Quantum Gravity: A Fluctuating Point of View

1 Introduction

One of the major challenges in theoretical physics is the unification of the standard model of particle physics (SM) with quantum gravity. Based on the classical Einstein–Hilbert action, gravity is perturbatively nonrenormalizable and hence cannot be expanded about a vanishing gravitational coupling, the Newton coupling. A very promising way out has been proposed by Weinberg [1], the asymptotic safety scenario. It draws from the theory of critical phenomena developed for investigating the phase structure of condensed matter and statistical systems. In the language of critical phenomena, the standard perturbation theory about a vanishing Newton coupling is an expansion about the free, Gaußian fixed point of the theory and fails since this fixed point is ultraviolet (UV)-repulsive in the relevant couplings. In turn, the asymptotic safety scenario builds upon the conjecture that quantum gravity also exhibits a nontrivial UV fixed point, the Reuter fixed point. This asymptotically safe fixed point should exhibit a finite-dimensional critical hypersurface, which renders the theory finite and predictive even beyond the Planck scale.

The method of choice for respective investigations is the renormalization group. Most investigations of asymptotically safe gravity have been performed with the functional renormalization group (fRG) in its form for the effective action [2]. The fRG approach to quantum gravity has been initiated by the seminal work [3], where the UV fixed point has been studied in the Einstein–Hilbert truncation. In this approximation, one retains only two couplings, the Newton coupling G_N and the cosmological constant $\text{Λ}$. Already this basic truncation exhibits a UV fixed point in four dimensions [3, 4]. This exciting finding has triggered a plethora of works for asymptotically safe gravity with and without matter. (We refer the reader to the textbooks [5, 6] and reviews [7–15]. For very recent accounts of the challenges for asymptotically safe gravity, see 16, 17. For generic reviews on the fRG, we refer to 18–27.)

The fRG approach to gravity centers around the quantum effective action of the theory $\text{Γ}[{\overline{g}}_{\mu \nu },h_{\mu \nu }]$, the quantum analog of the classical action. Here, ${\overline{g}}_{\mu \nu }$ is a generic metric background and the graviton field $h_{\mu \nu }$ accounts for quantum fluctuations about this background. The computation of the effective action $\text{Γ}[{\overline{g}}_{\mu \nu },h_{\mu \nu }]$ is tantamount to that of the path integral: the n-point correlation functions of the dynamical fluctuation field h are given by n derivatives of the effective action with respect to the correlation field, evaluated on the equations of motion, $h=0$ and $\overline{g}={\overline{g}}_{\text{EoM}}$, that is, on-shell. These correlation functions are nothing but the moments of the path integral and carry the dynamics of the quantum theory.

This seemingly introduces a background dependence of the approach. However, the approach has inherent on-shell background independence, also related to physical diffeomorphism invariance. Indeed, the background effective action $\text{Γ}\left[g_{\mu \nu }\right]=\text{Γ}[g_{\mu \nu },0]$ is diffeomorphism-invariant. The latter properties are the backbones of any quantum gravity approach, and their realization even within approximations is chiefly important.

The present review outlines the properties and results of the fRG approach to asymptotically safe quantum gravity in terms of background and fluctuation correlation functions of gravitons, shortly baptized the fluctuation approach to gravity. This approach is based on the observation that the dynamics of quantum gravity is encoded in the correlation functions of the fluctuation field h. Reliable computations of observables can only be done from these correlation functions. This situation calls for a systematic improvement of the standard background-field approximation. In this approximation, the correlation functions of the background metric and the fluctuation field are identified. We refrain from going into more details here, the underlying assumptions and challenges are discussed in Sections 5 and 6.

The fluctuation approach resolves these differences, and by now, it has matured enough to host a large number of results: this includes investigations of the Reuter fixed point in pure gravity in a rather elaborate truncation within a vertex expansion with momentum-dependent two-, three-, and four-point functions; the computation of the background-effective action for backgrounds with constant curvature; investigations of the stability of general gravity–matter systems; investigation of convergence properties of the expansion (apparent convergence); and a potential close perturbativeness of the asymptotically safe UV regime (effective universality). (We refer the reader to Section 8 for an explanation of the terminology and respective results.)

In Section 2, we discuss the general quantum field theory setting of quantum gravity, which we use for the fluctuation approach. This includes a discussion of the necessary gauge fixing and background independence of the approach. In Section 3, we discuss general parametrizations of the full metric in terms of a metric background and a fluctuation field. The preparation in Sections 2 and 3 allows us to introduce the fRG approach to quantum gravity in Section 4 as well as discussing the standard approximation used in the field, the background field approximation, in Section 5. The symmetry identities that relate the dynamical correlation functions of the fluctuation field and those of the background metric are discussed in Section 6. These symmetry identities imply the necessity to go beyond the background field approximation, and thus, we detail the fluctuation approach in Section 7. With the preparation of the sections before, we discuss the results of the fluctuation approach in Section 8 and close with a short conclusion and outlook in Section 9.

2 Quantum Field Theory Approach to Quantum Gravity

The present contribution discusses the advances and open problems of a quantum field theory approach to quantum gravity that is based on the computation of metric correlation functions or, more generally, correlation functions of operators in quantum gravity. Formally, such an approach is based on the existence of a path integral for quantum gravity, for example, defined by the integration over the space of all metrics $\left\{g_{\mu \nu }\right\}$ with a specific classical action for gravity $S_{\text{grav}}$, a standard choice being the Einstein–Hilbert action,

with the abbreviation $g=\text{det}{g}_{\mu \nu }$. In 1, we have introduced the Newton coupling G and the cosmological constant $\text{Λ}$. R stands for the Ricci scalar. In most works in the fRG approach, the theory is considered in its Euclidean version, which is indicated here by the missing minus sign in the square root of the determinant. The expectation value of a diffeomorphism invariant operator $\mathcal{O}\left[g_{\mu \nu }\right]$ is formally given by

Here and in the following, ^{^} indicates the fields that are integrated over. The formal definition 2 faces several problems. Some of them are standard problems of the quantization of gauge theories, and some of them are specific to quantum gravity. The latter problems include, for example, the lack of perturbative renormalizability of gravity for $S_{\text{EH}}$ [28–31], the apparent unitarity problems for higher derivative gravity a la Stelle [32–34], and the question whether the integration measure $\mathcal{D}\widehat{g}$ includes a sum over all topologies [35]. The latter question is also an eminent one in lattice gravity (see, e.g., 36–42). Note in this context that a general measure $\mathcal{D}\mu \left(\widehat{g}\right)$ can always be absorbed with a change of the gravity action in 2,

with a potentially nonlocal action $\text{Δ}{S}_{\text{grav}}$. In the fRG approach, the task of a finite definition of 2 and its computation is turned into the task of solving a flow equation for the quantum effective action $\text{Γ}[\overline{g},\varphi ]$. Here, ${\overline{g}}_{\mu \nu }$ is the background or reference metric, and ϕ are fluctuation fields, the expectation values of the fluctuation field operators $\widehat{\varphi }$. The latter includes the fluctuation field ${\widehat{h}}_{\mu \nu }$ of the metric, $g_{\mu \nu }=g_{\mu \nu }\left(\overline{g},h\right)$, and potential matter fields ${\varphi }_{\text{mat}}$ and auxiliary fields such as the ghosts $c_{\mu }$ of the gauge fixing in gravity,

In the case of further gauge fields, one may also use background fields for the gauge fields, which are suppressed here for the sake of convenience. A reparameterization $g_{\mu \nu }\left(\overline{g},h\right)$ seemingly introduces a background-metric dependence of the formulation. This is common to many approaches to quantum gravity due to the necessity of defining metric fluctuations. Accordingly, the question of background independence of the present approach is an eminent one and is discussed later. Here, we only want to mention the most common split between the background metric and the fluctuation field, the linear split,

This split also underlies most of the results discussed in Section 8. Note that from now on the lowering and raising of indices is done with the background metric $\overline{g}$, if not specified otherwise. Equation 5 emphasizes one specific problem with the background field approach in quantum gravity: while $g_{\mu \nu }$ and ${\overline{g}}_{\mu \nu }$ are metrics, their difference $h_{\mu \nu }=g_{\mu \nu }-{\overline{g}}_{\mu \nu }$ is not. Indeed $h_{\mu \nu }$ has no geometrical meaning at all. This is discussed in more detail in Section 3.

2.1 Gauge Fixing

In gauge theories such as gravity with the diffeomorphism (gauge) group or the simpler case of non-abelian gauge theories, the practical computation of observables 2 faces the gauge group redundancy in the path integral measure. While this redundancy is a finite-dimensional one within discrete lattice formulations, it is an infinite-dimensional one in functional approaches based on graviton correlation functions. In particular, it prohibits the straightforward definition of the propagator, which is key in most functional approaches.

Therefore, most of the latter approaches require a gauge fixing. (For a brief discussion of gauge-invariant functional approaches, see Section 6.3.) Put differently, we have to choose a parametrization of the theory. Typically, this is done with a linear gauge fixing for the fluctuation field $h_{\mu \nu }$ that carries the metric degrees of freedom,

A common gauge fixing condition $F_{\mu }$ is given by

where $\overline{\nabla }$ is the covariant derivative with the background metric ${\overline{g}}_{\mu \nu }$. The gauge fixing 7 is introduced in the path integral with the Faddeev–Popov trick and the Jacobi determinant of the respective reparameterization. The Faddeev–Popov determinant ${\text{Δ}}_{\text{FP}}$ can be rewritten in terms of a fermionic path integral with the ghost fields $c_{\mu }$ and ${\overline{c}}_{\mu }$. The ghost action related to 7 reads

with the Faddeev–Popov operator

Again, ${\text{Δ}}_{\text{FP}}$ is the covariant derivative with the background metric ${\overline{g}}_{\mu \nu }$, while $\nabla$ is that with the full metric $g_{\mu \nu }$. Note that $M_{\mu \nu }$ is linear in the fluctuation field h. The background metric ${\overline{g}}_{\mu \nu }$ cannot be avoided, and both gauge fixing and ghost action depend on it. This implies that also the quantum effective action depends on both metrics, the background metric ${\overline{g}}_{\mu \nu }$ and the full metric $g_{\mu \nu }\left(\overline{g},h\right)$, as we shall see later. Note however that the correlation functions of diffeomorphism-invariant operators and the solutions to the quantum equations of motion do not depend on the gauge fixing. Hence, they are background-independent as explained below.

2.2 Background Independence

Background independence of the construction is more than a formal property to aim for. We briefly recollect the standard arguments for background independence in the background field approach to quantum gauge field theories. We first restrict ourselves to pure gravity. Seemingly, background dependence of the path integral is introduced by gauge fixing such as 6 and the respective Faddeev–Popov determinant ${\text{Δ}}_{\text{FP}}$. The latter is the Jacobian of the reparameterization of the path integral in terms of gauge-fixed fields. We emphasize that the gauge fixing should be rather understood as a specific choice of field coordinates in the configuration space that facilitates the integration. The Faddeev–Popov trick is nothing but a convenient way to introduce these coordinates. In any case, it leads us to the expectation values of diffeomorphism-invariant operators defined in 2 for pure gravity with a path integral with the gauge-fixed action,

Note that integration over the diffeomorphism group (from the Faddeev–Popov trick) has been factored out in the numerator and denominator. This relies on the diffeomorphism-invariance of $S_{\text{grav}}$, $\mathcal{O}\left[\widehat{g}\right]$, and $\mathcal{D}\widehat{g}$. The full integration measure $\mathcal{D}\widehat{\varphi }$ in 10 now also includes the ghost fields, $\widehat{\varphi }=\left({\widehat{h}}_{\mu \nu },{\widehat{c}}_{\mu },{\widehat{\overline{c}}}_{\mu }\right)$. Naturally, the right-hand side in 10 is independent of the background field as the left-hand side trivially is (see 2). This background-metric independence is captured in the Nielsen or split Ward identity derived from taking a ${\overline{g}}_{\mu \nu }$ derivative of 10. Typically, one also subtracts the Dyson–Schwinger equation for $\langle \mathcal{O}\rangle$, which reads schematically

This leads us to the Nielsen identity for general diffeomorphism-invariant operators $\mathcal{O}\left[g\right]$ with

If solving the path integral within approximations, the check of the Nielsen identity 12 is crucial as it carries the physical background independence.

The identity 12 constitutes infinitely many relations for diffeomorphism-invariant correlation functions and can be rephrased in terms of derivatives of the effective action. Correlation functions are conveniently derived from the generating functional $Z[\overline{g},J]$ obtained by adding source terms for the fluctuation fields to the exponent in the path integral,

where $J=\left(J_{h_{\mu \nu }},J_{c_{\mu }},J_{{\overline{c}}_{\mu }},J_{\text{mat}}\right)$ and the normalization $\mathcal{N}$ is the denominator in 10. Lowering and rising the field indices are done with the metric ${\gamma }^{ab}$ in field space. (For details, see Supplementary Material.)

In 13, the action $S=S[\overline{g},\widehat{\varphi }]$ is the “classical” action of the gravity–matter system under consideration. The gauge fixing action $S_{\text{gf}}$ and the ghost action $S_{\text{gh}}$ of the full gravity–matter system may include further gauge fixings of gauge fields. Note that for gravity–matter systems, the “classical” action may not be based on the Einstein–Hilbert action of general relativity as discussed before. More generally also, the matter part may not simply be that of a standard renormalizable QFT in the presence of a dynamical metric background.

The generating functional $Z[\overline{g},J]$, or rather the Schwinger functional $\text{log}[\overline{g},J]$, generates connected n-point correlation functions of the fluctuation field with

where the indices $a_i$ stand for Lorentz and internal indices as well as species of fields. The subscript _con in 14 indicates the connected part of the correlation function. We have included a factor of $1/\sqrt{\overline{g}}$ in the definition of the functional derivative (see Supplementary Material). This cancels the $\sqrt{\overline{g}}$ factor in the space-time integral in the source term of 13. If instead, we had used $\sqrt{\widehat{g}}$ in the source term, derivatives with respect to the current J would generate infinite-order correlation functions.

Note that the generating functional 13 can be expressed with the right-hand side of 10 with the operator $\mathcal{O}=e^{{{\displaystyle \int }}^{\text{}}{\text{d}}^{4}x\sqrt{\overline{g}}{J}^a{\widehat{\varphi }}_a}$. However, this operator is neither diffeomorphism-invariant nor background-independent. For that reason, it cannot be mapped into a manifestly background-independent form such as 10. For $J=0$, we have $\mathcal{O}=1$ and $Z=1$, which is trivially background-independent. Accordingly, for $J\ne 0$, the gauge-fixed generating functional $Z[\overline{g},J]$ is background-dependent, as is the effective action $\text{Γ}[\overline{g},\varphi ]$,

For the relation 15, we have used that the effective action $\text{Γ}$ is the Legendre transformation of the Schwinger functional $W[\overline{g},J]=\text{log}Z[\overline{g},J]$. This leads to

with the fermion number $s_a=1$ for fermions and $s_a=0$ for bosons. Then, background-independence is achieved on the fluctuation field equations of motion (EoM) for $J[\overline{g},\varphi ]=0$. The on-shell vanishing of the currents entails that all diffeomorphism-invariant quantities are background-independent on-shell, and this independence is carried by 12.

An important consequence of background-independence is the equivalence of the solutions ${\overline{g}}_{\text{EoM}}\left[\varphi \right]$ to the fluctuation field EoM and the background field EoM,

with

If the fluctuation EoM holds, the current J is vanishing, and hence, the background EoM is nothing but the Nielsen identity 12. In turn, if the background EoM holds, the current J necessarily vanishes.

3 Field Parametrizations

So far, we have not specified the relation between the background metric and the full metric $g\left(\overline{g},h\right)$, which defines the role of the fluctuation field h. While most of the computations are done within the linear split 5, it is worth discussing the general case. This not only allows us to achieve a better understanding of the linear split but also allows us to discuss the challenges for manifestly diffeomorphism-invariant formulations.

The importance of the different splits for the path integral has been already mentioned in the context of the path integral measure (see the introduction of Section 2 around 2). In the flow equation approach to quantum gravity detailed in the next section (Section 4), the discussion of the path integral measure translates into that of the ordering of fluctuations: the fRG approach to quantum gravity is based on a Wilsonian successive integrating out of quantum fluctuations. In its form of a flow equation for the quantum effective action, $\text{Γ}[\overline{g},\varphi ]$ is has a simple form in terms of the full field-dependent fluctuation field propagator $G[\overline{g},\varphi ]$ of the theory (see 30). This is the connected part of the two-point function of the fluctuation field,

The definition 19 requires a gauge fixing (or reparameterization), as discussed in the previous section. Moreover, the Wilsonian cutoff regularizes the spectrum of the propagator. Consequently, the fRG approach crucially depends on the split of the full metric g into the background metric $\overline{g}$ and the fluctuation field h for two reasons:

Ordering of fluctuations: The quantum fluctuations of the fluctuation field h are successively integrated out and are ordered in terms of the background covariant Laplacian. Therefore, the meaning of this ordering depends on the chosen split.

Relevance of higher order correlations: The physics included with higher order correlation functions crucially depends on the chosen split. Thus, a different split orders quantum fluctuations differently. This leads to potentially qualitative differences for the convergence of a given approximation scheme.

In this section, we briefly introduce and discuss the different splits considered so far in the fRG approach to asymptotically safe quantum gravity.

3.1 Linear Split

We begin with the standard and simplest split, the linear split (see also 5). It is given by

The Jacobian of this transformation is unity, and the path integral measures agree. As mentioned before, with such a definition, the fluctuation field $h=g-\overline{g}$ is not a metric and has no geometrical meaning in the configuration space of metrics. Still, it is the natural choice as it facilitates explicit computations and the implementation of the quantization of the theory for a given classical action $S_{\text{grav}}$ on the space of metrics. Still, its lack of a geometrical interpretation makes it difficult to discuss the reparameterization invariance of the theory and the consequences of background independence. (For more details, see Section 6.) These intricacies have led to more elaborated splits based on the fiber bundle structure of the configuration space of metrics.

3.2 Exponential Split

In recent years, the exponential split has attracted some attention [43–59]. It is given by

The full metric is proportional to the exponential of the fluctuation field h indicating a Lie algebra nature of the fluctuation field h. Note that the parametrization 21 restricts the metric g, and in particular it does not allow for signature changes. Therefore, it is potentially not a reparameterization of the path integral in terms of integration over all metrics but a definition of another candidate for quantum gravity. Moreover, the assumption may change the integration. In summary, it is unclear whether a path integral with the exponential split and the measure $\mathcal{D}\widehat{h}$ describes the same quantum theory as that with the measure $\mathcal{D}\widehat{g}$. This parametrization is also linked to unimodular gravity (see, e.g., [60–66]).

3.3 Geometrical Split

We briefly describe the geometrical approach to quantum gravity pioneered by Vilkovisky and DeWitt (see, e.g., [67–70]). In the fRG approach to gravity, it has been discussed in 21, 71–75. It is a general framework, and all parametrizations used in the literature can be understood as different choices for the geometrical structure of the configuration space of metrics $g_{\mu \nu }$. This also allows for a better understanding of the Wilsonian integrating out of qua ntum fluctuations underlying the different splits.

In the linear split, as discussed in Section 3.1, the fluctuation field h neither is a metric nor does it have a geometrical interpretation in the configuration space $\text{Φ}$. In turn, in the geometrical approach, the fluctuation field is constructed such that it has a geometrical meaning. The background metric and the full metric are linked by geodesics with respect to a given connection in the configuration space. The Vilkovisky connection ${\text{Γ}}_{\text{V}}$ is a specifically useful one: it is constructed with the demand of maximal orthogonality between the diffeomorphism fiber in the configuration space and the base space. If such disentanglement is achieved, the path integral and the effective action only depend on the propagating degrees of freedom and the gauge redundancies are completely removed. This leads to the following conditions,

where

is the Riemannian metric on the quotient space $\text{Φ}/\mathcal{G}$, where $\mathcal{G}$ is the group of diffeomorphisms. This quotient space is labeled with capital Latin letters $A,B,C,\dots .$, while the diffeomorphism fiber is labeled with Greek letters $\alpha ,\beta ,\gamma ,\dots$. The full space is labeled with small Latin letters $a,b,c,\dots$. (For further details on the notation and the setup in the context of RG gravity, see e.g., 72.)

The background metric $\overline{g}$ and the full metric g are connected by a geodesic. With the Vilkovisky connection, the fluctuation field is a tangent vector on this geodesic at the background metric (Gaußian or geodesic normal coordinates). This is illustrated in Figure 1 and leads to

FIGURE 1. Illustration of the configuration space of metrics with the Vilkovisky connection. The background metric $\overline{g}$ and the full metric g are connected by geodesics. The fluctuation field $h^a$ is a tangent vector of these geodesics at the background metric. $h^A$ is the projection on the base space, while $h^{\alpha }$ is the projection on the diffeomorphism fiber. The effective action depends only on $h^A$ and not on $h^{\alpha }$.

The relation between g and $\overline{g}$ is nonpolynomial. Still, the Jacobian does not depend on the fluctuation field, and we have dropped it in 24. In this setting, it can be shown that the effective action $\text{Γ}$ only depends on the projection $h^A$ of the tangent vector $h^a$ onto the base space of the fiber bundle: $\text{Γ}=\text{Γ}[\overline{g},h^A]$. In turn, the projection of $h^a$ onto the diffeomorphism fiber, $h^{\alpha }$, drops out. Hence, the effective action is diffeomorphism-invariant as $h^A$ is a diffeomorphism scalar. Trivially, an infrared (IR) regularization of the $h^A$-path integral is diffeomorphism-invariant.

We close this section with some remarks on the implications of such a geometrical setup for “physical” gauge fixings, linear and exponential splits, and locality. The geometrical construction comes as close as possible to the definition of the configuration space of a gauge theory in terms of “physical” gauge-invariant fields and correlation functions. Such a parameterization is tantamount to a specific gauge fixing as already mentioned in Section 2.2. We may call such a gauge fixing “physical,” having in mind that it removes most of the redundancies related to the gauge group, in gravity that related to the diffeomorphism group. Note however that the terminology “physical gauge fixing” is not well-defined and also used differently in other contexts. In non-abelian gauge theories, the projection is unique and singles out the Landau–DeWitt gauge as the “physical” one. In gravity, one is left with a one-parameter family of gauges with the gauge fixing parameter β (see 7).

It is worth emphasizing that a gauge fixing condition for the geometrical field (or Gaußian normal field) $h^a$ is different from that for the fluctuation field h in the linear split. Only for specific choices of the latter, the maximal disentanglement of the geometrical construction is manifestly obtained. We also remark that the linear split is obtained by using a vanishing connection, hence entirely ignoring the geometrical structure of the configuration space. The exponential split simply uses the Riemannian part ${\text{Γ}}_g$ of the configuration space, hence ignoring the diffeomorphism group.

Finally, the geometrical construction with the Vilkovisky connection is highly nonlocal in configuration space, one of the ensuing problems being caustics and Gribov copies. This also raises the question of locality in the configuration space and that of momentum locality of the correlation functions of the geometrical fluctuation field h. The latter is discussed in detail in Section 7.4. Both locality issues highlight the challenges for manifest gauge- or diffeomorphism-invariant functional approaches to quantum gravity.

4 Flow Equation for Gravity

With the quantum field theory approach to quantum gravity outlined in the last sections, we are now in the position to discuss the flow equation approach to gravity. (For reviews, see 7–17, and for generic fRG reviews, see 18–27.) As already mentioned in the introduction of Section 3, the fRG approach to gravity is based on a successive integrating out of quantum fluctuations. Typically, this is done with an ordering of quantum fluctuations in momentum space: the regulator introduces a suppression of low-momentum fluctuations below an IR cutoff scale $p^2\mathrm{\lesssim }{k}^2$, and one RG step with $k\to k-\text{Δ}k$ relates to the integration of momentum modes $p^2\approx k^2$. In gravity, the implementation of such a momentum cutoff necessitates the choice of a background metric $\overline{g}$, and the (covariant) momenta are those related to the covariant Laplacian in the background metric, ${\text{Δ}}_{\overline{g}}$, with the spectral values $p_{\overline{g}}^2$.

Remarkably, the flow equation is insensitive to field reparameterizations of quantum gravity discussed in the last section or even physically different formulations: For the derivation, let us assume that a finite generating functional for correlation functions of the fluctuation field is given. In terms of a path integral, this is given by 13 with an assumed diffeomorphism-invariant regularization and renormalization procedure. More generally, such a finite generating functional is given by its defining property 14 under the assumption that these correlation functions are finite. Then, the flow equation can be readily derived without the necessity of referring to a specific representation of $Z[\overline{g},J]$ such as the path integral. (For detailed discussion, see 76.) The correlation functions of h depend on $\overline{g}$, as does the generating functional for $J\ne 0$ via the gauge fixing (see Section 2).

The flow equation for the effective action is derived from the IR regularized generating functional,

with a $\overline{g}$-dependent IR regulator $R_k$. Typically, the background dependence enters the regulator via a background Laplacian and background covariant derivatives. In flat space, the eigenvalues of the Laplacian are just momentum squared, $p^2$. As already discussed above, the regulator suppresses then IR momentum modes with $p^2\mathrm{\lesssim }{k}^2$. In turn, UV momentum modes with $p^2\mathrm{\gtrsim }{k}^2$ propagate freely, and the generating functional includes all quantum contributions generated by these modes.

It is convenient to write the regulator $R_k$ in terms of the classical or quantum dispersion of the field at hand,

where momentum-squared is counted in cutoff units. In these units, the IR regime is given by $x\mathrm{\lesssim }1$ and the UV regime by $x\mathrm{\gtrsim }1$. The tensor part $T_k^{ab}$ of the regulator is proportional to the classical or quantum dispersion of the field. Classically, it is the second derivative of the action with respect to the fields ${\varphi }^a$ and ${\varphi }^b$, that is, ${\left(S^{\left(2\right)}\right)}^{ab}\left(p\right)$. It carries the kinetic information about the field whose propagation is regularized. In turn, the dimensionless shape function$r_k$ specifies how the propagation is regularized. In most cases, the latter part is chosen such that the physical cutoff scales agree for all fields. This is typically achieved with identical (e.g., for several scalar or bosonic fields) or related shape functions (e.g., for scalars and Dirac fermions). It can be shown that such a choice also improves the convergence of generic expansion schemes [21, 77]. Moreover, $r_k$ has to be chosen such that the IR suppression of momentum modes and the UV decay of the regularization are guaranteed. These properties lead to the following asymptotics of the regulator shape function,

The first limit, 27a, guarantees the IR suppression of momentum modes. For example, for a scalar field in d dimensions with a quadratic dispersion $\propto p^2$, a regulator shape function $r_k(x\to 0)=1/x$ introduces a low momentum mass $k^2$ for this field. Indeed this is the common choice for the IR limit, but more singular choices work as well. Eq. 27a also entails that for $k\to \infty$, all momentum modes are suppressed and the theory approaches the UV-scaling regime. For asymptotically free theories, this is the classical theory, and for asymptotically safe theories, this is the nontrivial quantum UV theory.

The second limit, 27b, guarantees that the UV behavior of the theory is unchanged by the IR regularization. We shall see below that the limit in 27b has to be approached sufficiently fast. In our example of a scalar field in d dimensions, the regulator shape function has to decay with at least $\text{min}\left({\left(1/x\right)}^{d/2}1/x\right)$ for rendering the IR flows finite. This is also discussed later in more detail below 39. Note that the latter limit is that of a mass or Callan–Symanzik cutoff. Then, changing k changes a relevant parameter of the theory and hence changes the theory at all scales. Accordingly, the CallanSymanzik cutoff is not a local momentum cutoff. The limit 27b also has another implication: for $k\to 0$, the limit 27b holds for all momenta and the cutoff is removed from the theory. We remark that it is precisely this property, which is at stake for the Callan–Symanzik cutoff and similar ones.

Subject to the existence of a finite full generating functional $Z\left[\overline{g},J\right]$, the regularized generating functional $Z_k\left[\overline{g},J\right]$ is also finite (and smaller than $Z\left[\overline{g},J\right]$). The flow equation for the Schwinger functional $W_k\left[\overline{g},J\right]=\text{log}{Z}_k\left[\overline{g},J\right]$ is derived by taking the logarithmic k-derivative of 25. Schematically, this leads us to

where the RG-“time” t is defined with $t=\ln k/\Lambda$, and $\text{Λ}$ is some reference scale $\text{Λ}$. The trace sums over position space, Lorentz and internal indices, and species of fields. For the sake of a concise presentation, we have suppressed all space-time and internal indices including species of fields. We emphasize that for the explicit form of 28, the order of derivatives is important as J contains fermionic currents.

The term in parenthesis in 28 is nothing but the full two-point correlation functions of the theory: the first term is the connected part, that is, the scale-dependent propagator $G_k[\overline{g},\varphi ]$ of the theory (see 19). The second term is simply ${\varphi }^2$, the disconnected part. The scale-dependent effective action ${\text{Γ}}_k[\overline{g},\varphi ]$ is defined as the modified Legendre transformation of the Schwinger functional,

where $J=J\left[\overline{g},\varphi \right]$ is given by 16. The source term in (IV) depends on $\sqrt{\overline{g}}$, just as the source term in 13. Otherwise, the Legendre transform would not be linear in the mean field ϕ. Note also that the classical action of gravity may be unbounded, for example, in the case of the Einstein–Hilbert action. Then, the Legendre transformation is defined on a saddle point.

The flow equation for the effective action [2, 78, 79] follows straightforwardly from 28. The part proportional to ${\varphi }^2$ is canceled by the flow of the last term in (IV), and the flow of ${\text{Γ}}_k\left[\overline{g},\varphi \right]$ is given by

where $G_k\left[\overline{g},\varphi \right]$ is the full field-dependent propagator ${\delta }^2{W}_k/\delta J^2$, and the trace has been defined below 28. It now contains a relative minus sign for Graßmann-valued fields. With the definition of the Legendre transformation in (IV), the full propagator is given by

The flow equation for the effective action depends on the second derivative of the effective action with respect to the fluctuation fields, ${\text{Γ}}_k^{\left(\mathrm{0,2}\right)}\left[\overline{g},\varphi \right]$. The flow of the latter is derived from 30, with two derivatives with respect to the fluctuation field ϕ. This flow depends on itself and the vertices ${\text{Γ}}_k^{\left(\mathrm{0,3}\right)}\left[\overline{g},\varphi \right]$ and ${\text{Γ}}_k^{\left(\mathrm{0,4}\right)}\left[\overline{g},\varphi \right]$. This leads to a tower of coupled differential equations for the n-point vertices ${\text{Γ}}_k^{\left(n,m\right)}\left[\overline{g},\varphi \right]$, which is discussed in more detail in Section 7.1. We use the following notation for derivatives,

for general functionals of $\overline{g}$ and ϕ. The functional derivative in 32 includes a factor of $1/\sqrt{\overline{g}}$ (see Supplementary Material).

The different parameterizations of the metric field, discussed in Section 3, do not influence the flow equation for the effective action 31, and they only differ by their corresponding expansion schemes induced by the relations between metric and fluctuations (20, 21, 24). Still, from the viewpoint of diffeomorphism-invariance, the different parameterizations differ qualitatively. While the geometrical approach with the fluctuation field 24 by construction leads to a diffeomorphism-invariant effective action at all cutoff scales, diffeomorphism-invariance is broken in the linear split (20) and the exponential split 21 at a finite cutoff scale.

For all field parameterizations, a diffeomorphism-invariant effective action with one metric g is obtained at vanishing fluctuation graviton field $h=0$,

the background effective action. Its flow equation is given by 30, evaluated at vanishing fluctuation field $h=0$,

Importantly 34 is not closed: the right-hand side depends on ${\text{Γ}}_k^{\left(\mathrm{0,2}\right)}$, the two-point function of the fluctuation fields including the fluctuation graviton field h, while the left-hand side knows nothing about h. Hence, the information about ${\delta }^2{\Gamma }_k/\delta h^2$ has to be obtained separately.

5 Background Field Approximation

The background field approximation, introduced in [3, 80] for YangMills theory and gravity, respectively, is the most commonly used approximation in the fRG approach to quantum gravity (see the reviews 7–16). It elevates the diffeomorphism-invariance of the background effective action to that of the full effective action. To that end, we write the full effective action in an expansion about the background effective action in 33,

The gauge fixing term $S_{\text{gf}}$ is defined in 6 and ${\text{ΔΓ}}_k\left[\overline{g},\varphi =0\right]=0$. In the background field approximation, the last term in 35a is assumed to be negligible,

The underlying assumption is that the dynamics of a gauge theory is carried by gauge-invariant fluctuations, while ${\text{ΔΓ}}_k$ carries quantum deformations of the gauge fixing procedure and should not drive the dynamics. Then, derivatives with respect to $\overline{g}$ and h agree in the linear split and are related in a simple way in the other parameterizations via 21 and 24.

In the approximation 35 and with the linear split 20, the second derivatives of the effective action with respect to the background metric and the fluctuation field agree at $\varphi =0$ up to the gauge fixing term:

Inserting 35 into 30 leads us to a closed and diffeomorphism-invariant flow for the background effective action ${\text{Γ}}_k[g,\phi ]$.

5.1 Properties of the Background Approximation

It is the simple relation 36 and the manifest diffeomorphism-invariance of the approximation at all cutoff scales that make the background field approximation so attractive. A large amount of the results in asymptotically safe quantum gravity has been obtained in this approximation, and it is still the commonly used approximation in the field. This asks for independent checks of these results and its embedding in systematic expansion schemes that go beyond it. In the present work, we review the fluctuation approach (see Section 7), which includes the correlation functions of the fluctuation graviton field h. The results in the background field approximation are qualitatively in line with the results in the fluctuation approach discussed in Section 8. This confirms—in most cases—the underlying assumption 35b. Nonetheless, some words of caution are needed.

Despite its seeming manifest diffeomorphism-invariance, the background field approximation is at odds with diffeomorphism-invariance and background independence. To understand this counterintuitive remark, we recall some features of the background field formalism to standard quantum field theories, for example, the SM and QCD. The introduction of the background field to the gauge fixing allows defining a gauge-invariant background effective action. It is evident from its introduction that it is an auxiliary symmetry. The background field can even generate gauge-invariant background effective actions in theories that explicitly break gauge-invariance. This is clear from the construction of diffeomorphism-invariant background effective actions in gravity in the presence of a background-covariant momentum regulator. In a gauge-invariant theory without a cutoff, it can be shown that the physical gauge-invariance of the theory is carried by the fluctuation field in terms of nontrivial Ward– or Slavnov–Taylor identities. The underlying transformations are called quantum gauge/diffeomorphism transformations. This physical symmetry carries over to the auxiliary background gauge invariance via nontrivial Nielsen or split Ward identities. The latter encodes background independence of the theory and is introduced in Section 2.2. The Slavnov–Taylor and Nielsen identities for gravity are discussed in detail in Section 6.

In summary, only if the fluctuation correlation functions satisfy the nontrivial symmetry relations and the Nielsen identities, the auxiliary background gauge-invariance is physical. Then, it carries the underlying symmetry, and we have background independence.

5.2 Regulator Dependence of the Background Effective Action

In this section, we first argue that regulator choices within the general class defined with 26 and 27 can be used within the background field approximation to even change the (non-)existence or the nature of an asymptotically safe UV fixed point. This seems to casts some doubts on the reliability of results obtained in the background field approximation. However, we then show that the comparison with fluctuation results and the proper use of Nielsen identities (see Section 6) suffices to further restrict the general class of regulators such that it is adapted to the background field approximation.

The regulator term is the origin of the reliability problems of a naive use of the background field approximation within the fRG approach: it generates additional terms in ${\text{ΔΓ}}_k\left[\overline{g},h\right]$ in 35a via the background-metric dependence of the regulator. In the background field approximation 35b, this background-metric dependence is elevated to a dynamical one: in the approximation 36, the fluctuation two-point function ${\text{Γ}}_k^{\left(\mathrm{0,2}\right)}$ is computed from background-metric derivatives of the (integrated) flow with the exception of the gauge fixing term. These derivatives also hit the regulator. Accordingly, we have added dynamics via the choice of the regulator, and it remains to be proven in each application that this does not change the results qualitatively.

This has been discussed early on at the example of scalar theories and Yang–Mills theory in [81, 82]. In particular, it has been shown that the one-loop β-function in Yang–Mills theory can be changed from its universal result with regulator choices in the background field approximation. More precisely, it has been shown that for regulators, $R_k\left({\overline{\text{Δ}}}_s\right)$ with spin $s=1$ and spin $s=0$ covariant Laplacians ${\overline{\text{Δ}}}_s={\text{Δ}}_s\left(\overline{A}\right)$ the coefficient $Z_F$ of the $\text{tr}{F}^2$-term in the effective action runs at one loop as

(For details, refer to 82.) This spoils the universality of the one-loop β-function in the Yang–Mills theory. If one does not resort to the background field approximation, the correct one-loop β-function is obtained.

We now discuss the origin of this peculiar behavior. We follow the argument in [83] and for the general case including gravity (refer to [21, 72, 73, 83]). Simply put, we would like to show that the background effective action at a finite cutoff scale k and in particular in the limit $k\to \infty$ carries no physics without further restrictions of the regulator. We parameterize the regulator with

(see also 26). Note that in 38, we have introduced a $\overline{\nabla }$-dependent shape function, which is more general than the $x=\overline{\text{Δ}}/k^2$-dependent one (defined in 26). Still we use x in a slight abuse of notation for identifying the UV and IR limits. As already explained around 26, the shape function $r_k$ is a free function of the covariant derivative with the limits 27. In particular it has to decay in UV. With the parameterization 38, the flow equation 34 for the background effective action with $\overline{g}=g$ reads

From the first term on the right-hand side of the flow 39, we deduce that the UV limit of the shape function is constrained: $r_k(x\to \infty )\le 1/x^{d+\mathit{ϵ}}$ with $x=\text{Δ}/k^2$, as discussed below 27. In turn, the IR limit $x\to 0$ of $r_k$ can be singular without spoiling the finiteness of 39. In order to obtain a general background effective action, we simply demand that $r_k$ solves the differential equation,

This is a simple differential equation that admits a solution at least locally (in the flow time t). Note that the UV decay of $r_k$ also constrains the UV limit of $Y_k$ with ${\partial }_t{Y}_k(x\to \infty )\le 1/x^{d+\mathit{ϵ}}$. Inserting a shape function $r_k$ of 40 into 39, we arrive at

Equation 41 constrains the IR limit of the function $Y_k$: its flow ${\partial }_t{Y}_k$ has to be trace class for rendering the flow of the background effective action finite. If we also assume the trace-class property for $Y_k$, the order of t-derivative and trace can be swapped.

Apart from these trivial constraints, the choice of $Y_k(\nabla )$ is at our disposal. Integrating the flow 41 from some scale $\text{Λ}<k$, and taking the UV limit with $k\to \infty$ we arrive at

The term ${\text{Γ}}_{\text{Λ}}-\text{Tr}{Y}_{\text{Λ}}$ is k- and $\text{Λ}$-independent, and the latter property follows from RG-consistency: ${\partial }_{\text{Λ}}{\text{Γ}}_k\equiv 0$ for $k\ne \text{Λ}$ (see, e.g., 84). In the last relation in 42, we have assumed that the effective action is dominated by the UV term $\text{Tr}{Y}_k$. This assumption underlies most fixed-point analyses.

We emphasize that the result 42 is exact and no approximation has been applied. Equation 42 implies that without suitable restrictions on the regulator function $r_k$, the flow of the background effective action ${\text{Γ}}_k\left[g,\phi \right]$ (for large cutoff scales) has no physics content at all. Even at one- and two-loop order in perturbatively renormalizable theories, it does not reproduce universal results without further restrictions on the regulator.

The IR limit with $r_{k\to 0}=0$ puts a severe restriction onto $r_k$, which constrains the integrated flow together with the RG consistency at the initial cutoff scale $\text{Λ}$, ${\partial }_{\text{Λ}}{\text{Γ}}_{k=0}=0$. However, in the UV limit, the restriction

does effectively not restrict the UV scaling. The latter is dominated by the UV-relevant operators that satisfy 43 by definition. Note that so far, we have discussed the flow of the background effective action ${\text{Γ}}_k[g,\phi ]$ without resorting to approximations.

The above issues are already present for the full flow and emphasize the auxiliary nature of the background effective action at $k\ne 0$. In particular, no conclusion can be drawn from its regularity or singular behavior in the UV limit with $k\to \infty$. This situation is further complicated by the background field approximation. Then, the field dependence that originates from the regulator term is fed back into the flow equation as dynamical contributions. As we have discussed above, these contributions are ambiguous in particular in the UV limit. In conclusion, the background field approximation, while having the appeal of simplicity and seeming diffeomorphism-invariance, has to be applied with great care. To that end, we split the problems discussed above in their physics origin:

Physical diffeomorphism-invariance and background independence are carried by nontrivial Slavnov–Taylor and Nielsen identities of the fluctuation field.

The background field dependence of the regulator term is potentially dangerous in the UV and has to be separated.

A first step in the resolution of the issues of the background field dependence is to monitor the field-dependence that originates in the regulator. The related equation and discussion in the Yang–Mills theory and gravity can be found in 21, 72, 73, 82, 83, 85, 86. (For applications to gravity, see also 57, 87–90.) The equation that monitors this dependence is given by

Equation 44 allows to disentangle the background-metric dependence stemming from the regulator from the rest. In the Yang–Mills example from 37, it can be shown that the regulator-field dependence is responsible for a contribution $\left(1-n\right){\beta }_{{\alpha }_s,1-\text{loop}}$. Subtracting the contribution from the regulator-field dependence, the universal result is obtained. Indeed, even without an explicit computation, we can already infer from 44 that the universal 1-loop β-function of the dimensionless Yang–Mills coupling is achieved for IR regular regulators: the projection of the right-hand side of 44 on the dimensionless term proportional to $\text{tr}{F}_{\mu \nu }^2$ can only depend on the cutoff scale k in the presence of an additional scale. For IR-regular regulators, such a scale is absent and the k-derivative of 44 vanishes. In turn, IR-singular regulators implicitly introduce a further IR scale, and the k-derivative of 44 does not vanish. This explains the structure of the result in 37. We emphasize that the modification of the dynamics in the background field approximation via the regulator term is not restricted to IR-singular regulators. The latter fact is a peculiarity of the universal one-loop running of the dimensionless Yang–Mills coupling. In particular, we emphasize that for nonuniversal couplings and theories with dimensionful couplings such as gravity, the flow of 44 does not vanish for IR-regular regulators.

Based on this analysis it has been suggested in 81, 82, that within the background field approximation, the corresponding field-dependence should be subtracted before applying the approximation ${\text{Γ}}_k^{\left(\mathrm{0,2}\right)}\simeq {\text{Γ}}_k^{\left(\mathrm{2,0}\right)}$ for the right-hand side of the flow. This idea has been picked up by 91–93 for scalar theories, $f\left(R\right)$ gravity and gravity matter systems. (For more details, see Section 6.) These works are based on the relation 44, where one derivative with respect to the background is taken. To fully resolve ${\text{ΔΓ}}_k$ in 35a, a further field derivative of 44 is needed. Furthermore 44 does not comprise the full difference between h and $\overline{g}$ derivatives. While the background field correlation functions are diffeomorphism-covariant due to background diffeomorphism invariance, the fluctuation correlation functions satisfy difficult Slavnov–Taylor identities. This is well-known and well-studied (though not fully conclusively) in non-abelian gauge theories where one also has access to respective lattice results, for a recent review and related references [27]. In turn, the related analysis, while in high demand, is less advanced in quantum gravity (see also 16, 27). This is detailed in the next section.

6 Symmetry Identities

Physical observables are diffeomorphism-invariant and background-independent. The underlying symmetry is dynamical and is solely carried by the dynamical fluctuation fields. It is called quantum diffeomorphism invariance and reads

The background metric triggers an a priori auxiliary symmetry, the background diffeomorphism invariance. It is given by the transformation

Here, ${\mathrm{ℒ}}_{\omega }$ is the Lie derivative with respect to some vector field ${\omega }_{\mu }$, which reads for a rank-two tensor

Both tranformations, 45 and 46, generated diffeomorphism transformations on the full metric $g_{\mu \nu }$, so they do not differ on the functional of $g_{\mu \nu }$. Moreover, while 46 is an auxiliary symmetry, it still comprises the information of the dynamical quantum diffeomorphism symmetry 45 via the Nielsen identities. The latter carry the background independence of the theory.

Any fRG computation needs to introduce a gauge fixing and a regularization, which both apparently break diffeomorphism invariance and (on-shell) background independence. Thus, it is an important issue in the fRG approach to quantum gravity to discuss how these properties can be preserved in a nonperturbative computation. For each symmetry broken by the cutoff term, we can formulate a nontrivial modified symmetry identity, which captures the cutoff deformation of the underlying symmetry and smoothly approaches the unbroken symmetry identity at vanishing cutoff scale, $k=0$. We now first discuss how the Nielsen identities take care of background independence and afterward discuss quantum diffeomorphism invariance due to the Slavnov–Taylor identities. Note that also in discrete gravity models, the Ward identities play a crucial role (see 94) for a review of tensor models.

6.1 Background Independence

As discussed in Section 2.1, we always need to split the full metric into a background metric $\overline{g}$ and a fluctuation field h. This split introduces an additional symmetry given by all transformations of the background metric and of the fluctuation field that leave the full metric invariant.

For example, in the linear split 20, we have $\delta \overline{g}=-\delta h$. This symmetry is guaranteeing background independence since we can always find a transformation that changes the background according to our choice. This symmetry is broken off-shell by the gauge fixing and ghost action and further broken by the regularization on- and off-shell. The breaking of the symmetry is described by the Nielsen (or split Ward) identities [95, 96]. They encode the background independence of the physical observables and allow us to restore the symmetry at vanishing cutoff.

Let us first discuss the Nielsen identities without the regulator. The Ward identity for the effective action for any symmetry transformation $\mathcal{G}$ is given by

where $S_{\text{gf}}$ and $S_{\text{gh}}$ are defined as in 6 and 8. We apply this to the transformation of the metric split 48 and obtain the Nielsen identity $\text{NI}=0$, with

where $h_{\mu \nu }=\langle {\widehat{h}}_{\mu \nu }\rangle$, and the fluctuation field is understood as function of the full metric and the background metric $\widehat{h}\left(g,\overline{g}\right)$. For the linear split 20, we have $\delta {\widehat{h}}_{\rho \sigma }\left(x\right)/\delta {\overline{g}}_{\mu \nu }\left(y\right)=1/\sqrt{\overline{g}}\delta \left(x-y\right)1/2\left({\delta }_{\rho }^{\mu }{\delta }_{\sigma }^{\nu }+{\delta }_{\sigma }^{\mu }{\delta }_{\rho }^{\nu }\right)$ (see Supplementary Material), and thus,

The Nielsen identity for the exponential split 21 resembles 51: there is a nontrivial difference between the background-metric and fluctuation-field derivatives due to the gauge fixing and ghost terms. In 17, we have pointed out that at $k=0$, a solution of the background EoM is also a solution of the quantum EoM and vice versa. This implies together with 51 that the expectation value $\langle \left[{\delta }_{\overline{g}}-{\delta }_{\widehat{h}}\right]\left(S_{\text{gf}}+S_{\text{gh}}\right)\rangle$ needs to vanish on-shell. This is indeed nontrivial and does not happen off-shell.

In comparison, for the fully diffeomorphism-invariant Vilkovisky–DeWitt or geometrical effective action with the split given by 22, the dependence on the gauge fixing action and the ghost action is vanishing, and thus, the Nielsen identity reads

In contradistinction to the linear and exponential split, the $\overline{g}$ and h derivatives are directly related.

The Nielsen identities entail that for all metric splits, the effective action is not a function of the full metric g but depends separately on the background metric $\overline{g}$ and the fluctuation field h. Consequently, the effective action has no simple expansion in terms of diffeomorphism-invariant quantities in $g_{\mu \nu }$. Still, the Nielsen identities relate $\overline{g}$ and h derivatives such that on the solution of the Nielsen identities, the effective action carries background independence and only depends on one field.

So far, the analysis has been performed in the absence of the cutoff term, that is, at $k=0$. At finite k, the regulator term introduces a further breaking of the split symmetry 48. The Nielsen identities turn into modified Nielsen identities, $\text{mNI}=0$, that read for a general split

Note that in the last term in 53, only the metric fluctuation h contributes as the other fluctuation fields do not depend on the background metric. Furthermore, in the linear split, the last term is vanishing, and consequently, the mNI simplifies to

While some of the properties and consequences of the mNI are theory-dependent, most of them are generic, and much can be learned about applications in gravity from investigations in general theories: mNIs have been discussed in detail gravity, gauge theories, in scalar theories [3, 15, 21, 57, 72, 73, 81, 82, 87–93, 98–106].

There is an important qualitative difference between the breaking of the metric split symmetry 48 at finite k and at $k=0$. We have already discussed in Section 2.2 that the Nielsen identity at vanishing cutoff scale, $k=0$, encodes background independence, manifested in the equivalence of the solutions of the background and fluctuation EoMs, 17. At finite cutoff scale, $k\ne 0$, we necessarily have background dependence, as the quantum fluctuations have to be ordered in a specific background. This is also manifest in the missing equivalence of the background and fluctuation EoMs, the respective solutions do not agree,

(For a detailed discussion, see 97, 106, 107). The difference between the solutions can be parameterized by a term proportional to the regulator $R_k$, which is most easily seen in the modified Nielsen identity in the geometric approach, 52 and 53.

The difference between ${\overline{g}}_{\text{EoM}}^{\text{fluc}}$ and ${\overline{g}}_{\text{EoM}}^{\text{back}}$ was explicitly computed in 97, 107 for backgrounds with constant curvature. The ansatz for the background effective action is

where V is the space-time volume and $r=\overline{R}/k^2$ is the dimensionless background curvature. Thus, the background EoM becomes

which is displayed in the right panel of Figure 2 at the UV fixed point for different numbers of scalar fields $N_s$. The ansatz for the fluctuation one-point function reads

and thus, the quantum EoM is simply

This is shown in the left panel of Figure 2 at the UV fixed point for different numbers of scalar fields $N_s$.

FIGURE 2. Displayed are the potential of the one-point function and the derivate of the background potential for different numbers of scalar fields at the fixed point, as defined in 57 and 59. A zero in these functions indicates a solution to the quantum and background EoM, respectively. While the former always has two solutions, a minimum at negative curvature and a maximum at positive curvature, the latter shows no solution at all. The figures are taken from 97.

The background EoM does not display a solution in the whole investigated region, while the quantum EoM has two solutions, a minimum at negative curvature and a maximum at positive curvature. For a larger number of scalar fields, these two solutions merge. However, in this regime, the approximation lacks reliability due to large values of the graviton anomalous dimension. Importantly, Figure 2 manifests in explicit computation the difference between the background and quantum EoM, 55. The background EoM was also extensively investigated in the background field approximation with different choices of regulator and parameterization. For example, in 108, the linear split was used and a solution at large negative curvature was found. However, in 109, 110, two further solutions at positive curvature were found due to a different choice of the regulator. A solution at positive curvature was also found in 111 and with the exponential parameterization in 51.

In 106, a modification of the fRG equation was proposed. There, the effective action was defined as the Legendre transform of a normalized Schwinger functional, ${\widehat{W}}_k\left[\overline{g},J\right]=\log \left(Z_k\left[\overline{g},J\right]/Z_k[\overline{g},0]\right)$. This modification implies that the solutions to the quantum and background EoMs agree even at a finite cutoff scale. This does not imply that the modified effective action is background-independent at finite k since there are differences in the higher order correlation function. However, it allows for constructing improved background field approximations, which might allow resolving some tensions between background and fluctuation results.

6.2 From BRST to Diffeomorphism Invariance

While the auxiliary background diffeomorphism invariance 46 remains unbroken, the physical quantum diffeomorphism invariance 45 turns into a BRST symmetry due to the gauge fixing, which is then further broken by the regulator. The related symmetry identities are called (modified) Slavnov–Taylor identities [(m)STI] [112, 113]. They encode physical diffeomorphism invariance. We sketch the main ideas of the derivation and apply them to gravity.

In case of the linear gauge fixing condition 7, the generator of BRST transformation (or BRST operator) denoted by $\mathfrak{s}$, including the Nakanishi–Lautrup field ${\lambda }_{\mu }$, is given by

In (B), the vector field ${\omega }_{\mu }$ in the Lie derivative 47 is given by the ghost field, ${\omega }_{\mu }=c_{\mu }$. (For more details on the setup and the condensed notation used below, see 21). The Nakanishi–Lautrup field ${\lambda }_{\mu }$ transforms trivially under the BRST transformation, $\mathfrak{s}{\lambda }_{\mu }=0$. The classical gauge-fixed action including the gauge fixing and the ghost action is invariant under this transformation, $\mathfrak{s}\left(S_{\text{grav}}+S_{\text{gf}}+S_{\text{gh}}\right)=0$. Furthermore, $\mathfrak{s}$ is a nilpotent operator with ${\mathfrak{s}}^2=0$.

For the derivation of the STI, we include a source term $Q^a\mathfrak{s}{\widehat{\varphi }}_a$ for the BRST variations of the fields in the generating functional. The Schwinger functional now reads

where $S_{\text{tot}}=S_{\text{grav}}+S_{\text{gf}}+S_{\text{gh}}$. The STI follows from the BRST-invariance of generating functional,

The source term $J^a{\widehat{\varphi }}_a$ is the only BRST-variant term. The BRST operator $\mathfrak{s}$ commutes with bosonic sources and anti-commutes with fermionic sources. This leads us to $\mathfrak{s}{J}^a{\widehat{\varphi }}_a=J^a{\gamma }^b{}_a\mathfrak{s}{\widehat{\varphi }}_b$, where the metric ${{\gamma }^b}_a$ carries the minus sign for the fermionic terms (see Supplementary Material).

With these properties, we obtain the STI for the Schwinger functional,

This identity can be re-expressed in terms of the effective action. (See 21 for details.) Here, we just state the result for the STI in the absence of the cutoff term,

This equation is known as the quantum-master equation. The BRST variation of the effective action is given by $\delta \Gamma /\delta Q^a=\langle \mathfrak{s}{\widehat{\varphi }}_a\rangle$. These variations can be interpreted as generalized vertices of the theory.

Equation 64 encodes diffeomorphism invariance at $k=0$ where the regulator vanishes. At finite cutoff scale, an additional regulator contribution has to be taken into account, and we are led to the mSTI,

Some of the properties of the mSTI are theory-dependent, but most of them are generic: mSTIs in the presence and absence of background fields in gravity and gauge theories have been discussed in detail in [3, 15, 21, 22, 57, 72, 73, 81, 82, 87–93, 98–106, 114‒132].

In summary, we have three symmetries:

The auxiliary background diffeomorphism invariance 46, which remains unbroken.

The quantum diffeomorphism invariance 45, which describes physical diffeomorphism invariance. It is broken and encoded in the mSTI 65.

The split symmetry 48, which guarantees background independence. It is broken as well and encoded in the mNI 53.

The relations between background and fluctuation correlation functions are summarized in Figure 3. The relation between two fluctuation correlation functions can be expressed either with an mSTI or with a combination of mNI and background diffeomorphism invariance. However, it should be noted that in a truncated nonperturbative computation these two possibilities of relating fluctuation correlation function do not agree with each other. Nonetheless, it can be used to check the error of the truncation. (See Section 7.1 for more details.)

FIGURE 3. Displayed are the relations between background and fluctuation correlation functions in terms of symmetry identities. The background diffeomorphism symmetry 46 remains unbroken and trivially connects background correlation functions. The split symmetry (48) is encoded in the modified Nielsen identity (mNI) (53) and relates background correlation functions with fluctuation ones. The quantum diffeomorphism symmetry (45) is described by the modified Slavnov–Taylor identity (mSTI) 65 and relates fluctuation correlation functions. For the purpose of illustration, we have assumed that ${\text{Γ}}_k={\text{Γ}}_k[\overline{g},h,\phi ]$ depends on the background metric $\overline{g}$, the metric fluctuation h, and a scalar field φ. The notation ${\text{Γ}}_k^{\left(n_1,n_2,n_3\right)}$ is then defined as in 32.

Last but not least, the flow of mNI and the mSTI is proportional to itself, respectively. This is conveniently expressed in terms of the flow equation for composite operators, derived in [21, 128, 133]. Schematically, it reads

The operator ${\mathcal{O}}_k^{\left(\mathrm{0,2}\right)}$ is contracted with $G_k{\partial }_t{R}_k{G}_k$ in the trace. The set of composite operators ${\mathcal{O}}_k\left[\overline{g},\varphi \right]$ with the flow 66 includes general correlation functions $Z^{\left(0,n\right)}\left[\overline{g},J\left[\overline{g},\varphi \right]\right]$ with their disconnected parts as well as more general functions of the field-dependent source such as $J\left[\overline{g},\varphi \right]={\text{Γ}}_k^{\left(\mathrm{0,1}\right)}\left[\overline{g},\varphi \right]$. In the most general case of a functional with an explicit cutoff dependence, further terms enter 66 (see 21). An educative example is ${\text{Γ}}_k^{\left(\mathrm{0,1}\right)}$: inserting it into 66 leads to the fluctuation field derivative of the flow equation 30. An instructive example for the case of general correlation functions, and the necessity of including the disconnected terms is the full two-point function $G_{{\varphi }_1{\varphi }_2}+{\varphi }_1{\varphi }_2$. Equation 66 has been used in Yang–Mills theories for the traced Polyakov loop observables [134] and in gravity for the study of the renormalization and scaling of composite operators [135–138].Importantly, the set of composite operators $\left\{{\mathcal{O}}_k\right\}$ includes modified symmetry identities, that is, ${\text{Sym}}_k=\text{mSTI},\text{mNI},\dots$ (see 21 and also 82, 118, 119, 139). Hence, the flow of the symmetry identities reads schematically

Equation 67 implies that once we have solved these identities at a scale k, then the identities are satisfied at all scales. However, this only holds for untruncated flows or truncations that are compatible with 67. (More details can be found in Section 7.1.)

6.3 Challenges for Diffeomorphism-Invariant Flows

Gauge-invariant approaches to quantum field theories have received much attention over the decades both in perturbation theory and beyond. Such formulations also have met considerable challenges, except for lattice gauge theories that are based on link variables formulated in the gauge group. In turn, perturbation theory and nonperturbative functional approaches are based on correlation functions and in particular on the propagator of the algebra-valued gauge field. (For reviews on lattice approaches to quantum gravity see, e.g., 140–144).

Gauge-invariant functional formulations are based either on gauge-invariant or gauge-covariant variables such as the geometrical formulation, the field strength formulation, or Wilson line formulations similar to lattice gauge theories. Implementations in the flow equation approach range from generalized Polchinski equations with gauge-covariant kernels for the Wilson effective action [145–156] and its recent manifestations [157–159], over the geometrical or Vilkovisky–DeWitt flows for the effective action [21, 71–74], to a recent suggestion for a gauge-invariant flow for the effective action [160–164].

Most of these approaches rely explicitly or implicitly on the definition of projection operators on the subspace of the dynamical degrees of freedom. Typically, this is achieved by a gauge fixing, but the notation of a projection is far more versatile. The appropriate definition of this projection and the respective geometrical structure of the configuration space is at the root of the geometrical construction. This has been discussed in Sections 3.3, 6.2, and 6.3, and we refer to the discussions there. The notable nonlocality of the projections both in field space as well as momentum space is an inherent property of the construction of gauge-invariant subspaces. Consequently, it should be considered an inherent feature of such a construction. This inherent nonlocality may be buried in functional self-consistency relations, but it is present explicitly or implicitly without any doubt.

In any case, the situation calls for self-consistency checks of the final formulations of gauge-invariant or diffeomorphism-invariant flows. This necessity has been discussed already in 165: there the terminology of complete and consistent flows was introduced. The former flows generate all quantum fluctuations from a given classical action, while the latter flows generate a well-defined subset of quantum fluctuations from a given—partial—effective action. A well-known example for the latter is thermal flows, which only generate thermal fluctuations from the full quantum effective action at vanishing temperature. In 165, 166, an important and simple consistency check for flow equations has been suggested: any complete flow equation must generate the complete perturbation theory upon iteration from the given classical action. While one-loop perturbation theory in the fluctuation field is trivially achieved within one-loop exact flow equations, two-loop perturbation theory provides a nontrivial necessary, while not sufficient, consistency check.

These checks for diffeomorphism-invariant fRG approaches have been passed for the Wilsonian approach [145–156] or are trivial for the geometrical effective action approach [21, 71–74]. It is a highly relevant and interesting question how the more recent proposals [158–164] fare in such a self-consistency check. Respective investigations either confirm the completeness of the approaches or may show their consistency, that is, they may integrate out a well-defined subset of quantum fluctuations. Finally, for potentially consistent flows, such an investigation may enable the construction of nontrivial two-loop consistent extensions. We emphasize that such an extension does not simply pass a two-loop test but more importantly allows for two-loop resummed nonperturbative approximations. The latter set of approximations certainly live up to the self-consistency of other state-of-the-art computations in asymptotically safe quantum gravity, while having the benefit of inherent diffeomorphism invariance.

7 Fluctuation Approach

In the last sections, we have detailed the need for an fRG approach to quantum gravity that goes beyond the background field approximation and that allows satisfying the nontrivial symmetry identities, the mSTI 65 and the mNI 53. For general metrics $g_{\mu \nu }$, this requires to solve the flow equation 30 for the two-field action ${\text{Γ}}_k[\overline{g},h]$. It is already a formidable task for the one-field flow in the background field approximation discussed in Section 5. Indeed, already in scalar theories, one has to resort to approximations such as the derivative expansion or the vertex expansion, and this is no different in gravity. As already discussed, while the quantum dynamics of asymptotically safe gravity is generated and carried by the fluctuation correlation functions, it is the diffeomorphism-invariant background effective action $\text{Γ}\left[g\right]$ that allows for a more direct physics interpretation. The latter is extracted from the flow 34 that solely depends on the fluctuation two-point function ${\text{Γ}}^{\left(\mathrm{0,2}\right)}[g,0]$. The flow of the latter depends on higher order fluctuation correlation functions (see Section 7.1).