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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

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 S[g,matter] be any local action, where g = (gαβ)1≤α,β≤d 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:

| ∫ -d--S[ω2g,matter ]|| = dx √g-g αβδS-[g,matter-]. (2.13 ) dω2 ω=1 δgαβ
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 -- ZN√ gR (g), the parameter that becomes inessential may be taken as its prefactor, i.e. may be identified with the inverse of Newton’s constant, via
− 1 d-−-2- d−2 ZN = 2d − 3 Vol(S )GNewton =: cdGNewton. (2.14 )
The normalization factor cd, d ≥ 4 [196], is chosen such that the coefficient in the nonrelativistic force law, as computed from √ -- ZN gR (g) + Lmatter, equals GNewton Vol(Sd− 2). For d = 2,3 a different normalization has to be adopted.

The physics interpretation of the inessential parameter ω is that it also sets the absolute momentum or spectral scale. To see this we can think of gαβ as a reference metric in the background field formalism. For example for the spectral values ν(g) of the covariant Laplacian Δg associated with gαβ one has

2 −2 ν(ω g ) = ω ν(g), (2.15 )
since Δ ω2g = ω −2Δg. The spectral values play the role of a covariant momentum squared. Indeed, if the metric is taken dimensionless, ν(g) carries dimension 2 (since Δ g does) and for a flat metric gαβ = η αβ they reduce to 2 − ν(η) = k, for plane waves labeled by k. From Equation (2.15View Equation) 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 ω, the inverse of Newton’s constant Z −1 = c G N d Newton, 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 GNewton has significance only relative to the running coefficient of some reference operator. The most commonly used choice is a cosmological constant term ∫ √ -- Λ dx g. Indeed

GNewton Λd−d2 = const τ(μ)2∕d (2.16 )
is dimensionless and invariant under constant rescalings of the metric [116Jump To The Next Citation Point]. One usually switches to dimensionless parameters via
2−d d λ(μ ) cdGNewton = μ gN (μ), Λ = 2μ -----, (2.17 ) gN(μ)
where μ is some dimension one parameter which will be taken as ‘renormalization group time’. The Einstein–Hilbert action then reads
μd−2 ∫ √ -- ------ dx g[R(g) − 2μ2λ (μ)]. (2.18 ) gN (μ)
Being dimensionless one expects the running of gN (μ) and λ(μ ) to be governed by flow equations without explicit μ dependence
∂ ∂ μ --gN = γg(gN,λ ), μ---λ = βλ(gN,λ ). (2.19 ) ∂μ ∂ μ
For the essential parameter τ(μ) = gN (μ )λ(μ)(d−2)∕2 obtained from Equation (2.16View Equation) this gives
[ ] ∂ γg d − 2βλ μ ---τ = τ ---+ -------- . (2.20 ) ∂μ gN 2 λ
Within an asympotically safe Quantum Gravidynamics this should be an asymptotically safe coupling, i.e. 
μ0s≤uμp≤∞τ (μ) < ∞, lμim→∞τ (μ) = τ∗ < ∞, (2.21 )
where here 0 < τ∗ < ∞. Given Equation (2.21View Equation) there are two possibilities. First, the various scheme choices are such that the parameters gN(μ) and λ (μ) are both nonsingular and approach finite values g∗ and λ∗ for μ → ∞. Second, the scheme choices are such that one of them becomes singular and the other vanishes for μ → ∞. Usually the first possibility is chosen; then the gN(μ) flow defined by the first equation in Equation (2.19View Equation) has all the properties required for an essential asymptotically safe coupling. This ‘nonsingular parametric representation’ of the τ (μ ) coupling flow is advantageous for most purposes.

The second possibility is realized when inserting a singular solution of the equation for gN(μ) into the equation for λ(μ). 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

1 [GNewton]d−2 = MPl ≈ 1.4 × 1019 GeV, (2.22 )
amounts to working with Planck units. From Equation (2.17View Equation) it then follows that
( ) ( ) μ d−2 μ d−2 gN(μ) = cd ---- , (d − 2)cd ---- = γg(g,λ). (2.23 ) MPl MPl
We may assume that the second equation has a local solution gN(μ ) = f(λ,μ∕MPl ). Reinserting into the λ equation gives a flow equation
-∂- ^ μ ∂k λ(μ) = βλ(λ, μ∕MPl ), (2.24 )
which now explicitly depends on μ. Writing similarly ˜τ∗ := τ∗(f(λ,μ ∕MPl),λ ) the condition defining the τ(μ ) fixed point becomes
| ^β ||= − 2 λ. (2.25 ) λ ˜τ∗
Both formulations are mathematically equivalent to the extent the inversion formula gN (μ) = f(λ,μ ∕MPl ) is globally defined. For definiteness we considered here the cosmological constant term as a reference operator, but the principle clearly generalizes.

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 [206Jump To The Next Citation Point], uses 1∕p4 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 [206Jump To The Next Citation Point]. However the 1∕p4 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 [94Jump To The Next Citation Point]. The idea is to try to maintain a 2 1∕p 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 [10Jump To The Next Citation Point] 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 (2.32View Equation).

The general classical action in d dimensions containing up to four derivatives of the metric reads

∫ [ ] √ -- --1-- -1- 2 ω-- 2 θ- S = dx g Λ − c G R + 2sC − 3sR + sE ∫ [ d N ] = dx √g-- Λ − --1--R + zR2 + yR R αβ + xR Rαβγδ . (2.26 ) cdGN αβ αβγδ
Here 2 C is the square of the Weyl tensor, E is the integrand of the Gauss–Bonnet term, and a total derivative term ∇2R has been omitted. The sign of the C2 coupling, s > 0, 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 s,ω,θ is chosen for later convenience; we follow the conventions of [59Jump To The Next Citation Point]. The parameters in the second line are related to those in the first by
sx = 1-+ θ, sy = − --2---− 4θ, sz = − ω-+ θ + -------1------. (2.27 ) 2 d − 2 3 (d − 1)(d − 2)
In d = 4 the Gauss–Bonnet term is negligible; however if dimensional regularization is used, d ⁄= 4, it is crucial to keep the term. For d = 3 both E and C2 vanish.

The perturbative quantization of Equation (2.26View Equation) proceeds as usual. Gauge fixing and ghost terms are added and the total action is expanded in powers of hαβ = gαβ − δαβ. Due to the crucial C2 term the gauge-fixed propagator read off from the quadratic part of the full action has a characteristic 4 1∕p falloff in d = 4,

( ) 1 1 1 1 G (p ) ∼ -2--2----2--= --2 -2-− -2-----2 , (2.28 ) p (p + m ) m p p + m
where m is the mass of a “wrong sign” propagating spin 2 mode [206207Jump To The Next Citation Point]. Also spin 0 modes with the “wrong sign” may occur depending on the coefficient of the R2 term [20749Jump To The Next Citation Point]. 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 [59Jump To The Next Citation Point19], proper time cut-off [83], zeta function [18]. The resulting one-loop flow equations in d = 4 for s,ω, θ are agreed upon and read
2 d-- 133- 2 (4π) μ dμs = − 10 s , 2 -d- 25-+-1098-ω-+-200ω2- (4π ) μdμ ω = − 60 s, (2.29 ) (4π)2μ-d-θ = 7(56 −-171θ)s. d μ 90
These equations have a trivial fixed point s = 0, ω = const ∗ ∗, θ = const ∗, and a nontrivial fixed point s∗ = 0, √ ----- ω∗ = − (549 ± 7 6049 )∕200, θ∗ = 56∕171. Importantly the 2 C coupling s is asymptotically free.

To describe the flow of the Newton and cosmological constants one switches to the dimensionless parameters gN and λ as in Section 2.3.1. The result obtained in Berrodo–Peixoto and Shapiro [59] via dimensional regularization reads in our conventions

2 γg = 2gN − --1---3 +-26ω-−-40ω--sgN − γ1g2 , (4π )2 12 ω N 1 [1 + 20ω2 1 + 86ω + 40ω2 ] (2.30 ) β λ = − 2 λ +----2- ----2----s2 + ---------------sλ + γ1λgN. (4π) 8ω gN 12 ω
The function γ1 depends on the choice of gauge, but in the combination (2.16View Equation), i.e. gNλ here, it drops out. The somewhat surprising 1 ∕gN term in βλ comes from a counterterm proportional to the volume but not to Λ. Observe that whenever γ1 is independent of gN and s, the flow equations (2.20View Equation, 2.30View Equation) are compatible with the existence of a non-Gaussian fixed point,
2 g∗N = ---, (2.31 ) γ1
with λ ∗ unspecified. A recent study [54Jump To The Next Citation Point] uses a specific momentum space cutoff ℛk and evaluates the Γ k effective average action to one loop using known heat kernel coefficients. The resulting flow equations are of the above form with 2 γ1 = γ1(ω) = (83 + 70ω + 8ω )∕(18π ); further two additional terms in βλ are found which fix also the value of λ∗ in terms of γ1. 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 ℛk-dependent but nonzero results for their coefficients.

The flow equations (2.20View Equation, 2.30View Equation) of course also admit the Gaussian fixed point ∗ gN = 0 = λ ∗, 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 (2.31View Equation) 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 g∗N in Equation (2.31View Equation) is always non-universal, the anomalous dimension η = γ ∕g − 2 N g N is exactly − 2 at the fixed point (2.31View Equation). The general argument for the dimensional reduction of the residual interactions outlined after Equation (1.5View Equation) can thus already be based on PT alone! Second the result (2.31View Equation) 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 [157Jump To The Next Citation Point] to compare the perturbative one-loop flow (2.20View Equation, 2.30View Equation) with the linearization of the (gN,λ ) 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 γg, βλ reduce to ratios of polynomials in gN, λ [136Jump To The Next Citation Point]. Upon expansion to quadratic order one finds

μ -d-g = 2g − --1--g2 + O (g2λ ), dμ N N (4π)2 N N (2.32 ) μ-d-λ = − 2λ + --1---gN-(1 + 2λ ) + 5---1---g2N + O (g2Nλ). dμ (4π )2 2 12(4π )4
This is of the form (2.30View Equation) at s = 0 with γ = 1∕(4π )2 1, and two additional terms in the second equation. The nontrival fixed point for gN remains of the form (2.31View Equation), while the one for λ is best seen in the τ = gN λ evolution equation, 2 2 4 3 2 μdτ∕d μ = (1 − 8λ )gN∕ (48 (4π) ) + O (λ gN ).

The most important drawback of the perturbatively renormalizable theories based on Equation (2.26View Equation) are the problems with unitarity entailed by the propagator (2.28View Equation). As already mentioned these problem are absent in an alternative perturbative formulation where a 2 1∕p type propagator is used throughout [94Jump To The Next Citation Point]. We now describe this construction in slightly more detail following the presentation in [10Jump To The Next Citation Point].

Starting from the d = 4 Lagrangian − --1- √gR (g) cdGN without cosmological constant the one-loop divergencies come out in dimensional regularization as [210]

1 √ -( 1 7 ) ---2------- g ----R2 + ---RαβR αβ . (2.33 ) 8π (4 − d ) 120 20
They can be removed in two different ways. One is by adding new couplings so that a higher derivative action of the form (2.26View Equation) arises with parameters
( ) ( ) d−4 1 1 d−4 1 7 Λ = 0, x = 0, zB = μ z − ---2----------- , yB = μ y − ---2----------(.2.34 ) 8π (4 − d )120 8π (4 − d) 20
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 z,y. The unphysical singularities are of order 1∕z, 1∕y. The second option to remove Equation (2.33View Equation) is by a singular field redefinition
c G 1 ( 11 ) gαβ ↦→ gαβ + --2-d-N------- − 7R αβ + --gαβR . (2.35 ) 8π (4 − d) 20 3
This restores the original √gR- (g) Lagrangian up to two- and higher loop contributions. However this feature is specific to one loop. As shown in [95222] at two loops there is a divergence proportional to R αβγδR αβρσRρσγδ, which cannot be absorbed by a field redefinition. A counterterm proportional to it must thus be added to √gR- (g). Importantly, when re-expanded in powers of h = g − δ α β αβ αβ, this counterterm, however, produces only terms quadratic in h that are proportional to the Ricci tensor or the Ricci scalar. These can be removed by a covariant field redefinition, so that the intial 1∕p2 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
∑ L = − --1--√gR- (g ) + (c G )2d−id−d g L (g), (2.36 ) cdGN d N i i i≥1
where Li(g) are local currvature invariants of mass dimension − di, the gi are dimensionless couplings, and the power of cdGN (normalized as in Equation (2.15View Equation)) gives each term in the sum mass dimension − d.

Let us briefly recap the power counting and scaling dimensions of local curvature invariants. These are integrals ∫ Pi[g] = ddxLi (g ) over densities Li(g) which are products of factors of the form ∇ α1 ...∇ αl−4R αl−3...αl, suitably contracted to get a scalar and then multiplied by √g--. One easily checks L (ω2g ) = ωsiL (g) i i, ω > 0, with s = d − 2p − q i, 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 Pi(g) if g is taken dimensionless. For the mass dimension di of the associated coupling ui in a product uiPi [g] one thus gets di = si = d − 2p − q. For example, the three local invariants in Equation (1.14View Equation) have mass dimensions − d0 = − d, − d1 = − (d − 2), − d = − (d − 4) 2, respectively. There are three other local invariants with mass dimension − (d − 4), namely the ones with integrands 2 αβγδ αβ 2 C = R R αβγδ − 2R Rαβ + R ∕3 (the square of the Weyl tensor), αβγδ αβ 2 E = R R αβγδ − 4R R αβ + R (the generalized Euler density), and ∇2R. 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 (2.26View Equation)).

A generic term in Pi will be symbolically of the form ∇qRp, where all possible contractions of the 4p + q indices may occur. Since the Ricci tensor is schematically of the form R = ∇2f + O (f2), the piece in Pi quadratic in f is of the form ∇q+4Rp −2f2. The coefficient of f2 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 (2.36View Equation) to terms with − d = − d + 2p + q i, p ≥ 3, and the propagator derived from it will remain of the 2 1∕p type to all loop orders. This suggests that Equation (2.36View Equation) 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 [94Jump To The Next Citation Point].

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 = μterm. In other words the couplings in Equation (2.36View Equation) 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 (2.36View Equation) are power counting irrelevant with respect to the 1∕p2 propagator. Equivalently all but a few couplings ui(μ) = μdigi(μ) have non-negative mass dimensions di ≥ 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 (2.36View Equation) these power counting irrelevant couplings with di < 0 are crucial for the absorption of infinities and thus are converted into practically relevant ones. In the context of Equation (2.36View Equation) 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 + R2 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 [31Jump To The Next Citation Point] 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.

A geometry can be described by picking a representative ˆgαβ described by a d(d − 1)∕2 parametric metric. Here ˆgαβ can be specified by picking an explicit parameterization or by imposing a gauge fixing condition (F ∘ ˆg)α = 0. Typical choices are a harmonic gauge condition with respect to some reference metric connection, or a proper time gauge (F ∘ ˆg)α = nα, for a fixed timelike co-vector nα. Once ˆgαβ has been fixed, the push forward with a generic diffeomorphism V will generate the associated orbit,

∂V-γ-∂V-δ gαβ(x ) = (V ∗ˆg)αβ(x) = ∂x α ∂xβ ˆgγδ(V (x)). (2.37 )
The functional integral over the metrics gαβ should factorize into one over the geometries ˆg and one over the diffeomorphisms,
𝒟g αβ = J (ˆg)𝒟g αβ 𝒟V α, (2.38 )
with a V independent Jacobian J(ˆg) = J (g). To compute the Jacobian one views the space of metrics locally as a pseudo-Riemannian manifold (ℳ, 𝒢) and uses the fact that the Jacobian J (g ) at g ∈ ℳ only contains information about the cotangent space ℋT (g ) at g. 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 ℳ. In analogy with the finite-dimensional case 1 = ∫ dμ (δx ) exp {iδxαg (x)δxβ} x 2 αβ, dμ (δx ) = ∘det-g(x-)∕ (2 π)d∕2 x, the measure 𝒟g αβ can then be defined by imposing a normalization condition in terms of a Gaussian functional integral on the cotangent space
∫ { i } 1 = 𝒟f αβ exp -(f,f)T , (2.39 ) 2
where the cotangent space ℋT (g) to ℳ at g is equipped with the metric
∫ ⟨f,f ⟩ = dx√ −-gf (x )𝒢αβ,γδ(g)f (x), T αβ γδ 1 (2.40 ) 𝒢 αβ,γδ(g) = --(g αγgβδ + g αδg βγ + Cg αβg γδ), 2
and the 1-forms f αβ := δgαβ span ℋT (g) (with ‘T’ mnemonic for “tensor”). The measure defined by Equation (2.39View Equation) will be formally diffeomorphism invariant provided the metric on ℳ is covariant. Requiring the metric to be ultralocal in addition fixes 𝒢 to be of the above “deWitt” form up to an overall normalization and the undetermined constant C. The latter determines the signature of the metric on ℳ, it is of Riemannian type for C > − d∕2, Lorentzian for C < − d∕2, and degenerate for C = − d∕2. Here we take C ⁄= − d∕2 but leave C unspecified otherwise.

On the tangent space the parameterization (2.37View Equation) amounts to

fαβ = (Lv )αβ + (V ∗ ˆf)αβ, (2.41 )
where Vα (x ) = xα + vα(x) + O (v2) and vα∂ α is the associated vector field. Further the Lie derivative ℒvg αβ =:(Lv)αβ is for the given ˆg regarded a linear map L from vectors to symmetric tensors. Its kernel are the Killing vectors of g, while the kernel of its adjoint † L describes the genuine variations in the geometry. Here
∫ † √ -- αβ L : ℋT (g) −→ ℋV (g), (v,v)V := dx g vα(x)g (x)vβ(x ) (2.42 )
maps symmetric tensors to (co-)vectors, the space of which is equipped with the obvious invariant inner product. With the normalization (Lv, Lv)T = (v, L†Lv )V the vector Laplacian L†L is given by
† β [ β 2 β β ] (L L)α = − 2 δα ∇ + (1 + C )∇ α∇ + Rα (g ) . (2.43 )
Under mild technical conditions L †L will have a well-defined inverse, (L †L )−1 : ℋV (g) → ℋV (g). This allows one to replace Equation (2.41View Equation) by a decomposition orthogonal with respect to ( , )T, viz. 
′ ˆ ′ : † −1 † ˆ fαβ = (Lv )αβ + (P V ∗f )αβ, v = v + (L L) L f , (2.44 ) P := 𝟙 − L(L †L)−1L †, P 2 = P, P L = 0.
Using 𝒟f = J(ˆg)𝒟v 𝒟 ˆf and Equation (2.39View Equation) the computation of the Jacobian in Equation (2.38View Equation) then reduces to that of two Gaussians
∫ { } ∫ { } i † i 1 = J (ˆg) 𝒟v exp --(v, L Lv )V) 𝒟 fˆαβ exp -(fˆ,Pfˆ)T) , (2.45 ) 2 2
which, suitably regularized, we take as the definition of J(ˆg). If ˆg is defined through a gauge fixing condition (F ∘ ˆg)α = 0 the result (2.45View Equation) can be rewritten as [31Jump To The Next Citation Point]
† − 1∕2 J(ˆg) = [deVt (F ∘ F )] deVt(F ∘ L). (2.46 )
The subscript V denotes a vector determinant defined by detV W = exp {− ∑ 1 TrV(1 − W )k} k≥1 k, where W maps vectors to vectors and the trace refers to the inner product in Equation (2.42View Equation). We remark that the second factor in Equation (2.46View Equation) is the Faddeev–Popov determinant for the gauge (F ∘ fˆ)α = 0, while the first factor is an ˆfαβ 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:

−1∫ iS [g] ∫ iS[ˆg] [VolDi ff] 𝒟 gαβ e ↦→ 𝒟 ˆgαβ det(F ∘ L )(ˆg) e . (2.47 ) ℳ ℳ ∕Diff V
Here we omitted the normalization factor and for illustration included the factor exp(iS[g]) (with an invariant action, S[V ˆg] = S [ˆg] ∗) 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 (2.47View Equation) can only serve as a useful guideline. There are two ways to proceed from Equation (2.47View Equation). 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 (2.47View Equation) 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 [31142Jump To The Next Citation Point149]. In Equation (2.37View Equation) then ˆgαβ is written as σ ⊥ e gαβ, where now ⊥ gαβ is subject to a gauge condition ⊥ (F ∘ g )α = 0. On the cotangent space this leads to a York-type decomposition [235] replacing (2.41View Equation), where the variations fσ of the conformal factor and that of the tracefree part fα⊥β of ˆfαβ describe the variations of the geometry, while the tracefree part, (LTFv )αβ := ∇ αvβ + ∇ βvα − 2gαβ∇ γvγ d, and the trace part of the Lie derivative (Lv )αβ describe the gauge variations. Writing ⊥ 𝒟f αβ = 𝒟f σ𝒟f αβ𝒟v the computation of the Jacobian proceeds as above and leads to Equation (2.47View Equation) with the following replacements: 𝒟 ˆg αβ is replaced with 𝒟g ⊥ 𝒟σ αβ, L with LTF, and ˆg with eσg⊥ in the integrand. By studying the dependence of σ ⊥ detV(F ∘ L )(e g ) on the conformal factor it has been shown in [142Jump To The Next Citation Point] 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:

  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 [gαβ,¯gαβ,...] as described in Appendix B. Here gαβ is interpreted as an initially source-dependent “expectation value of the quantum metric”, ¯gαβ 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” gαβ, ¯gαβ are necessarily the best choice.

The use of an initially generic background geometry ¯gαβ 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 [156Jump To The Next Citation Point157Jump To The Next Citation Point]. An effective action implicitly specifies an expectation functional 𝒪 ↦→ ⟨𝒪 ⟩¯g (“a state”) which depends parameterically on the background metric. The background metric is then self-consistently identified with the expectation value of the metric

⟨gαβ⟩¯g [g] = ¯gαβ, (2.48 ) ∗
in a way that encodes information about the full quantum dynamics. Importantly this self-consistent background ¯g∗[g]αβ is no longer prescribed externally, and to the extent one has access to nontrivial solutions ¯gαβ = ¯g∗[g ]αβ, Equation (2.49View Equation) gives rise to a formulation with a ‘state-dependent dynamically adjusted reference metric’. The functional g ↦→ ¯g∗[g] is defined by
δ ----Γ [g,¯g,... ] = 0 iff ¯gαβ = g¯∗[g]αβ. (2.49 ) δgαβ
We postpone the question how solutions of Equation (2.49View Equation) can be found. Note that after the identification has been made ¯Γ [g,...] := Γ [g,¯g [g],...] ∗ is a functional of a single metric only, which obeys
δ¯Γ [g,...] --------- = 0. (2.50 ) δgαβ
Vertex functions are defined by functional differentiation at fixed background with subsequent identification (2.48View Equation), i.e. 
(n) δ δ || Γ (x1,...,xn;g) := ------... ------Γ [g,g¯, ...]| . (2.51 ) δg(x1) δg(xn) ¯g=¯g∗[g]
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 (2.48View Equation), but their precise physics significance remains to be understood.

The condition (2.49View Equation) is equivalent to the vanishing of the extremizing sources αβ J ∗ [g,¯g,...] in the definition of Legendre transform (see Appendix B). Evidently Equation (2.49View Equation) also amounts to the vanishing of the one-point functions in Equation (2.51View Equation). Usually the extremizing sources J α∗β[g,¯g,...] are constructed by formal inversion of a power series in f¯αβ := gαβ − ¯gαβ. Then f¯αβ = 0 always is a solution of Equation (2.49View Equation) and the functional g ↦→ ¯g [g] ∗ is simply the identity. In this case the self-consistent background coincides with the naive prescribed background. To find nontrivial solutions of Equation (2.49View Equation) 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 (2.49View Equation) is a nontrivial problem. The interpretation via Equation (2.48View Equation) suggests an indirect characterization, namely those solutions of Equation (2.49View Equation) should be regarded as physical which come from physically acceptable states [157Jump To The Next Citation Point].

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 φi = ⟨χi⟩, is proportional to the minimum value of the Hamiltonian H in that part of the Hilbert space spanned by normalizable states |ψ⟩ satisfying i i ⟨ψ |χ |ψ⟩ = φ. A similar interpretation holds formally for the various background effective actions [50Jump To The Next Citation Point]. 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 (2.53View Equation) 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 (2.48View Equation) or similar objects computed from the vertex functions (2.51View Equation) 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 [202215]. 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 φ [58],

∫ ∫ iS[g,φ] ∘ -----∘ ----- G (R ) = 𝒟g 𝒟 φ e dxdy g(x ) g(y)φ (x )φ(y)δ(Σg (x,y) − R), (2.52 )
where Σg(x,y ) is the minimal geodesic distance between the the points x and y. The first integral is the heuristic geometry and matter functional integral; all configurations are taken into account which produce the given geodesic distance R. A nontrivial prediction of the present scenario is that if Equation 2.52View Equation) is based on an asymptotically safe functional measure, a log(R ) behavior for R → 0 is expected [157Jump To The Next Citation Point].

If one wants to probe the functional measure over geometries only, an interesting operator insertion is the trace of the heat kernel [115Jump To The Next Citation Point118Jump To The Next Citation Point8Jump To The Next Citation Point],

∫ ∫ iS[g] ∘ ----- G(T ) = 𝒟g e dx g (x )exp(T Δg )(x, x). (2.53 )
Here Δg := gαβ∇ α∇ β = √g-−1∂α (√gg- αβ∂β) is the Laplace–Beltrami operator, and the heat kernel exp (TΔ )(x,x′) g associated with it is the symmetric (in x,x ′) bi-solution of the heat equation ∂T K = ΔgK with initial condition ′ ′ limT →0 exp (T Δg )(x,x ) = δ (x, x ). The T → ∞ limit will then probe the large scale structure of the typical geometries in the measure and the T → 0 limit will probe the micro aspects. Both expressions (2.52View Equation, 2.53View Equation) 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 T → ∞ behavior of G (T ) = ⟨Pg (T)⟩ is like that in flat space, −d∕2 ⟨Pg(T )⟩ ∼ T for T → ∞, is an example for a (rather weak) selection criterion for states [157Jump To The Next Citation Point]. 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 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 dμk,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 (2.52View Equation) or Equation (2.53View Equation) 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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