The results from the 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 -type propagator that is used. The same happens when (Einstein or a higher derivative) gravity is coupled to a large number of matter fields and a expansion is performed. A nontrivial fixed point is found that goes hand in hand with a -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 (perturbative 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 expansions: In the non-gravitational examples of perturbatively non-renormalizable field theories with a non-Gaussian fixed point the non-Gaussian fixed point can be viewed as a ‘remnant’ of an asymptotically free fixed point in a lower-dimensional version of the theory. It is thus natural to ask how gravity behaves in this respect. In spacetime dimensions Newton’s constant is dimensionless, and formally the theory with the bare action 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 dimensions and then studying the behavior near . The resulting “-expansion” amounts to a double expansion in the number of ‘graviton’ loops and in the dimensionality parameter . Typically dimensional regularization is used, in which case the UV divergencies give rise to the usual poles in . Specific for gravity are however two types of complications. The first one is due to the fact that is topological at , which gives rise to additional “kinematical” poles of order 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 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.

For the reference operator various choices have been adopted (we follow the discussion in Kawai et al. [118, 116, 117, 3] with the conventions of [117]):

(i) a cosmological constant term ,

(ii) monomials from matter fields which are quantum mechanically non-scale invariant in ,

(iii) monomials from matter fields which are quantum mechanically scale invariant in ,

(iv) the conformal mode of the metric itself in a background field expansion.

All choices lead to a flow equation of the form

but the coefficient depends on the choice of the reference operator [118]. For all there is a nontrivial fixed point with a one-dimensional unstable manifold. In other words is an asymptotically safe coupling in dimensions, and the above rule of thumb suggests that this a remnant of a nontrivial fixed point in with respect to which is asymptotically safe (see Section 1.4 for the renormalization group terminology).Technically the non-universality of arises from the before-mentioned kinematical poles. In the early papers [86, 53, 227] the Choice i was adopted giving , or if free matter of central charge 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 comes out as , where , respectively. Here is the scaling dimension of the reference operator, and again free matter of central charge 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 and , and that reference operators with 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 , the conformal factor , and a part which is traceless, . Specifically is inserted into the Einstein–Hilbert action; propagators are defined (after gauge fixing) by the terms quadratic in and , and vertices correspond to the higher order terms. This procedure turns out to have a number of advantages. First the conformal mode is renormalized differently from the modes and can be viewed as defining a reference operator in itself; in particular the coefficient comes out as . Second, and related to the first point, the system has a well-defined -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 .

(b) Evidence from perturbation theory and large : Modifications of the Einstein–Hilbert action where fourth derivative terms are included are known to be perturbatively renormalizable [206, 83, 19, 59]. A convenient parameterization is

Here is a constant such that , is the square of the Weyl tensor, and is the integrand of the Gauss–Bonnet term. In the latter is negligible, unless dimensional regularization is used. The sign of the crucial coupling is fixed by the requirement that the Euclidean functional integral is damping. The one-loop beta functions for the (non-negative) couplings, , , , are known and on the basis of them these couplings are expected to be asymptotically safe. In particular is asymptotically free, . The remaining couplings and are made dimensionless via , , where is the renormalization scale. At these flow equations are compatible with the existence of a non-trivial fixed point for Newton’s constant, . The value of is highly nonuniversal but it cannot naturally be made to vanish, i.e. the nontrivial and the trivial fixed point, , do not merge. The rationale for identifying a nontrivial fixed point by perturbative means is explained in Section 2.2. The benign renormalizability properties seen in this framework are mostly due to the type propagator, at the expense of unphysical propagating modes.The action (1.7) can be supplemented by a matter action, containing a large number, , of free matter fields. One can then keep the product fixed, retain the usual normalization of the matter kinetic terms, and expand in powers of . Renormalizability of the resulting ‘large expansion’ then amounts to being able to remove the UV cutoff order by order in the formal series in . 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 scalar fields, Dirac fields, and Maxwell fields.

Starting from the Einstein–Hilbert action the high energy behavior of the usual -type propagator gets modified. To leading order in 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 to , in four dimensions. In dimensions the resulting expansion is believed to be renormalizable in the sense that the UV cutoff can strictly be removed order by order in without additional (counter) terms in the Lagrangian. In the same is presumed to hold provided an extra 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 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 , and . Its unstable manifold is three dimensional, i.e. all couplings are asymptotically safe. Repeating the computation in dimensions the fixed point still exists and (taking into account the different UV regularization) corresponds to the large (central charge) limit of the fixed point found the expansion.

These results have recently been confirmed and extended by Percacci [169] using the heat kernel expansion. In the presence of scalar fields, Dirac fields, and Maxwell fields, the flow equations for and come out to leading order in as

One sees that the coupling is always asymptotically free, and that Newton’s constant has a nontrivial fixed point, , which is positive if the number of matter fields is not too large.As a caveat one should add that the -type propagators occuring both in the perturbative and in the large 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 reduction.

(c) Evidence from symmetry reductions: Here one considers the usual gravitational functional integral but restricts it from “4-geometries modulo diffeomorphisms” to “4-geometries constant along a foliation modulo diffeomorphisms”. This means that instead of the familiar foliation of geometries one considers a foliation in terms of two-dimensional hypersurfaces and performs the functional integral only over configurations that are constant as one moves along the stack of two-surfaces. Technically this constancy condition is formulated in terms of two commuting vectors fields , , that are Killing vectors of the class of geometries considered, . For definiteness we consider here only the case where both Killing vectors are spacelike. From this pair of Killing vector fields one can form the symmetric matrix . Then (with the components of and ) defines a metric on the orbit space which obeys and . The functional integral is eventually performed over metrics of the form

where the components of a metric tensor are parameterized by the independent functions in and . Each of these functions is constant along the stack of two-surfaces but may be arbitrarily rough within a two-surface.In the context of the asymptotic safety scenario the restriction of the functional integral to metrics of the form (1.9) is a very fruitful one:

- 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 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 truncation. The renormalization of the 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.

Two additional bonus features are: In this sector the explicit construction of Dirac observables is feasible (classically and presumably also in the quantum theory). Finally a large class of matter couplings is easily incorporated.

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” , 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.

Technically the renormalization is done by borrowing covariant background field techniques from Riemannian sigma-models (see [84, 110, 201, 57, 220, 162]). In the particular application here the sigma-model perturbation theory is partially nonperturbative from the viewpoint of a graviton loop expansion as not all of the metric degrees of freedom are Taylor expanded in the bare action (see Section 3.2). This together with the field reparameterization invariance blurs the distinction between a perturbative and a non-perturbative treatment of the gravitational modes. The renormalization can be done to all orders of sigma-model perturbation theory, which is ‘not-really-perturbative’ for the gravitational modes. It turns out that strict cutoff independence can be achieved only by allowing for infinitely many essential couplings. They are conveniently combined into a generating functional , which is a positive function of one real variable. Schematically the renormalized action takes the form [154]

Here is a metric of the form (1.9), is the Einstein–Hilbert action evaluated on it, and is the generating coupling function evaluated on the renormalized area radius field . Higher derivative terms are not needed in this subsector for the absorption of counter terms; the “other second derivative terms” needed are known explicitly.This “coupling functional” is scale dependent and is subject to a flow equation of the form

where is the renormalization scale and is the ‘running’ generating functional. To preclude a misunderstanding let us stress that the function changes with , irrespective of the name of the argument, not just its value on , say. Interestingly a closed formula for the beta function (or functional) in Equation (1.11) can be found [154, 155]. The resulting flow equation is a nonlinear partial integro-differential equation and difficult to analyze. The fixed points however are easily found. Apart from the degenerate ‘Gaussian’ one, , there is a nontrivial fixed point . For the Gaussian fixed point a linearized stability analysis is empty, the structure of the quadratic perturbation equation suggests that it has both attractive and repulsive directions in the space of functions . For the non-Gaussian fixed point a linearized stability analysis is non-empty and leads to a system of linear integro-differential equations. It can be shown [155] that all linearized perturbations decay for , which is precisely what Weinberg’s criterion for asymptotic safety asks for. Moreover the basic propagator used is free from unphysical poles. Applying the criterion described in Section 1.3 this strongly suggests that a continuum limit exist for the reduced Quantum Gravidynamics beyond approximations (like the sigma-model perturbation theory/partially nonperturbative graviton expansion used to compute Equation (1.11)). See [158] for a proposed ‘exact’ bootstrap construction, whose relation to a truncated functional integral however remains to be understood.In summary, in the context of the 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 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.

(d) Evidence from truncated flows of the effective average action: The effective average action is a generating functional generalizing the usual effective action, to which it reduces for . Here depends on the UV cutoff and an additional scale , indicating that in the defining functional integral roughly the field modes with momenta in the range have been integrated out. Correspondingly gives back the bare action and is the usual quantum effective action, in the presence of the UV cutoff . The modes in the momentum range are omitted or suppressed by a mode cutoff ‘action’ , and one can think of as being the conventional effective action but computed with a bare action that differs from the original one by the addition of ; specifically

A summary of the key properties of the effective average action (1.12) can be found in Appendix C. Here we highlight that from the regularized functional integral defining an (‘exact’) functional renormalization group equation (FRGE) can be derived. Schematically it has the form , where the “right-hand-side” involves the Hessian of with respect to the dynamical fields. The FRGE itself (that is, its right-hand-side) carries no explicit dependence on the UV cutoff, or one which can trivially be removed. However the removal of the UV regulator implicit in the definition of is nontrivial and is related to the traditional UV renormalization problem (see Section 2.2). Whenever massless degrees of freedom are involved, also the existence of the limit of is nontrivial and requires identification of the proper infrared degrees of freedom. In the present context we take this for granted and focus on the UV aspects.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 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). 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 obeys an ‘exact’ FRGE, which is a new computational tool in quantum gravity not limited to perturbation theory. In practice is replaced in this equation with a independent functional interpreted as . The assumption that the ‘continuum limit’ 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 (where right-hand-side now also refers to the Hessian of ). The structure of the solutions of this cut-off independent FRGE should be such that they can plausibly be identified with . Presupposing the ‘infrared safety’ in the above sense, a necessary condition for this is that the limits and exist. Since the first limit probes whether 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 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 which coincides with at some scale . The point is that only for very special ‘fine tuned’ initial functionals will the associated solution of the cutoff independent FRGE exist globally [157]. The existence of the 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 . We refer to Section 2.2 for a more detailed discussion of this issue.

In practice of course a nonlinear functional differential equation is very difficult to solve. To make the FRGE computationally useful the space of functionals is truncated typically to a finite-dimensional one of the form

where the are given ‘well-chosen’ – local or nonlocal – functionals of the fields (among them the expectation value of the metric in the case of gravity) and the are numerical parameters that carry the scale dependence. For ’s obeying a non-redundancy condition, the play the role of essential couplings which have been normalized to have vanishing mass dimension by taking out a power . The original FRGE then can be converted into a system of nonlinear ordinary differential equations for these couplings. In the case of gravity the following ansatz has been made [133, 131]: where is the metric and is the associated curvature scalar. The flow pattern displays a number of remarkable properties. Most importantly a non-Gaussian fixed point exists (first found in [204] based on [179] and extensively corroborated in [205, 133, 131, 136, 39]). Within the truncation (1.14) a three-dimensional subset of initial data is attracted to the fixed point under the reversed flow where the fixed point couplings , , are finite and positive and no blow-up occurs in the flow for large . Moreover unphysical propagating modes appear to be absent. Again this adheres precisely to the asymptotic safety criterion. Some of the trajectories with initial data in the unstable manifold cannot quite be extended to , due to (infrared) singularities. This problem is familiar from nongravitational theories and is presumably an artifact of the truncation. In the vicinity of the fixed point, on the other hand, all trajectories show remarkable robustness properties against modifications of the mode cutoff scheme (see Section 4.3) which provide good reasons to believe that the structural aspects of the above results are not an artifact of the truncation used. The upshot is that there is a clear signal for asymptotic safety in the subsector (1.13), obtained via truncated functional renormalization flow equations.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 , puts certain constraints on the allowed number of copies. When a self-interacting scalar is coupled non-minmally via , one finds a fixed point , (whose values are with matched normalizations the same as in the pure gravity computation) while all self-couplings vanish, , . In the vicinity of the fixed point a linearized stability analysis can be performed; the admixture with and then lifts the marginality of , which becomes marginally irrelevant [171, 170]. The running of and 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.

http://www.livingreviews.org/lrr-2006-5 |
© Max Planck Society and the author(s)
Problems/comments to |