The Asymptotic Safety Scenario in Quantum Gravity
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DOI: 10.12942/lrr-2006-5
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- Niedermaier, M. & Reuter, M. Living Rev. Relativ. (2006) 9: 5. doi:10.12942/lrr-2006-5
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Abstract
The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. The evidence from symmetry truncations and from the truncated flow of the effective average action is presented in detail. A dimensional reduction phenomenon for the residual interactions in the extreme ultraviolet links both results. For practical reasons the background effective action is used as the central object in the quantum theory. In terms of it criteria for a continuum limit are formulated and the notion of a background geometry self-consistently determined by the quantum dynamics is presented. Self-contained appendices provide prerequisites on the background effective action, the effective average action, and their respective renormalization flows.
1 Introduction and Survey
The search for a physically viable theory of quantized gravitation is ongoing; in part because the physics it ought to describe is unknown, and in part because different approaches may not ‘approach’ the same physics. The most prominent contenders are string theory and loop quantum gravity, with ample literature available on either sides. For book-sized expositions see for example [97, 177, 112, 199]. The present report and [157] describe a circle of ideas which differ in several important ways from these approaches.
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.
Relate micro- and macro-physics of the gravitational field through a renormalization flow.
As the basic physics premise stipulate that the physical degrees of freedom in the extreme ultraviolet interact predominantly antiscreening.
Based on this premise benign renormalization properties in the ultraviolet are plausible. The resulting “Quantum Gravidynamics” can then be viewed as a peculiar quasi-renormalizable field theory based on a non-Gaussian fixed point.
In the extreme ultraviolet the residual interactions appear two-dimensional.
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 10^{30} 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 n^{2} observables are injective functions of n relevant couplings each; then n measurements will determine the couplings, leaving n^{2} − 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/p^{2} type propagator. As first advocated by Gomis and Weinberg [94] the use of a 1/p^{2} 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/p^{2} 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/p^{4} 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/p^{2} 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/p^{2} 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/p^{4} 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/(p^{4} ln p^{2}), 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 x^{2} 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/p^{4}-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/p^{4}-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 +ϵexpansions: 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.
- (i)
a cosmological constant term \(\int {{d^{2 + \epsilon}}x\sqrt g}\),
- (ii)
monomials from matter fields which are quantum mechanically non-scale invariant in d = 2,
- (iii)
monomials from matter fields which are quantum mechanically scale invariant in d = 2,
- (iv)
the conformal mode of the metric itself in a background field expansion.
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 + 6Δ_{0} − c)/(24π), where Δ_{0} = 1/2, 1, respectively. Here Δ_{0} 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 Δ_{0} = 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/p^{2}-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/p^{2} to 1/(p^{4} ln p^{2}), 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 C^{2} 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/p^{4}-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.
The restricted functional integral inherits the perturbative non-renormalizability (with finitely many relevant couplings) from the full theory.
It takes into account the crucial ‘spin-2’ aspect, that is, linear and nonlinear gravitational waves with two independent polarizations per spacetime point are included.
It goes beyond the Eikonal approximation [209, 71] whose dynamics can be understood via a related 2 + 2 decomposition [113, 73].
Based on heuristic arguments the dynamics of full Quantum Gravidynamics is expected to be effectively two-dimensional in the extreme ultraviolet with qualitative properties resembling that of the 2 + 2 truncation. The renormalization of the 2 + 2 truncation can thus serve as a prototype study and its asymptotic safety probes the self-consistency of the scenario.
For the restricted functional integral the full infinite-dimensional renormalization group dynamics can be studied; it reveals both a Gaussian and a non-Gaussian fixed point, where the properties of the latter are compatible with the existence of a non-perturbative continuum limit.
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 λ_{0} and κ_{0} then lifts the marginality of λ_{4}, which becomes marginally irrelevant [171, 170]. The running of κ_{0} and λ_{0} 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.
(C1) strictly independent of the UV cutoff,
(C2) independent of the choice of the coarse graining operation (within a certain class), and
(C3) invariant under point transformations of the fields.
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.
(PTC1) Existence of a formal continuum limit. This means, the removal of the UV cutoff is possible and the renormalized physical quantities are independent of the scheme and of the choice of interpolating fields — all in the sense of formal power series in the loop counting parameter. The perturbative beta functions always have a trivial (Gaussian) fixed-point but may also have a nontrivial (non-Gaussian) fixed point.
(PTC2) The dimension of the unstable manifold of the (Gaussian or non-Gaussian) fixed point as computed from the perturbative beta functions equals the number of independent essential couplings.
(FRGC1) The solution of the FRG equation admits (for fine tuned initial data Γ_{initial} at some k = k_{initial}) a global solution Γ_{k}, i.e. one that can be extended both to k → ∞ and to k → 0 (where the latter limit is not part of the UV problem in itself).
(FRGC2) The functional derivatives of lim_{k→0} Γ_{k} (vertex functions) meet certain requirements which ensure stability/positivity/unitarity.
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.
The choice of couplings has to be based on observables; this will pin down the physically relevant notion of positivity/unitarity.
The number of essential or relevant couplings is not a-priori finite.
What matters is not so much the dimension of the unstable manifold than how observables depend on the relevant couplings.
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.
1.4.1 Dynamical triangulations
The framework closest in spirit to the present one are discretized approaches to the gravitational functional integral, where a continuum limit in the statistical field theory sense is aimed at. See [138] for a general review and [8] for the dynamical triangulations approach. Arguably the most promising variant of the latter is the causal dynamical triangulations approach by Ambjørn, Jurkiewicz, and Loll [5]. In this setting the formal four dimensional quantum gravity functional integral is replaced by a sum over discrete geometries, Z = ∑_{T}m(T)e^{iS(T)}. The geometries T are piecewise Minkowskian and selected such that they admit a Wick rotation to piecewise Euclidean geometries. The edge lengths in the spatial and the temporal directions are ℓ_{space} = a^{2} and ℓ_{time} = −αa^{2}, where a sets the discretization scale and α > 0 is an adjustable parameter. The flip α ↦ −α defines a Wick rotation under which the weights in the partition function become real: \({e^{iS(T)}} \mapsto {e^{- {S_{eucl}}(T)}}\). For α = −1 the usual expressions for the action used [8] in equilateral Euclidean dynamical triangulations are recovered, but the sum is only over those Euclidean triangulations T which lie in the image of the above Wick rotation. The weight factor m(T) is the inverse of the order of the automorphism group of the triangulation, i.e. 1 for almost all of them. With these specifications the goal is to construct a continuum limit by sending a → 0 and the number N of simplices to infinity, while adjusting the two bare parameters (corresponding to Newton’s constant and a cosmological constant) in S_{eucl}(T) as well as the overall scale of Z. Very likely, in order for such a continuum limit to exist and to be insensitive against modifications of the discretized setting, a renormalization group fixed point in the coupling flow is needed. Assuming that the system indeed has a fixed point, this fixed point would by construction have a nontrivial unstable manifold, and ideally both couplings would be asymptotically safe, thereby realizing the strong asymptotic safety scenario (using the terminology of Section 1.3). Consistent with this picture and the previously described dimensional reduction phenomenon for asymptotically safe functional measures (see also Section 2.4), the microscopic geometries appear to be effectively two-dimensional [6, 7].
Despite these similarities there are (for the time being) also important differences. First the discretized action depends on two parameters only and it is hoped that a renormalized trajectory can be found by tuning only these two parameters. Since in dynamical triangulations there is no naive (classical) continuum limit, one cannot directly compare the discretized action used with a microscopic action in the previous sense. Conceptually one can assign a microscopic action to the two parametric measure defined by the causal dynamical triangulations by requiring that combined with the regularized kinematical continuum measure (see Section 2.3.3) it reproduces the same correlation functions in the continuum limit. The microscopic action defined that way would presumably be different from the Einstein-Hilbert action, but it would still contain only two tunable parameters. In other words the hope is that the particular non-naive discretization procedure gets all but two coordinates of the unstable manifold automatically right. A second difference concerns the role of averages of the metric. The transfer matrix used in [8] is presumed to have a unique ground state for both finite and infinite triangulations. Expectation values in a reconstructed Hilbert space will refer to this ground state and hence be unique for a given operator. A microscopic metric operator does not exist in a dynamical triangulations approach but if one were to define coarse grained variants, their expectation value would have to be unique. In contrast the field theoretical formulations based on a background effective action allow for a large class of averaged metrics.
1.4.2 Loop quantum gravity
The term loop quantum gravity is by now used for a number of interrelated formulations (see [199] for a guide). For definiteness we confine our comparative remarks to the original canonical formulation using loop (holonomy) variables.
Here a reformulation of general relativity in terms of Ashtekar variables (A, E) is taken as a starting point, where schematically A and E are defined on a three-dimensional time slice and are conjugate to each other, {A, E} = δ, with respect to the canonical symplectic structure (see [199, 15]). From A one can form holonomies (line integrals along loops) and from E one can form fluxes (integrals over two-dimensional hypersurfaces) without using more than the manifold structure. The Poisson bracket {A, E} = δ is converted into a Poisson algebra for the holonomy and the flux variables. Two basic assumptions then govern the transition to the quantum theory: First the Poisson bracket {A, E} = δ is replaced by a commutator [A, E] = iħδ and is subsequently converted into an algebraic structure among the holonomy and flux variables. Second, representations of this algebra are sought on a state space built from multiple products of holonomies associated with a graph (spin network states). The inner product on this space is sensitive only to the coincidence or non-coincidence of the graphs labeling the states (not to their embedding into the three-manifold). Based on a Gelfand triple associated with this kinematical state space one then aims at the incorporation of dynamics via a (weak) solution of the Hamiltonian constraint of general relativity (or a ‘squared’ variant thereof). To this end one has to transplant the constraint into holonomy and flux variables so that it can act on the above state space. This step is technically difficult and the results obtained do not allow one to address the off-shell closure of the constraint algebra, an essential requirement emphasized in [151].
As far as comparison with Quantum Gravidynamics is concerned, important differences occur both on a kinematical and on a dynamical level, even if a variant of Gravidynamics formulated in terms of the Ashtekar variables (A, E) was used [156]. Step one in the above quantization procedure keeps the right-hand-side of the commutator [A, E] = iħδ free from dynamical information. In any field theoretical framework, on the other hand, one would expect the right-hand-side to be modified: minimally (if A and E are multiplicatively renormalized) by multiplication with a (divergent) wave function renormalization constant, or (if A and E are nonlinearly renormalized) by having δ replaced with a more general, possibly field dependent, distribution. Stipulation of unmodified canonical commutation relations might put severe constraints on the allowed interactions, as it does in quantum field theories with a sufficiently soft ultraviolet behavior. (We have in mind here “triviality” results, where e.g. for scalar quantum field theories in dimensions d ≥ 4 a finite wave function renormalization constant goes hand in hand with the absence of interaction [24, 77]). A second marked difference to Quantum Gravidynamics is that in Loop Quantum Gravity there appears to be no room for the distinction between fine grained (‘rough’) and coarse grained (‘smooth’) geometries. The inner product used in the second of the above steps sees only whether the graphs of two spin network states coincide or not, but is insensitive to the ‘roughness’ of the geometry encoded initially in the (A, E) pair. This information appears to be lost [151]. In a field theory the geometries would be sampled according to some underlying measure and the typical configurations are very rough (non-differentiable). As long as the above ‘holonomy inner product’ on such sampled geometries is well defined and depends only on the coincidence or non-coincidence of the graphs the information about the measure according to which the sampling is done appears to be lost. Every measure will look the same. This property seems to match the existence of a preferred diffeomorphism invariant measure [14] (on a space generated by the holonomies) which is uniquely determined by some natural requirements. The typical A configurations are also of distributional type [14, 140]. This uniqueness translates into the uniqueness of the associated representation of the holonomy-flux algebra (which rephrases the content of the original [A, E] = iħδ algebra). In a field theory based on the (A, E) variables, on the other hand, there would be a cone of regularized measures which incorporate dynamical information and on which the renormalization group acts.
Another difference concerns the interplay between the dynamics and the canonical commutation relations. In a field theory the moral from Haag’s theorem is that “the choice of the representation of the canonical commutation relations is a dynamical problem” [99]. Further the inability to pick the ‘good’ representation beforehand is one way to look at the origin of the divergencies in a canonically quantized relativistic field theory. (To a certain extent the implications of Haag’s theorem can be avoided by considering scattering states and spatially cutoff interactions; in a quantum gravity context, however, it is unclear what this would amount to.) In contrast, in the above holonomy setting a preferred representation of the holonomy-flux algebra is uniquely determined by a set of natural requirements which do not refer to the dynamics. The dynamics formulated in terms of the Hamiltonian constraint thus must be automatically well-defined on the above kinematical arena (see [151, 173] for a discussion of the ambiguities in such constructions). In a field theoretical framework, on the other hand, the constraints would be defined as composite operators in a way that explicitly requires dynamical information (fed in through the renormalized action). So the constraints and the space on which they act are dynamically correlated. In loop quantum gravity, in contrast, both aspects are decoupled.
Finally, the microscopic action for asymptotically safe Quantum Gravidynamics is very likely different from the Einstein-Hilbert action and thus not of second order. This changes the perspective on a canonical formulation considerably.
1.4.3 String theory
String theory provides a possible context for the unification of known and unknown forces including quantum gravity [97, 177, 112]. As far as quantum gravity is concerned the point of departure is the presupposition that the renormalization problem for the quantized gravitational field is both insoluble and irrelevant. Presently a clearly defined dynamical principle that could serve as a substitute seems to be available only for so-called perturbative first quantized string theory, to which we therefore confine the following comparative comments.
In this setting certain two-dimensional (supersymmetric) conformal field theories are believed to capture (some of) the ‘ultimate degrees of freedom of Nature’. The attribute ‘perturbative’ mostly refers to the fact that a functional integral over the two-surfaces on which the theories are defined is meant to be performed, too, but in a genus expansion this gives rise to a divergent and not Borel summable series. (In a non-perturbative formulation aimed at degrees of freedom corresponding to other extended objects are meant to occur and to cure this problem.) For the relation to gravity it is mainly the bosonic part of the conformal field theories which is relevant, so we take the 2D fermions to be implicitly present in the following without displaying them.
Since arguments presented after Equation (1.5) suggest a kind of ‘dimensional reduction’ to d = 2, one might be tempted to see this as a vindication of string theory from the present viewpoint. However string theory’s very departure was the presupposition that no fixed point exists for the gravitational functional integral. Moreover in string theory the sigma-model fields relate the worldsheet to a (4 + 6-dimensional) target manifold with a prescribed metric (or pairs thereof related by T-duality). The at least perturbatively known dynamics of the sigma-model fields does not appear to simulate the functional integral over metrics (see Equation (1.16)). The additional functional integral over Euclidean worldsheet geometries is problematic in itself and leaves unanswered the question how and why it successfully captures or replaces the ultaviolet aspects of the original functional integral, other than by definition. In the context of the asymptotic safety scenario, on the other hand, the presumed reduction to effectively two-dimensional propagating degrees of freedom is a consequence of the renormalization group dynamics, which in this case acts like a ‘holographic map’. This holographic map is of course not explicitly known, nor is it off-hand likely that it can be described by some effective string theory. A more immediate difference is that Quantum Gravidynamics does not require the introduction of hitherto unseen degrees of freedom.
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/p^{4}-type propagator used. The alternative perturbative framework already mentioned, namely to use a 1/p^{2}-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 + R^{2} 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 R^{2} 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 + R^{2} 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 10^{30} 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 Renormalizing the Non-Renormalizable
The modern view on renormalization has been shaped by Kadanoff and Wilson. The more familiar perturbative notion of renormalizability is neither sufficient (e.g. ϕ^{4} theory in d = 4) nor necessary (e.g. Gross-Neveu model in d = 3) for renormalizability in the Kadanoff-Wilson sense. For the convenience of the reader we summarize the main ideas in Appendix A, which also serves to introduce the terminology. The title of this section is borrowed from a paper by Gawedzki and Kupiainen [88].
In the present context the relevance of a Kadanoff-Wilson view on renormalization is two-fold: First it allows one to formulate the notion of renormalizability without reference to perturbation theory, and second it allows one to treat to a certain extent renormalizable and non-renormalizable on the same footing. The mismatch between the perturbative non-renormalizability of the Einstein-Hilbert action and the presumed asymptotic safety of a functional measure constructed by other means can thus be systematically explored.
In a gravitational context also the significance of renormalizabilty is less clear cut, and one should presumably go back to the even more fundamental property for which renormalizability is believed to be instrumental, namely the existence of a genuine continuum limit, roughly in the sense outlined in Section 1.3. Since rigorous results based on controlled approximations are unlikely to be obtained in the near future, we describe in the following criteria for the plausible existence of a genuine continuum limit based on two uncontrolled approximations: renormalized perturbation theory and the functional renormalization group approach. Such criteria are ‘implicit wisdom’ and are hardly ever spelled out. In the context of Quantum Gravidynamics, however, the absence of an obvious counterpart of the correlation length and non-renormalizability of the Einstein-Hilbert action makes things more subtle. In Sections 2.1 and 2.2 we therefore try to make the implicit explicit and to formulate critera for the existence of a genuine continuum limit which are applicable to Quantum Gravidynamics as well.
In Section 2.3 we describe the renormalization problem for Quantum Gravidynamics and in Section 2.4 the dimensional reduction phenomenon outlined before.
For a summary of basic renormalization group concepts we refer to Appendix A and for a review of the renormalization group for the effective average action to Appendix C.
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 P_{2}(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 g_{1} 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 g_{1}). 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 g_{1} can then be identified asymptotically and lie in the unstable manifold of the fixed point g_{1} = 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 g_{1} 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 g_{1} is regular with lim_{μ→∞}g_{1}(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:
(PTC1) Existence of a formal continuum limit, i.e. removal of the UV cutoff is possible and the renormalized physical quantities are independent of the scheme and of the choice of interpolating fields, all in the sense of formal power series in the loop counting parameter.
(PTC2) The perturbative beta functions have a Gaussian or a non-Gaussian fixed point and the dimension of its unstable manifold (as computed from the perturbative beta functions) equals the number of independent essential couplings. Equivalently, all essential couplings are asymptotically safe in perturbation theory.
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 p^{2} ≈ k^{2}, 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:
(FRGC1) A solution of the cutoff independent FRGE (2.12) which exists globally in k (for all 0 ≤ k ≤ ∞) can reasonably be identified with the continuum limit of the effective average action lim_{Λ→∞} Γ_{λ,k} [ϕ] constructed by other means. For such a solution lim_{k→0} Γ_{k}[ϕ] is the full quantum effective action and lim_{k→∞} Γ_{k}[ϕ] = S_{*}[ϕ] is the fixed point action.
(FRGC2) For a unitary relativistic quantum field theory positivity/unitarity must be tested retroactively from the functional derivatives of lim_{k→0} Γ_{k}[ϕ].
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.
2.3.1 The role of Newton’s constant
2.3.2 Perturbation theory and higher derivative theories
By higher derivative theories we mean here gravitational theories whose bare action contains, in addition to the Einstein-Hilbert term, scalars built from powers of the Riemann tensor and its covariant derivatives. In overview there are two distinct perturbative treatments of such theories.
The first one, initiated by Stelle [206], uses 1/p^{4} type propagators (in four dimensions) in which case a higher derivative action containing all (three) quartic derivative terms can be expected to be power counting renormalizable. In this case strict renormalizability with only 4 (or 5, if Newton’s constant is included) couplings can be achieved [206]. However the 1/p^{4} type propagators are problematic from the point of view of unitarity.
An alternative perturbative treatment of higher derivative theories was first advocated by Gomis-Weinberg [94]. The idea is to try to maintain a 1/p^{2} type propagator and include all (infinitely many) counterterms generated in the bare action. Consistency requires that quadratic counterterms (those which contribute to the propagator) can be absorbed by field redefinitions. As shown by Anselmi [10] this is the case either in the absence of a cosmological constant term or when the background spacetime admits a metric with constant curvature.
We now present both of these perturbative treatments in more detail. A putative matching to a nonperturbative renormalization flow is outlined in Equation (2.32).
The flow equations (2.20, 2.30) of course also admit the Gaussian fixed point \({\rm{g}}_{\rm{N}}^{\ast} = 0 = \lambda\), and one may be tempted to identify the ‘realm’ of perturbation theory (PT) with the ‘expansion’ around a Gaussian fixed point. As explained in Section 2.1, however, the conceptual status of PT referring to a non-Gaussian fixed point is not significantly different from that referring to a Gaussian fixed point. In other words there is no reason to take the perturbative non-Gaussian fixed point (2.31) any less serious than the perturbative Gaussian one. This important point will reoccur in the framework of the 2 + 2 truncation in Section 3, where a non-Gaussian fixed point is also identified by perturbative means.
The fact that a non-Gaussian fixed point can already be identified in PT is important for several reasons. First, although the value of \({\rm{g}}_{\rm{N}}^{\ast}\) in Equation (2.31) is always non-universal, the anomalous dimension η_{N} = γ_{g}/g_{N} − 2 is exactly −2 at the fixed point (2.31). The general argument for the dimensional reduction of the residual interactions outlined after Equation (1.5) can thus already be based on PT alone! Second the result (2.31) suggests that the interplay between the perturbative and the nonperturbative dynamics might be similar to that of non-Abelian gauge theories, where the nonperturbative dynamics is qualitatively and quantitatively important mostly in the infrared.
The most important drawback of the perturbatively renormalizable theories based on Equation (2.26) are the problems with unitarity entailed by the propagator (2.28). As already mentioned these problem are absent in an alternative perturbative formulation where a 1/p^{2} type propagator is used throughout [94]. We now describe this construction in slightly more detail following the presentation in [10].
Let us briefly recap the power counting and scaling dimensions of local curvature invariants. These are integrals P_{i}[g] = ∫ d^{d}x L_{i}(g) over densities which are products of factors of the form \({\nabla _{{\alpha _1}}} \ldots {\nabla _{{\alpha _{l - 4}}}}{R_{{\alpha _{l - 3 \cdots}}{\alpha _l}}}\), suitably contracted to get a scalar and then multiplied by \(\sqrt g\). One easily checks \({L_i}\left({{\omega ^2}h} \right) = {\omega ^{{s_i}}}{L_i}\left(g \right),\omega > 0\), with s_{i} = d − 2p − q, where p is the total power of the Riemann tensor and q is the (necessarily even) total number of covariant derivatives. This scaling dimension matches minus the mass dimension of P_{i}(g) if g is taken dimensionless. For the mass dimension d_{i} of the associated coupling u_{i} in a product u_{i}P_{i}[g] one thus gets d_{i} = s_{i} = d − 2p − q. For example, the three local invariants in Equation (1.14) have mass dimensions −d_{0} = −d, −d_{1} = −(d − 2), −d_{2} = −(d − 4), respectively. There are three other local invariants with mass dimension −(d − 4), namely the ones with integrands C^{2} = R^{αβγδ}R_{αβγδ} − 2R^{αβ}R_{αβ} + R^{2}/3 (the square of the Weyl tensor), E = R^{αβγδ}R_{αβγδ} − 4R^{αβ}R_{αβ} + R^{2} (the generalized Euler density), and ∇^{2}R. Then there is a set of dimension − (d − 6) local invariants, and so on. Note that in d = 4 the integrands of the last two of the dimensionless invariants are total divergencies so that in d = 4 there are only 4 local invariants with non-positive mass dimension (see Equation (2.26)).
A generic term in P_{i} will be symbolically of the form ∇^{q}R^{p}, where all possible contractions of the 4p + q indices may occur. Since the Ricci tensor is schematically of the form R = ∇^{2}f + O(f^{2}), the piece in P_{i} quadratic in f is of the form ∇^{q+4}R^{p−2}f^{2}. The coefficient of f^{2} is a tensor with 4 free indices and one can verify by inspection that the possible index contractions are such that the Ricci tensor or Ricci scalar either occurs directly, or after using the contracted Bianchi identity. In summary, one may restrict the sum in Equation (2.36) to terms with −d_{i} = −d + 2p + q, p ≥ 3, and the propagator derived from it will remain of the 1/p^{2} type to all loop orders. This suggests that Equation (2.36) will give rise to a renormalizable Lagrangian. A proof requires to show that after gauge fixing and ghost terms have been included all counter terms can be chosen local and covariant and has been given in [94].
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 are likely to blow up likewise at some (high) energy scale μ = μ_{term}. In other words the couplings in Equation (2.36) are presumably not all asymptotically safe.
Let us add a brief comment on the relevant-irrelevant distinction in this context, if only to point out that it is no longer useful. Recall from Section 1.3 that the notion of a relevant or irrelevant coupling applies even to flow lines not connected to a fixed point. This is the issue here. All but a few of the interaction monomials in Equation (2.36) are power counting irrelevant with respect to the 1/p^{2} propagator. Equivalently all but a few couplings \({u_i}(\mu) = {\mu ^{{d_i}}}{{\rm{g}}_i}(\mu)\) have non-negative mass dimensions d_{i} ≥ 0. These are the only ones not irrelevant with respect to the stability matrix Θ computed at the perturbative Gaussian fixed point. However in Equation (2.36) these power counting irrelevant couplings with d_{i} < 0 are crucial for the absorption of infinities and thus are converted into practically relevant ones. In the context of Equation (2.36) we shall therefore discontinue to use the terms relevant/irrelevant.
Comparing both perturbative constructions one can see that the challenge of Quantum Gravidynamics lies not so much in achieving renormalizability, but to reconcile asymptotically safe couplings with the absence of unphysical propagating modes. This program is realized in Section 3 for the 2 + 2 reduction; the results of Section 4 for the R + R^{2} type truncation likewise are compatible with the absence of unphysical propagating modes.
In order to realize this program without reductions or truncations, a mathematically controllable nonperturbative definition of Quantum Gravidynamics is needed. Within a functional integral formulation this involves the following main steps: definition of a kinematical measure, setting up a coarse graining flow for the dynamical measures, and then probing its asymptotic safety.
2.3.3 Kinematical measure
For a functional integral over geometries even the kinematical measure, excluding the action dependent factor, is nontrivial to obtain. A geometric construction of such a measure has been given by Bern, Blau, and Mottola [31] generalizing a similar construction in Yang-Mills theories [20]. It has the advantage of separating the physical and the gauge degrees of freedom (at least locally in field space) in a way that is not tied to perturbation theory. The functional integral aimed at is one over geometries, i.e. equivalence classes of metrics modulo diffeomorphisms. For the subsequent construction the difference between Lorentzian and Riemannian signature metrics is inessential; for definiteness we consider the Lorenzian case and correspondingly have an action dependence exp iS[g] in mind.
In the above discussion we did not split off the conformal factor in the geometries. Doing this however only requires minor modifications and was the setting used in [31, 142, 149]. In Equation (2.37) then ĝ_{αβ} is written as \({e^\sigma}g_{\alpha \beta}^ \bot\), where now \(g_{\alpha \beta}^ \bot\) is subject to a gauge condition (F ο g^{⊥})_{α} = 0. On the cotangent space this leads to a York-type decomposition [235] replacing (2.41), where the variations f_{σ} of the conformal factor and that of the tracefree part \(\hat f_{\alpha \beta}^ \bot\) of \({{\hat f}_{\alpha \beta}}\) describe the variations of the geometry, while the tracefree part, \({\left({{L^{{\rm{TF}}}}\upsilon} \right)_{\alpha \beta}}: = {\nabla _\alpha}{\upsilon _\beta} + {\nabla _\beta}{\upsilon _\alpha} - {2 \over d}{g_{\alpha \beta}}{\nabla ^\gamma}{\upsilon _\gamma}\), and the trace part of the Lie derivative (Lv)_{αβ} describe the gauge variations. Writing \({\mathcal D}{f_{\alpha \beta}} = {\mathcal D}{f_\sigma}{\mathcal D}f{\bot \over {\alpha \beta}}{\mathcal D}\upsilon\) the computation of the Jacobian proceeds as above and leads to Equation (2.47) with the following replacements: \({\mathcal D}{{\hat g}_{\alpha \beta}}\) is replaced with \({\mathcal D}g{\bot \over {\alpha \beta}}{\mathcal D}\sigma\), L with L^{TF}, and ĝ with e^{σ}g^{⊥} in the integrand. By studying the dependence of det_{V}(F ο L)(e^{σ}g^{⊥}) on the conformal factor it has been shown in [142] that in the Gaussian approximation of the Euclidean functional integral the instability associated with the unboundedness of the Euclidean Einstein-Hilbert action is absent, due to a compensating contribution from the determinant. It can be argued that this mechanism is valid also for the interacting theory. From the present viewpoint however the (Euclidean or Lorentzian) Einstein-Hilbert action should not be expected to be the proper microscopic action. So the “large field” or “large gradient” problem has to be readdressed anyhow in the context of Quantum Gravidynamics. Note also that once the conformal mode of the metric has been split off the way how it enters a microscopic or an effective action is no longer constrained by power counting considerations. See [12] for an effective dynamics for the conformal factor only.
Once a kinematical measure on the equivalence classes of metrics (or other dynamical variables) has been defined, the construction of an associated dynamical measure will have to rest on renormalization group ideas. Apart from the technical problems invoved in setting up a computationally useful coarse graining flow for the measure on geometries, there is also the apparent conceptual problem how diffeomorphism invariance can be reconciled with the existence of a scale with respect to which the coarse graining is done. However no problem of principle arises here. First, similar as in a lattice field theory, where one has to distinguish between the external lattice spacing and a dynamically generated correlation length, a distinction between an external scale parameter and a dynamically co-determined resolution scale has to be made. A convenient way to achieve compatibility of the coarse graining with diffeomorphism invariance is by use of the background field formalism. The initially generic background metric serves as a reference to discriminate modes, say in terms of the spectrum of a covariant differential operator in the background metric (see Section 4.1). Subsequently the background is self-consistenly identified with the expectation value of the quantum metric as in the discussion below.
The functional integral over “all geometries” should really be thought of as one over “all geometries subject to suitable boundary conditions”. Likewise the action is meant to include boundary terms which indirectly specify the state of the quantum system.
- 1.
The choice of couplings has to be based on observables; this will pin down the physically relevant notion of positivity/unitarity.
- 2.
The number of essential or relevant couplings is not a-priori finite.
- 3.
What matters is not so much the dimension of the unstable manifold than how observables depend on the relevant couplings.
2.3.4 Effective action and states
Unfortunately, at present little is known about generic quantum gravity observables, so that the functional averages whose expansion would define physical couplings are hard to come by. For the time being we therefore adopt a more pragmatic approach and use as the central object to formulate the renormalization flow the background effective action Γ[g_{αβ}, ḡ_{αβ},…] as described in Appendix B. Here g_{αβ} is interpreted as an initially source-dependent “expectation value of the quantum metric”, ḡ_{αβ} is an initially independently prescribed “background metric”, and the dots indicate other fields, conjugate to sources, which are inessential for the following discussion. For clarities sake let us add that it is not assumed that the metric exists as an operator, or that the metric-like “conjugate sources” g_{αβ}, ḡ_{αβ} are necessarily the best choice.
The condition (2.49) is equivalent to the vanishing of the extremizing sources \(J_{\ast}^{\alpha \beta}\left[ {g,\bar g, \ldots} \right]\) in the definition of Legendre transform (see Appendix B). Evidently Equation (2.49) also amounts to the vanishing of the one-point functions in Equation (2.51). Usually the extremizing sources \(J_{\ast}^{\alpha \beta}\left[ {g,\bar g, \ldots} \right]\) are constructed by formal inversion of a power series in \({{\bar f}_{\alpha \beta}}: = {g_{\alpha \beta}} - {{\bar g}_{\alpha \beta}}\). Then \({{\bar f}_{\alpha \beta}}=0\) always is a solution of Equation (2.49) and the functional g ↦ ḡ_{*} [g] is simply the identity. In this case the self-consistent background coincides with the naive prescribed background. To find nontrivial solutions of Equation (2.49) one has to go beyond the formal series inversions and the uniqueness assumptions usually made.
Due to the highly nonlocal character of the effective action the identification of physical solutions of Equation (2.49) is a nontrivial problem. The interpretation via Equation (2.48) suggests an indirect characterization, namely those solutions of Equation (2.49) should be regarded as physical which come from physically acceptable states [157].
The notion of a state is implicitly encoded in the effective action. Recall that the standard effective action, when evaluated at a given time-independent function ϕ^{i} = 〈χ^{i}〉, is proportional to the minimum value of the Hamiltonian H in that part of the Hilbert space spanned by normalizable states ∣ψ〉 satisfying 〈ψ∣χ^{i}∣ψ〉 = ϕ^{i}. A similar interpretation holds formally for the various background effective actions [51]. In conventional quantum field theories there is a clear-cut notion of a ground state and of the state space based on it. In a functional integral formulation the information about the state can be encoded in suitable boundary terms for the microscopic action. Already in quantum field theories on curved but non-dynamical spacetimes a preferred vacuum is typically absent and physically acceptable states have to be selected by suitable conditions (like, for example, the well-known Hadamard condition in the case of a Klein-Gordon field). In quantum gravity the formulation of analogous selection criteria is an open problem. As a tentative example we mention the condition formulated after Equation (2.53) below. On the level of the effective action one should think of Γ as a functional of both the selected state and of the fields. The selected state will indirectly (co-)determine the space of functionals on which the renormalization flow acts. For example the type of nonlocalities which actually occur in Γ should know about the fact that Γ stems from a microscopic action suited for the appropriate notion of positivity and from a physically acceptable state.
2.3.5 Towards physical quantities
Finally one will have to face the question of what generic physical quantities are and how to compute them. Although this is of course a decisive issue in any approach to quantum gravity, surprisingly little work has been done in this direction. In classical general relativity Dirac observables do in principle encode all intrinsic properties of the spacetimes, but they are nonlocal functionals of the metric and implicitly refer to a solution of the Cauchy problem. In a canonical formulation quantum counterparts thereof should generate the physical state space, but they are difficult to come by, and a canonical formulation is anyhow disfavored by the asymptotic safety scenario. S-matrix elements with respect to a self-consistent background (2.48) or similar objects computed from the vertex functions (2.51) might be candidates for generic physical quantities, but have not been studied so far.
In a lattice field theory the discretized functional measure typically generates an intrinsic scale, the (dimensionless) correlation length ξ, which allows one to convert lattice distances into a physical standard of length, such that say, ξ lattice spacings equal 1 fm. A (massive) continuum limit is eventually defined by sending ξ to infinity in a way such that physical distances (number of lattice spacings)/ξ fm are kept fixed and a ‘nonboring’ limit arises. In a functional measure over geometries dμ_{k,Λ}(g), initially defined with an UV cutoff Λ and an external scale parameter k, it is not immediate how to generalize the concept of a correlation length. Exponents extracted from the decay properties of Equation (2.52) or Equation (2.53) are natural candidates, but the ultimate test of the fruitfulness of such a definition would lie in the successful construction of a continuum limit. In contrast to a conventional field theory it is not even clear what the desired/required properties of such a continuum system should be. The working definitions proposed in Section 1.3 tries to identify some salient features.
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.
2.4.1 Scaling of fixed point action
One sees that in the fixed point regime \({{\rm{g}}_i}(k)\sim{\rm{g}}_i^ {\ast}\) the k-dependence enters only through the combination k^{2}g_{αβ}, a kind of self-similarity. This simple but momentous fact eventually underlies all the subsequent arguments. It is ‘as if’ in the fixed point regime only a rescaled metric \({{\tilde g}_{\alpha \beta}} = {k^2}{g_{\alpha \beta}}\) entered which carries dimension two. This has consequences for the ‘effective dimensionality’ of Newton’s constant: Recall that conventionally the Ricci scalar term, \(\int {dx} \,\sqrt g R(g)\), has mass dimension 2 − d in d dimensions. Upon substitution \({g_{\alpha \beta}} \mapsto {{\tilde g}_{\alpha \beta}}\) one quickly verifies that \(\int {dx} \,\sqrt {\tilde g} R(\tilde g)\) is dimensionless. Its prefactor, i.e. the inverse of Newton’s constant, then can be taken dimensionless — as it is in two dimensions. Compared to the infrared regime it looks ‘as if’ Newton’s constant changed its effective dimensionality from d − 2 to zero, i.e. at the fixed point there must be a large anomalous dimension η_{N} = 2 − d.
Formally what is special about the Einstein-Hilbert term is that the kinetic (second derivative) term itself carries a dimensionful coupling. To avoid the above conclusion one might try to assign the metric a mass dimension 2 from the beginning (i.e. not just in the asymptotic regime). However this would merely shift the effect from the gravity to the matter sector, as we wish to argue now.
In addition to the dimensionful metric \({{\tilde g}_{\alpha \beta}}: = {k^2}{g_{\alpha \beta}}\), we introduce a dimensionful vielbein by \({{\tilde E}_\alpha}^m: = kE_\alpha ^m\), if \({g_{\alpha \beta}} = {E_\alpha}^m{E_\beta}^n{\eta _{mn}}\) is the dimensionless metric. With respect to a dimensionless metric \(\int {dx\sqrt g R(g)}\) has mass dimension 2 − d in d dimensions, while the mass dimensions d_{χ} of a Bose field χ and that d_{ψ} of a Fermi field ψ are set such that their kinetic terms are dimensionless, i.e. d_{χ} = (d − 2)/2 and d_{ψ} = (d − 1)/2. Upon substitution \({g_{\alpha \beta}} \mapsto {{\tilde g}_{\alpha \beta}}\) the gravity part \(\int {dx\sqrt {\tilde g} R(\tilde g)}\) becomes dimensionless, while the kinetic terms of a Bose and Fermi field pick up a mass dimension of d − 2 and d − 1, respectively. This means their wave function renormalization constants Z_{χ}(k) and Z_{ψ}(k) are now dimensionful and should be written in terms of dimensionless parameters as Z_{χ}(k) = k^{d−2}/g_{χ}(k) and Z_{ψ}(k) = k^{d−1}/g_{ψ}(k), say. For the dimensionless parameters one expects finite limit values \({\lim\nolimits _{k \rightarrow \infty}}{{\rm{g}}_\chi}(k) = {\rm{g}}_\chi ^ {\ast} > 0\) and \(\mathop {\lim}\nolimits_{k \to \infty} {{\rm{g}}_\psi}(k) = {\rm{g}}_\psi ^0 > 0\), since otherwise the corresponding (free) field would simply decouple. Defining the anomalous dimension as usual. η_{χ} = −k∂_{k} ln Z_{χ} and η_{ψ} = −k∂_{k} ln Z_{ψ}, the argument presented after Equation (1.5) can be repeated and gives that \(\eta _{\rm{\chi}}^{\ast} = 2 - d\), \(\eta _\psi ^{\ast} = 1 - d\) for the fixed point values, respectively. The original large momentum behavior 1/p^{2} for bosons and 1/p for fermions is thus modified to a 1/p^{d} behavior in the fixed point regime, in both cases.
This translates into a logarithmic short distance behavior which is universal for all (free) matter. Initially the propagators used here should be viewed as “test propagators”, in the sense that one transplants the information in the η’s derived from the gravitational functional integral into a conventional propagator on a (flat or curved) background spacetime. Since the short distance asymptotics is the same on any (flat or curved) reference spacetime, this leads to the prediction anticipated in Section 2.3: The short distance behavior of the quantum gravity average of the geodesic two-point correlator (2.52) of a scalar field should be logarithmic.
On the other hand the universality of the logarithmic short distance behavior in the matter propagators also justifies to attribute the phenomenon to a modification in the underlying random geometry, a kind of “quantum equivalence principle”.
2.4.2 Anomalous dimension at non-Gaussian fixed point
The “anomalous dimension argument” has already been sketched in the introduction. Here we present a few more details and relate it to Section 2.4.1.
The fact that a large anomalous dimension occurs at a non-Gaussian fixed point was initially observed in the context of the 2 + ϵ expansion [116, 117] and later in computations based on the effective average action [133, 131]. The above argument shows that no specific computational information enters.
Let us emphasize that in general an anomalous dimension is not related to the geometry of field propagation and in a conventional field theory one cannot sensibly define a fractal dimension by looking at the high momentum behavior of a two-point function [125]. What is special about gravity is ultimately that the propagating field itself defines distances. One aspect thereof is the universal way matter is affected, as seen in Section 2.4.1. In contrast to an anomalous dimension in conventional field theories, this “quantum equivalence principle” allows one to attribute a geometric significance to the modified short distance behavior of the test propagators, see Section 2.4.4.
2.4.3 Strict renormalizability and 1/p^{4} propagators
With hindsight the above patterns are already implicit in earlier work on strictly renormalizable gravity theories. As emphasized repeatedly the benign renormalizability properties of higher derivative theories are mostly due to the use of 1/p^{4} type propagator (in d = 4 dimensions). As seen in Section 2.3.2 this 1/p^{4} type behavior goes hand in hand with asymptotically safe couplings. Specifically for the dimensionless Newton’s constant g_{N} it is compatible with the existence of a nontrivial fixed point (see Equation (2.31)). This in turn enforces anomalous dimension η_{N} = −2 at the fixed point which links back to the 1/p^{4} type propagator.
Similarly in the 1/N expansion [216, 217, 203] a nontrivial fixed point goes hand in hand with a propagator whose high momentum behavior is of the form 1/(p^{4} ln p^{2}) in four dimensions, and formally 1/p^{d} in d dimensions. In position space this amounts to a ln x^{2} behavior, once again.
2.4.4 Spectral dimension and scaling of fixed point action
The scaling (2.55) of the fixed point action also allows one to estimate the behavior of the spectral dimension in the ultraviolet. This leads to a variant [157] of the argument first used in [135, 134]).
Consider the quantum gravity average 〈P_{g}(T)〉 over the trace of the heat kernel P_{g}(T) in a class of states to be specified later. Morally speaking the functional average is over compact closed d-dimensional manifolds \(({\mathcal M},g)\), and the states are such that they favor geometries which are smooth and approximately flat on large scales.
For T → 0 one has an asymptotic expansion \({P_g}(T) \sim {(4\pi T)^{- d/2}}\sum\nolimits_{n \geq 0} {{T^n}} \int {dx} \sqrt g {a_n}(x)\), where the a_{n} are the Seeley-deWitt coefficients. These are local curvature invariants, \({a_0} = 1,\;{a_1} = {1 \over 6}R(g)\), etc. The series can be rearranged so as to collect terms with a fixed power in the curvature or with a fixed number of derivatives [225, 17]. Both produces nonlocal curvature invariants. The second rearrangement is relevant when the curvatures are small but rapidly varying (so that the derivatives of the curvatures are more important then their powers). The leading derivative terms then are given by \({P_g}(T) \sim {(4\pi T)^{- d/2}}[V(g) + T\int {dx\sqrt {g{a_1}}} + {T^2}{N_2}(T) + \ldots ]\), where N_{2}(T) is a known nonlocal quadratic expression in the curvature tensors (see e.g. [225, 17] for surveys). The T → ∞ behavior is more subtle as also global information on the manifold enters. For compact manifolds a typical behavior is P_{g}(T) ∼ (4πT)^{−d/2}[1 + O(exp(−cT))], where the rate of decay c of the subleading term is governed by the smallest non-zero eigenvalue.
We assume now that the states considered are such that the T → ∞ behavior of 〈P_{g}(T)〉 is like that in flat space, i.e. 〈P_{g}(T)〈 ∼ T^{−d/2} for T → ∞. This is an indirect characterization of a class of states which favor geometries that are smooth and almost flat on large scales [157]. (A rough analogy may be the way how the short-distance Hadamard condition used for free QFTs in curved spacetime selects states with desirable stability properties.) Recall that in a functional integral formulation the information about the state can be encoded in suitable boundary terms added to the microscopic action. The effective action used in a later stage of the argument is supposed to be one which derives from a microscopic action in which suitable (though not explicitly known) boundary terms encoding the information about the state have been included.
In summary, the asymptotic safety scenario leads to the specific (theoretical) prediction that the (normally powerlike) short distance singularities of all free matter propagators are softened to logarithmic ones — normally a characteristic feature of massless Klein-Gordon fields in two dimensions. In quantum gravity averages like Equation (2.52) this leads to the expectation that they should scale like G(R) ∼ ln R, for R → 0. On the other hand this universality allows one to shuffle the effect from matter to gravity propagators. This justifies to attribute the effect to a modification in the underlying random geometry. The average of heat of the heat kernel, G(T) in Equation (2.53), then scales like T^{−d/4}. This means the spectral dimension of the random geometries probed by a certain class of “macroscopic” states equals d/2, which (notably!) equals 2 precisely in d = 4 dimensions.
- 1.
It should be two-dimensional and self-interacting, the latter because of the non-Gaussian nature of the original fixed point.
- 2.
It should not be a conformal field theory in the usual sense, as the extreme UV regime in the original theory is reached from outside the critical surface (“massive continuum limit”).
- 3.
It should have degrees of freedom which can account for the antiscreening behavior presumed to be responsible for the asymptotically safe stabilization of the UV properties.
Note that in principle the identification of such a UV field theory is a well-posed problem. Presupposing that the functional integral has been made well-defined and through suitable operator insertions data for its extreme UV properties have been obtained, for any proposed field theory with the Properties 1–3 one can test whether or not these data are reproduced.
3 Asymptotic Safety from Dimensional Reduction
The systems investigated in this section can be looked at in two ways. First as prototype field theories which have the qualitative Properties 1–3 tentatively identified at the end of the last Section 2.4.4 as characteristics which an effective field theory description of the extreme UV regime of Quantum Gravidynamics should have. Second, they can be viewed as a symmetry reduction of the gravitational functional integral whose embedding into the full theory is left open for the time being. Technically one starts off from the usual gravitational functional integral but restricts it from “4-geometries modulo diffeomorphisms” to “4-geometries constant along a 2 + 2 foliation modulo diffeomorphisms”. This means instead of the familiar 3+1 foliation of geometries one considers a foliation in terms of two-dimensional hypersurfaces Σ and performs the functional integral only over configurations that are constant as one moves along the stack of two-surfaces. The same can be done with the functional integral over matter configurations.
The truncation can be motivated in various ways. It is complementary to the Eikonal sector and describes gravity with collinear initial data in a sense explained later on. It takes into account the crucial ‘spin 2’ aspect, that is, linear and nonlinear gravitational waves are included in this sector and treated without further approximations. Asymptotic safety in this sector is arguably a necessary condition for asymptotic safety of the full theory. Finally, as already mentioned, the sector can serve as a test bed for the investigation of the renormalization structures needed once the extreme UV regime of has been reached.
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).
3.1.1 Gravity theories
3.1.2 2-Killing vector reduction
Concerning the reduction, we consider here only the case when both Killing vector fields K_{1}, K_{2} 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]).
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 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 sl_{2}, i.e. [h, e] = −2e. Together the metric (3.11) always has dim Ḡ + 2k + 2 Killing vectors.
In contrast the last sl_{2} 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.
- 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.
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.
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.
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.
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).
3.1.3 Hamiltonian formulation
For the computation of the Poisson algebra it is convenient to put ρ on-shell throughout (as its equation of motion ∂^{μ}∂_{μ}ρ = 0 is trivially solved) and to interpret the improvement terms in Equation (3.15) (here, those linear in the canonical variables) such that second time derivatives are eliminated. As expected, the \({{\mathcal H}_1}\) generates infinitesimal spatial reparameterizations and the covariance of the fields is a merely kinematical property. Explicitly a spatial density d(x) of weight s transforms as \(\left\{{{{\mathcal H}_1}\left(x \right),\,d\left(y \right)} \right\} = {\partial _1}d\delta \left({x - y} \right) - sd\left(y \right)\delta{\prime}\left({x - y} \right)\), the right-hand-side being the infinitesimal version of \(d(x) \rightarrow \tilde d({\tilde x}) = [{f{\prime}({\tilde x})}]^{- s}d({f({\tilde x})})\), under \(x \rightarrow \tilde x = {f^{- 1}}\left(x \right)\). The canonical momenta π_{ρ}, π_{σ}, π_{i} are spatial densities of weight s = 1, while \({{\mathcal H}_0},\,{{\mathcal H}_1}\), and e^{σ} are densities of weight 2. The Hamiltonian constraint on the other hand resumes its usual kinematical-dynamical double role.
3.1.4 Lapse and shift in 2D gravity theories
Here we collect some useful formulas for 2D gravity theories in a lapse/shift parameterization of the metric, taken from [156]. As a byproduct we obtain a closed expression for the current K_{μ} of the Euler density \(\sqrt \gamma {R^{\left(2 \right)}}\left(\gamma \right) = - {\partial _\mu}{K_\mu}\) expressed in terms of the metric only. See [62] for a discussion. Our curvature conventions are the ones used throughout, the metric γ_{μν} has eigenvalues (−, +).
This provides an explicit though noncovariant expression for the current K^{μ} in terms of the metric. Related formulas either invoke the zweibein or use an explicit parameterization. The one given in [62] is based on an SL(2, ℝ) type parameterization of \({\gamma _{\mu \nu}}/\sqrt {- \gamma}\) and is equivalent to Equation (3.22). Compared to Equation (2.11) in [62] a curl term ϵ^{μν}∂_{ν}ϕ has been added which allows one to express K^{μ} solely in terms of the metric. Another advantage of Equation (3.22) is that the separation in dynamical and nondynamical variables is manifest: K^{μ} is a function of det γ and the combination \(\sqrt {- \gamma} {\gamma ^{\mu \nu}}\) only; the former is the dynamical variable, the latter can be parameterized in terms of the lapse and shift functions. They can be anticipated to be nondynamical in that no time derivatives of lapse and shift appear in K^{μ}, as is manifest from Equation (3.21).
3.1.5 Symmetries and currents
It may be worthwhile to point out what is trivial and what is nontrivial about the relations (3.29). Once the expression for the current \({{\mathcal J}_\mu}\) is known it is trivial to verify its conservation using the definition of the potentials χ and \(\tilde \rho\). Since \({{\mathcal H}_0}\) generates time translations on the basis fields ρ, σ, φ^{i} the associated conserved charge Poisson commutes with \({{\mathcal H}_0}\) (and trivially with \({{\mathcal H}_1}\)) and thus qualifies as a genuine Dirac observable. What is nontrivial about Equation (3.29) is that a Dirac observable can be constructed explicitly in a way that does not require a solution of the Cauchy problem. The potentials χ, \(\tilde \rho\) are only defined on-shell but one does not need to know how they are parameterized by initial data. In stark contrast the known abstract construction principles for Dirac observables in full general relativity always refer to a solution of the Cauchy problem (see [70] for a recent account). The bonus feature of the 2-Killing vector reduction that allows for this feat is the existence of a solution generating group [90, 91] (“Geroch group”) and, related to it, the existence of a Lax pair. The latter allows one to convert the Cauchy problem into a linear singular integral equation [4, 103] (which is still nontrivial to solve) and at the same time it underlies the techniques used to find an infinite set of nonlocal conserved currents of which the one in Equation (3.29) is the lowest (least nonlinear) one.
In the quantum theory, a construction of observables from first principles has not yet been achieved. Existence of a quantum counterpart of the first charge (3.29) would already be a very nontrivial indication for the quantum integrability of the systems. For its construction the procedure of Lüscher [139] could be adopted. Independent of this, the ‘exact’ bootstrap formulation of [158] shows that the existence of a ‘complete’ set of quantum obeservables is compatible with the quantum integrability of the system.
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.
3.2.1 Collinear gravitons
For the special case of the Einstein-Rosen waves an on-shell formulation suffices and a related study linking the ‘graviton modes’ to the ‘Einstein-Rosen modes’ can be found in [23].
In summary we conclude that the 2-Killing vector subsector comprises the gravitational self-interaction of collinear gravitons, in the same sense as the full Einstein-Hilbert action describes the self-interaction of non-collinear gravitons.
3.2.2 Dirac versus covariant quantization
The choice (3.47) is adapted to a covariant quantization. Here \({\hat \gamma _{\mu \nu}}{e^{\bar \sigma}}\) is a generic (off-shell) background metric with again the conformal mode \(\bar \sigma\) split off. The fluctuation field \(f_{\mu \nu}^T\) is trace-free with respect to the background \({\bar \gamma ^{\mu \nu}}f_{\mu \nu}^T = 0\). Then Equation (3.45) describes a unimodular gravity theory \(\left({\det \,{e^{{f^T}}} = 1} \right)\) and the original metric is parameterized by \(f_{\mu \nu}^T\) and \({f_\sigma}: = \sigma - \bar \sigma\) as \({\gamma _{\mu \nu}} = {\bar \gamma _{\mu \rho}}{\left({{e^{{f^T}}}} \right)^\rho}_\nu {e^{{f_\sigma}}}\). In a covariant formulation the degrees of freedom in f^{T} would be promoted to propagating ones by adding a gauge fixing term to the action (3.45). The associated Faddeev-Popov determinant is designed to cancel out their effect again. In the case at hand this is clearly roundabout as the gauge-frozen Lagrangian (3.45) is already nondegenerate.
This setting can be promoted to a generalization of the one presented in Section 3.1.2 to generic backgrounds. In the terminology of Section B.2 one then gets a non-geodesic background-fluctuation split, which treats the nonpropagating lapse and shift degrees of freedom on an equal footing with the others. In order to contrast it with the geodesic background-fluctuation split for the propagating modes used later on, we spell out here the first few steps of such a procedure.
In all cases one sees that the procedure outlined has two drawbacks. First, the split (3.49) ignores the special status of the lapse and shift degrees of freedom in γ_{μν}; all components are expanded. We know, however, that there must be two infinite series built from the components of \({\bar \gamma _{\mu \nu}}\) and f_{μν} that enter the left-hand-side of Equation (3.53) anyhow linearly. Concerning the lower 2 × 2 block in Equation (3.48) both the linear and the York-type decomposition will only keep the ISO(2) symmetry more or less manifest. The nonlinear realization of the SL(2, ℝ) symmetry then has to be restored through Ward identites, iteratively in a perturbative formulation or otherwise in a nonperturbative one.
In the following we shall adopt the following remedies. The fact that the lapse and shift degrees of freedom in γ_{μν} enter the left-hand-side of Equation (3.53) linearly of course just means that they are the Lagrange multipliers of the constraints in a Hamiltonian formulation. The linearity can thus be exploited either by a gauge fixing with respect to these variables before expanding, giving rise to a proper time formulation, or by directly adopting a Dirac quantization prodecure. By and large both should be eqaivalent; in [154, 155] a direct Dirac quantization was used, and we shall describe the results in the next two Sections 3.3 and 3.4. With the lapse and shift in Equation (3.14) ‘gone’ one only needs to perform a background-fluctuation split only for the remaining propagating fields Δ, ψ, ρ, σ.
To cope with the second of the before-mentioned drawbacks we equip — following deWitt and Vilkovisky — this space of propagating fields with a pseudo-Riemannian metric and perform a normal-coordinate expansion around a (‘background’) point with respect to it. This leads to the formalism summarized in Section B.2.2. The pseudo-Riemannian metric on the space of propagating fields can be read off from Equation (3.16) and converts the gauge-frozen but nondegenerate Lagrangian into that of a (pseudo-)Riemannian nonlinear-sigma model. The renormalization theory of these systems is well understood and we summarize the aspects needed here in Section B.3. In exchange for the gauge-freezing one then has to define quantum counterparts of the constraints (3.15) as renormalized composite operators. This will be done in Equations (3.100) ff.
- 1.
Lapse and shift viewed as infinite series in the fluctuation field are not expanded. Only the metric degrees of freedom other than lapse and shift are expanded.
- 2.Through the use of the background effective action formalism the expectation of the quantum metric 〈g_{αβ}〉 and the background \({\bar g_{\alpha \beta}}\) are related by the conditionOne does not expand around a solution of the classical field equations.$${{\delta {\Gamma _{\rm{B}}}[\langle {g_{\alpha \beta}}\rangle;{{\bar g}_{\alpha \beta}}]} \over {\delta \langle {g_{\alpha \beta}}\rangle}} = 0.$$(3.55)
- 3.
Through the use of a geodesic background-fluctuation split on the space of propagating fields the resulting background effective action is in principle invariant under arbitary local reparameterizations of the propagating fields. Among those the ones associated with isometries or conformal isometries on field space are of special interest and give rise to Ward identities associated with the Noether currents (3.26) and the conformal currents (3.27). The latter are built into the formalism, and do not have to be imposed order by order.
3.2.3 Conformal factor
Before turning to the quantum theory of these warped product sigma-models we briefy discuss the status of the conformal factor instability in a covariant formulation of the 2 + 2 truncation. As emphazised by Mazur and Mottola [142] in linearized Euclidean quantum Einstein Gravity (based on the Euclidean version of the action (3.30)) there is really no conformal factor instability. The f ∂^{2}f kinetic term in the second part of Equation (3.30) with the wrong sign receives an extra contribution from the measure which after switching to gauge invariant variables renders both the Gaussian functional integral over the conformal factor and that for the physical degrees of freedom well-defined. They also gave a structural argument why this should be so even on a nonlinear level: As one can see from a canonical formulation the conformal factor in Einstein gravity is really a constrained degree of freedom and should not have a canonically conjugate momentum.
In the 2 + 2 truncation we shall use a Lorentzian functional integral defined through the sigma-model perturbation theory outlined above. So a conformal factor instability proper associated with a Euclidean functional integral anyhow does not arise. Nevertheless it is instructive to trace the fate of the incriminated f ∂^{2}f term.
From the York-type decomposition (3.49, 3.50) one sees that \({f_\sigma} + {f_\rho} = {1 \over 2}\left({{f^\mu}_\mu + {f^a}_a} \right) = :{1 \over 2}f\) plays the role of the (gauge-variant) conformal factor. The wrong sign kinetic term is indeed still present in the second part of Equation (3.34) and f also appears through a dilaton type coupling in the \(f_{\mu \nu}^T{\partial _\mu}{\partial _\nu}\left({k - f} \right)\) term. In 2D however \(f_{\mu \nu}^T\) has no propagating degrees of freedom and the term could be taken care of promoting \(f_{\mu \nu}^T\) to a dynamical degree of freedom via gauge fixing and then cancelling the effect by a Faddeev-Popov determinant. As already argued before it is better to avoid this and look at the remaining propagating degrees of freedom directly. They simplify when reexpressed in terms of f_{ρ} = (f + k)/4 and f_{σ} = (f − k)/4, viz. \({1 \over 4}\partial k\partial f + {1 \over 8}{\left({\partial k} \right)^2} - {3 \over 8}{\left({\partial f} \right)^2} = - 4\partial {f_\sigma}\partial {f_\rho} - 2{\left({\partial {f_\rho}} \right)^2}\). This occurs here on the linearized level but comparing with Equation (3.16) one sees that the same structure is present in the full gauge-frozen action. We thus consider from now on directly the corresponding terms proportional to −∂ρ∂(σ + ½ ln ρ). By a local redefinition of σ one can eliminate the term quadratic in ρ and in dimensional regularization used later no Jacobian arises. One is left with a −∂ρ∂σ term which upon diagonalization gives rise to one field whose kinetic term has the wrong sign. However ρ is a dilaton type field which multiplies all of the self-interacting positive energy scalars in the first term of Equation (3.16), and the dynamics of this mode turns out to be very special (see Section 3.3). Heuristically this can be seen by viewing the σ field in the Lorentzian functional integral simply as a Lagrange multiplier for a δ(∂^{2}ρ) insertion. The remaining Lorentzian functional integral would allow for a conventional Wick rotation with a manifestly bounded Euclidean action. We expect that roughly along these lines a non-perturbative definition of the functional integral for Equation (3.16) could be given, which would clearly be one without any conformal factor instability. Within the perturbative construction used in Section 3.3 the special status of the ∂^{2}ρ field, viewed as a renormalized operator, can be verified. Since the system is renormalizable only with infinitely many couplings, the functional dependence on ρ in the renormalized Lagrangian and in the ∂^{2}ρ field has to be ‘deformed’ in a systematic way; however this does not affect the principle aspect that no instability occurs.
Finally, let us briefly comment on the role of Newton’s constant and of the cosmological constant in the 2 + 2 truncations. The gravity part of the action (3.10) or (3.45) arises from evaluating the Einstein-Hilbert action S_{eh} on the class of metrics (3.6). The constant 1/λ in Equation (3.10, 3.45) can be identified with d^{2}y/g_{N}, i.e. with Newton’s constant per unit volume of the orbits. As such λ is an inessential parameter and its running is defined only relative to a reference operator. For the 2 + 2 truncations it turns out that the way how the action (3.10) depends on ρ has to be modified in a nontrivial and scale dependent way by a function h(·) (see Equation (3.56) below) in order to achieve strict cut-off independence. This modification amounts to the inclusion of infinitely many essential couplings, only the overall scale of h(·) remains an inessential parameter. It is thus convenient not to renormalize this overall scale and to treat λ in Equation (3.56) as a loop counting parameter.
A similar remark applies to the cosmological constant. Adding a cosmological constant term to the Ricci scalar term results in a Λ ρe^{σ} type addition to Equation (3.56) below. In the quantum theory one is again forced to replace ρ with an scale dependent function f(ρ) in order to achieve strict cutoff independence [156]. The cosmological constant proper can be identified with the overall scale of the function f(·). The function f is subject to a non-autonomous flow equation, triggered by h, but if its initial value is set to zero it remains zero in the course of the flow [156]. To simplify the exposition we thus set f ≡ 0 from the beginning and omit the cosmological constant term in the following. It is however a nontrivial statement that this can be be done in a way compatible with the renormalization flow.
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.
- 1.
First, the scalar fields (Δ, ψ, ρ, σ) in Equation (3.56) parameterize a 4D spacetime metric with 2 Killing vectors (not the position of a string in target space) while the target space metric ɧ here (see Equation (3.60) below) has 4 Killing vectors. It is auxiliary and not interpreted as a physical spacetime metric. From the viewpoint of “strings in curved spacetime” the system (3.56) (without matter), on the other hand, describes strings moving on a spacetime with 4 Killing vectors and signature (+, +, +, −).
- 2.
The aim in the renormalization process here is to preserve the conformal geometry in target space, not conformal invariance on the worldsheet (base space) Σ. To achieve this one needs metric dependent diffeomorphisms in target space which, as mentioned before, neither need to be nor have been considered before in the context of Riemannian sigma-models.
- 3.
As a consequence of Difference 2 the renormalized fields ρ and σ become scale dependent and their renormalization flow backreacts on the coupling flow (see Equations (3.75, 3.84) below). This aspect is absent if one naively specializes the renormalization theory of a generic Riemannian sigma-model to a target space geometry which is a warped product (see [221]).
- 4.
As will become clear later in the class of warped product sigma-models considered here the Weyl anomaly is overdetermined at the fixed point of the coupling flow. In contrast to a generic Riemannian sigma-model one is therefore not free to adjust the renormalized target space metric ɧ_{ij}(ϕ) such that the Weyl anomaly vanishes and the system is a conformally invariant 2D field theory.
- 5.
The renormalization flow in Riemannian sigma-models is of the form \(\mu {d \over {d\mu}}\mathfrak{h}_{ij}=\beta_{ij}\left(\mathfrak{h}\right)\), where ɧ is the renormalized generating coupling functional (“target space metric”) with the renormalized quantum fields ϕ inserted. Conceptually the highly nonlinear but local β_{ij}(ɧ) on the right-hand-side thus is a (very special) composite operator, whose finiteness is guaranteed by the construction (see Section B.3). The fact that this very special composite operator is finite does of course not entail that any other nonlinear composite operator built from ϕ or ɧ_{ij}(ϕ) is finite (without introducing additional counterterms). For example ϕ^{2} or a curvature combination of ɧ_{ij}(ϕ) not occuring in β_{ij}(ɧ) is simply not defined off-hand. This is true no matter how ϕ ↦ ɧ_{ij}(ϕ) is chosen, so the folklore that one can restrict attention to functionals ɧ for which the trace or Weyl anomaly vanishes and get a “finite” QFT is incorrect (see [201] for a discussion). Moreover the Weyl anomaly is itself a (very special) composite operator and the condition for its vanishing is not equivalent to a partial differential equation of the same form for any classical metric. By expanding the quantum fields ϕ around a classical background configuration one can convert the condition for a vanishing Weyl anomaly into a condition formulated in terms of a classical metric [220]. However beyond lowest order (that is, beyond the Ricci term) nonlocal terms are generated, and the resulting cumbersome equations are rarely used. As a consequence beyond leading order (beyond Ricci flatness modulo an improvement term) most of the “consistent string backgrounds” (defined by ad-hoc replacing the composite operator ɧ(ϕ) by a classical metric in the formula for the Weyl anomaly as a composite operator) are actually not consistent, in the sense that the corresponding metric re-interpreted (ad-hoc) as one with the quantum fields re-inserted does not guarantee the vanishing of the Weyl anomaly in its operator form.
- 6.
Even the Ricci flow equations arising at lowest order have the property that for a generic smooth target space metric the flow is often singular towards the ultraviolet [52]. For generic target spaces the Riemannian sigma-models are therefore unlikely to give rise to genuine (not merely effective) quantum field theories.
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−2}F(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.
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):
We can put this result into the context the general discussion in Section 2 and arrive at the following conclusion:
3.5 Conclusion
With respect to the non-Gaussian fixed point h^{beta}(·) all couplings in the generating functional h(·) are asymptotically safe. All symmetry reduced gravity systems satisfy the Criteria (PTC1) and (PTC2) to all loop orders of sigma-model perturbation theory. As explained in Section 3.2 from the viewpoint of the graviton loop expansion the distinction between a perturbative and a non-perturbative treatment is blurred here.
It is instructive to compare these properties to that of the Gaussian fixed point. The Gaussian fixed point of the flow (3.83) is best understood in analogy to the Gaussian fixed-point of a conventional nonlinear sigma-model. For a G/H nonlinear sigma-model with Lagrangian \(L = - {1 \over {2{\rm{g}_0}}}{\mathfrak{m}_{ij}}\left(\varphi \right){\partial ^\mu}{\varphi ^i}{\partial _\mu}{\varphi ^j}\) (with m satisfying Equation (3.12)) the beta function \({\beta _{G/H}}\left({{\rm{g}_0}} \right) = {\rm{g}_0}\sum\nolimits_{l \leq 1} {l{\zeta _l}} {\left({{{{\rm{g}_0}} \over {2\pi}}} \right)^l}\) has only the trivial zero \(\rm{g}_0^{\ast} = 0\). As g_{0} → 0 the renormalized Lagrangian blows up, but in an expansion around m_{ij}(φ) = δ_{ij} one can see that for g_{0} → 0 the interaction terms vanish. In this sense the fixed point \(\rm{g}_0^{\ast} = 0\) is Gaussian. This holds irrespective of the sign of ζ_{1}, which however determines the stability properties of the flow. The stability ‘matrix’ vanishes so that the linearized stability analysis is empty. By direct inspection of the differential equation one sees that the unstable manifold of \(\rm{g}_0^{\ast}\) is one-dimensional for ζ_{1} > 0 (typical for G/H compact) and empty for ζ_{1} < 0 (typical for G/H noncompact). Indeed, \(- \mu {d \over {d\mu}}{{\rm{\bar g}}_0} = {{{\zeta _1}} \over {2\pi}}{\rm{\bar g}}_0^2 + O\left({{\rm{\bar g}}_0^3} \right)\), and if one insists on ḡ_{0} ≥ 0 for positivity-of-energy reasons, the flow will be attracted to \(\rm{g}_0^{\ast} = 0\) for μ → ∞ iff ζ_{1} > 0. In particular for ζ_{1} < 0 these models are, based on the Criterion (PTC2) of Section 2, not expected to have a genuine continuum limit.
Comparison: noncompact G/H sigma-model vs. dimensionally reduced gravity theory with G/H coset.
G/H sigma-model | dimensionally reduced gravity with G/H |
---|---|
renormalizable | non-renormalizable |
one essential coupling | ∞ essential couplings |
g_{0} | function h(·) |
flow is formally infrared free | flow is asymptotically safe |
formal trivial fixed point | non-trivial fixed point |
g_{0} = 0 | h^{beta}(·) |
formally IR stable | UV stable |
trace anomalous | trace anomaly vanishes |
The comparison highlights why the above conclusion is surprising and significant. While the noncompact G/H sigma-models are renormalizable with just one relevant coupling (denoted by g in the table), at least in the known constructions they do not have a fixed point at which they are conformally invariant. Their gravitational counterparts require infinitely many relevant couplings for their UV renormalization. This infinite coupling flow has a nontrivial UV fixed point at which the theory is conformally invariant. Most importantly the stability properties of the renormalization flow are reversed (compared to the flow of g) for all of the infinitely many relevant couplings. As there appears to be no structural reason for this surprising reversal in the reduced theory itself, we regard it as strong evidence for the existence of an UV stable fixed point for the full renormalization group dynamics.
- 1.
The systems inherit the lack of standard perturbative renormalizability from the full theory. A cut-off independent quantum theory can be achieved at the expense of introducing infinitely many couplings combined into a generating function h(·) of one variable.
- 2.
The argument of this function is the ‘area radius’ field ρ associated with the two Killing vectors. The field ρ is (nonlinearly) renormalized but no extra renormalizations are needed to define arbitrary powers thereof.
- 3.
A universal formula for the beta functional for h and hence for the infinitely many couplings contained in it can be given. The flow possesses a Gaussian as well as a non-Gaussian fixed point. With respect to the non-Gaussian fixed point all couplings in h are asymptotically safe.
- 4.
At the fixed point the trace anomaly vanishes and the quantum constraints (well-defined as composite operators) \([\![{{\mathcal H}_0}]\!],\,[\![{{\mathcal H}_1}]\!]\) can in principle be imposed. The linear combinations \([\![{{\mathcal H}_0} \pm {{\mathcal H}_1}]\!]\) are expected to generate commuting copies of a centrally extended conformal algebra acting on an indefinite metric Hilbert space.
- 5.
Despite the conformal invariance at the fixed point there is a scale dependent local parameter, whose scale dependence is governed by the beta function of the G/H sigma-model without coupling to gravity.
So far we considered the renormalization of the symmetry reduced theories in its own right, leaving the embedding into the full Quantum Gravidynamics open. The proposed relation to qualitative aspects of the Quantum Gravidynamics in the extreme UV has already been mentioned. Here we offer some tentative remarks on the embedding otherwise. The constructions presented in this section can be extended to 2 + ϵ dimensions in the spirit of an ϵ-expansion. At the same time this mimics quantum aspects of the 1-Killing vector reduction. One finds that the qualitative features of the renormalization flow — non-Gaussian fixed point and asymptotic safety — are still present [156]. A cosmological ‘constant’ term can likewise be included and displays a similar pattern as outlined at the end of Section 3.2. The advantage of this setting is that the UV cutoff can strictly be removed, which is hard to achieve with a nonperturbative technique. The extension of these results from a quasi-perturbative analysis to a nonperturbative one, ideally via controlled approximations, is an important open problem. The same holds for the analysis of the 1-Killing vector reduction, which holds the potential for cosmological applications. These truncations can be viewed as complementary to the ‘hierarchical’ truncations used in Section 4: A manifest truncation is initially imposed on the functional integral, but the infinite coupling renormalization flow can then be studied in great detail, often without further approximations.
4 Asymptotic Safety from the Effective Average Action
As surveyed in the introduction, important evidence for the asymptotic safety scenario comes from the truncated flow of the effective average action, as computed from its functional renormalization flow equation (FRGE). In this section we use the term Quantum Einstein Gravity to refer to a version of Quantum Gravidynamics where the metric is used as the dynamical variable. The key results have been outlined in the introduction. Here we present in more detail the effective average action for gravity, its flow equation, and the results obtained from its truncations.
4.1 The effective average action for gravity and its FRGE
The effective average action is a scale dependent variant Γ_{k} of the usual effective action Γ, modified by a mode-cutoff k, such that Γ_{k} can be interpreted as describing an ‘effective field theory at scale k’. For non-gauge theories a self-contained summary of this formalism can be found in Appendix C. In the application to gauge theories and gravity two conceptual problems occur.
First the standard effective action is not a gauge invariant functional of its argument. For example if in a Yang-Mills theory one gauge-fixes the functional integral with an ordinary gauge fixing condition like \({\partial ^\mu}A_\mu ^a = 0\), couples the Yang-Mills field \(A_\mu ^a\) to a source, and constructs the ordinary effective action, the resulting functional \(\Gamma \left[ {A_\mu ^a} \right]\) is not invariant under the gauge transformations of \({A_\mu ^a}\). Although physical quantities extracted from \(\Gamma \left[ {A_\mu ^a} \right]\) are expected to be gauge invariant, the noninvariance is cumbersome for renormalization purposes. The second problem is related to the fact that in a gauge theory a “coarse graining” based on a naive Fourier decomposition of \(A_\mu ^a\left(x \right)\) is not gauge covariant and hence not physical. In fact, if one were to gauge transform a slowly varying \(A_\mu ^a\left(x \right)\) with a parameter function ω(x) with a fast x-variation, a gauge field with a fast x-variation would arise, which however still describes the same physics.
Both problems can be overcome by using the background field formalism. The background effective action generally is a gauge invariant functional of its argument (see Appendix B). The second problem is overcome by using the spectrum of a covariant differential operator built from the background field configuration to discrimate between slow modes (small eigenvalues) and fast modes (large eigenvalues) [187]. This sacrifices to some extent the intuition of a spatial coarse graining, but it produces a gauge invariant separation of modes. Applied to a non-gauge theory it amounts to expanding the field in terms of eigenfunctions of the (positive) operator −∂^{2} and declaring its eigenmodes ‘long’ or ‘short’ wavelength depending on whether the corresponding p^{2} is smaller or larger than a given k^{2}.
This is the strategy adopted to define the effective average action for gravity [179]. In short: The effective average action for gravity is a variant of the background effective action \(\Gamma \left[ {\left\langle {{f_{\alpha \beta}}} \right\rangle ,\,{\sigma ^\alpha},\,{{\bar \sigma}_\alpha};\,{{\bar g}_{\alpha \beta}}} \right]\) described in Appendix B (see Equations (B.48, B.51)), where the bare action is modified by mode cutoff terms as in Appendix C, but with the mode cutoff defined via the spectrum of a covariant differential operator built from the background metric. For convenience we quickly recapitulate the main features of the background field technique here and then describe the modifications needed for the mode cutoff.
In the next step the initial bare action should be replaced by one involving a mode cutoff term. In the background field technique the mode cutoff should be done in a way that preserves the invariance under the background gauge transformations (4.2). We now first present the steps leading to the scale dependent effective average action \({\Gamma _k}\left[ {\left\langle {{g_{\alpha \beta}}} \right\rangle - {{\bar g}_{\alpha \beta}},\,{\sigma ^\alpha},\,{{\bar \sigma}_\alpha};\,{{\bar g}_{\alpha \beta}}} \right]\) in some detail and then present the FRGE for it. The functional integrals occuring are largely formal; for definiteness we consider the Euclidean variant where the integral over Riemannian geometries is intended. The precise definition of the generating functionals is not essential here, as they mainly serve to arrive at the gravitional FRGE. The latter provides a novel tool for investigating the gravitational renormalization flow.
This concludes the definition of the effective average action and its various specializations. We now present its key properties.
4.1.1 Properties of the effective average action
- 1.The effective average action is invariant under background field diffeomorphismswhere all its arguments transform as tensors of the corresponding rank. This is a direct consequence of the corresponding property of W_{k} in Equation (4.3)$${\Gamma _k}[\Phi + {{\mathcal L}_\nu}\Phi ] = {\Gamma _k}[\Phi ],\quad \quad \Phi \,: = \left\{{\left\langle {{g_{\alpha \beta}}} \right\rangle ,{{\overline g}_{\alpha \beta}},{\sigma ^\alpha},{{\overline \sigma}_\alpha}} \right\},$$(4.16)where \({{\mathcal L}_v}\) is the Lie derivative of the respective tensor type. Here Equation (4.16) is obtained as in Equation (B.47) of Appendix B, where the background covariance of the mode cutoff terms (4.13) is essential. Further the measure \({\mathcal D}{f_{\alpha \beta}}\) is assumed to be diffeomorphism invariant.$${W_k}[{\mathcal J} + {{\mathcal L}_\nu}{\mathcal J}] = {W_k}[{\mathcal J}],\quad \quad {\mathcal J}\,: = \left\{{{J^{\alpha \beta}},{j^\alpha},{{\overline j}_\alpha},{{\bar g}_{\alpha \beta}}} \right\},$$(4.17)
- 2.It satisfies the functional integro-differential equationwhere S_{tot} ≔ S + S_{gf} + S_{gh} (with S_{gf} and S_{gh} minus the first two terms in the exponent of Equation (4.4) and \({C_k}: = C_k^{{\rm{grav}}} + C_k^{{\rm{gh}}}\).$$\begin{array}{*{20}c} {\exp \{- {\Gamma _k}[\bar f,\sigma ,\bar \sigma ;\bar g]\} = \int {{\mathcal D}f\,{\mathcal D}C\,{\mathcal D}\bar C\,\exp \left\{{- {S_{{\rm{tot}}}}[f,C,\bar C;\bar g]} \right. - {C_k}[f - \bar f,C - \sigma ,\bar C - \bar \sigma ]} \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {\left. {+ \int {dx} \sqrt {\bar g} \left[ {({f_{\alpha \beta}} - {{\bar f}_{\alpha \beta}}){{\delta {\Gamma _k}} \over {\delta {{\bar f}_{\alpha \beta}}}} + ({C^\alpha} - {\sigma ^\alpha}){{\delta {\Gamma _k}} \over {\delta {\sigma ^\alpha}}} + ({C_\alpha} - {{\bar \sigma}_\alpha}){{\delta {\Gamma _k}} \over {\delta {{\bar \sigma}_\alpha}}}} \right]} \right\},} \\ \end{array}$$(4.18)
- 3.The k-dependence of the effective average action is governed by an exact FRGE. Following the same lines as in the scalar case one arrives at [179]Here \(\Gamma _k^{\left(2 \right)}\) denotes the Hessian of Γ_{k} with respect to the dynamical fields \(\bar f,\,\sigma ,\,\bar \sigma\) at fixed ḡ. It is a block matrix labeled by the fields \({\varphi _i}: = \left\{{{{\bar f}_{\alpha \beta}},\,{\sigma ^\alpha},\,{{\bar \sigma}_\alpha}} \right\}\),$$\begin{array}{*{20}c} {k{\partial _k}{\Gamma _k}[\bar f,\sigma ,\bar \sigma ;\bar g] = {1 \over 2}{\rm{Tr}}\left[ {\left({\Gamma _k^{(2)} + {{\hat{\mathcal R}}_k}} \right)_{\bar f\,\bar f}^{- 1}{{\left({k{\partial _k}{{\hat{\mathcal R}}_k}} \right)}_{\bar f\,\bar f}}} \right]\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad} \\ {- {1 \over 2}{\rm{Tr}}\left[ {\left\{{\left({\Gamma _k^{(2)} + {{\hat{\mathcal R}}_k}} \right)_{\bar \sigma \sigma}^{- 1} - \left({\Gamma _k^{(2)} + {{\hat{\mathcal R}}_k}} \right)_{\sigma \bar \sigma}^{- 1}{{\left({k{\partial _k}{{\hat{\mathcal R}}_k}} \right)}_{\bar \sigma \sigma}}} \right\}} \right].} \\ \end{array}$$(4.19)(In the ghost sector the derivatives are understood as left derivatives.) Likewise, \({\hat {\mathcal R}_k}\) is a block matrix with entries \(\left({{{\hat {\mathcal R}}_k}} \right)_{\bar f\bar f}^{\alpha \beta \gamma \delta}: = {\kappa ^2}{\mathcal R}_k^{{\rm{grav}}}{\left[ {\bar g} \right]^{\alpha \beta \gamma \delta}}\) and \({\hat {\mathcal R}_{\bar \sigma \sigma}} = \sqrt 2 \hat {\mathcal R}_k^{{\rm{gh}}}\left[ {\bar g} \right]\). Performing the trace in the position representation it includes an integration \(\int {dx\sqrt {\bar g\left(x \right)}}\) involving the background volume element. For any cutoff which is qualitatively similar to Equation (4.14, 4.15) the traces on the right-hand-side of Equation (4.19) are well convergent, both in the infrared and in the ultraviolet. By virtue of the factor \(k{\partial _k}{{\hat {\mathcal R}}_k}\), the dominant contributions come from a narrow band of generalized momenta centered around k. Large momenta are exponentially suppressed.$$\tilde \Gamma _k^{(2)ij}(x,y)\,: = {1 \over {\sqrt {\overline g (x)\overline g (y)}}}{{{\delta ^2}{{\tilde \Gamma}_k}} \over {\delta {\varphi _i}(x)\,\delta {\varphi _j}(y)}}.$$(4.20)
The conceptual status and the use of the gravitational FRGE (4.20) is the same in the scalar case discussed in Section 2.2. Its perturbative expansion should reproduce the traditional non-renormalizable cutoff dependencies starting from two-loops. In the context of the asymptotic safety scenario the hypothesis at stake is that in an exact treatment of the equation the cutoff dependencies entering through the initial data get reshuffled in a way compatible with asymptotic safety. The Criterion (FRGC1) for the existence of a genuine continuuum limit discussed in Section 2.3 also applies to Equation (4.20). In brief, provided a global solution of the FRGE (4.20) can be found (one which exists for all 0 ≤ k ≤ ∞), it can reasonably be identified with a renormalized effective average action lim_{Λ→∞} Γ_{Λ,k} constructed by other means. The intricacies of the renormalization process have been shifted to the search for fine-tuned initial functionals for which a global solution of Equation (4.20) exists. For such a global solution lim_{k→0} Γ_{k} then is the full quantum effective action and lim_{k→∞} Γ_{k} = S_{*} is the fixed point action. As already noted in Section 2.3 the appropriate positivity requirement (FRGC2) remains to be formulated; one aspect of it concerns the choice of \({{\mathcal Z}_k}\) factors in Equation (4.13) and will be discussed below.
The background gauge invariance of Γ_{k} expressed in Equation (4.16) is of great practical importance. It ensures that if the initial functional does not contain non-invariant terms, the flow implied by the above FRGE will not generate such terms. In contrast locality is not preserved of course; even if the initial functional is local the flow generates all sorts of terms, both local and nonlocal, compatible with the symmetries.
For the derivation of the flow equation it is important that the cutoff functionals in Equation (4.13) are quadratic in the fluctuation fields; only then a flow equation arises which contains only second functional derivatives of Γ_{k}, but no higher ones. For example using a cutoff operator involving the Laplace operator of the full metric g_{αβ} = ḡ_{αβ} + f_{αβ} would result in prohibitively complicated flow equations which could hardly be used for practical computations.
For most purposes the reduced effective average action (4.9) is suffient and it is likewise background invariant, \({{\bar \Gamma}_k}\left[ {g + {{\mathcal L}_v}g} \right] = {{\bar \Gamma}_k}\left[ g \right]\). Unfortunately \({{\bar \Gamma}_k}\left[ {{{\bar g}_{\alpha \beta}}} \right]\) does not satisfy an exact FRGE, basically because it contains too little information. The actual RG evolution has to be performed at the level of the functional \({\Gamma _k}\left[ {\left\langle g \right\rangle ,\,\bar g,\,\sigma ,\,\bar \sigma} \right]\). Only after the evolution one may set \(\left\langle g \right\rangle = \bar g,\,\sigma = 0,\,\bar \sigma = 0\). As a result, the actual theory space of Quantum Einstein Gravity in this setting consists of functionals of all four variables, \(\left\langle {{g_{\alpha \beta}}} \right\rangle ,\,{{\bar g}_{\alpha \beta}},\,{\sigma ^\alpha},\,{{\bar \sigma}_\alpha}\), subject to the invariance condition (4.9). Since \(\Gamma _k^{\left(2 \right)}\) involves derivatives with respect to \({{\bar f}_{\alpha \beta}}\) at fixed ḡ_{αβ} it is clear that the evolution cannot be formulated in terms of \({{\bar \Gamma}_k}\) alone.
- 4.\({\Gamma _k} = \left[ {\bar f,\,\sigma ,\,\bar \sigma ;\,\bar g} \right]\) approaches for k → 0 the background effective action of Appendix B, since \({\mathcal R}_k^{{\rm{grav}}},\,{\mathcal R}_k^{{\rm{gh}}}\) vanish for k → 0. The k → ∞ limit can be infered from Equation (4.18) by the same reasoning as in the scalar case (see Appendix C). This givesNote that the bare initial functional Γ_{k} includes the gauge fixing and ghost actions. At the level of the functional \({\bar \Gamma _k}\left[ g \right]\) Equation (4.21) reduces to \({\lim\nolimits_{k \rightarrow \infty}}{\bar \Gamma _k}\left[ g \right] = S\left[ g \right]\).$$\underset{k \rightarrow \infty}{\lim} \,{\Gamma _k}[\overline f ,\sigma ,\overline \sigma ;\overline g ] = {S_{{\rm{tot}}}}[\overline f ,\sigma ,\overline \sigma ;\overline g ].$$(4.21)
- 5.The effective average action satisfies a functional BRST Ward identity which reflects the invariance of S_{tot} under the BRST transformationsHere ϵ is an anti-commuting parameter. Since the mode cutoff action C_{k} is not BRST invariant, the Ward identity differs from the standard one by terms involving \({\mathcal R}_k^{{\rm{grav}}},\,{\mathcal R}_k^{{\rm{gh}}}\). For the explicit form of the identity we refer to [179].$$\begin{array}{*{20}c} {{\delta _\epsilon}{f_{\alpha \beta}} = \epsilon {\kappa ^{- 2}}{{\mathcal L}_C}({{\overline g}_{\alpha \beta}} + {f_{\alpha \beta}}),\quad \quad {\delta _\epsilon}{{\overline g}_{\alpha \beta}} = 0,\quad \quad \quad \quad} \\ {{\delta _\epsilon}{C^\alpha} = \epsilon {\kappa ^{- 2}}{C^\beta}{\partial _\beta}{C^\alpha},\quad \quad \quad \quad {\delta _\epsilon}{{\overline C}_\alpha} = \epsilon {{(\alpha \kappa)}^{- 1}}{Q_\alpha}.} \\ \end{array}$$(4.22)
- 6.Initially the vertex (or 1-PI Greens) functions are given by multiple functional derivatives of \({\Gamma _k}\left[ {\bar f,\,\sigma ,\,\bar \sigma ;\,\bar g} \right]\) with respect to \(\bar f,\,\sigma ,\,\bar \sigma\) at fixed ḡ and settingafter differentiation. The resulting multi-point functions \(\Gamma _k^{\left(n \right)}\left({{x_1},\, \ldots \,,\,{x_n};\,g} \right)\) are k-dependent functionals of the (k-independent) \(\left\langle {{g_{\alpha \beta}}} \right\rangle = {{\bar g}_{\alpha \beta}}\). For extremizing sources obtained by formal series inversion the condition (4.23) automatically switches of the sources in Equation (4.7); for the ghosts this is consistent with Γ_{k} having ghost number zero. Note that the one-point function \(\Gamma _k^{\left(1 \right)}\left({{x_1};\,g} \right) \equiv 0\) vanishes identically.$${\overline f _{\alpha \beta}} = \left\langle {{f_{\alpha \beta}}} \right\rangle = 0,\quad \quad {\sigma ^\alpha} = \left\langle {{C^\alpha}} \right\rangle = 0,\quad \quad {\overline \sigma _\alpha} = \left\langle {{{\overline C}_\alpha}} \right\rangle = 0$$(4.23)An equivalent set of vertex functions should in analogy to the Yang-Mills case [68, 42, 67] be obtained by differentiating \({\Gamma _k}\left[ {\bar g,\,\sigma ,\,\bar \sigma} \right]: = {\Gamma _k}\left[ {0,\,\sigma ,\,\bar \sigma ;\,\bar g} \right] \equiv {\Gamma _k}\left[ {\bar g,\,\bar g,\,\sigma ,\,\bar \sigma} \right]\) with respect to ḡ. Specifically for \({\sigma ^\alpha} = {\bar \sigma _\alpha} = 0\) one gets multipoint functions \(\bar \Gamma _k^{\left(n \right)}\left({{x_1},\, \ldots \,,\,{x_n}} \right)\), with the shorthand (4.9). The solutions (ğ_{k})_{αβ} ofplay an important role in the interpretation of the formalism (see Section4.2).$${{\delta {{\overline \Gamma}_k}} \over {\delta {g_{\alpha \beta}}}}[{\overset \vee g_k}] = 0$$(4.24)
The precise physics significance of the multipoint functions \(\Gamma _k^{\left(n \right)}\) and \(\bar \Gamma _k^{\left(n \right)}\) remains to be understood. One would expect them to be related to S-matrix elements on a self-consistent background but, for example, an understanding of the correct infrared degrees of freedom is missing.
In gravity the situation may be more subtle. First, consider the case where ϕ is some normal mode of \({{\bar f}_{\alpha \beta}}\) and that it is an eigenfunction of \(\Gamma _k^{\left(2 \right)}\) with eigenvalue \(Z_k^\phi {p^2}\), where p^{2} is a positive eigenvalue of some covariant kinetic operator, typically of the form \(- {{\bar \nabla}^2} + R\)-terms. If \(Z_k^\phi > 0\) the situation is clear, and the rule discussed in the context of scalar theories applies: One chooses \({\mathcal Z} = Z_k^\phi\) because this guarantees that for the low momentum modes the running inverse propagator \(\Gamma _k^{\left(2 \right)} + {{\mathcal R}_k}\) becomes \(Z_k^\phi \left({{p^2} + {k^2}} \right)\), exactly as it should be.
More tricky is the question how \({{\mathcal Z}_k}\) should be chosen if \(Z_k^\phi\) is negative. If one continues to use \({{\mathcal Z}_k} = Z_k^\phi\), the evolution equation is perfectly well defined because the inverse propagator \(- \left\vert {Z_k^\phi} \right\vert\left({{p^2} + {k^2}} \right)\) never vanishes, and the traces of Equation (4.19) are not suffering from any infrared problems. In fact, if we write down the perturbative expansion for the functional trace, for instance, it is clear that all propagators are correctly cut off in the infrared, and that loop momenta smaller than k are suppressed. On the other hand, if we set \({{\mathcal Z}_k} = - Z_k^\phi\), then \(- \left\vert {Z_k^\phi} \right\vert\left({{p^2} - {k^2}} \right)\) introduces a spurios singularity at p^{2} = k^{2}, and the cutoff fails to make the theory infrared finite. This choice of \({{\mathcal Z}_k}\) is ruled out therefore. At first sight the choice \({{\mathcal Z}_k} = - Z_k^\phi\) might have appeared more natural because only if \({{\mathcal Z}_k} > 0\) the factor \(\exp \left({- {C_k}} \right) \sim \exp \left({- \int \phi {{\mathcal R}_k}\phi} \right)\) is a damped exponentially which suppresses the low momentum modes in the usual way. For the other choice \({{\mathcal Z}_k} = + Z_k^\phi < 0\) the factor \(\exp \left\{{\int {\left\vert {{{\mathcal R}_k}} \right\vert{\phi ^2}}} \right\}\) is a growing exponential instead and, at least at first sight, seems to enhance rather than suppress the infrared modes. However, as suggested by the perturbative argument, this conclusion is too naive perhaps.
At least formally the construction of the effective average action can be repeated for Lorentzian signature metrics. Then one deals with oscillating exponentials e^{iS}, and for arguments like the one leading to Equation (4.21) one has to employ the Riemann-Lebesgue lemma. Apart from the obvious substitutions, \({\Gamma _k} \rightarrow - i{\Gamma _k},\,{{\mathcal R}_k} \rightarrow - i{{\mathcal R}_k}\), the evolution equation remains unaltered. For \({{\mathcal Z}_k} = Z_k^\phi\) it has all the desired features, and \(Z_k^\phi < 0\) seems not to pose any special problem, since \(\exp \left\{{\pm i\int {\phi \left\vert {{{\mathcal R}_k}} \right\vert\phi}} \right\}\) for either sign leads to an IR suppression.
For finite k the Euclidean FRGE is perfectly well-defined even if \({{\mathcal Z}_k} < 0\), while the status of the Euclidean functional integral with its growing exponential seems problematic. In principle there exists the possibility of declaring the FRGE the primary object. If a global solution to it exists this functional might define a consistent quantum theory of gravity even though the functional integral per se does not exist. As noted in Section 3.4 the inclusion of ‘other desirable’ features might then be more difficult, though.
It might be, and there exist indications in this direction [131, 132], that for the exact RG flow \(\Gamma _k^{\left(2 \right)}\) is always a positive operator (one with positive spectrum) along physically relevant RG trajectories. Then \(Z_k^\phi > 0\) for all modes and the problem does not arise. If this contention is correct, the \(Z_k^\phi < 0\) phenomenon which is known to occur in certain truncations would be an artifact of the approximations made.
In [133, 131] a slightly more general variant of the construction described here has been employed. In order to facilitate the calculation of the functional traces in the FRGE (4.19) it is helpful to employ a transverse-traceless (TT) decomposition of the metric: \({f_{\alpha \beta}} = f_{\alpha \beta}^T + {{\bar \nabla}_\alpha}{V_\beta} + {{\bar \nabla}_\beta}{V_\alpha} + {{\bar \nabla}_\alpha}{{\bar \nabla}_{\beta \sigma}} - {d^{- 1}}{{\bar g}_{\alpha \beta}}{{\bar \nabla}^2}\sigma + {d^{- 1}}{{\bar g}_{\alpha \beta}}\phi = {{\hat f}_{\alpha \beta}} + {d^{- 1}}{{\bar g}_{\alpha \beta}}\phi\). Here \(f_{\alpha \beta}^T\) is a transverse traceless tensor, V_{α} a transverse vector, and σ and ϕ are scalars. In this framework it is natural to formulate the cutoff in terms of the component fields appearing in the TT decomposition \({C_k} \sim \int {f_{\alpha \beta}^T} {{\mathcal R}_k}{f^{{T^{\alpha \beta}}}} + \int {{V_\alpha}{{\mathcal R}_k}} {V^\alpha} + \ldots\). This cutoff is referred to as a cutoff of “type B”, in contradistinction to the “type A” cutoff described above, \({C_k} \sim \int {{f_{\alpha \beta}}} {{\mathcal R}_k}{f^{\alpha \beta}}\). Since covariant derivatives do not commute, the two cutofs are not exactly equal even if they contain the same shape function \({{\mathcal R}^{\left(0 \right)}}\). Thus, comparing type A and type B cutoffs is an additional possibility for checking scheme (in)dependence [133, 131].
4.2 Geometries at different resolution scales
In this section we elaborate on the interpretation of the effective average action formalism in a gravitational context. Specifically we argue that Γ_{k} encodes information about ‘quantum geometries’ at different resolution scales.
Recall that the effective average action Γ_{k} may be regarded as the standard effective action where the bare action has been modified by the addition of the mode cutoff term C_{k}. Every given (exact or truncated) renormalization group (RG) trajectory can be viewed as a collection of effective field theories {Γ_{k}, 0 ≤ k ≤ ∞}. In this sense a single fundamental theory gives rise to a double infinity of effective theories — one for each trajectory and one for each value of k. As explained in Appendix C, the motivation for this construction is that one would like to be able to ‘read off’ part of the physics contents of the theory simply by inspecting the effective action relevant to the problem under consideration. If the problem has only one scale k, the values of the running couplings and masses in Γ_{k} may be treated approximately as classical parameters.
We now select a state which favors geometries that are smooth and almost flat on large scales as in Section 2.3. We can think of this state as a background dependent expectation functional \({\mathcal O} \mapsto {\left\langle {\mathcal O} \right\rangle _{\bar g}}\) where the background has been self-consistently adjusted through the condition \({\langle {{g_{\alpha \beta}}} \rangle_{\bar g_{\ast}[ g]}} = {\bar g_{\alpha \beta}}\) (see Equation (2.48)). This switches off the source, and any fixed point of the map g ↦ ḡ_{*}[g] gives a particular solution to Equation (4.26) at k = 0, implicitly referring to the underlying state [156].
In terms of the effective average action \({{\bar \Gamma}_k}\) the state should implicitly determine a family of solutions {ğ_{k}, 0 ≤ k ≤ ∞}. Structures on a scale k_{1} are best described by \({\Gamma _{{k_1}}}\). In principle one could also use \({\Gamma _{{k_2}}}\) with k_{2} ≠ k_{1} to describe structures at scale k_{1} but then a further functional integration would be needed. It is natural to think of the family {ğ_{k}, 0 ≤ k ≤ ∞} as describing aspects of a “quantum spacetime”. By a “quantum spacetime” we mean a manifold equipped with infinitely many metrics; in general none of them will be a solution of the Einstein field equations. One should keep in mind however that the quantum counterpart of a classical spacetime is characterized by many more data than the metric expectation values ğ_{k} alone, in particular by all the higher functional derivatives of Γ_{k} evaluated at ğ_{k}. The second derivative for instance evaluated at ğ_{k} is the inverse graviton propagator in the background ğ_{k}. Note that all these higher multi-point functions probe aspects of the underlying quantum state.
By virtue of the effective field theory properties of Γ_{k} the interpretation of the metrics ğ_{k} is as follows. Features involving a typical scale k_{1} are best described by \({\Gamma _{{k_1}}}\). Hence \(\overset{\vee}{g_{k_{1}}}\) is the average metric detected in a (hypothetical) experiment which probes aspects of the quantum spacetimes with typical momenta k_{1}. In more figurative terms one can think of ğ_{k} as a ‘microscope’ whose variable ‘resolving power’ is given by the energy scale k.
This picture underlies the discussions in [135, 134] where the quantum spacetimes are viewed as fractal-like and the qualitative properties of the spectral dimension (2.53) have been derived. We refer to Section 2.4 and [135, 134] for detailed expositions. The fractal aspects here refer to the generalized ‘scale’ transformations k → 2k, say. Moreover a scale dependent metric ğ_{k} associates a resolution dependent proper length to any (k-independent) curve. The k-dependence of this proper length can be thought of as analogous to the well-known example that the length of the coast line of England depends on the size of the yardstick used.
Usually the resolving power of a microscope is characterized by a length scale ℓ defined as the smallest distance of two points the microscope can distinguish. In the above analogy between the effective average action and a “microscope” the resolving power is implicitly given by the mass scale k and it is not immediate how k relates to a distance. One would like to know the minimum proper distance ℓ(k_{1}) of two points which can be distinguished in a hypothetical experiment with a probe of momentum k_{1}, effectively described by the action \({\Gamma _{{k_1}}}\). Conversely, if one wants to ‘focus’ the microscope on structures of a given proper length ℓ one must know the k-value corresponding to this particular value of ℓ. For non-gravitational theories in flat Euclidean space one has ℓ(k) ≈ π/k, but in quantum gravity the relation is more complicated.
Given a family of solutions {ğ_{k}, 0 ≤ k ≤ ∞} with the above interpretation the construction of a candidate for ℓ(k) proceeds as follows [135, 134, 184]: One considers the spectral problem of the (tensor) Laplacian − Δ_{ğk} associated with ğ_{k}. To avoid technicalities inessential for the discussion we assume that all geometries in the family {ğ_{k}} are compact and closed. The spectrum of − Δ_{ğk} will then be discrete; we write \(- {\Delta _{{{\check g}_k}}}{\phi _n}\left({{{\check g}_k}} \right) = {{\mathcal E}_n}\left({{{\check g}_k}} \right){\phi _n}\left({{{\check g}_k}} \right),\,n \in \mathbb{N}\), for the spectral problem. As indicated, both the spectral values \({{\mathcal E}_n}\left(\overset{\vee}{g_{k}}\right)\) and the eigenfunctions ϕ_{n}(ğ_{k}) will now depend on k.
Since \({\bar \Gamma _k}\left[ g \right]\) depends on the choice of the mode-cutof scheme so will the solutions of Equation (4.26), and hence the resolving power ℓ(k; x). It can thus not be identified with the resolving power of an actual experimental set up, but is only meant to provide an order of magnitude estimate. The scheme independence of the resolution which can be achieved in an actual experimental set up would in this picture arise because the scheme dependence in the trajectory cancels against that in the ℓ versus k relation.
This concludes our presentation of the effective average action formalism for gravity. In the next Section 4.3 we will use the FRGE for Γ_{k} as a tool to gain insight into the gravitational renormalization flow.
4.3 Truncated flow equations
Approximate computations of the effective average action can be done in a variety of ways: by perturbation theory, by saddle point approximations of the functional integral, or by looking for approximate solutions of the FRGE. A nonperturbative method of the latter type consists in truncating the underlying functional space. Using an ansatz for Γ_{k} where k-independent local or nonlocal invariants are multiplied by running parameters, the FRGE (4.19) can eventually be converted into a system of ordinary differential equations for these parameters. In this section we outline how the conversion is done in principle.
We should mention that apart from familiarity and the retroactive justification through the results described later on, there is no structural reason to single out the truncations (4.34, 4.35). Even the truncated coarse graining flow in Equation (4.31) will generate all sorts of terms in \({{\bar \Gamma}_k}\left[ g \right]\), the only constraint comes from general covariance. Both local and nonlocal terms are induced. The local invariants contain monomials built from curvature tensors and their covariant derivatives, with any number of tensors and derivatives and of all possible index structures. The form of typical nonlocal terms can be motivated from a perturbative computation of Γ_{k}; an example is \(\int {{d^4}x\sqrt g {R_{\alpha \beta \gamma \delta}}\ln (- {\nabla ^2}){R^{\alpha \beta \gamma \delta}}}\). Since Γ_{k} approaches the ordinary effective action Γ for k → 0 it is clear that such terms must generated by the flow since they are known to be present in Γ. For an investigation of the non-ultraviolet properties of the theory, the inclusion of such terms is very desirable but it is still beyond the calculational state of the art (see however [180]).
The main technical complication comes from evaluating the functional trace on the right-hand-side of the flow equation (4.31) to the extent that one can match the terms against those occuring on the left-hand-side. We shall now illustrate this procedure and its difficulties in the case of the Einstein-Hilbert truncation (4.34) in more detail.
Upon inserting the ansatz (4.33) into the partially truncated flow equation (4.31) it should eventually give rise to a system of two ordinary differential equations for Z_{Nk} and \({{\bar \lambda}_k}\). Even in this simple case their derivation is rather technical, so we shall focus on matters of principle here. In order to find k∂_{k}Z_{Nk} and \(k{\partial _k}{{\bar \lambda}_k}\) it is sufficient to consider (4.31) for g_{αβ} = ḡ_{αβ}. In this case the left-hand-side of the flow equation becomes \(2{k^2}\int {dx} \sqrt g \left[ {- R(g)k{\partial _k}{Z_{Nk}} + 2k{\partial _k}({Z_{Nk}}{{\bar \lambda}_k})} \right]\). The right-hand-side contains the functional derivatives of Γ^{(2)}; in their evaluation one must keep in mind that the identification g_{αβ} = ḡ_{αβ} can be used only after the differentiation has been performed at fixed ḡ_{αβ}. Upon evaluation of the functional trace the right-hand-side should then admit an expansion in terms of invariants P_{α}[g], among them \(\int {\sqrt g}\) and \(\int {\sqrt g} R(g)\). The projected flow equations are obtained by extracting the k-dependent coefficients of these two terms and discarding all others. Equating the result to the left-hand-side and comparing the coefficients of \(\int {\sqrt g}\) and \(\int {\sqrt g} R\), the desired pair of coupled differential equations for Z_{Nk} and \({\bar \lambda _k}\) is obtained.
In principle the isolation of the relevant coefficients in the functional trace on the right-hand-side can be done without ever considering any specific metric g_{αβ} = ḡ_{αβ}. Known techniques like the derivative expansion and heat kernel asymptotics could be used for this purpose, but their application is extremely tedious usually. For example, because the operators \(\Gamma _k^{(2)}\) and \(\Gamma _k^{(2)}\) and \({{\mathcal R}_k},\;{\mathcal R}_k^{{\rm{gh}}}\), are typically of a complicated non-standard type so that no efficient use of the tabulated Seeley-deWitt coefficients can be made. Fortunately all that is needed to extract the desired coefficients is to get an unambiguous signal for the invariants they multiply on a suitable subclass of geometries g = ḡ. The subclass of geometries should be large enough to allow one to disentangle the invariants retained and small enough to really simplify the calculation.
For the Einstein-Hilbert truncation the most efficient choice is a family of d-spheres S^{d}(r), labeled by their radius r. For those geometries ∇_{ρ}R_{αβγδ} = 0, so they give a vanishing value on all invariants constructed from g = ḡ containing covariant derivatives acting on curvature tensors. What remains (among the local invariants) are terms of the form \(\int {\sqrt g P} (R)\), where P is a polynomial in the Riemann tensor with arbitary index contractions. To linear order in the (contractions of the) Riemann tensor the two invariants relevant for the Einstein-Hilbert truncation are discriminated by the S^{d}(r) metrics as they scale differently with the radius of the sphere: \(\int {\sqrt g \sim {r^d},\int {\sqrt g R(g) \sim {r^{d - 2}}}}\). Thus, in order to compute the beta functions of λ_{k} and Z_{Nk} it is sufficient to insert an S^{d}(r) metric with arbitrary r and to compare the coefficients of r^{d} and r^{d−2}. If one wants to do better and include the three quadratic invariants ∫ R_{αβ}R^{αβ}, and ∫ R^{2}, the family S^{d}(r) is not general enough to separate them; all scale like r^{d−4} with the radius.
At this point the operator under the first trace on the right-hand-side of Equation (4.31) has become block diagonal, with the \(\hat f\hat f\) and ϕϕ blocks given by Equation (4.40). Both block operators are expressible in terms of the Laplacian ∇^{2}, in the former case acting on traceless symmetric tensor fields, in the latter on scalars. The second trace in Equation (4.31) stems from the ghosts; it contains (4.41) with ∇^{2} acting on vector fields.
With the derivation of the system (4.45) we managed to find an approximation to a two-dimensional projection of the FRGE flow. Its properties and the domain of applicability or reliability of the Einstein-Hilbert truncation will be discussed in Section 4.4. It will turn out that there are important qualitative features of the truncated coupling flow (4.45) which are independent of the cutoff scheme, i.e. independent of the function \({{\mathcal R}^{(0)}}\). The details of the flow pattern on the other hand depend on the choice of the function \({{\mathcal R}^{(0)}}\) and hence have no intrinsic significance.
By construction the normalized cutoff function \({{\mathcal R}^{(0)}}(u),u = {p^2}/{k^2}\), u = p^{2}/k^{2}, in Equation (C.21) describes the “shape” of \({{\mathcal R}_k}({p^2})\) in the transition region where it interpolates between the prescribed behavior for p^{2} ≪ k^{2} and k^{2} ≫ p^{2}, respectively. It is referred to as the shape function therefore.
Above we illustrated the general ideas and constructions underlying truncated gravitational RG flows by means of the simplest example, the Einstein-Hilbert truncation (4.34). The flow equations for the R^{2} truncation are likewise known in closed form but are too complicated to be displayed here. These ordinary differential equations can now be analyzed with analytical and numerical methods. Their solution reveals important evidence for asymptotic safety. Before discussing these results in Section 4.4 we comment here on two types of possible generalizations.
Concerning generalizations of the ghost sector truncation, beyond Equation (4.29) no results are available yet, but there is a partial result concerning the gauge fixing term. Even if one makes the ansatz (4.33) for Γ_{k}[g, ḡ] in which the gauge fixing term has the classical (or more appropriately, bare) structure one should treat its prefactor as a running coupling: α = α_{k}. After all, the actual “theory space” of functionals \(\Gamma \left[ {g,\bar g,\sigma ,\bar \sigma} \right]\) contains “\(\bar \Gamma\)-type” and “gauge-fixing-type” actions on a completely symmetric footing. The beta function of α has not been determined yet from the FRGE, but there is a simple argument which allows us to bypass this calculation.
In nonperturbative Yang-Mills theory and in perturbative quantum gravity α = α_{k} = 0 is known to be a fixed point for the α evolution. The following heuristic argument suggests that the same should be true beyond perturbation theory for the functional integral defining the effective average action for gravity. In the standard functional integral the limit α → 0 corresponds to a sharp implementation of the gauge fixing condition, i.e. exp(−S_{gf}) becomes proportional to δ[Q_{α}]. The domain of the \({\mathcal D}{f_{\alpha \beta}}\) integration consists of those f_{αβ}’s which satisfy the gauge fixing condition exactly, Q_{α} = 0. Adding the infrared cutoff at k amounts to suppressing some of the f_{αβ} modes while retaining the others. But since all of them satisfy Q_{α} = 0, a variation of k cannot change the domain of the f_{αβ} integration. The delta functional δ[Q_{α}] continues to be present for any value of k if it was there originally. As a consequence, α vanishes for all k, i.e. α = 0 is a fixed point of the α evolution [137].
In other words we can mimic the dynamical treatment of a running α by setting the gauge fixing parameter to the constant value α = 0. The calculation for α = 0 is more complicated than at α = 1, but for the Einstein-Hilbert truncation the α-dependence of β_{g} and β_{λ}, for arbitrary constant α, has been found in [133]. The R^{2} truncations could be analyzed only in the simple α = 1 gauge, but the results from the Einstein-Hilbert truncation suggest the UV quantities of interest do not change much between α = 0 and α = 1 [133, 131].
Up to now we considered pure gravity. As far as the general formalism is concerned, the inclusion of matter fields is straightforward. The structure of the flow equation remains unaltered, except that now \(\Gamma _k^{(2)}\) and \({{\mathcal R}_k}\) are operators on the larger space of both gravity and matter fluctuations. In practice the derivation of the projected FRG equations can be quite formidable, the main difficulty being the decoupling of the various modes (diagonalization of \(\Gamma _k^{(2)}\)) which in most calculational schemes is necessary for the computation of the functional traces.
Various matter systems, both interacting and non-interacting (apart from their interaction with gravity) have been studied in the literature. A rather detailed analysis of the fixed point has been performed by Percacci et al. In [72, 171, 170] arbitrary multiplets of free (massless) fields with spin 0, 1/2, 1 and 3/2 were included. In [170] a fully interacting scalar theory coupled to gravity in the Einstein-Hilbert approximation was analyzed, with a local potential approximation for the scalar self-interaction. A remarkable finding is that in a linearized stability analysis the marginality of the quartic self-coupling is lifted by the quantum gravitational corrections. The coupling becomes marginally irrelevant, which may offer a new perspective on the triviality issue and the ensued bounds on the Higgs mass. Making the number of matter fields large O(N), the matter interactions dominate at all scales and the nontrivial fixed point of the 1/N expansion [216, 217, 203] is recovered [169].
4.4 Einstein-Hilbert and R^{2} truncations
In this section we review the main results obtained in the effective average action framework via the truncated flow equations of the previous Section 4.3. To facilitate comparison with the original papers we write here G_{k} = g_{N}/(16π) for the running Newton constant; unless stated otherwise the results refer to d = 4.
4.4.1 Phase portrait of the Einstein-Hilbert truncation
The RG flow is dominated by two fixed points (g_{*},λ_{*}): a Gaussian fixed point (GFP) at g_{*} = λ_{*} = 0, and a non-Gaussian fixed point (NGFP) with g_{*} > 0 and λ_{*} > 0. There are three classes of trajectories emanating from the NGFP: Trajectories of Type Ia and IIIa run towards negative and positive cosmological constants, respectively, and the single trajectory of Type IIa (“separatrix”) hits the GFP for k → 0. The short-distance properties of Quantum Einstein Gravity are governed by the NGFP; for k → ∞, in Figure 2 all RG trajectories on the half-plane g > 0 run into this fixed point — its unstable manifold is two-dimensional. Note that near the NGFP the dimensionful Newton constant vanishes for k → ∞ according to G_{k} = g_{k}/k^{2} ≈ g_{*}/k^{2} → 0. The conjectured nonperturbative renormalizability of Quantum Einstein Gravity is due to this NGFP: If it was present in the untruncated RG flow it could be used to construct a microscopic quantum theory of gravity by taking the limit of infinite UV cutoff along one of the trajectories running into the NGFP, implying that the theory does not develop uncontrolled singularities at high energies [227].
The trajectories of Type IIIa cannot be continued all the way down to the infrared (k = 0) but rather terminate at a finite scale k_{term} > 0. (This feature is not resolved in Figure 2.) At this scale the β-functions diverge. As a result, the flow equations cannot be integrated beyond this point. The value of k_{term} depends on the trajectory considered. The trajectory terminates when the dimensionless cosmological constant reaches the value λ = 1/2. This is due to the fact that the functions \(\Phi _n^p(w)\) and \(\tilde \Phi _n^p(w)\) — for any admissible choice of \({{\mathcal R}^{(0)}}\) — have a singularity at w = −1, and because w = −2λ in all terms of the β-functions. In Equations (4.50) the divergence at λ = 1/2 is seen explicitly. The phenomenon of terminating RG trajectories is familiar from simpler theories, such as Yang-Mills theories. It usually indicates that the truncation becomes insufficient at small k.
4.4.2 Evidence for asymptotic safety — Survey
Here we collect the evidence for asymptotic safety obtained from the Einstein-Hilbert and R^{2} truncations, Equation (4.34) and Equation (4.35), respectively, of the flow equations in Section 4.2 [133, 131].
The details of the flow pattern depend on a number of ad-hoc choices. It is crucial that the properties of the flow which point towards the asymptotic safety scenario are robust upon alterations of these choices. This robustness of the qualitative features will be discussed in more detail below. Here let us only recapitulate the three main ingredients of the (truncated) flow equations that can be varied: The shape functions \({{\mathcal R}^{(0)}}\) in Equation (4.31) can be varied, the gauge parameter α in Equation (4.33) can be varied, and the vector and transversal parts in the traceless tensor modes can be treated differently (type A and B cutoffs).
Picking a specific value for the gauge parameter has a somewhat different status than the other two choices. The truncations are actually one-parameter families of truncations labelled by α; in a more refined treatment α = α_{k} would be a running parameter itself determined by the FRGE.
In practice the shape function \({{\mathcal R}^{(0)}}\) was varied within the class (4.51) of exponential cutoffs and a similar one-parameter class of cutoffs with compact support [133, 131]. Changing the cutoff function C_{k} at fixed k may be thought of as analogous to a change of scheme in perturbation theory.
- 1.
Existence of a non-Gaussian fixed point: The NGFP exists no matter how \({{\mathcal R}^{(0)}}\) and α are chosen, both for type A and B cutoffs.
- 2.
Positive Newton constant: While the position of the fixed point is scheme dependent (see below), all cutoffs yield positive values of g_{*} and λ_{*}. A negative g_{*} would have been problematic for stability reasons, but there is no mechanism in the flow equation which would exclude it on general grounds. This feature is preserved in the R^{2} truncation.
- 3.
Unstable manifold of maximal dimension: The existence of a nontrivial unstable manifold is crucial for the asymptotic safety scenario. The fact that the unstable manifold has (for d = 4) its maximal dimension (at least in the vicinity of the fixed point) indicates that the set of curvature invariants retained is dynamically natural. Again this is (in d = 4) a bonus feature not built into the flow equations. It holds for both the Einstein-Hilbert and the R^{2} truncation.
- 4.
Smallness of R^{2}coupling: Also with the generalized truncation the fixed point is found to exist for all admissible cutoffs. It is quite remarkable that ν_{*} is always significantly smaller than λ_{*} and g_{*}. Within the limited precision of our calculation this means that in the three-dimensional parameter space the fixed point practically lies on the (λ, g)-plane with ν = 0, i.e. on the parameter space of the pure Einstein-Hilbert truncation.
We proceeded to discuss various aspects of the evidence for asymptotic safety in more detail, namely the structure of unstable manifold and the robustness of the qualitative features of the flow. Finally we offer some comments on the full FRGE dynamics.
4.4.3 Structure of the unstable manifold
As explained in Section 2.1 it is often convenient to set t ≔ ln k_{0}/k (which is to be read as lnΛ/k − ln k_{0}/Λ in the presence of an ultraviolet cutoff Λ) and ask “where a coarse graining rajectory comes from” by formally sending t to −∞ (while the coarse graining flow is in the direction of increasing t). The tangent space to the unstable manifold has its maximal dimension if all the essential couplings taken into account hit the fixed point as t is sent to −∞: The fixed point is ultraviolet stable in the direction opposite to the coarse graining. This is the case for both the Einstein-Hilbert truncation and the R^{2} truncation, as we shall describe now in more detail.
Solving the full, nonlinear flow equations numerically [181] shows that the asymptotic scaling region where the linearization (4.56) is valid extends from k = ∞ down to about k ≈ m_{P1} with the Planck mass defined as \({m_{{\rm{P}}1}} = G_0^{- 1/2}\). Here m_{P1} marks the lower boundary of the asymptotic scaling region. We set k_{0} = m_{P1} so that the asymptotic scaling regime extends from about t = 0 to t = −∞.
- 1.
The ν-components of Re V and Im V are tiny. Hence these two vectors span a plane which virtually coincides with the (λ, g) subspace at ν = 0, i.e. with the parameter space of the Einstein-Hilbert truncation. As a consequence, the Re C- and Im C-normal modes are essentially the same trajectories as the “old” normal modes already found without the R^{2}-term. Also the corresponding ϑ′- and ϑ″-values coincide within the scheme dependence.
- 2.
The new eigenvalue ϑ_{3} introduced by the R^{2}-term is significantly larger in modulus than ϑ′. When a trajectory approaches the fixed point from below (t → −∞), the “old” normal modes ∝ Re C, Im C are proportional to exp(−ϑ′t), but the new one is proportional to exp(−ϑ_{3}t), so that it decays much more quickly. For every trajectory running into the fixed point, i.e. for every set of constants (Re C, Im C, C_{3}), we find therefore that once − t is sufficiently large the trajectory lies entirely in the Re V-Im V subspace, i.e. in the ν = 0-plane practically.
Due to the large value of −ϑ_{3}, the new scaling field is ‘very relevant’. However, when we start at the fixed point (t = −∞) and raise t it is only at the low energy(!) scale k ≈ m_{P1} (t ≈ 0) that exp(− ϑ_{3}t) reaches unity, and only then, i.e. far away from the fixed point, the new scaling field starts growing rapidly.
- 3.
Since the matrix Θ is not symmetric its eigenvectors have no reason to be orthogonal. In fact, one finds that V^{3} lies almost in the Re V-Im V-plane. For the angles between the eigenvectors given above we obtain ∢(Re V, Im V) = 102.3°, ∢(Re V, V^{3}) = 100.7°, ∢(Im V, V^{3}) = 156.7°. Their sum is 359.7 ° which confirms that Re V, Im V, and V^{3} are almost coplanar. Therefore as one raises t and moves away from the fixed point so that the V^{3} scaling field starts growing, it is again predominantly the \(\int {dx\sqrt g}\) and \(\int {dx\sqrt g R(g)}\) invariants which get excited, but not \(\int {dx\sqrt g R{{(g)}^2}}\).
4.4.4 Robustness of qualitative features
As explained before the details of the coupling flow produced by the various truncations of Equation (4.19) depend on the choice of the cutoff action (\({{\mathcal R}^{(0)}}\), type A vs. B) and the gauge parameter α. Remarkably the qualitative properties of the flow, in particular those features pointing towards the asymptotic safety scenario are unchanged upon alterations of the computational scheme. Here we discuss these robustness properties in more detail. The degree of insensitivity of quantities expected to be “universal” can serve as a measure for the reliability of a truncation.
We begin with the very existence of a non-Gaussian fixed point. Importantly, both for type A and type B cutoffs the non-Gaussian fixed point is found to exists for all shape functions \({\mathcal R}_s^{(0)}\). This generalizes earlier results in [205]. Indeed, it seems impossible to find an admissible mode-cutoff which destroys the fixed point in d = 4. This is nontrivial since in higher dimensions (d ≳ 5) the fixed point exists for some but does not exist for other mode-cutoffs [181] (see however [79]).
Within the Einstein-Hilbert truncation also a RG formalism different from (and in fact much simpler than) that of the average action was used [39]. The fixed point was found to exist already in a simple RG improved 1-loop calculation with a proper time cutoff.
We take this as an indication that the fixed point seen in the Einstein-Hilbert [204, 133, 136, 39] and the R^{2} truncations [131] is the projection of a genuine fixed point and not just an artifact of an insufficient truncation.
Support for this interpretation comes from considering the product g_{*}λ_{*} of the fixed point coordinates. Recall from Section 2.3.2 that the product g(k)λ(k) is a dimensionless essential coupling invariant under constant rescalings of the metric [116]. One would expect that this combination is also more robust with respect to scheme changes.
In summary, the qualitative properties listed above (ϑ′, ϑ_{3} < 0, −ϑ_{3} ≫ −ϑ′, etc.) hold for all cutoffs. The ϑ’s have a stronger scheme dependence than g_{*}λ_{*}, however. This is most probably due to having neglected further relevant operators in the truncation so that the Θ matrix we are diagonalizing is still too small.
Finally one can study the dimension dependence of these results. The beta functions produced by the truncated FRGE are continuous functions of the spacetime dimension d and it is instructive to analyze them for d ≠ 4. This was done for the Einstein-Hilbert truncation in [181, 79], with the result that the coupling flow is quantitatively similar to the 4-dimensional one for not too large d. The robustness features have been explored with varios cutoffs with the result that the sensitity on the cutoff parameters increases with increasing d. In [181] a strong cutoff dependence was found for d larger than approximately 6, for two versions of the sharp cutoff (with s = 1, 30) and for the exponential cutoff with s = 1. In [79] a number of different cutoffs were employed and no sharp increase in sensitivity to the cutoff parameters was reported for d ≤ 10.
This concludes our analysis of the robustness properties of the truncated RG flow. For further details the reader is referred to Lauscher et al. [133, 131, 132]. On the basis of these robustness properties we believe that the non-Gaussian fixed point seen in the Einstein-Hilbert and R^{2} truncations is very unlikely to be an artifact of the truncations. On the contrary there are good reasons to view it as the projection of a fixed point of the full FRGE dynamics. It is especially gratifying to see that within the scheme dependence the additional R^{2}-term has a quantitatively small impact on the location of fixed point and its unstable manifold.
In summary, we interpret the above results and their mutual consistency as quite nontrivial indications supporting the conjecture that 4-dimensional Quantum Einstein Gravity possesses a RG fixed point with precisely the properties needed for its asymptotic safety.
4.4.5 Comments on the full FRGE dynamics
The generalization of the previous results to more complex truncations would be highly desirable, but for time being it is out of computational reach. We therefore add some comments on what one can reasonably expect to happen.
Let us first briefly recall the scaling pattern based on the perturbative Gaussian fixed point. As described in Section 3.3 in a perturbative construction of the effective action the divergent part of the ℓ loop contribution is always local and thus can be added as a counter term to a local bare action S[g] = ∑_{i}u_{i}P_{i}[g], where the sum is over local curvature invariants P_{i}[g]. The scaling pattern of the monomials P_{i}[g] with respect to the perturbative Gaussian fixed point will thus reflect those of the I_{i}[g] in the effective action and vice versa. As explained in Section 2.3 the short-distance behavior of the perturbatively defined theory will be dominated by the P_{i}’s with the largest number of derivatives acting upon g_{αβ}. In a local invariant containing the Riemann tensor to the pth power and q covariant derivatives acting on it, the number of derivatives acting on g_{αβ} is 2p + q. If one starts with just a few P_{i}’s and performs loop calculations one discovers that higher P_{i}’s are needed as counter terms. As a consequence the high energy behavior is dominated by the bottomless chain of invariants with more and more derivatives.
As already argued in Section 2.3 in an asymptotically safe Quantum Gravidynamics the situation is different. The absence of a blow-up in the couplings is part of the defining property. The dominance of the high energy behavior by the bottomless chain of high derivative local invariants is replaced with the expectation that all invariants should be about equally important in the extreme ultraviolet.
Clearly the above argument can be generalized to action functionals depending on all the fields g_{αβ}, ḡ_{αβ}, σ^{α}, \({\bar \sigma _\alpha}\). Also the choice of field variables is inessential and the argument should carry over to other types of flow equations. It suggests that there could indeed be a fixed point action \({S_*}\left[ g \right] = {\rm{li}}{{\rm{m}}_{k \rightarrow \infty}}{\bar \Gamma _k}\left[ g \right] = {\lim\nolimits_{k \rightarrow \infty}}{\Gamma _k}\left[ {g,g,0,0} \right]\) which is well-defined when expressed in terms of dimensionless quantities and which describes the extreme ultraviolet dynamics of Quantum Einstein Gravity. By construction the unstable manifold of this fixed point action would be nontrivial.
Without further insight unfortunately little can be said about its dimension. Among the local invariants in Equation (4.57) arguably only a finite number should be relevant. This is because the power counting dimensions d_{i} < 0 of the infinite set of irrelevant local invariants may receive large positive corrections which makes them relevant with respect to the NGFP. An example for this phenomenon is provided by the \({I_2}\left[ g \right]: = \int {{d^4}x\sqrt g R{{(g)}^2}}\) invariant. It is power counting marginal (d_{2} = 0) but with respect to the NGFP the scaling dimension of the associated dimensionless coupling is shifted to a large positive value \(d_2^{{\rm{NGFP}}} = - {\vartheta _3}\) of O(10). Nevertheless it seems implausible that this will happen to couplings with arbitrarily large negative power counting dimension as correspondingly large corrections would be required.
Also scalar modes like the conformal factor have vanishing power counting dimension. The way how such dimensionless scalars enter the effective action is then not constrained by the above ‘implausibility’ argument. An unconstrained functional occurance however opens the door to a potentially infinite-dimensional unstable manifold.
Another core issue are of course the positivity properties of the Quantum Einstein Gravity defined through the FRGE. As already explained in Section 1.5 the notorious problems with positivity and causality which arise within standard perturbation theory around flat space in higher-derivative theories of Lorentzian gravity are not an issue in the FRG approach. For example if Γ_{k} is of the R + R^{2} 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 this is irrelevant, however, 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 only that \(\Gamma _k^{(2)} + {{\mathcal R}_k}\) is a positive operator for all k. In the exact theory this is believed to be the case.
A rather encouraging first result in this direction comes from the R^{2} truncation [131]. In the FRG formalism the problem of the higher derivative ghosts is to some extent related to the negative \(Z_k^\phi\) factors discussed in Section 2.1. It was found that, contrary to the Einstein-Hilbert truncation, the R^{2} truncation has only positive \(Z_k^\phi\) factors in the fixed point regime k ≥ m_{P1}. Hence in this truncation the existence of ‘safe’ couplings appears to be compatible with the absence of unphysical propagating modes, as required by the scenario.
5 Conclusions
- 1.
Consolidating the existence of a non-Gaussian fixed point and that of an asymptotically safe coupling flow. This may be done in various formalisms, field variables, and approximations.
- 2.
Clarifying the microstructure of the geometries, identification of the antiscreening degrees of freedom, and the role of the ultraviolet cutoff.
- 3.
Clarifying the physically adequate notion of unitarity and its interplay with Areas 1 and 2.
- 4.
Characterization of generic observables and working out sound consequences for the macro-physics.
Our conventions are: \({\nabla _i}{\upsilon ^k} = {\partial _i}{\upsilon ^k} + {\Gamma ^k}_{ij}{\upsilon ^j}\), with \({\Gamma ^k}_{ij} = {1 \over 2}{\mathfrak h^{kl}}[{\partial _j}{\mathfrak h_{il}} + {\partial _i}{\mathfrak h_{jl}} - {\partial _l}{\mathfrak h_{ij}}]\). The Riemann tensor is defined by \(({\nabla _i}{\nabla _j} - {\nabla _j}{\nabla _i}){\upsilon ^k} = {R^k}_{lij}{\upsilon ^l}\), so that \({R^k}_{lij} = {\partial _i}{\Gamma ^k}_{li} - {\partial _i}{\Gamma ^k}_{li} + {\Gamma ^k}_{lm}{\Gamma ^m}_{lj} - {\Gamma ^k}_{jm}{\Gamma ^m}_{li}\). The Ricci tensor is \({R_{ij}} = {R^m}_{imj}\).
Acknowledgements
We wish to thank E. Seiler for a critical reading of the manuscript and for suggesting various improvements on the field theoretical parts. M.N. would like to thank C. Bervellier, D. Litim, R. Loll, and A. Niemi for comments and correspondence. M.R. wishes to thank A. Bonnano, O. Lauscher, F. Saueressig, J. Schwindt and H. Weyer for fruitful collaborations on the work summarized here, and R. Percacci and C. Wetterich for many discussions.