Before turning to renormalization aspects proper, let us describe the special role of Newton’s constant in a diffeomorphism invariant theory with a dynamical metric. Let be any local action, where is the metric and the “matter” fields are not scaled when the metric is. Scale changes in the metric then give rise to a variation of the Lagrangian which vanishes on shell:essential (i.e. a genuine coupling) becomes inessential (i.e. can be changed at will by a redefinition of the fields). The running of this parameter, like that of a wave function renormalization constant, has no direct significance. If the pure gravity part contains the usual Ricci scalar term , the parameter that becomes inessential may be taken as its prefactor, i.e. may be identified with the inverse of Newton’s constant, via , is chosen such that the coefficient in the nonrelativistic force law, as computed from , equals . For a different normalization has to be adopted.
The physics interpretation of the inessential parameter is that it also sets the absolute momentum or spectral scale. To see this we can think of as a reference metric in the background field formalism. For example for the spectral values of the covariant Laplacian associated with one has
Being inessential the quantum field theoretical running of has significance only relative to the running coefficient of some reference operator. The most commonly used choice is a cosmological constant term . Indeed. One usually switches to dimensionless parameters via
The second possibility is realized when inserting a singular solution of the equation for into the equation for . This naturally occurs when working in Planck units. One makes use of the fact that an inessential parameter can be frozen at a prescribed value. Specifically fixing
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 , uses 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 (or , if Newton’s constant is included) couplings can be achieved . However the 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 . The idea is to try to maintain a 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  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 general classical action in dimensions containing up to four derivatives of the metric reads. The parameters in the second line are related to those in the first by
The perturbative quantization of Equation (2.26) proceeds as usual. Gauge fixing and ghost terms are added and the total action is expanded in powers of . Due to the crucial term the gauge-fixed propagator read off from the quadratic part of the full action has a characteristic falloff in ,[206, 207]. Also spin 0 modes with the “wrong sign” may occur depending on the coefficient of the term [207, 49]. The one-loop counterterm (minus the divergent part of the effective action) has been computed by a number of authors using different regularizations: dimensional regularization [59, 19], proper time cut-off , zeta function . The resulting one-loop flow equations in for are agreed upon and read
To describe the flow of the Newton and cosmological constants one switches to the dimensionless parameters and as in Section 2.3.1. The result obtained in Berrodo–Peixoto and Shapiro  via dimensional regularization reads in our conventions uses a specific momentum space cutoff and evaluates the effective average action to one loop using known heat kernel coefficients. The resulting flow equations are of the above form with ; further two additional terms in are found which fix also the value of in terms of . The difference in these non-universal terms can be understood  from the fact that dimensional regularization discards quadratic and quartic divergencies, while a momentum space cutoff gives -dependent but nonzero results for their coefficients.
The flow equations (2.20, 2.30) of course also admit the Gaussian fixed point , 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 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 in Equation (2.31) is always non-universal, the anomalous dimension is exactly 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.
It is instructive  to compare the perturbative one-loop flow (2.20, 2.30) with the linearization of the flow obtained from the FRGE framework described in Section 4. In the so-called Einstein–Hilbert truncation using an optimed cutoff and a limiting version of the gauge-fixing parameter, the ‘beta’ functions , reduce to ratios of polynomials in , . Upon expansion to quadratic order one finds
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 type propagator is used throughout . We now describe this construction in slightly more detail following the presentation in .
Starting from the Lagrangian without cosmological constant the one-loop divergencies come out in dimensional regularization as [95, 222] at two loops there is a divergence proportional to , which cannot be absorbed by a field redefinition. A counterterm proportional to it must thus be added to . Importantly, when re-expanded in powers of , this counterterm, however, produces only terms quadratic in that are proportional to the Ricci tensor or the Ricci scalar. These can be removed by a covariant field redefinition, so that the intial type propagator does not receive corrections. A simple argument  shows that this property also holds for all higher order counterterms that can be expected to occur. Explicitly, consider a Lagrangian of the form
Let us briefly recap the power counting and scaling dimensions of local curvature invariants. These are integrals over densities which are products of factors of the form , suitably contracted to get a scalar and then multiplied by . One easily checks , , with , where is the total power of the Riemann tensor and is the (necessarily even) total number of covariant derivatives. This scaling dimension matches minus the mass dimension of if is taken dimensionless. For the mass dimension of the associated coupling in a product one thus gets . For example, the three local invariants in Equation (1.14) have mass dimensions , , , respectively. There are three other local invariants with mass dimension , namely the ones with integrands (the square of the Weyl tensor), (the generalized Euler density), and . Then there is a set of dimension local invariants, and so on. Note that in the integrands of the last two of the dimensionless invariants are total divergencies so that in there are only local invariants with non-positive mass dimension (see Equation (2.26)).
A generic term in will be symbolically of the form , where all possible contractions of the indices may occur. Since the Ricci tensor is schematically of the form , the piece in quadratic in is of the form . The coefficient of 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 , , and the propagator derived from it will remain of the 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 .
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 and would be unphysical for . In this case the computed results for physical quantities are likely to blow up likewise at some (high) energy scale . 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 propagator. Equivalently all but a few couplings have non-negative mass dimensions . These are the only ones not irrelevant with respect to the stability matrix computed at the perturbative Gaussian fixed point. However in Equation (2.36) these power counting irrelevant couplings with 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 reduction; the results of Section 4 for the 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.
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  generalizing a similar construction in Yang–Mills theories . 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 in mind.
A geometry can be described by picking a representative described by a parametric metric. Here can be specified by picking an explicit parameterization or by imposing a gauge fixing condition . Typical choices are a harmonic gauge condition with respect to some reference metric connection, or a proper time gauge , for a fixed timelike co-vector . Once has been fixed, the push forward with a generic diffeomorphism will generate the associated orbit,
On the tangent space the parameterization (2.37) amounts to
In summary one arrives at the following proposal for a kinematical measure over geometries:
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 is written as , where now is subject to a gauge condition . On the cotangent space this leads to a York-type decomposition  replacing (2.41), where the variations of the conformal factor and that of the tracefree part of describe the variations of the geometry, while the tracefree part, , and the trace part of the Lie derivative describe the gauge variations. Writing the computation of the Jacobian proceeds as above and leads to Equation (2.47) with the following replacements: is replaced with , with , and with in the integrand. By studying the dependence of on the conformal factor it has been shown in  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  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.
After a coarse graining flow for the dynamical measures has been set up the crucial issue will be whether or not it has a fixed point with a nontrivial finite-dimensional unstable manifold, describing an interacting system. In this case it would define an asymptotically safe functional measure in the sense defined in Section 1.3. For the reasons explained there the existence of an asymptotically safe functional integral masure is however neither necessary nor sufficient for a physically viable theory of Quantum Gravidynamics. For the latter a somewhat modified notion of a safe functional measure is appropriate which incorporates the interplay between couplings and observables:
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 as described in Appendix B. Here is interpreted as an initially source-dependent “expectation value of the quantum metric”, 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” , are necessarily the best choice.
The use of an initially generic background geometry has the advantage that one can define propagation and covariant mode-cutoffs with respect to it. A background effective action of this type has an interesting interplay with the notion of a state [156, 157]. An effective action implicitly specifies an expectation functional (“a state”) which depends parameterically on the background metric. The background metric is then self-consistently identified with the expectation value of the metric
The condition (2.49) is equivalent to the vanishing of the extremizing sources 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 are constructed by formal inversion of a power series in . Then always is a solution of Equation (2.49) and the functional 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 .
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 , is proportional to the minimum value of the Hamiltonian in that part of the Hilbert space spanned by normalizable states satisfying . A similar interpretation holds formally for the various background effective actions . 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 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 should know about the fact that stems from a microscopic action suited for the appropriate notion of positivity and from a physically acceptable state.
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.
For the time being a pragmatic approach is to consider quantities which are of interest in a quantum field theory on a fixed but generic geometry and then perform an average over geometries with the measure previously constructed. On a perturbative level interesting possible effects have been studied in [202, 215]. On a nonperturbative level this type of correlations have been discussed mostly in discretized formulations but the principle is of course general. To fix ideas we note the example of a geodesic two point correlator of a scalar field ,.
If one wants to probe the functional measure over geometries only, an interesting operator insertion is the trace of the heat kernel [115, 118, 8],. The states obeying it should favor geometries that are smooth and almost flat on large scales.
In a lattice field theory the discretized functional measure typically generates an intrinsic scale, the (dimensionless) correlation length , which allows one to convert lattice distances into a physical standard of length, such that say, lattice spacings equal . A (massive) continuum limit is eventually defined by sending to infinity in a way such that physical distances are kept fixed and a ‘nonboring’ limit arises. In a functional measure over geometries , initially defined with an UV cutoff and an external scale parameter , 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.
© Max Planck Society and the author(s)