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:

As a consequence one of the coupling parameters which in the absence of gravity would be 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 The normalization factor , [196], 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

since . The spectral values play the role of a covariant momentum squared. Indeed, if the metric is taken dimensionless, carries dimension (since does) and for a flat metric they reduce to , for plane waves labeled by . From Equation (2.15) one sees that rescaling of the metric and rescaling of the spectral values amout to the same thing. Since the former parameter is inessential the latter is too. Hence in a theory with a dynamical metric the three (conceptually distinct) inessential parameters – overall scale of the metric , the inverse of Newton’s constant , and the overall normalization of the spectral/momentum values – are in one-to-one correspondence. For definiteness we take Newton’s constant as the variant under consideration.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

is dimensionless and invariant under constant rescalings of the metric [116]. One usually switches to dimensionless parameters via where is some dimension one parameter which will be taken as ‘renormalization group time’. The Einstein–Hilbert action then reads Being dimensionless one expects the running of and to be governed by flow equations without explicit dependence For the essential parameter obtained from Equation (2.16) this gives Within an asympotically safe Quantum Gravidynamics this should be an asymptotically safe coupling, i.e. where here . Given Equation (2.21) there are two possibilities. First, the various scheme choices are such that the parameters and are both nonsingular and approach finite values and for . Second, the scheme choices are such that one of them becomes singular and the other vanishes for . Usually the first possibility is chosen; then the flow defined by the first equation in Equation (2.19) has all the properties required for an essential asymptotically safe coupling. This ‘nonsingular parametric representation’ of the coupling flow is advantageous for most purposes.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

amounts to working with Planck units. From Equation (2.17) it then follows that We may assume that the second equation has a local solution . Reinserting into the equation gives a flow equation which now explicitly depends on . Writing similarly the condition defining the fixed point becomes Both formulations are mathematically equivalent to the extent the inversion formula is globally defined. For definiteness we considered here the cosmological constant term as a reference operator, but the principle clearly generalizes.

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 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 [206]. 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 [94]. 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 [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 general classical action in dimensions containing up to four derivatives of the metric reads

Here is the square of the Weyl tensor, is the integrand of the Gauss–Bonnet term, and a total derivative term has been omitted. The sign of the coupling, , is fixed by the requirement that the Euclidean functional integral is damping. The metric is Euclidean to facilitate comparison with the original literature. The parameterization of the coefficients by couplings is chosen for later convenience; we follow the conventions of [59]. The parameters in the second line are related to those in the first by In the Gauss–Bonnet term is negligible; however if dimensional regularization is used, , it is crucial to keep the term. For both and vanish.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 ,

where is the mass of a “wrong sign” propagating spin 2 mode [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 [83], zeta function [18]. The resulting one-loop flow equations in for are agreed upon and read These equations have a trivial fixed point , , and a nontrivial fixed point , , . Importantly the coupling is asymptotically free.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 [59] via dimensional regularization reads in our conventions

The function depends on the choice of gauge, but in the combination (2.16), i.e. here, it drops out. The somewhat surprising term in comes from a counterterm proportional to the volume but not to . Observe that whenever is independent of and , the flow equations (2.20, 2.30) are compatible with the existence of a non-Gaussian fixed point, with unspecified. A recent study [54] 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 [54] 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 [157] 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 , [136]. Upon expansion to quadratic order one finds

This is of the form (2.30) at with , and two additional terms in the second equation. The nontrival fixed point for remains of the form (2.31), while the one for is best seen in the evolution equation, .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 [94]. We now describe this construction in slightly more detail following the presentation in [10].

Starting from the Lagrangian without cosmological constant the one-loop divergencies come out in dimensional regularization as [210]

They can be removed in two different ways. One is by adding new couplings so that a higher derivative action of the form (2.26) arises with parameters The renormalizability of the resulting theory is mostly due to the modified propagator which can be viewed as a resummed graviton propagator in a power series in . The unphysical singularities are of order , . The second option to remove Equation (2.33) is by a singular field redefinition This restores the original Lagrangian up to two- and higher loop contributions. However this feature is specific to one loop. As shown in [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 [10] shows that this property also holds for all higher order counterterms that can be expected to occur. Explicitly, consider a Lagrangian of the form where are local currvature invariants of mass dimension , the are dimensionless couplings, and the power of (normalized as in Equation (2.15)) gives each term in the sum mass dimension .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 [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 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 [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 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,

The functional integral over the metrics should factorize into one over the geometries and one over the diffeomorphisms, with a independent Jacobian . To compute the Jacobian one views the space of metrics locally as a pseudo-Riemannian manifold and uses the fact that the Jacobian at only contains information about the cotangent space at . Moreover picking coordinates on the cotangent space the Jacobian produced by a linear change of coordinates in the cotangent space will coincide with the Jacobian induced by a corresponding nonlinear change of coordinates on . In analogy with the finite-dimensional case , , the measure can then be defined by imposing a normalization condition in terms of a Gaussian functional integral on the cotangent space where the cotangent space to at is equipped with the metric and the 1-forms span (with ‘T’ mnemonic for “tensor”). The measure defined by Equation (2.39) will be formally diffeomorphism invariant provided the metric on is covariant. Requiring the metric to be ultralocal in addition fixes to be of the above “deWitt” form up to an overall normalization and the undetermined constant . The latter determines the signature of the metric on , it is of Riemannian type for , Lorentzian for , and degenerate for . Here we take but leave unspecified otherwise.On the tangent space the parameterization (2.37) amounts to

where and is the associated vector field. Further the Lie derivative is for the given regarded a linear map from vectors to symmetric tensors. Its kernel are the Killing vectors of , while the kernel of its adjoint describes the genuine variations in the geometry. Here maps symmetric tensors to (co-)vectors, the space of which is equipped with the obvious invariant inner product. With the normalization the vector Laplacian is given by Under mild technical conditions will have a well-defined inverse, . This allows one to replace Equation (2.41) by a decomposition orthogonal with respect to , viz. Using and Equation (2.39) the computation of the Jacobian in Equation (2.38) then reduces to that of two Gaussians which, suitably regularized, we take as the definition of . If is defined through a gauge fixing condition the result (2.45) can be rewritten as [31] The subscript denotes a vector determinant defined by , where maps vectors to vectors and the trace refers to the inner product in Equation (2.42). We remark that the second factor in Equation (2.46) is the Faddeev–Popov determinant for the gauge , while the first factor is an independent normalization factor. Within perturbation theory the above construction is equivalent to the familiar BRST formulation with ghosts.In summary one arrives at the following proposal for a kinematical measure over geometries:

Here we omitted the normalization factor and for illustration included the factor (with an invariant action, ) that would specify the dynamical measure. The kinematical measure on the right-hand-side can also directly be verified to be diffeomorphism invariant and is hence well-defined on the equivalence classes. Of course the latter presupposes an invariant regulator which is why Equation (2.47) can only serve as a useful guideline. There are two ways to proceed from Equation (2.47). One would work with an noninvariant regulator, maintain the original notion of diffeomorphisms, and use conventional field theoretical techniques to restore the diffeomorphism invariance through Ward identites at the end. Alternatively one can replace the right-hand-side of Equation (2.47) directly with a discretized version, in which case of course diffeomorphism invariance cannot be tested on this level. Both strategies are complementary and have been widely used. For completeness let us also mention at this point the well-known feature of functional integrals that once the regulator is removed the kinematical measure and the action factor do not have a mathematical meaning individually. In an interacting theory this is also related to the renormalization problem.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 [235] 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 [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.

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:

- 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.

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

in a way that encodes information about the full quantum dynamics. Importantly this self-consistent background is no longer prescribed externally, and to the extent one has access to nontrivial solutions , Equation (2.49) gives rise to a formulation with a ‘state-dependent dynamically adjusted reference metric’. The functional is defined by We postpone the question how solutions of Equation (2.49) can be found. Note that after the identification has been made is a functional of a single metric only, which obeys Vertex functions are defined by functional differentiation at fixed background with subsequent identification (2.48), i.e. The set of these vertex functions in principle contains the same information as the original functional measure including the state. One would expect them to be related to S-matrix elements on a self-consistent background (2.48), but their precise physics significance remains to be understood.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 [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 , 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 [50]. 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 [58],

where is the minimal geodesic distance between the the points and . The first integral is the heuristic geometry and matter functional integral; all configurations are taken into account which produce the given geodesic distance . A nontrivial prediction of the present scenario is that if Equation 2.52) is based on an asymptotically safe functional measure, a behavior for is expected [157].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],

Here is the Laplace–Beltrami operator, and the heat kernel associated with it is the symmetric (in ) bi-solution of the heat equation with initial condition . The limit will then probe the large scale structure of the typical geometries in the measure and the limit will probe the micro aspects. Both expressions (2.52, 2.53) are here only heuristic, in particular normalization factors have been omitted and the functional measure over geometries would have to be defined as previously outlined. The condition that the behavior of is like that in flat space, for , is an example for a (rather weak) selection criterion for states [157]. 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.

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