse grained geometries directly on observables. In the background field formalism the distinction is made with respect to an initially prescribed but generic background geometry which after the functional integral is performed (entirely or in a certain mode range) gets related to the expectation value of the quantum metric by a consistency condition involving the full quantum dynamics.

Q3 Doesn’t such a non-perturbative renormalizability scenario require a hidden enhanced symmetry?

A3 Improved renormalizability properties around a given fixed point are indeed typically rooted in symmetries. A good example is QCD in a lightfront formulation where gauge invariance is an ‘emergent phenomenon’ occuring only after an infinite reduction of couplings [174]. In the case of Quantum Gravidynamics, the symmetry in question would be one that becomes visible only around the non-Gaussian fixed point. If it exists, its identification would constitute a breakthrough. From the Kadanoff-Wilson view of renormalization it is however the fixed point which is fundamental — the enhanced symmetry properties are a consequence (see the notion of generalized symmetries in [236, 160]).

Q4 Shouldn’t the proposed anti-screening be seen in perturbation theory?

A4 Maybe, maybe not. Presently no good criterion for antiscreening in this context is known. For the reasons explained in Section 1.1 it should not merely be identified with the sign of the dominant contribution to some beta function. The answer to the above question will thus depend somewhat on the identification of the proper degrees of freedom and the quantitiy considered.

One is via the 2 → 2 scattering amplitude. The coefficient ζ_scatt was computed initially by Donoghue and later by Khriplovich and Kirilin; the result considered definite in [32] is \({\zeta _{{\rm{scatt}}}} = {{41} \over {10\pi}}\). It decomposes into a negative vertex and triangle contributions \({\zeta _\upsilon} = {{105} \over {3\pi}}\), and a just slightly larger positive remainder \({\zeta _{{\rm{scatt}}}} - {\zeta _\upsilon} = {{117.3} \over {3\pi}}\) coming from box, seagull, and vacuum polarization diagrams.

Another option is to consider corrections to the Schwarzschild metric. Different sets of diagrams have been used for the definition [119, 33] and affect the parameterization (in-)dependence and other properties of the corrections. Both choices advocated lead to ζ_metric < 0, which amounts to antiscreening.

Let us also mention alternative definitions of an effective Newton potential via Wilson lines in Regge calculus [100] or by resummation of scalar matter loops [226]. The latter gives rise to an “antiscreening” Yukawa type correction of the form \(V\left(r \right) = - {G \over {4\pi r}}\left({1 - {e^{- r\sqrt {\zeta G}}}} \right)\), with ζ > 0. Via \(V(r) = \int {{{{d^3}k} \over {{{(2\pi)}^3}}}{e^{i\vec k \cdot \vec x}}G(k)/{{\vec k}^2}}\) it can be interpreted as a running Newton constant \(G(k) = G/(1 + \zeta {{\vec k}^2})\).

Each of the issues raised clearly deserves much further investigation. For the time being we conclude however that the asymptotic safety scenario is conceptually self-consistent. It remains to assemble hard computational evidence for the existence of a non-Gaussian fixed point with a nontrival and regular unstable manifold. This task will be taken up in Sections 3 and 4.

2.1 Perturbation theory and continuum limit

Perturbatively renormalizable field theories are a degenerate special case of the Wilson-Kadanoff framework. The main advantage of perturbation theory is that the UV cutoff Λ can be removed exactly and independently of the properties of the coupling flow. The existence of a Λ → ∞ limit with the required Properties (PTC1) can often be rigorously proven, in contrast to most nonperturbative techniques where this can only be established approximately by assembling evidence. With Criterion (PTC1) satisfied, the coupling flow then can be studied in a second step and used to probe whether or not the Criterion (PTC2) for the existence of a genuine continuum limit as anticipated in Section 1.3 is also satisfied. The main disadvantage of perturbation theory is that everything is initially defined as a formal power series in the loop counting parameter. Even if one trades the latter for a running coupling, the series in this coupling remains a formal one, typically non-convergent and not Borel-summable. It is generally believed, however, that provided Criterion (PTC2) is satisfied for a perturbative Gaussian fixed point, the series is asymptotic to the (usually unknown) exact result. In this case the perturbative analysis should indicate the existence of a genuine continuum limit based on an underlying Gaussian fixed point proper. Our main reason for going through this in some detail below is to point out that in a situation with several couplings the very same rationale applies if the perturbative fixed point is a non-Gaussian rather than a Gaussian one.

By construction the perturbative beta functions have a fixed point at \({\rm{g}}_i^{\ast} = 0\), which is called the perturbative Gaussian fixed point. Nothing prevents them from having other fixed points, but the Gaussian one is built into the construction. This is because a free theory has vanishing beta functions and the couplings \({{\rm{g}}_i} = {u_i}{\mu ^{- {d_i}}}\) have been introduced to parameterize the deviations from the free theory with action S_*,μ. Not surprisingly the stability matrix Θ _ij = ∂β _i /∂g _j ∣_g*=0 of the perturbative Gaussian fixed point just reproduces the information which has been put in. The eigenvalues come out to be −d _i modulo corrections in the loop coupling parameter, where −d _i are the mass dimensions of the corresponding interaction monomials. For the eigenvectors one finds a one-to-one correspondence to the unit vectors in the ‘coupling direction’ g _i , again with power corrections in the loop counting parameter. One sees that the couplings u _i not irrelevant with respect to the stability matrix Θ computed at the perturbative Gaussian fixed point are the ones with mass dimensions d _i ≥ 0, i.e. just the power counting renormalizable ones.

Nevertheless, except for some special cases, it is difficult to give a mathematically precise meaning to the ‘∼’ in Equation (2.5). Ideally one would be able to prove that perturbation theory is asymptotic to the (usually unknown) exact answer for the same quantity. For lattice theories on a finite lattice this is often possible; the problems start when taking the limit of infinite lattice size (see [159] for a discussion). In the continuum limit a proof that perturbation theory is asymptotic has been achieved in a number of low-dimensional quantum field theories: the superrenormalizable P₂(ϕ) and \(\phi _3^4\) theories [69, 43] and the two-dimensional Gross-Neveu model, where the correlation functions are the Borel sum of their renormalized perturbation expansion [87, 89]. Strong evidence for the asymptotic correctness of perturbation theory has also been obtained in the O(3) nonlinear sigma-model via the form factor bootstrap [22]. In four or higher-dimensional theories unfortunately no such results are available. It is still believed that whenever the above g₁ is asymptotically free in perturbation theory, that the corresponding series is asymptotic to the unknown exact answer. On the other hand, to the best of our knowledge, a serious attempt to establish the asymptotic nature of the expansion has never been made, nor are plausible strategies available. The pragmatic attitude usually adopted is to refrain from the attempt to theoretically understand the domain of applicability of perturbation theory. Instead one interprets the ‘∼’ in Equation (2.5) as an approximate numerical equality, to a suitable loop order L and in a benign scheme, as long as it works, and attributes larger discrepancies to the ‘onset of nonperturbative physics’. This is clearly unsatisfactory, but often the best one can do. Note also that some of the predictive power of the QFT considered is wasted by this procedure and that it amounts to a partial immunization of perturbative predictions against (experimental or theoretical) refutation.

So far the discussion was independent of the nature of the running of \(\bar \lambda (\mu, {\Lambda _{{\rm{beta}}}})\) (which was traded for g₁). The chances that the vague approximate relation ‘∼’ in Equation (2.5) can be promoted to the status of an asymptotic expansion are of course way better if \(\bar \lambda (\mu, {\Lambda _{{\rm{beta}}}})\) is driven towards \(\bar \lambda = 0\) by the perturbative flow. Only then is it reasonable to expect that an asymptotic relation of the form (2.5) holds, linking the perturbative Gaussian fixed point to a genuine Gaussian fixed point defined by nonperturbative means. The perturbatively and the nonperturbatively defined coupling g₁ can then be identified asymptotically and lie in the unstable manifold of the fixed point g₁ = 0. On the other hand the existence of a Gaussian fixed point with a nontrivial unstable manifold is thought to entail the existence of a genuine continuum limit in the sense discussed before. In summary, if g₁ is traded for a running \(\bar \lambda (\mu, {\Lambda _{{\rm{beta}}}})\), a perturbative criterion for the existence of a genuine continuum limit is that the perturbative flow of g₁ is regular with lim_μ→∞g₁(μ) = 0. Since the beta functions of the other couplings are formal power series in λ without constant coefficients, the other couplings will vanish likewise as g1 → 0, and one recovers the local quadratic action S_*,μ[χ] at the fixed point. The upshot is that the coupling with respect to which the perturbative expansion is performed should be asymptotically free in perturbation theory in order to render the existence of a nonperturbative continuum limit plausible.

Summarizing: In perturbation theory the removal of the cutoff can be done independently of the properties of the coupling flow, while in a non-perturbative setting both aspects are linked. Only if the coupling flow computed from the perturbative beta functions meets certain conditions is it reasonable to assume that there exists an underlying non-perturbative framework to whose results the perturbative series is asymptotic. Specifically we formulate the following criterion:

2.2 Functional flow equations and UV renormalization

The technique of functional renormalization group equations (FRGEs) does not rely on a perturbative expansion and has been widely used for the computation of critical exponents and the flow of generalized couplings. For a systematic exposition of this technique and its applications we refer to the reviews [146, 21, 166, 229, 29]. Here we shall mainly use the effective average action Γ _k and its ‘exact’ FRGE. We refer to Appendix C for a summary of this formulation, and discuss in this section how the UV renormalization problem presents itself in an FRGE [157].

In typical applications of the FRG the ultraviolet renormalization problem does not have to be addressed. In the context of the asymptotic safety scenario this is different. By definition the perturbative series in a field theory based on an asymptotically safe functional measure has a dependence on the UV cutoff which is not strictly renormalizable (see Section 1.3). The perturbative expansion of an FRGE must reproduce the structure of these divergencies. On the other hand in an exact treatment or based on different approximation techniques a reshuffling of the cutoff dependence is meant to occur which allows for a genuine continuum limit. We therefore outline here how the UV renormalization problem manifests itself in the framework of the functional flow equations. The goal will be to formulate a criterion for the plausible existence of a genuine continuum limit in parallel to the one above based on perturbative indicators.

For finite cutoffs (Λ, k) the trace of the right-hand-side of Equation (2.11) will exist as the potentially problematic high momentum parts are cut off. In slightly more technical terms, since the product of a trace-class operator with a bounded operator is again trace-class, the trace in Equation (2.11) is finite as long as the inverse of \(\Gamma _{\Lambda, k}^{(2)}\left[ \phi \right] + {{\mathcal R}_{\Lambda, k}}\) defines a bounded operator. For finite UV cutoff one sees from the momentum space version of Equation (B.2) in Section 3.4 that this will normally be the case. The trace-class property of the mode cutoff operator (for which Equation (2.10) is a sufficient condition) also ensures that the trace in Equation (2.11) can be evaluated in any basis, the momentum space variant displayed in the second line is just one convenient choice.

Importantly the FRGE (2.11) is independent of the bare action S _λ , which enters only via the initial condition Γ_λ,λ = S _λ (for large Λ). In the FRGE approach the calculation of the functional integral for Γ_λ,k is replaced by the task of integrating this RG equation from k = Λ, where the initial condition Γ_λ,λ = S _λ is imposed, down to k = 0, where the effective average action equals the ordinary effective action Γ _λ .

All this has been for a fixed UV cutoff Λ. The removal of the cutoff is of course the central theme of UV renormalization. In the FRG formulation one has to distinguish between two aspects: first, removal of the explicit Λ dependence in the trace on the right-hand-side of Equation (2.11), and second removal of the UV cutoff in Γ_λ,k itself, which was needed in order to make the original functional integral well-defined.

The first aspect is unproblematic: The trace is manifestly finite as long as the inverse of \(\Gamma _{\Lambda, k}^{(2)}\left[ \phi \right] + {{\mathcal R}_{\Lambda, k}}\) defines a bounded operator. If now \(\Gamma _{\Lambda, k}^{(2)}\left[ \phi \right]\) is independently known to have a finite and nontrivial limit as Λ → ∞, the explicit Λ dependence carried by the \({{\mathcal R}_{\Lambda, k}}\) term is harmless and the trace always exists. Roughly this is because the derivative kernel \(k{\partial _k}{{\mathcal R}_{\Lambda, k}}\) has support mostly on a thin shell around p² ≈ k², so that the (potentially problematic) large p behavior of the other factor is irrelevant (cf. Appendix C.2).

The second aspect of course relates to the traditional UV renormalization problem. Since Γ_λ,k came from a regularized functional integral it will develop the usual UV divergencies as one attempts to send Λ to infinity. The remedy is to carefully adjust the bare action S _λ [ϕ] — that is, the initial condition for the FRGE (2.11) — in such a way that functional integral — viz. the solution of the FRGE — is asymptotically independent of Λ. Concretely this could be done by fine-tuning the way how the parameters u _α (Λ) in the expansion S _λ [χ] = ∑ _α u _α (Λ)P _α [χ] depends on Λ. However the FRGE method in itself provides no means to find the proper initial functional S _λ [χ]. Identification of the fine-tuned S _λ [χ] lies at the core of the UV renormalization problem, irrespective of whether Γ_λ,k is defined directly via the functional integral or via the FRGE. Beyond perturbation theory the only known techniques to identify the proper S _λ start directly from the functional integral and are ‘constructive’ in spirit (see [195, 48]). Unfortunately four-dimensional quantum field theories of interest are still beyond constructive control.

One may also ask whether perhaps the cutoff-dependent FRGE (2.11) itself can be used to show that a limit lim_Λ→∞ Γ_Λ,k[ϕ] exists. Indeed using other FRGEs and a perturbative ansatz for the solution has lead to economic proofs of perturbative renormalizability, i.e. of the existence of a formal continuum limit in the sense of Criterion (PTC1) discussed before (see [200, 123]). Unfortunately so far this could not be extended to construct a nonperturbative continuum limit of fully fledged quantum field theories (see [145] for a recent review of such constructive uses of FRGEs). For the time being one has to be content with the following if…then statement:

So far the positivity or unitarity requirement has not been discussed. From the (Osterwalder-Schrader or Wightman) reconstruction theorems it is known how the unitarity of a quantum field theory on a flat spacetime translates into nonlinear conditions on the multipoint functions. Since the latter can be expressed in terms of the functional derivatives of Γ _k , unitarity can in principle be tested retroactively, and is expected to hold only in the limit k → 0. Unfortunately this is a very indirect and retroactive criterion. One of the roles of the bare action S_Λ[χ] = Γ_λ,λ[χ] is to encode properties which are likely to ensure the desired properties of lim_k→0 Γ _k [ϕ]. In theories with massless degrees of freedom the k → 0 limit is nontrivial and the problem of gaining computational control over the infrared physics should be separated from the UV aspects of the continuum limit as much as possible. However the k → 0 limit is essential to probe stability/positivity/unitarity.

One aspect of positivity is the convexity of the effective action. The functional equations (2.11, 2.12) do in itself “not know” that Γ _k is the Legendre transform of a convex functional and hence must be itself convex. Convexity must therefore enter through the inital data and it will also put constraints on the choice of the mode cutoffs. Good mode cutoffs are characterized by the fact that \(\Gamma _k^{(2)} + {{\mathcal R}_k}\) has positive spectral values for all k (cf. Equation (C.14)). If no blow-up occurs in the flow the limit \({\lim\nolimits_{k \to \infty}}\Gamma _k^{(2)}\) will then also have non-negative spectrum. Of course this presupposes again that the proper initial conditions have been identified and the role of the bare action is as above.

For flat space quantum field theories one expects that S _λ [χ] must be local, i.e. a differential polynomial of finite order in the fields so as to end up with an effective action \({\lim\nolimits_{k \to \infty}}{\lim\nolimits_{\Lambda \to \infty}}{\Gamma _{\Lambda, k}}[\phi ]\) describing a local/microcausal unitary quantum field theory.

For convenient reference we summarize these conclusions in the following criterion:

We add some comments:

Since the FRGE (2.12) is a differential equation in k, an initial functional Γ_initial[ϕ] has to be specified for some 0 < k_initial ≤ ∞, to generate a local solution near k = k_initial. The point is that for ‘almost all’ choices of Γ_initial[ϕ] the local solution cannot be extended to all values of k. Finding the rare initial functionals for which this is possible is the FRGE counterpart of the UV renormalization problem. The existence of the k → 0 limit is itself not part of the UV problem; in conventional quantum field theories the k → 0 limit is however essential to probe unitarity/positivity/stability.

It is presently not known whether the above criterion can be converted into a theorem. Suppose for a quantum field theory on the lattice (with lattice spacing Λ⁻¹) the effective action \(\Gamma _{\Lambda, k}^{{\rm{latt}}}[\phi ]\) has been constructed nonperturbatively from a transfer operator satisfying reflection positivity and that a continuum limit \({\lim\nolimits_{\Lambda \to \infty}}\Gamma _{\Lambda, k}^{{\rm{latt}}}[\phi ]\) is assumed to exist. Does it coincide with a solution Γ _k [ϕ] of Equation (2.12) satisfying the Criteria (FRGC1) and (FRGC2)? Note that this is ‘only’ a matter of controlling the limit, for finite Λ also \(\Gamma _{\Lambda, k}^{{\rm{latt}}}\) will satisfy the flow equation (2.11).

For an application to quantum gravity one will initially only ask for Criterion (FRGC1), perhaps with even only a partial understanding of the k → 0 limit. As mentioned, the k → 0 limit should also be related to positivity issues. The proper positivity requirement replacing Criterion (FRGC2) yet has to be found, however some constraint will certainly be needed. Concerning Criterion (FRGC1) the premise in the if…then statement preceeding Equation (2.12) has to be justified by external means or taken as a working hypothesis. In principle one can also adopt the viewpoint that the quantum gravity counterpart of Equation (2.12) discussed in Section 4 simply defines the effective action for quantum gravity whenever a solution meets Criterion (FRGC1). The main drawback with this proposal is that it makes it difficult to include information concerning Criterion (FRGC2). However difficult and roundabout a functional integral construction is, it allows one to incorporate ‘other’ desirable features of the system in a relatively transparent way.

We shall therefore also in the application to quantum gravity assume that a solution Γ _k of the cutoff independent FRGE (2.12) satisfying Criterion (FRGC1) comes from an underlying functional integral. This amounts to the assumption that the renormalization problem for Γ_k,Λ defined in terms of a functional integral can be solved and that the limit lim_Λ→∞ Γ_k,Λ can be identified with Γ _k . This is of course a rather strong hypothesis, however its self-consistency can be tested within the FRG framework.

To this end one truncates the space of candidate continuum functionals \(\Gamma _{k}^{{\rm{trunc}}}[\phi ]\) to one where the initial value problem for the flow equation (2.12) can be solved in reasonably closed form. One can then by ‘direct inspection’ determine the initial data for which a global solution exists. Convexity of the truncated \({\lim\nolimits_{k \to 0}}\Gamma _k^{{\rm{trunc}}}[\phi ]\) can serve as guideline to identify good truncations. If the set of these initial data forms a nontrivial unstable manifold of the fixed point \(S_{\ast}^{{\rm{trunc}}}[\phi ] = {\lim\nolimits_{k \to \infty}}\Gamma _k^{{\rm{trunc}}}[\phi ]\), application of the above criterion suggests that \(\Gamma _k^{{\rm{trunc}}}\) can approximately be identified with the projection of the continuum limit (lim_Λ→∞ Γ_λ,k)^trunc of some Γ_λ,k computed by other means. The identification can only be an approximate one because in the \(\Gamma _k^{{\rm{trunc}}}\) evolution one first truncates and then evolves in k, while in (lim_Λ→∞ Γ_λ,k [ϕ])^trunc one first evolves in k and then truncates. Alternatively one can imagine to have replaced the original dynamics by some ‘hierarchical’ (for want of a better term) approximation implicitly defined by the property that \({\left({{{\lim}_{\Lambda \to \infty}}\Gamma _{\Lambda, k}^{{\rm{hier}}}[\phi ]} \right)^{{\rm{trunc}}}} = {\lim\nolimits_{\Lambda, k}}\Gamma _{\Lambda, k}^{{\rm{hier}}}[\phi ]\) (see [75] for the relation between a hierarchical dynamics and the local potential approximation). The existence of an UV fixed point with a nontrivial unstable manifold for \(\Gamma _k^{{\rm{trunc}}}\) can then be taken as witnessing the renormalizability of the ‘hierarchical’ dynamics.

2.3 Towards Quantum Gravidynamics

The application of renormalization group ideas to quantum gravity has a long history. In accordance with the previous discussion, we focus here on the aspects needed to explain the apparent mismatch between the perturbative non-renormalizability and the presumed nonperturbative renormalizability. In fact, when looking at higher derivative theories renormalizability can already be achieved on a perturbative level in several instructive ways.

2.4 Dimensional reduction of residual interactions in UV

As highlighted in the introduction an important qualitative feature of an asymptotically safe functional integral can be inferred without actually evaluating it, namely that in the extreme ultraviolet the residual interactions appear two-dimensional. There are a number of interconnected heuristic arguments for this phenomenon which we present here.

3.1 2 + 2 truncation of Einstein gravity + matter

In accordance with the general picture the renormalization flow will also dictate here to a certain extent the form of the renormalized actions. As mentioned the truncated 2 + 2 functional integral turns out to inherit the lack of perturbative renormalizability (with finitely many couplings) from the gravitational part of the full functional integral. However the restricted functional integral is more benign insofar as it is possible to preserve the conformai geometry in field space and insofar as no higher derivative terms are required for the absorption of cutoff dependencies. The strategy is similar as in the perturbative treatment of the full theory advocated by Gomis and Weinberg [94]: One works with a propagator free of unphysical poles and takes into account all counter terms enforced, but only those. (For the reasons explained in Section 2.3.2 we deliberatly avoid using the ‘relevant/irrelevant’ terminology here.) To emphasize the fact that no higher derivative terms are needed we shall refer to the quantum theory defined that way as the symmetry truncation of Quantum Einstein Gravity. We anticipate this fact in the following by taking a classical gravity + matter action as a starting point which is quadratic in the derivatives only (see Equation (3.1) below).

3.1.2 2-Killing vector reduction

Concerning the reduction, we consider here only the case when both Killing vector fields K₁, K₂ are spacelike everywhere and commuting. The other signature (one timelike and one spacelike Killing vector field) is most efficiently treated by relating it to the spacelike case via an Abelian T-duality transformation (see [154]). Alternatively one can perform the reduction in two steps and perform a suitable Hodge dualization in-between (see [45, 56]).

Thus, from here on we take \({K_a} = K_a^\alpha {\partial _\alpha}\), a = 1, 2, to be Killing vector fields on the Lorentzian manifold \(({\mathcal M},g)\) that are spacelike everywhere and commuting: \({L_{{K_a}}}{g_{\alpha \beta}} = 0\) and [K₁, K₂] = 0. Their Lorentzian norms and inner product form a symmetric 2 × 2 matrix M = (M _ab )_1≤a,b≤2. For the resulting three scalar fields on the 4D spacetime it is convenient to adopt a lapse-shift type parameterization. This gives

$${M_{ab}}: = {g_{\alpha \beta}}K_a^\alpha K_b^\beta = {K_a} \cdot {K_b},\quad \quad M = :{\rho \over \Delta}\left({\begin{array}{*{20}c} {{\Delta ^2} + {\psi ^2}} & \psi \\ \psi & 1 \\ \end{array}} \right).$$

(3.3)

In the general relativity literature the fields Δ > 0 and ψ are known as the real and imaginary parts of the “Ernst potential”. The parameterization (3.3) is chosen such that the (non-negative) “area element” det M is the square of one of the fields. Taking the positive square root one has

$$\rho : = \sqrt {{K_1} \cdot {K_1}\,{K_2} \cdot {K_2} - {{({K_1} \cdot {K_2})}^2}} \geq 0.$$

(3.4)

By definition the metric g is left unchanged along the flow lines of the Killing vector fields. We denote the space of orbits by Σ. A projection operator onto the (co-)tangent space to each point in Σ is given by \({\gamma _\alpha}^\beta : = {\delta _\alpha}^\beta - {M^{ab}}{K_{a\alpha}}{K_{a\alpha}}{K_b}^\beta\), where M ^ab are the components of M⁻¹ and \({K_{a\alpha}}: = {g_{\alpha \beta}}K_\alpha ^\beta\). The associated Lorentzian metric \({\gamma _{\alpha \beta}}: = {\gamma _\alpha}^{{\alpha\prime}}{\gamma _\beta}^{{\beta\prime}}{g_{{\alpha\prime}{\beta\prime}}} = {g_{\alpha \beta}} - {M^{ab}}{K_{a\alpha}}{K_{b\beta}}\) satisfies

$${{\mathcal L}_{{K_a}}}\,{\gamma _{\alpha \beta}} = 0,\quad \quad K_a^\alpha {\gamma _{\alpha \beta}} = 0.$$

(3.5)

Since \({\gamma _\alpha}^\beta\) is a projector of rank two, the metric γ _αβ has three independent components (not accounting for diffeomorphism redundancies). Generally one can show [90, 91] that there exists a one-to-one correspondence between tensor fields on the “orbit space” (Σ, γ) and tensor fields on (\(({\mathcal M},g)\), g) with vanishing Lie derivative along \({K_a}^\alpha\) and which are “completely orthogonal” to \({K_a}^\alpha\). This will be used for the matter fields below. Given γ subject to Equation (3.5) and M one can reconstruct the original metric tensor as

$${g_{\alpha \beta}} = {\gamma _{\alpha \beta}} + {M^{ab}}{K_{a\alpha}}{K_{b\beta}}.$$

(3.6)

The 10 components are parameterized by the 3 + 3 independent functions in γ and M. Each of these functions is constant along the flow lines of the Killing vector fields but may vary arbitrarily within Σ.

We deliberately refrained from picking coordinates so far to emphasize the geometric nature of the reduction. As usual however the choice of adapted coordinates is advantageous. We now pick (“Killing”) coordinates in which K _a acts as ∂/∂y ^a , for a = 1, 2. In these coordinates the components of γ and M are independent of y¹ and y² and thus are functions of the remaining (nonunique “non-Killing”) coordinates x⁰ and x¹ only. We write γ _μν (x), μ = 0, 1, for the components of γ in such a coordinate system. Since both Killing vectors are spacelike, γ _μν has eigenvalues (−, +) and can be brought into the form γ _μν (x) = e^σ(x)η _μν , by a change of the non-Killing coordinates, where η is the metric of flat 1 + 1-dimensional Minkowski space. This can be taken to define σ. Alternatively one can introduce σ by

$${\partial _\rho} \cdot {\partial _\rho} = \Delta (x){\gamma ^{\mu \nu}}(x)\,{\partial _\mu}\rho \,{\partial _\nu}\rho = \Delta (x){e^{- \sigma (x)}}{\partial ^\mu}\rho \,{\partial _\mu}\rho.$$

(3.7)

On the left-hand-side is the (coordinate-independent) Lorentzian norm of the gradient of the 4D scalar field ρ; on the right-hand-side we evaluated this norm in the Killing coordinates where it must be proportional to γ ^μν (x) ∂ _μ ρ∂ _ν ρ. Adjusting also the non-Killing coordinates then gives the rightmost expression in Equation (3.7), which could also be taken to define σ. The upshot is that the most general 4D metric with two commuting Killing vectors is parameterized by four scalar fields, ρ, σ and Δ, ψ. In the adapted coordinates the 4D metric then reads

$$d{S^2} = {e^\sigma}[ - {(d{x^0})^2} + {(d{x^1})^2}] + {\rho \over \Delta}{(d{y^1} + \psi d{y^2})^2} + \rho \Delta {(d{y^2})^2}.$$

(3.8)

As already mentioned, the fields Δ and ψ are known as the real and the imaginary part of the “Ernst potential”; we shall refer to ρ as the “area radius” associated with the two Killing vectors, and to σ as the conformal factor. To motivate the latter note that a Weyl transformation g _αβ → e ^ω g _αβ of the 4D metric compatible with the 2-Killing vectors amounts to the simultaneous rescalings γ _μν (x) → e^ω(x)γ _μν (x) and ρ(x) → e^ω(x)ρ(x).

The matter content in Equation (3.1) consists of the k Abelian gauge fields and the sigma-model scalars \({{\bar \varphi}^i},\,i = 1, \ldots, \bar n\). For the scalars the reduction is trivial, and simply amounts to considering configurations constant in the Killing coordinates. For the gauge fields it turns out that the 4k components of \(B_\alpha ^{\hat \iota},\hat \iota = 1, \ldots, k\), î = 1, …,k, give rise to 2k fields A ^I , I = 1, …, 2k, which transform as scalars under a change of the non-Killing coordinates (x⁰, x¹). In brief this comes about as follows. The field equation ∇ _α (μF ^αβ − ν *F ^αβ ) = 0 for the gauge fields in Equation (3.1) can be interpreted as the Bianchi identity for a field strength G _αβ = ∂ _α C _β − ∂ _β C _α derived from dual potentials \(C_\alpha ^{\hat \iota},\hat \iota = 1, \ldots, k\). We can take one of the Killing vectors, say K = K₁, and built 2k spacetime scalars by contraction \({B^{\hat \iota}}: = B_\alpha ^{\hat \iota}{K^\alpha}\) and \({C^{\hat \iota}}: = C_\alpha ^{\hat \iota}{K^\alpha}\). Reduction with respect to the other Killing vector K₂ just requires that these scalars are constant in the corresponding Killing coordinate y². The dependence on y¹ is constrained by gauge invariance. If \(B_\alpha ^{\hat \iota} \mapsto B_\alpha ^{\hat \iota} + {\partial _\alpha}{b^{\hat \iota}}\) and \(C_\alpha ^i \mapsto C_\alpha ^i + {\partial _\alpha}{c^i}\), the scalars change by a term K ^α ∂ _α b ^î and K ^α ∂ _α c ^î , respectively, and hence are invariant under y¹ independent gauge transformations. Thus, if the 4D gauge potentials and their duals, together with the corresponding transformations are taken to be independent of y¹, y², a set of gauge invariant scalars B ^î (x) and C ^î (x) arises. As a matter of fact a constant remnant of the gauge transformations remains and gives rise to a symmetry of the reduced system (see the discussion after Equation (3.11) below). We arrange the 2k fields B ^î , C ^î in a column vector A ^I , I = 1,…,2k. For convenience we summarize the field content of the 2-Killing vector subsector of Equation (3.1) in Table 1.

Table 1

Field content of the 2-Killing vector subsector of the gravity theories 3.1.

4D fields	fields in 2-Killing subsector
g αβ metric	Δ,ψ,ρ,σ
\(B_\alpha ^{\hat i}\) Abelian gauge fields	A I , I = 1,…,2k
\({\bar \varphi ^i}\) KK scalars	\({\overline \varphi ^i},\,i = 1,\,{.}{.}{.}\,\overline n\)

We combine all but ρ and σ into an \(n: = 2 + \bar n + 2k\) components scalar field

$$\varphi = (\Delta, \,\psi, \,{\bar \psi ^1}, \ldots, {\bar \psi ^{\bar n}}, \ldots, \,{A^1}, \ldots, \,{A^{2k}})$$

(3.9)

on the 2D orbit space with metric γ _μν and coordinates (x⁰, x¹). A lengthy computation (which is best done in a two step procedure; see [45, 56]) gives the form of the action (3.1) on the field configurations compatible with the two Killing vectors. The result is

$$S = {1 \over {2\lambda}}\int {{d^2}x\,\rho \sqrt \gamma [2R(\gamma) + {\gamma ^{\mu \nu}}{\rho ^{- 2}}{\partial _\mu}\rho {\partial _\nu}\rho - {\gamma ^{\mu \nu}}{{\mathfrak{m}}_{ij}}(\varphi)\,{\partial _\mu}{\varphi ^i}\,{\partial _\nu}{\varphi ^j}]}.$$

(3.10)

Here λ is the gravitational constant per unit volume of the internal space. One sees that the reduced action has the form of a 2D nonlinear sigma-model non-minimally coupled to 2D gravity via the area radius ρ of the two Killing vectors. The target space of the sigma-model has dimension \(n = 2 + \bar n + 2k\), we take Equation (3.9), viewed as a column, as field coordinates. With the normalization \({\left\langle {{{\bar J}_{\hat \alpha}},\,{{\bar J}_{\hat \beta}}} \right\rangle _{\bar {\rm{g}}}} = \bar {\mathfrak{m}_{ij}}\left({\bar \varphi} \right){\partial _{\hat \alpha}}{\bar \varphi ^i}{\partial _{\hat \beta}}{\bar \varphi ^j}\) the metric then comes out as

Open image in new window

(3.11)

Here ϒ and \({{\mathcal V}_{\rm{c}}}\) are as in Equation (3.2) and q > 0 is a parameter used to adjust normalizations. The metric (3.11) has Riemannian signature (if the reduction was performed with respect to one spacelike and one timelike Killing vector it had 2k negative eigenvalues).

We briefly digress on the isometries of Equation (3.11). By virtue of the Ḡ invariance of the action m has dim Ḡ Killing vectors of which \(\bar n = \dim \bar G/\bar H\) are algebraically independent. Interestingly, the action (3.10) is also invariant under \(A \mapsto A + a,\,\psi \mapsto \psi - {q \over 2}{A^T}\Upsilon a\), with a constant 2k column a. These symmetries can be viewed as residual gauge transformations; note however that a compensating transformation of the gravitational potential ψ is needed. Finally constant translations in ψ and scale transformations \(\left({\Delta, {\mkern 1mu} \psi, {\mkern 1mu} \bar \varphi, {\mkern 1mu} A} \right) \mapsto \left({s\Delta, s\psi, {\mkern 1mu} \bar \varphi, {\mkern 1mu} {s^{1/2}}A} \right),{\mkern 1mu} s > 0\), are obvious symmetries of the action. The associated Killing vectors e, h of m generate a Borel subalgebra of sl₂, i.e. [h, e] = −2e. Together the metric (3.11) always has dim Ḡ + 2k + 2 Killing vectors.

In contrast the last sl₂ generator f is only a Killing vector of m under certain conditions on \(\bar G/\bar H\). If these are satisfied a remarkable ‘csymmetry enhancement’ takes place in that m is the metric of a much larger symmetric space G/H, where G is a non-compact real form of a simple Lie group with dim G = dim Ḡ + 4k + dim SL(2). The point is that if f exists as Killing vector its commutator with the gauge transformations is nontrivial and yields 2k additional symmetries (generalized “Harrison transformations”). Since m always has dim Ḡ + 2k + 2 Killing vectors, the additional 1 + 2k then match the dimension of G. For the number of dependent Killing vectors, i.e. the dimension of the putative maximal subgroup H ⊂ G one expects \(\dim \,H = \dim \,\bar H + 1 + 2k\). Indeed under the conditions stated the symmetric space G/H exists and is uniquely determined by \(\bar G/\bar H\) (and the signature of the Killing vectors). See [45] for a complete list. Evidently the gauge fields are crucial for the symmetry enhancement. Among the systems in [45] only pure gravity has k = 0.

From now on we restrict attention to the cases where such a symmetry enhancement takes place. The scalars φ ⁱ can then be arranged into a coset nonlinear sigma-model whose \(2k + \bar n + 2\)-dimensional target space is of the form G/H. Here G is always a simple noncompact Lie group and H a maximal subgroup; the coset is a Riemannian space with metric m _ij (φ). Being the metric of a symmetric space m enjoys the properties

$${\nabla _m}{R_{ijkl}}({\mathfrak{m}}) = 0,\quad \quad {R_{ij}}({\mathfrak{m}}) = {\zeta _1}\,{{\mathfrak{m}}_{ij}},$$

(3.12)

which will be important later on. By construction G/H contains as subcosets the space SL(2, ℝ)/SO(2) on which the gravitational potentials are coordinates and the original coset \(\bar G/\bar H\) of the scalars \(\bar \varphi\):

$${\rm{SL(2,}}\,{\mathbb R})/{\rm SO}(2) \subset G/H \supset \bar G/\bar H.$$

(3.13)

A brief list of examples of 4D theories (3.1) and the cosets G/H they give rise to in Equation (3.10) is: Pure gravity in 4D corresponds to SL(2, ℝ)/SO(2), Einstein-Maxwell theory gives rise to SU(2,1)/S[U(2) × U(1)], a 4D Einstein-dilaton-axion theory gives a coset SO(3, 2)/SO(3) × SO(2), and the reduction of N = 8 supergravity leads to a bosonic sector with E₈₍₊₈)/SO(16) coset.

The reduced classical field theories (3.10) have some remarkable properties which we discuss now:

1. 1.

   First, the field equations and the symplectic structure derived from Equation (3.11) coincide with the restriction of the field equations and the symplectic structure derived from Equation (3.1). In fact, this is a general feature of this type of symmetry reductions, and can be understood in terms of the “principle of symmetric criticality” [76].

    
2. 2.

   The field equations are classical integrable in the sense that they can be written as the compatibility condition of a pair of first order matrix-valued differential operators, depending on a free complex parameter. As a consequence large classes of solutions can be constructed analytically (see the books [26, 121, 98] and [103, 4] for detailed expositions).

    
3. 3.

   The classical integrability also entails that an infinite number of nonlocal conserved charges can be constructed explicitly. These are generalizations of the Luscher-Pohlmayer charges for the O(N) model. Moreover, these charges Poisson commute with the Hamiltonian and the diffeomorphism constraint that arise in a Hamiltonian (“Arnowitt-Deser-Misner-type”) analysis of the covariant system (3.11). In other words, an infinite system of Dirac observables can be constructed explicitly as functionals of the metric and the matter fields! Given the fact that (apart from mass and angular momentum) not a single Dirac observable is known explicitly in full general relativity, this is a most remarkable feature.

    
4. 4.

   The system captures the crucial “spin-two” aspect of gravity. For example without matter the classical solutions comprise various types of (nonlinear) gravitational waves with two independent polarizations (per spacetime point).

    
5. 5.

   In conformal gauge, γ _μν (x) = e^σ(x)η _μν , the curvature term in Equation (3.11) is proportional to ∂ ^μ ρ∂ _μ σ. Upon diagonalization, ∂ ^μ (ρ + σ)∂ _μ (ρ + σ) − ∂ ^μ (ρ−σ)∂ _μ (ρ − σ), this is proportional to a sum of two standard kinetic terms, one of which invariably has the ‘wrong sign’. Since a Weyl rescaling g _αβ ↦ e^ω(x)g _αβ of the metrics (3.3) amounts to ρ ↦ e ^ω ρ, σ ↦ σ + ω this appears to reflect a conformal factor instability of the 4D gravitational action(s). Upon closer inspection it signals the absence of a genuine instability (see Section 3.2).

    

Taken together these properties make the 2-Killing vector reductions a compelling laboratory to study the quantum aspect of the gravitational field.

3.2 Collinear gravitons, Dirac quantization, and conformal factor

In this section, taken from [156], we discuss a number of structural issues of the 2 + 2 truncations and advocate that, as far as the investigation of the renormalization properties is concerned, the use of a proper time or Dirac quantization is the method of choice. We begin by describing what conventional graviton perturbation theory looks like in this sector.

3.3 Tamed non-renormalizability

For the reasons explained in the Appendices B.2.2 and B.3 we now study the quantum theory based on the sigma-model Lagrangian (3.16) in the setting of the covariant background field expansion. Since this is a well-tested formalism (see Section B.3) it has the additional advantage that any unexpected findings cannot be blamed on the use of an untested formalism. Technically it is also convenient to use dimensional regularization and minimal subtraction. The analysis can then be done to all orders of sigma-model perturbation theory.

Our first goal thus is to construct the infinite cut-off limit of the background effective action \({\lim\nolimits _{\Lambda \rightarrow \infty}}{\Gamma _\Lambda}\left[ {\left\langle {{g_{\alpha \beta}}} \right\rangle, \,{{\bar g}_{\alpha \beta}}} \right]\) in the covariant background field formalism to all orders of the loop expansion. It turns out that this can be done only if infinitely many essential couplings are allowed, so even the trunctated functional integral based on Equation (3.16) is not renormalizable in the strict sense. However, once one allows for infinitely couplings strict cutoff independence (Λ → ∞) can be achieved. Remarkably, for Equation (3.1) the generalized beta function for a generating functional of these couplings can be found in closed form (see Equation (3.88) below). This allows one to study their RG flow in detail and to prove the existence of a non-Gaussian UV stable fixed point. One also finds a Gaussian fixed point which is not UV stable.

As described above in the sigma-model perturbation theory we use dimensional regularization and minimal subtraction. Counter terms will then have poles in d − 2 rather than containing positive powers of the cutoff Λ. The role of the scale k is played by the renormalization scale μ, the fields and the couplings at the cutoff scale are called the “bare” fields and the “bare” couplings, while the fields and couplings at scale μ are referred to as “renormalized”. The fact that the ansatz (3.56) ‘works’ is expressed in the following result:

Result (Generalized renormalizability) [154, 155]:

A subscript ‘_B’ denotes the bare fields while the plain symbols refer to the renormalized ones and similarly for h. Notably no higher order derivative terms are enforced by the renormalization process; strict cutoff independence can be achieved without them. However the fact that h_B(·) and h(·) differ marks the deviation from conventional renormalizability. The pre-factor h(ρ)/ρ also has a physical interpretation: To lowest order it is for pure gravity the conformal factor in a Weyl transformation g _αβ (x) → e^ω(ρ(x)) g _αβ (x) a generic four-dimensional metric g _αβ (x) with two Killing vectors in adapted coordinates.

The derivation of this result is based on a reformulation of the class of QFTs based on Equation (3.56) as a Riemannian sigma-model in the sense of Friedan [84]. This is a class of two-dimensional QFTs which is also (perturbatively) renormalizable only in a generalized sense, namely by allowing for infinitely many relevant couplings. The generating functional for these couplings in this case is a (pseudo-)Riemannian metric ɧ on a “target manifold \({\mathcal M}\)” of arbitrary dimension D and field coordinates \(\phi :\Sigma \rightarrow {\mathcal M}\), where Σ is the two-dimensional “base manifold”. The renormalization theory of these systems is well understood. A brief summary of the results relevant here is given in Appendix B.3.

The systems (3.56) can be interpreted as Riemannian sigma models where the target manifold of a special class of “warped products” (see Equation (3.59) below) and the fields are ϕ = (φ, ρ, σ). The relation between the quantum theory of these Riemannian sigma-models and the QFT based on Equation (3.56) will roughly be that one performs an infinite reduction of couplings in a sense similar to [236, 160, 174]. The generating functional ɧ _ij (ϕ) is parameterized by \({{D\left({D - 1} \right)} \over 2}\) functions of D variables, while the generating functional h in Equation (3.56) amounts to one function of one variable. Thus “\({{D\left({D - 1} \right)} \over 2} \times D \times \infty\)” many couplings are reduced to “1 × ∞” many couplings. As always in a reduction of couplings the nontrivial point is that this reduction can be done in a way compatible with the RG dynamics. The original construction in [236, 160] was in the context of strictly renormalizable QFTs with a finite number of relevant couplings. In a QFT with infinitely many relevant couplings (QCD in a lightfront formulation) the reduction principle was used by Perry-Wilson [174]. A general study of an ‘infinite reduction’ of couplings has been performed in [11].

The reduction technique used here is different, but essentially Equation (3.68) below plays the role of the reduction equation. Apart from the different derivation and the fact that the reduction is performed on the level of generating functionals, the main difference to a usual reduction is that Equation (3.68) also involves nonlinear field redefinitions without which the reduction could not be achieved here. The reduction equation (3.68) thus mixes field redefinitions and couplings. From the viewpoint of Riemannian sigma-models this amounts to the use of metric dependent diffeomorphisms on the target manifold, a concept neither needed nor used in the context of Riemannian sigma-models otherwise.

The situation changes drastically if one considers Riemannian sigma-models where the target manifold is one the warped products (3.59) below. The Problem 5 is absent on the basis of the following Non-renormalization Lemma, the Problem 6 is evaded because the Ricci-type flow arising at first order is constant [61] while to higher orders the asymptotic safety property to be described strikes:

Non-renormalization Lemma [154]:

The field ρ is nonlinearly renormalized but once it is renormalized arbitrary powers thereof (defined by multiplication pointwise on the base manifold) are automatically finite, without the need of additional counterterms. In terms of the normal product defined in Appendix B.3. [F(ρ)] = μ^d−2F(ρ) for an arbitary (analytic) function F.

Needless to say that the same is not true for σ or any other of the quantum fields φ ⁱ . As a consequence of this Non-renormalization Lemma the renormalization flow equations for the generating functional h (the counterpart of ɧ) can be consistently interpreted as an equation for a classical field, which we also denote by ρ since the quantum field can be manipulated as if it was a classical field. The resulting flow equations then take the form of a recursive system of nonlinear partial integro-differential equations, which are studied in Section 3.4.

We now describe the derivation of these results in outline; the full details can be found in [154, 155]. The class of warped product target manifolds relevant for Equation (3.56) is of the form

$$({\mathcal M},\,{\mathfrak h}) = (G/{H_h} \times {\mathcal R},\,{{\mathfrak{m}}\,_h} \times \mathfrak{r}),$$

(3.59)

where m is the metric (3.11) on the symmetric space G/H, h = h(ρ) is the ‘warp factor’, and \({\mathcal R}\) is a flat two-dimensional space with Lorentzian metric given by the lower 2 × 2 block in the metric

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(3.60)

If ζ₁ is the scalar curvature of the G/H normalized as in Equation (3.12) the metric (3.60) has scalar curvature

$$R({\mathfrak h}) = {{{\varsigma _1}\,\dim \,G/H} \over {h(\rho)}}$$

(3.61)

so that the warp function parameterizes the inverse curvature radius of the target space.

Here we combined the field vector (3.9) with ϕⁿ⁺¹ ≔ σ to a \(D = 2 + \bar n + 2k + 2\)-dimensional vector ϕ = (φ¹,…, φ ⁿ , ρ, σ), and the metric (3.60) refer to this coordinate system. Further a, b are real parameters kept mainly to illustrate that they drop out in the quantities of interest. The metric is chosen such that for the parameter values a = −1, b = −1 the Lagrangian (3.56) can be written in the form

$${L_h}(\varphi, \,\rho, \sigma) = - {1 \over {2\lambda}}{{\mathfrak{h}}_{ij}}(\phi){\partial ^\mu}{\phi ^i}{\partial _\mu}{\phi ^j}.$$

(3.62)

In addition to the Killing vectors associated with m the metric (3.60) possesses two conformal Killing vectors \({{\bf{t}}_ +} = \rho {\partial _\rho} - {a \over {2b}}{\partial _\sigma}\) and \({\bf{d}} = - \rho \,\ln \,\rho {\partial _\rho} + \left({\sigma + {a \over b}\ln \,\rho} \right){\partial _\sigma}\), which together with t₋ = ∂ _σ generate the isometries of ℝ^1,1, i.e. [t₊, t₋] = 0, [d, t_±] = ±t_±. Conversely any metric with these conformal isometries can be brought into the above form. Each Killing vector of ɧ of course gives rise to a Noether current; the conformal Killing vectors t₊ and d give rise to currents analogous to those in Equation (3.27). The counterpart of on-shell the relations (3.27) is ∂ ^μ C _μ = ρ∂ _ρ ln h · L, ∂ ^μ D _μ = −ln ρ ρ∂ _ρ ln h · L. Upon quantization Λ in Equation (3.62) plays the role of the loop counting parameter. In dimensional regularization (∫ d²x → ∫ d ^d x) the l-loop counter terms contain poles of order v ≤ l in (2 − d). We denote the coefficient of the v-th order pole by \(T_{ij}^{\left({\nu, \, l} \right)}\left(\mathfrak{h} \right)\). In principle the higher order pole terms are determined recursively by the residues \(T_{ij}^{\left({1, \, l} \right)}\left(\mathfrak{h} \right)\) of the first order poles. Taking the consistency of the cancellations for granted one can focus on the residues of the first oder poles, which we shall do throughout. One can show [154] that they have the following structure:

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(3.63)

It should be stressed that this is not trivially a consequence of the block-diagonal form of Equation (3.60), rather the properties (3.12) enter in an essential way.

Combining Equations (3.60, 3.65, 3.66) and (3.63) one finds that the first order poles cancel in the renormalized Lagrangian iff the following “reduction condition” holds:

$${{\mathcal L}_\Xi}{{\mathfrak h}_{ij}} + H(\rho, \lambda){{\mathfrak h}_{ij}} = \lambda T_{ij}^{(1)}({\mathfrak h}/\lambda),$$

(3.68)

where \({T^{\left(1 \right)}}\left(\mathfrak{h} \right) = \sum\nolimits_{l \geq 1} {{{\left({{1 \over {2\pi}}} \right)}^l}{T^{\left({1,\,l} \right)}}} \left(\mathfrak{h} \right)\) and \({{\mathcal L}_\Xi}\mathfrak{h}\) is the Lie derivative of ɧ, \(\mathfrak{h},{{\mathcal L}_\Xi}{\mathfrak{h}_{ij}}: = {\left({{{\mathcal L}_\Xi}\mathfrak{h}} \right)_{ij}} = {\Xi ^k}{\partial _k}{\mathfrak{h}_{ij}} + {\partial _i}{\Xi ^k}{\mathfrak{h}_{kj}} + {\partial _j}{\Xi ^k}{\mathfrak{h}_{ki}}\). The ρ-dependence of H marks the deviation from conventional renormalizability. Guided by the structure of Equations (3.60) and (3.63) we search for a solution with \({\Xi ^j} = \left({0,\, \ldots, \,0,\,{\Xi ^\rho}\left({\rho, \,\lambda} \right),\,{\Xi ^\sigma}\left({\rho, \,\lambda} \right)} \right)\), where here and later on we also use ρ = n +1, σ = n + 2 for the index labeling. The Lie derivative term with this Ξ ^j is

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(3.69)

The reduction condition (3.68) then is equivalent to a simple system of differential equations whose solution is

$$\begin{array}{*{20}c} {H(h/\lambda) = - {1 \over {h(\rho, \lambda)}}\rho {\partial _\rho}\left[ {h(\rho, \lambda){{{\Xi ^\rho}(h/\lambda)} \over \rho}} \right],}\\ {{\Xi ^\rho}(h/\lambda) = - \rho \int \nolimits^\rho {{du} \over u}{B_\lambda}\left({{\lambda \over {h(u,\lambda)}}} \right),\quad \quad \quad \quad \quad}\\ {{\Xi ^\sigma}(h/\lambda) = - {a \over {2b\rho}}{\Xi ^\rho}(h/\lambda) + {1 \over {2b}}\int \nolimits^\rho {{du} \over u}S(u,\lambda).}\\ \end{array}$$

(3.70)

Here we set

$${B_\lambda}(\lambda): = \sum\limits_{l \geq 1} {{\zeta _l}{{\left({{\lambda \over {2\pi}}} \right)}^l},} \quad \;\;S(\rho, \lambda): = n\sum\limits_{l \geq 1} {{{\left({{\lambda \over {2\pi}}} \right)}^l}{h^{- l}}{S_l}(\rho),}$$

(3.71)

and slightly adjusted the notation to stress the functional dependence on h/Λ. Possibly Λ-dependent integration constants have been absorbed into the lower integration boundaries of the integrals. Throughout these solutions should be read as shorthands for their series expansions in Λ with h of the form (3.67). For example

$${\Xi ^\rho}(\rho, \lambda) = {\lambda \over {2\pi}}{{{\zeta _1}} \over p}{\rho ^{- p + 1}} + {\left({{\lambda \over {2\pi}}} \right)^2}\rho \int\nolimits_\rho ^\infty {{{du} \over {{u^{2p + 1}}}}[{\zeta _2} - {\zeta _1}{h_1}(u)] + O({\lambda ^3}){.}}$$

(3.72)

This completes the renormalization of the Lagrangian L _h . The nonlinear field redefinitions alluded to in Equation (3.57) are explicitly given by Equation (3.70). The function ρ → h(ρ) plays the role of a generating function of an infinite set of essential couplings. In principle it could be expanded with respect to a basis of μ-independent functions of ρ with μ-dependent coefficients, the couplings. Technically the fact that these couplings are essential (in sense defined in the introduction) follows from Equations (3.27). Since ρ is a nontrivial function on the base manifold, the Lagrangian is a total divergence on shell if and only if h(ρ) = ρ ^p , or h(ρ) = ln ρ ^p . The first case corresponds to the classical Lagrangian (3.16), the second case was studied (in a different context) by Tseytlin [221]. In the case h(ρ) = ρ ^p the identity pL = ∂ ^μ C _μ reflects the fact that the overall scale of the metric is an inessential parameter (see Appendix A). The renormalization flow associated with the coupling functional h will be studied in the next Section 3.4.

3.4 Non-Gaussian fixed point and asymptotic safety

Another intriguing property of Equation (3.85) can be seen from the second line in Equation (3.84). The first two terms correspond to the beta function of a G/H sigma-model without coupling to 2D gravity (i.e. with nondynamical ρ and σ). The last term is crucial for all the subsequent properties of the flow (3.83). Comparing with Equation (3.75) one sees that it describes a backreaction of the scale dependent area radius ρ on the coupling flow, which is mediated by the quantum dynamics of the other fields. We shall return to this point below.

The origin of this feature is the seeming violation of scale invariance on the level of the renormalized action. Recall from after Equation (3.62) that ∂ ^μ C _μ = ρ∂ _ρ ln h · L _h , so that for h^beta(ρ) ≠ ρ ^p the action is no longer scale invariant. However this is precisely the property which allows one to cancel the (otherwise) anomalous term in the trace of the would-be energy momentum tensor, as discussed before, rendering the system conformally invariant at the fixed point. Due to the lack of naive scale invariance on the level of the renormalized action the dynamics of quantum gravidynamics is different from that of quantum general relativity, in the sector considered, even at the fixed point. The moral presumably generalizes: The form of the (bare and/or renormalized) action may have to differ from the Einstein-Hilbert action in order to incorporate the physics properties aimed at.

Result (UV stability):

The situation is illustrated in Figure 1. The proof of this result is somewhat technical and can be found in [155].

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We can put this result into the context the general discussion in Section 2 and arrive at the following conclusion: