Gauge invariant renormalizability of quantum gravity111 Invited chapter for
the Section ”Effective Quantum Gravity” of the ”Handbook of Quantum Gravity”
(Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer Singapore, expected in
2023).

P.M. Lavrov(a)222E-mail address<EMAIL_ADDRESS>and I.L.
Shapiro(b)333E-mail address<EMAIL_ADDRESS>, On leave from Tomsk
State Pedagogical University, Russia

(a) Center of Theoretical Physics, Tomsk State Pedagogical University

Kievskaya St. 60, 634061, Tomsk, Russia

(b) Departamento de Física, Universidade Federal de Juiz de Fora,

Juiz de Fora, MG, 36036-900, Brazil

> Abstract. The current understanding of renormalization in quantum gravity
> (QG) is based on the fact that UV divergences of effective actions in the
> covariant QG models are covariant local expressions. This fundamental
> statement plays a central role in QG and, therefore, it is important to
> prove it for the widest possible range of the QG theories. Using the
> Batalin-Vilkovisky technique and the background field method, we elaborate
> the proof of gauge invariant renormalizability for a generic model of
> quantum gravity that is diffeomorphism invariant and does not have
> additional, potentially anomalous, symmetries.
>
> Keywords: Quantum gravity, BRST, Batalin-Vilkovisky technique, background
> field method, renormalizability
>
> MSC: 83C45, 81T13, 81T15, 81T70

##  1 Introduction

All the fundamental interactions that exist in Nature (electroweak, strong and
gravitational) are described in terms of gauge theories [1]. Thus, the
quantization of gauge theories that provides important insights into the
quantum properties of the fundamental forces plays an important role in this
process. General relativity (GR) and most of its extensions are based on the
general covariance principle. In the models where this symmetry does not hold
at the classical level (let us note that there is a special Chapter in this
Section, devoted to the Hořava, or Hořava - Lifshitz, gravity models, in which
the covariance is violated), this symmetry is supposed to restore in the limit
corresponding to observations. Thus, the basic models of QG have to be based
on the covariant classical theory.

At the classical level, the coordinate transformations change the metric (or
other fields, describing gravity, e.g., independent connection, or tetrad,
etc) to the equivalent one, from a physical viewpoint. The situation is the
same as in other gauge theories, with the coordinate invariance playing the
role of gauge symmetry. In this review, we discuss only the QG models in which
gravity is described by the metric. Thus, we are leaving aside such
interesting issues as first-order (Palatini) formalism in QG [2], teleparallel
gravity [3], etc. Let us note, by passing, that the quantum aspects of the
teleparallel gravity are not well elaborated.

In quantum theory, the gauge symmetry results in degeneracy and the
difficulties in the immediate application of quantum mechanics, in both
Hamiltonian (canonical) and the Lagrangian (covariant) formalisms. In this
Section, we concentrate our attention on the Lagrangian approach and the path
integral technique, which is an essential ingredient of covariant quantization
and provides economic way of getting the Feynman rules directly from the
classical Lagrangian.

The relevant question is what happens with the diffeomorphism invariance at
the quantum level. In particular, is it true that this symmetry holds after
the loop corrections are taken into account? Or, in a more particular way,
does the general covariance hold in the counterterms that cancel the
divergences of the effective action in the model of QG? In what follows, we
shall see that the answer to both questions may be positive for a variety of
the QG models.

Since QG represents a particular type of gauge theories, the quantization of
gravity should follow the same rules as in other gauge theories. The covariant
quantization of gauge theories has made a long way starting from the famous
work of Feynman [4] where the non-unitarity of $S$-matrix in Yang-Mills [5]
and gravity theories have been found within the approach based on a naive
quantization procedure. This procedure was developed in quantum
electrodynamics (QED), which is an Abelian gauge theory. Later on, consistent
covariant quantization of the Yang-Mills theories has been found in the papers
by Faddeev and Popov [6], and DeWitt [7]. The functional integration measure
that appears in the course of quantization [7], can be encoded into an
additional functional integration over the so-called ghost fields or Faddeev-
Popov ghosts [6]. The Faddeev-Popov method operates with the complete (also
called total) quantum action which is a sum of the initial classical action,
the ghost action, and the gauge fixing action. The total action should be used
for the construction of generating functional of Green functions.

The first proof of the gauge invariant renormalizability in the Yang-Mills
theories has been given by ’t Hooft and Veltman in [8, 9], that was an
important contribution to creating the quantum field theory (QFT) foundations
of the Standard Model of particle physics and its extensions. These works have
been awarded the 1999 Nobel Prize in Physics “for elucidating the quantum
structure of electroweak interactions in physics”. The proof of
renormalizability was based on the basic Ward identities, which was a very
complicated approach. Soon after that, there was developed new powerful
techniques of exploring non-Abelian gauge symmetries at the quantum level. One
of the most remarkable discoveries in this respect was that the total Faddeev-
Popov-DeWitt action is invariant under global supersymmetry called the BRST
symmetry (named by Becchi, Rouet, Stora, and Tyutin) [10, 11, 12]). The
nilpotent BRST charge plays a crucial role in the construction of the physical
state space of any Yang-Mills - type theory allowing to define the gauge-
independent and unitary $S$-matrix [13].

The most efficient way to explore the gauge symmetry at the quantum level is
the Batalin-Vilkovisky technique, that is a natural extension of the DeWitt –
Faddeev-Popov [7, 6] procedure and the Becchi-Rouet-Stora and Tyutin (BRST)
symmetry [11, 12]. The Batalin-Vilkovisky approach can be most successfully
applied in the theory of the Yang-Mills type when the generators of gauge
transformations form a closed algebra. We shall see, in the next sections,
that QG, with the diffeomorphism invariance as a gauge symmetry, belongs to
this class of theories. Another relevant restriction is that there should not
be other symmetries that can “compete” with the diffeomorphism invariance at
the quantum level. A simple criterium to see whether an additional symmetry is
really spoiling the renormalizability is to check whether there is a
regularization preserving both symmetries at the same time. In all known
cases, the lack of such symmetry means there are quantum anomalies. A typical
example is the local conformal symmetry or the Weyl symmetry. It is expected
that the anomaly violates the renormalizability of the theories of QG
(contrary to the QFT in curved spacetime) with this symmetry at higher loop
orders. However, since this is an advanced (and not very well-elaborated
subject, we shall not be concerned about it in the present review. Thus, in
what follows we assume the use of dimensional regularization and the absence
of conformal symmetry.

The Batalin-Vilkovisky technique implies introducing additional objects called
antifields. With antifields, the BRST symmetry and the corresponding equations
for the effective action (Zinn-Justin equation [14] and Slavnov-Taylor
identities in Yang-Mills theory [8, 15, 16]) become more straightforward to
analyze. The gauge invariant renormalizability guarantees that in all orders
of loop expansion for the quantum effective action, one can control
deformations of the generators of gauge transformations, which leave such an
action invariant. In the background field method, one can maintain general
covariance of the divergent part of effective action when the mean quantum
fields, ghosts, and antifields are switched off.

The traditional (and correct) view of the difficulty in quantizing the
gravitational field is that the quantum GR is not a renormalizable theory,
while the renormalizable version of the theory includes fourth derivatives in
the action [17] and therefore it is not unitary. In the last decades, this
simple two-side story was getting more complicated, with the new models of
superrenormalizable gravity, both polynomial [18] and non-polynomial [19, 20,
21]. Typically, these models intend to resolve the conflict between non-
renormalizability and non-unitarity by introducing the finite or infinite
amounts of derivatives compared to the fourth-derivative model [17].

The main advantage of the non-polynomial models is that the tree level
propagator may have the unique physical pole corresponding to the massless
graviton. At the same time, the dressed propagator has, typically, an infinite
(countable) amount of the ghost-like states with complex poles [22] and hence
the questions about physical contents and quantum consistency of such a theory
remains open, especially taking into account the problems with reflection
positivity [23, 24].

On the other hand, within the polynomial model, one can prove the unitarity of
the $S$-matrix within the Lee-Wick approach [25] to quantum gravity in four
[26] and even higher dimensional space-times [27]. Furthermore, it is possible
to make explicit one-loop calculations [28] which provide exact beta-functions
in these theories due to the superrenormalizability of the theory. In the part
of stability, the existing investigations concerned special backgrounds,
namely cosmological [29, 30] and black hole cases [31, 32, 33]. While the
black hole results do not look conclusive, the results for the cosmological
backgrounds provide a good intuitive understanding of the problem of stability
in the gravity models with higher derivative ghosts.

Independent on the efforts in better understanding the role of ghosts and
instabilities in both polynomial and non-polynomial models, it would be useful
to have formal proof that, at the quantum level, these theories are
renormalizable or superrenormalizable. The first proof was given for the
fourth derivative quantum gravity [17], with some simplifications and
generalizations achieved in [34, 35] as an application of a general approach
[36]. Recently, there was a renewed interest in this subject, in particular,
the proofs of the general-covariant renormalizability in the general types of
local [37] models of QG and even a more general Batalin-Vilkovisky-based proof
of a gauge invariant renormalizability in the general models of quantum
gravity, which may include higher derivatives, including polynomial or non-
polynomial models [38]. In what follows, we shall closely follow the last
reference, just adding a few more details.

The review is organized as follows. In Sec. 2, we formulate the Batalin-
Vilkovisky formalism combined with the background field method in QG. In Sec.
3, this formalism is applied to the formal proof of renormalizability in the
model of QG of the general form. On top of that, we discuss the gauge fixing
independence of the vacuum functional and of the gravitational $S$-matrix.
Sec. 4 contains the short historic note, the extended list of the main
references (which certainly remains incomplete owing to the limited size of
the review) and a brief discussion of the requirements on the gauge theory
which enable one to use the Batalin-Vilkovisky formalism. In Sec. 5, the gauge
fixing in the higher derivative models is discussed. Finally, in Sec. 6, we
draw our conclusions.

Condensed DeWitt’s notations [39] are used in the review. Right and left
derivatives of a quantity $f$ with respect to the variable $\varphi$ are
denoted as $\frac{\delta_{r}f}{\delta\varphi}$ and
$\frac{\delta_{l}f}{\delta\varphi}$, correspondingly. The Grassmann parity and
the ghost number of a quantity $A$ are denoted by $\varepsilon(A)$ and ${\rm
gh}(A)$, respectively. The reader can see Eq. (22) for the definition, in the
last case. For the sake of generality, we perform all general considerations
for an arbitrary spacetime dimension $D$. Let us note that this is different
from the previous Chapter of this Section of the Handbook, where it was
assumed $D=4$. The condensed notation for the space-time integral in $D$
dimensions, $\int dx=\int d^{D}x$ is used throughout the text of the present
Chapter.

##  2 Quantum gravity in the background field formalism

Our purpose is to explore the gravitational theory based on an arbitrary
action of a Riemann’s metric $\,S_{0}(g)$, where $\,g=g_{\mu\nu}(x)$. In what
follows we usually omit the indices in the arguments of functions or
functionals. The action is assumed invariant under the general coordinate
transformations,

$\displaystyle x^{\prime\,\mu}=x^{\prime\,\mu}(x),\qquad
g^{\prime}_{\mu\nu}(x^{\prime})\,=\,\frac{\partial
x^{\alpha}}{\partial{x^{\prime}}^{\mu}}\frac{\partial
x^{\beta}}{\partial{x^{\prime}}^{\nu}}\,g_{\alpha\beta}(x).$ (1)

The standard examples of the theories of our interest are Einstein’s gravity
(GR) with a cosmological constant term,

$\displaystyle S_{EH}(g)$ $\displaystyle=$
$\displaystyle-\,\frac{1}{\kappa^{2}}\int
dx\sqrt{-g}\,\big{(}R+2\Lambda\big{)}$ (2)

and a general version of higher derivative gravity. The corresponding action
is described in the previous Chapter [40],

$\displaystyle S(g)$ $\displaystyle\,=\,$ $\displaystyle S_{EH}(g)\,+\,\int
dx\sqrt{-g}\,\bigg{\\{}R^{\mu\nu\alpha\beta}\Pi_{1}\Big{(}\frac{\square}{M^{2}}\Big{)}R_{\mu\nu\alpha\beta}$
(3)
$\displaystyle+\,\,R^{\mu\nu}\Pi_{2}\Big{(}\frac{\square}{M^{2}}\Big{)}R_{\mu\nu}\,+\,R\Pi_{3}\Big{(}\frac{\square}{M^{2}}\Big{)}R\,\,+\,\,{\cal
O}\big{(}R_{...}^{3}\big{)}\bigg{\\}}.$

Here $\Pi_{1,2,3}$ are polynomial or non-polynomial form factors in the
quadratic in curvatures part of the Lagrangian and the last term represents
non-quadratic in curvature terms. In quantum theory, action (3) may lead to
the theory which is non-renormalizable, renormalizable or superrenormalizable,
depending on the choice of the functions $\Pi_{1,2,3}(x)$ and the non-
quadratic terms.

The dimensional parameter $M^{2}$ in the form factors $\,\Pi_{1,2,3}\,$ is a
universal mass scale at which the QG effect becomes relevant. For instance,
$M^{2}$ can be the square of the Planck mass, but there may be other options,
including multiple scale models, as analyzed in [41]. For the analysis
presented below, these (otherwise important) details are irrelevant since the
unique necessary feature is that the action $S(g)$ is diffeomorphism
invariant.

In the infinitesimal form, the transformations (1) read

$\displaystyle{x^{\prime}}^{\mu}=x^{\mu}+\xi^{\mu}(x)\quad\Longrightarrow\quad
g^{\prime}_{\mu\nu}(x)=g_{\mu\nu}(x)+\delta g_{\mu\nu}(x),$ (4)

where

$\displaystyle\delta
g_{\mu\nu}(x)=-\xi^{\sigma}(x)\partial_{\sigma}g_{\mu\nu}(x)-g_{\mu\sigma}(x)\partial_{\nu}\xi^{\sigma}(x)-g_{\sigma\nu}(x)\partial_{\mu}\xi^{\sigma}(x).$
(5)

On top of (5), we also need the transformation rule for vector fields
$A_{\mu}(x)$ and $A^{\mu}(x)$,

$\displaystyle\delta
A_{\mu}(x)\,=\,-\,\xi^{\sigma}(x)\partial_{\sigma}A_{\mu}(x)-A_{\sigma}(x)\partial_{\mu}\xi^{\sigma}(x),$
(6) $\displaystyle\delta
A^{\mu}(x)\,=\,-\,\xi^{\sigma}(x)\partial_{\sigma}A^{\mu}(x)+A^{\sigma}(x)\partial_{\sigma}\xi^{\mu}(x).$
(7)

The invariance of the action $\,S_{0}(g)\,$ under the transformations (5) can
be expressed in the form of Noether identity

$\displaystyle\int dx\,\,\frac{\delta S_{0}(g)}{\delta
g_{\mu\nu}(x)}\,\,\delta g_{\mu\nu}(x)\,=\,0.$ (8)

Let us present the transformations (5) in the form

$\displaystyle\delta g_{\mu\nu}(x)\,=\,\int
dy\,\,R_{\mu\nu\sigma}(x,y;g)\,\xi^{\sigma}(y),$ (9)

where the generators of gauge transformations of the metric tensor
$g_{\mu\nu}$ with gauge parameters $\xi^{\sigma}(x)$ are

$\displaystyle R_{\mu\nu\sigma}(x,y;g)$ $\displaystyle=$
$\displaystyle-\delta(x-y)\partial_{\sigma}g_{\mu\nu}(x)-g_{\mu\sigma}(x)\partial_{\nu}\delta(x-y)$
(10) $\displaystyle-\,\,g_{\sigma\nu}(x)\partial_{\mu}\delta(x-y).$

The algebra of generators (also called algebra of gauge transformations), can
be easily verified to be

$\displaystyle\int du\,\bigg{[}\frac{\delta R_{\mu\nu\sigma}(x,y;g)}{\delta
g_{\alpha\beta}(u)}\,R_{\alpha\beta\gamma}(u,z;g)\,\,-\,\,\frac{\delta
R_{\mu\nu\gamma}(x,z;g)}{\delta
g_{\alpha\beta}(u)}\,R_{\alpha\beta\sigma}(u,y;g)\bigg{]}$
$\displaystyle\qquad\qquad\qquad=\,\,-\,\int
du\,\,R_{\mu\nu\lambda}(x,u;g)\,F^{\lambda}_{\sigma\gamma}(u,y,z),$ (11)

where

$\displaystyle F^{\lambda}_{\alpha\beta}(x,y,z)$ $\displaystyle=$
$\displaystyle\delta(x-y)\;\delta^{\lambda}_{\beta}\;\frac{\partial}{\partial
x^{\alpha}}\delta(x-z)\,-\,\delta(x-z)\;\delta^{\lambda}_{\alpha}\;\frac{\partial}{\partial
x^{\beta}}\delta(x-y),\quad$ $\displaystyle F^{\lambda}_{\alpha\beta}(x,y,z)$
$\displaystyle=$ $\displaystyle-\,F^{\lambda}_{\beta\alpha}(x,z,y)$ (12)

are structure functions of the gauge algebra which do not depend on the metric
$g_{\mu\nu}$.

The algebra (11) holds independently of the form of the covariant action.
Thus, any covariant theory of gravity is a gauge theory with closed gauge
algebra and with the structure functions independent of the fields (metric
tensor, in this case). Thus, QG is a theory of the Yang-Mills type.

Starting from this point, the analysis of renormalization is pretty much
standard, but it is important to introduce one more important element in our
considerations. It proves useful to perform quantization of gravity not on the
Minkowski space-time with the constant metric tensor $\eta_{\mu\nu}$, but on
undefined external background, represented by the metric tensor
${\bar{g}}_{\mu\nu}(x)$. Introducing an arbitrary background metric provides
serious advantages by using the background field method, as we shall see in
what follows. The standard references on the background field method in QFT
are [7, 42, 43]. One can see also recent discussions of the method for the
gauge theories in [44, 45, 46, 47]) and specifically for QG in [37].

In the background field method, the metric $g_{\mu\nu}(x)$ is replaced by the
sum

$\displaystyle
g_{\mu\nu}(x)\,\,\,\longrightarrow\,\,\,{\bar{g}}_{\mu\nu}(x)+h_{\mu\nu}(x),\quad\,\,\mbox{such
that}\qquad S_{0}(g)\,\,\,\longrightarrow\,\,\,S_{0}({\bar{g}}+h).\quad$ (13)

Here $h_{\mu\nu}(x)$ is called quantum metric and is regarded as an
integration variable in the functional integrals defining the generating
functionals of Green functions.

The action $S_{0}({\bar{g}}+h)$ is a functional of two variables ${\bar{g}}$
and $h$ and has an additional symmetry corresponding to the transformations

$\displaystyle\delta{\bar{g}}_{\mu\nu}=\epsilon_{\mu\nu}\qquad\mbox{and}\qquad\delta
h_{\mu\nu}=-\epsilon_{\mu\nu},$ (14)

with arbitrary symmetric tensor functions
$\epsilon_{\nu\mu}=\epsilon_{\mu\nu}=\epsilon_{\mu\nu}(x)$. In particular,
this implies an ambiguity in defining the gauge transformations for
${\bar{g}}$ and $h$. To fix this arbitrariness, we require that the
transformation of our interest has the right flat limit when
${\bar{g}}_{\mu\nu}(x)$ is replaced by $\eta_{\mu\nu}$. In this way, one can
easily show that the gauge transformation of the quantum metric $h_{\mu\nu}$,
in the presence of external (fixed) background ${\bar{g}}_{\mu\nu}$, has the
form

$\displaystyle\delta h_{\mu\nu}(x)\,=\,\int
dy\,R_{\mu\nu\sigma}(x,y;{\bar{g}}+h)\xi^{\sigma},$ (15)

while $\delta{\bar{g}}_{\mu\nu}(x)=0$ and the action remains invariant,
$\delta{S}_{0}({\bar{g}}+h)=0$.

Because of the similarity with the Yang-Mills field, the Faddeev-Popov
quantization procedure is quite standard (see, e.g., the previous Chapter of
this Section for the specific QG details) and the resulting action
$S_{FP}=S_{FP}(\phi,{\bar{g}})$ has the form [6]

$\displaystyle
S_{FP}\,=\,S_{0}({\bar{g}}+h)+S_{gh}(\phi,{\bar{g}})+S_{gf}(\phi,{\bar{g}}),$
(16)

where the contents of $\phi$ depends on the choice of the QG model (3), as
explained below. In the presence of a background metric, the ghost action has
the form

$\displaystyle S_{gh}(\phi,{\bar{g}})\,=\,\int
dxdydz\sqrt{-{\bar{g}}(x)}\;{\bar{C}}^{\alpha}(x)\,H_{\alpha}^{\beta\gamma}(x,y;{\bar{g}},h)\,R_{\beta\gamma\sigma}(y,z;{\bar{g}}+h)\,C^{\sigma}(z),\quad$
(17)

with the notation

$\displaystyle
H_{\alpha}^{\beta\gamma}(x,y;{\bar{g}},h)=\frac{\delta\chi_{\alpha}(x;{\bar{g}},h)}{\delta
h_{\beta\gamma}(y)}.$ (18)

Furthermore, $S_{gf}({\bar{g}},h)$ is the gauge fixing action

$\displaystyle S_{gf}(\phi,{\bar{g}})=\int
dx\sqrt{-{\bar{g}}(x)}\;B^{\alpha}(x)\chi_{\alpha}(x;{\bar{g}},h).$ (19)

which corresponds to the singular gauge condition. For the non-singular gauge
condition the action has the form

$\displaystyle S_{gf}(\phi,{\bar{g}})=\int
dx\sqrt{-{\bar{g}}(x)}\;\Big{[}B^{\alpha}(x)\chi_{\alpha}(x;{\bar{g}},h)+\frac{1}{2}B^{\alpha}(x){\bar{g}}_{\alpha\beta}(x)B^{\beta}(x)\Big{]}.$
(20)

The reader can easily identify (20) as the particular version of the general
gauge fixing choice discussed in the previous Chapter [40] and also note that
(19) is a degenerate version of (20). In what follows, we shall use the form
(19), which is most useful for exploring the general features of
renormalization. Finally, $\chi_{\alpha}(x;{\bar{g}},h)$ are the gauge fixing
functions. It is worth mentioning that the generalization to a more
complicated non-singular gauge fixing, required in the higher derivative
theories, is considered in Sec. 5.

Now we are in a position to introduce an important notation used in (16),

$\displaystyle\phi\,=\,\\{\phi^{i}\\}\,\,=\,\,\big{\\{}h_{\mu\nu},\,B^{\alpha},\,C^{\alpha},\,{\bar{C}}^{\alpha}\big{\\}},$
(21)

which is the full set of quantum fields including quantum metric, Faddeev-
Popov ghost, anti-ghost and the Nakanishi-Lautrup auxiliary fields
$B^{\alpha}$.

We will need two sets of numbers characterizing the quantum fields. The
Grassmann parity of the fields $\phi^{i}$ will be denoted as
$\varepsilon(\phi^{i})=\varepsilon_{i}$. By definition, for ghost and anti-
ghost we have $\varepsilon(C^{\alpha})=\varepsilon({\bar{C}}^{\alpha})=1$,
while for the auxiliary fields $B^{\alpha}$ and the quantum metric we have
$\varepsilon(B^{\alpha})=\varepsilon(h_{\mu\nu})=0$.

On top of this, another conserved quantity called ghost number, can be defined
for the same fields as

$\displaystyle{\rm gh}(C^{\alpha})=1,\qquad{\rm
gh}({\bar{C}}^{\alpha})=-1\qquad\mbox{and}\qquad{\rm gh}(B^{\alpha})={\rm
gh}(h_{\mu\nu})=0.$ (22)

Consider the gravitational BRST transformations, which were introduced in [48,
17, 49]. For any admissible choice of gauge fixing functions
$\chi_{\alpha}(x;{\bar{g}},h)$ action (16) is invariant under global
supersymmetry (BRST symmetry) [10, 12],

$\displaystyle\delta_{B}h_{\mu\nu}(x)=\int
dyR_{\mu\nu\alpha}(x,y;{\bar{g}}+h)C^{\alpha}(y)\mu,$
$\displaystyle\delta_{B}B^{\alpha}(x)=0,$
$\displaystyle\delta_{B}C^{\alpha}(x)=-\,C^{\sigma}(x)\partial_{\sigma}C^{\alpha}(x)\mu,$
(23) $\displaystyle\delta_{B}{\bar{C}}^{\alpha}(x)=B^{\alpha}(x)\mu,$

where $\mu$ is a constant Grassmann parameter, such that $\mu^{2}=0$. It is a
remarkable fact that the BRST transformations (23) do not depend on gauge
fixing condition. One can present the BRST transformations (23) in the form

$\displaystyle\delta_{B}\phi^{i}(x)\,=\,R^{i}(x;\phi,{\bar{g}})\,\mu,$ (24)

where
$R^{i}=\big{\\{}R^{(h)}_{\mu\nu},\,R_{(B)}^{\,\alpha},\,R_{(C)}^{\,\alpha},\,R_{({\bar{C}})}^{\,\alpha}\big{\\}}$
and

$\displaystyle R^{(h)}_{\,\mu\nu}(x;\phi,{\bar{g}})\,=\,\int
dy\,R_{\mu\nu\sigma}(x,y;{\bar{g}}+h)\,C^{\sigma}(y),$ (25) $\displaystyle
R_{(B)}^{\alpha}(x;\phi,{\bar{g}})\,=\,0,$ $\displaystyle
R_{(C)}^{\alpha}(x;\phi,{\bar{g}})\,=\,-\,C^{\sigma}(x)\partial_{\sigma}C^{\alpha}(x),$
$\displaystyle R_{({\bar{C}})}^{\alpha}(x;\phi,{\bar{g}})\,=\,B^{\alpha}(x).$

It is easy to see that, in all cases,
$\varepsilon(R^{i}(x;\phi,{\bar{g}}))=\varepsilon_{i}+1$.

There is an important nilpotency property of the BRST transformations (23),
playing a significant role in the proof of gauge independence of the
$S$-matrix on quantum level. To discuss this property, it is useful to
introduce the BRST-operator, $\widehat{s}$, defined by its action of the
fields

$\displaystyle\delta_{B}\phi^{i}=\hat{s}\phi^{i}\mu,\qquad\varepsilon(\hat{s})=1.$
(26)

Let us formulate the proof of nilpotency using the condensed DeWitt’s
notations. Acting twice on all the quantum fields and using the explicit form
(23), we get

$\displaystyle\hat{s}^{2}B^{\alpha}\,=\,\hat{s}(\hat{s}B^{\alpha})=\hat{s}0=0,$
$\displaystyle\hat{s}^{2}{\bar{C}}^{\alpha}\,=\,\hat{s}(\hat{s}{\bar{C}}^{\alpha})=\hat{s}B^{\alpha}=0,$
$\displaystyle\hat{s}^{2}C^{\alpha}=\hat{s}(\hat{s}C^{\alpha})=-\hat{s}(C^{\sigma}\partial_{\sigma}C^{\alpha})=C^{\sigma}\partial_{\sigma}(\hat{s}C^{\alpha})-(\hat{s}C^{\sigma})\partial_{\sigma}C^{\alpha}$
$\displaystyle\qquad\;\;\,\,=\,-\,C^{\sigma}\partial_{\sigma}(C^{\rho}\partial_{\rho}C^{\alpha})+C^{\rho}\partial_{\rho}C^{\sigma}\partial_{\sigma}C^{\alpha}$
$\displaystyle\qquad\;\;\,\,=\,-\,C^{\sigma}\partial_{\sigma}C^{\rho}\partial_{\rho}C^{\alpha}-C^{\sigma}C^{\rho}\partial_{\sigma}\partial_{\rho}C^{\alpha}+C^{\rho}\partial_{\rho}C^{\sigma}\partial_{\sigma}C^{\alpha}$
$\displaystyle\qquad\;\;\,\,=\,-\,C^{\sigma}C^{\rho}\partial_{\sigma}\partial_{\rho}C^{\alpha}=0,$
$\displaystyle\hat{s}^{2}h_{\mu\nu}\,=\,\hat{s}(\hat{s}h_{\mu\nu})=\hat{s}(R_{\mu\nu\sigma}C^{\sigma})=(\hat{s}R_{\mu\nu\sigma})C^{\sigma}+R_{\mu\nu\sigma}(\hat{s}C^{\sigma})$
$\displaystyle\qquad\;\;\;\,\,=\,-\partial_{\sigma}(\hat{s}g_{\mu\nu})C^{\sigma}-\partial_{\sigma}g_{\mu\nu}(\hat{s}C^{\sigma})-(\hat{s}g_{\mu\sigma})\partial_{\nu}C^{\sigma}$
$\displaystyle\qquad\quad-\,g_{\mu\sigma}\partial_{\nu}(\hat{s}C^{\sigma})-(\hat{s}g_{\sigma\nu})\partial_{\mu}C_{\sigma}-g_{\sigma\nu}\partial_{\mu}(\hat{s}C^{\sigma})$
$\displaystyle\qquad\quad=\,\partial_{\sigma}\partial_{\rho}g_{\mu\nu}C^{\rho}C^{\sigma}+\partial_{\rho}g_{\mu\nu}\partial_{\sigma}C^{\rho}C^{\sigma}+\partial_{\rho}g_{\mu\nu}C^{\sigma}\partial_{\sigma}C^{\rho}$
$\displaystyle\qquad\quad+\,\partial_{\sigma}g_{\mu\rho}\partial_{\nu}C^{\rho}C^{\sigma}+\partial_{\sigma}g_{\mu\rho}C^{\sigma}\partial_{\nu}C^{\rho}+\partial_{\sigma}g_{\rho\nu}\partial_{\mu}C^{\rho}C^{\sigma}+\partial_{\sigma}g_{\rho\nu}C^{\sigma}\partial_{\nu}C^{\rho}$
$\displaystyle\qquad\quad+\,g_{\mu\rho}\partial_{\sigma}\partial_{\nu}C^{\rho}C^{\sigma}+g_{\mu\rho}C^{\sigma}\partial_{\sigma}\partial_{\nu}C^{\rho}+g_{\rho\nu}\partial_{\sigma}\partial_{\mu}C^{\rho}C^{\sigma}+g_{\rho\nu}C^{\sigma}\partial_{\sigma}\partial_{\mu}C^{\rho}$
$\displaystyle\qquad\quad+\,g_{\mu\rho}\partial_{\sigma}C^{\rho}\partial_{\nu}C^{\sigma}+g_{\mu\rho}\partial_{\nu}C^{\sigma}\partial_{\sigma}C^{\rho}+g_{\rho\nu}\partial_{\sigma}\partial_{\mu}C^{\rho}C^{\sigma}+g_{\rho\nu}\partial_{\mu}C^{\sigma}\partial_{\sigma}C^{\rho}$
$\displaystyle\qquad\quad+\,g_{\rho\sigma}\partial{\mu}C^{\rho}\partial_{\nu}C^{\sigma}+g_{\rho\sigma}\partial_{\nu}C^{\sigma}\partial_{\mu}C^{\rho}=0,$
(27)

where the anti-commutativity of ghost fields and the subsequent relations

$\displaystyle\partial_{\rho}\partial_{\sigma}\phi C^{\sigma}C^{\rho}=0,$ (28)

valid for an arbitrary function $\phi$, were used.

The BRST invariance of the action $S_{FP}$ can be expressed as

$\displaystyle\int
dx\,\,\frac{\delta_{r}S_{FP}}{\delta\phi^{i}(x)}\,\,R^{i}(x;\phi,{\bar{g}})\,=\,0.$
(29)

A compact and useful form of the invariance property (29) is called Zinn-
Justin equation [14]. To get it, one has to introduce the set of additional
field variables $\phi^{*}_{i}(x)$ and extend the action. The new fields
$\phi^{*}_{i}(x)$ are defined to have Grassmann parities opposite to the
corresponding fields $\,\phi^{i}(x)$, namely
$\,\varepsilon(\phi^{*}_{i})=\varepsilon_{i}+1$. The extended action
$S=S(\phi,\phi^{*},{\bar{g}})$ has the form

$\displaystyle S\,\,=\,\,S_{FP}\,+\,\int
dx\;\phi^{*}_{i}(x)\,R^{i}(x;\phi,{\bar{g}}).$ (30)

It is easy to note that the new fields $\phi^{*}_{i}(x)$ play the role of
sources to the BRST generators (25). With this addition, the relation (29)
gets the form of the Zinn-Justin equation for the action (30),

$\displaystyle\int
dx\,\,\frac{\delta_{r}S}{\delta\phi^{i}(x)}\,\frac{\delta_{l}S}{\delta\phi^{*}_{i}(x)}\,=\,0.$
(31)

It is worth remembering that using the left and right derivatives in the last
equation is relevant due to the nontrivial Grassmann parities of the involved
quantities.

Let us introduce the terminology of the Batalin-Vilkovisky formalism [50, 51].
The sources $\,\phi^{*}_{i}(x)\,$ are called antifields. Another fundamental
notion is the antibracket for two arbitrary functionals of fields and
antifields, $F=F(\phi,\phi^{*})$ and $G=G(\phi,\phi^{*})$, which is defined as

$\displaystyle(F,G)\,=\,\int
dx\,\bigg{(}\frac{\delta_{r}F}{\delta\phi^{i}(x)}\frac{\delta_{l}G}{\delta\phi^{*}_{i}(x)}-\frac{\delta_{r}F}{\delta\phi^{*}_{i}(x)}\frac{\delta_{l}G}{\delta\phi^{i}(x)}\bigg{)},$
(32)

The antibracket obeys the following properties:

1) Grassmann parity relations

$\displaystyle\varepsilon((F,G))=\varepsilon(F)+\varepsilon(G)+1=\varepsilon((G,F));$
(33)

2) Generalized antisymmetry property

$\displaystyle(F,G)=-(G,F)(-1)^{(\varepsilon(F)+1)(\varepsilon(G)+1)};$ (34)

3) Leibniz rule

$\displaystyle(F,GH)=(F,G)H+(F,H)G(-1)^{\varepsilon(G)\varepsilon(H)};$ (35)

4) Generalized Jacobi identity

$\displaystyle((F,G),H)(-1)^{(\varepsilon(F)+1)(\varepsilon(H)+1)}+{\sf
cycle}(F,G,H)\equiv 0.$ (36)

In terms of the antibracket, Zinn-Justin equation (31) can be written as

$\displaystyle(S,S)\,=\,0,$ (37)

which is the classical master equation of Batalin-Vilkovisky formalism [50,
51]. In what follows, we shall see that this equation can be generalized to
the quantum domain. This generalization is extensively used to analyse
renormalization in QG.

The formulation of the quantum gauge theory starts from the generating
functional of Green functions in the form of functional integral

$\displaystyle Z(J,{\bar{g}})\,=\,\int
d\phi\;\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{FP}(\phi,{\bar{g}})+J\phi\big{]}\Big{\\}}\,=\,\exp\Big{\\{}\frac{i}{\hbar}W(J,{\bar{g}})\Big{\\}},$
(38)

where $W(J,{\bar{g}})$ is the generating functional of connected Green
functions. In (38), the DeWitt notations are

$\displaystyle J\phi=\int dx\,J_{i}(x)\phi^{i}(x),\quad\mbox{where}\,\,\quad
J_{i}(x)=\big{\\{}J^{\mu\nu}(x),J^{(B)}_{\alpha}(x),{\bar{J}}_{\alpha}(x),J_{\alpha}(x)\big{\\}}\qquad$
(39)

are external sources for the fields (21). The Grassmann parities and ghost
numbers of these sources satisfy the relations

$\displaystyle\varepsilon(J_{i})=\varepsilon(\phi^{i}),\qquad{\rm
gh}(J_{i})={\rm gh}(\phi^{i}).$ (40)

Let us start a detailed analysis of the generating functionals and their gauge
dependencies from the generating functional (38). Later on, to complete the
program to prove gauge invariance of renormalization in QG, we shall introduce
a more general object $Z(J,\phi^{*},{\bar{g}})$, additionally depending on the
antifields $\phi^{*}$. This extended definition will be given below, and now
we work out a more simple case, just to introduce the necessary notions in a
more transparent way.

As a first step, consider the vacuum functional $Z_{\Psi}({\bar{g}})$, that
corresponds to the choice of the gauge fixing functional (29) in the presence
of external metric ${\bar{g}}$,

$\displaystyle Z_{\Psi}({\bar{g}})$ $\displaystyle=$ $\displaystyle\int
d\phi\;\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{0}({\bar{g}}+h)+\Psi(\phi,{\bar{g}}){\hat{R}}(\phi,{\bar{g}})\big{]}\Big{\\}}\,=\,\exp\Big{\\{}\frac{i}{\hbar}W_{\Psi}({\bar{g}})\Big{\\}},\qquad$
(41)

where we introduced the operator

$\displaystyle{\hat{R}}(\phi,{\bar{g}})=\int
dx\,\frac{\delta_{r}}{\delta\phi^{i}(x)}R^{i}(x;\phi,{\bar{g}})$ (42)

and the fermionic gauge fixing functional $\,\Psi(\phi,{\bar{g}})$,

$\displaystyle\Psi(\phi,{\bar{g}})=\int
dx\sqrt{-{\bar{g}(x)}}\;{\bar{C}}^{\alpha}\chi_{\alpha}(x;{\bar{g}},h).$ (43)

Taking into account (42) and (43), the expression in the exponential in the
integrand of (41) is nothing but $\frac{i}{\hbar}S_{FP}(\phi,{\bar{g}})$.
Thus, the expression (41) becomes

$\displaystyle Z_{\Psi}({\bar{g}})$ $\displaystyle=$ $\displaystyle\int
d\phi\;\exp\Big{\\{}\frac{i}{\hbar}S_{FP}(\phi,{\bar{g}})\Big{\\}},$ (44)

which is just (38) without the source term in the exponential.

The main advantage of using the fermionic gauge fixing functional
$\,\Psi(\phi,{\bar{g}})$ is that it is a scalar function that contains the
information about the gauge fixing function $\chi_{\alpha}(x;{\bar{g}},h)$. In
particular, the possible change of the gauge fixing can be explored by
evaluating $Z_{\Psi+\delta\Psi}$, that is the modified vacuum functional
corresponding to $\Psi(\phi,{\bar{g}})+\delta\Psi(\phi,{\bar{g}})$. Here,
$\delta\Psi(\phi,{\bar{g}})$ is an infinitesimal functional with odd Grassmann
parity. Besides this requirement, $\delta\Psi(\phi,{\bar{g}})$ maybe an
arbitrary variation corresponding to the change of
$\chi_{\alpha}(x;{\bar{g}},h)$ in Eq. (43).

Taking into account (44), with the new term we get the vacuum functional with
the variation of the gauge fixing,

$\displaystyle Z_{\Psi+\delta\Psi}({\bar{g}})=\int
d\phi\;\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{FP}(\phi,{\bar{g}})+\delta\Psi(\phi,{\bar{g}}){\hat{R}}(\phi,{\bar{g}})\big{]}\Big{\\}}.$
(45)

The next step is to make the change of quantum variables $\phi^{\;\\!i}$ in
the form of BRST transformations (23), but with replacing the constant
parameter $\mu$ by a functional $\mu=\mu(\phi,{\bar{g}})$,

$\displaystyle\phi^{i}(x)\,\,\,\,\longrightarrow\,\,\,\,\phi^{\prime
i}(x)=\phi^{i}(x)+R^{i}(x;\phi,{\bar{g}})\mu(\phi,{\bar{g}})=\phi^{i}(x)+\Delta\phi^{i}(x).\quad$
(46)

In what follows we shall use compact notations for the generators
$R^{i}(x;\phi,{\bar{g}})=R^{i}(x)$ and for the field
$\mu(\phi,{\bar{g}})=\mu$. Owing to the linearity of BRST transformations and
the absence of derivatives acting on the parameter $\mu$, the total action
$S_{FP}(\phi,{\bar{g}})$ remains invariant under (46) even for the non-
constant $\mu$. It is easy to check that the Jacobian of transformations (46)
reads [52] 444Let us note that the Jacobian of the transformations (46) can be
evaluated exactly [53, 54] but we will not reproduce this calculation here.

$\displaystyle J\,=\,J(\phi,{\bar{g}})\,=\,\exp\Big{\\{}\int
dx(-1)^{\varepsilon_{i}}M^{i}_{\;i}(x,x)\Big{\\}},$ (47)

where matrix $M^{i}_{\;j}(x,y)$ has the form

$\displaystyle
M^{i}_{\;j}(x,y)\,=\,\frac{\delta_{r}\Delta\phi^{i}(x)}{\delta\phi^{j}(y)}\,=\,(-1)^{\varepsilon_{j}+1}\frac{\delta_{r}\mu}{\delta\phi^{j}(y)}R^{i}(x)\,-\,(-1)^{\varepsilon_{j}(\varepsilon_{i}+1)}\frac{\delta_{l}R^{i}(x)}{\delta\phi^{j}(y)}\,\mu.\quad$
(48)

In the theories of the Yang-Mills type because of the antisymmetry of the
structure constants, there is the following relation:

$\displaystyle\int
dx\,(-1)^{\varepsilon_{i}}\,\frac{\delta_{l}R^{i}(x)}{\delta\phi^{i}(x)}\,=\,0.$
(49)

This relation can be verified using the definitions (25). Using (49), from
(47) and (48) it follows that

$\displaystyle J=\exp\\{-\mu(\phi,{\bar{g}}){\hat{R}}(\phi,{\bar{g}})\\},$
(50)

where we used the operator (42).

Choosing the functional $\mu$ in the form

$\displaystyle\mu\,=\,\frac{i}{\hbar}\delta\Psi(\phi,{\bar{g}}),$ (51)

one can observe that the described change of variables in the functional
integral completely compensates the modification in the expression (45)
compared to the initial formula (44). Thus, we arrive at the gauge
independence of the vacuum functional

$\displaystyle Z_{\Psi}({\bar{g}})\,\,=\,\,Z_{\Psi+\delta\Psi}({\bar{g}}).$
(52)

The last identity can be written as the vanishing variation of the vacuum
functionals $Z$ or for $W$, as introduced in (41),

$\displaystyle\delta_{\Psi}Z({\bar{g}})\,=\,0\qquad\mbox{or}\qquad\delta_{\Psi}W({\bar{g}})\,=\,0.$
(53)

The invariance (52) shows that we can omit the label $\Psi$ in the definition
of the generating functionals (38).

The importance of the invariance (52) is related to the equivalence theorem by
Kallosh and Tyutin [55]. This QFT theorem states that if the vacuum functional
is invariant, the $S$-matrix does not depend on the gauge fixing choice (see
also [56, 57]). In the QG case, the existence of the $S$-matrix requires that
the background metric ${\bar{g}}_{\mu\nu}$ admits asymptotic states. In
particular, this requirement is satisfied if there is a flat Minkowski metric
in some extremes, such that, in these regions, one can form asymptotic $in$
and $out$ states for the gravitons or, more generally, for the gravitational
perturbations over the flat spacetime. If this condition is satisfied, the
invariance (52) implies that the $S$-matrix in the QG theory does not depend
on the gauge fixing. One can say that if the theory and the physical situation
to which this theory is applied, admit the construction of the $S$-matrix, the
last will be independent on the choice of the gauge fixing conditions.

It is remarkable that we can make such a strong statement for an arbitrary
model of QG, even without requiring renormalizability of the theory or the
locality of the classical action. Let us stress that the statement formulated
above is valid only within the conventional perturbative approach to QFT, or
to the QG, as particular case. On the other hand, the situation may be
completely different in other approaches. The discussion of these situations
(including the asymptotic safety scenario in QG) and further references can be
found in Ref. [38].

On another hand, in QG gravity the $S$-matrix can be hardly seen as the object
of central interest. One of the reasons is that the gravitational fields in
the asymptotic states should be free and this cannot be provided in the
framework of general relativity. On the other hand, the main applications of
QG are assumed in cosmology and black hole physics. And in both cases, there
are no direct relations to the scattering problems and the $S$-matrix does not
look the most appropriate object to explore. Taking these points into account,
it may be more advantageous to work with the effective action, taking control
of its non-universality, i.e., the dependence on the gauge fixing and, more
generally, on the parametrization of quantum fields. Let us now see how this
can be done.

The effective action $\Gamma(\Phi,{\bar{g}})$ is defined by means of Legendre
transformation,

$\displaystyle\Gamma(\Phi,{\bar{g}})\,=\,W(J,{\bar{g}})\,-\,J_{i}\Phi^{i},$
(54)

where $\Phi=\\{\Phi^{i}\\}$ are mean fields and $J_{i}$ are the solutions of
the equations

$\displaystyle\frac{\delta W(J,{\bar{g}})}{\delta
J_{i}}\,=\,\Phi^{i}\quad\mbox{and}\quad J_{i}\Phi^{i}=\int
dx\;\\!J_{i}(x)\Phi^{i}(x).$ (55)

In terms of effective action, the property (53) means the on shell
independence on the gauge fixing condition, and can be cast in the form

$\displaystyle\delta_{\Psi}\Gamma(\Phi,{\bar{g}})\Bigg{|}_{\frac{\delta\Gamma(\Phi,{\bar{g}})}{\delta\Phi}=\;\\!0}=\,\,\,0,$
(56)

i.e., the effective action evaluated on its extremal does not depend on gauge.

Until now, we did not assume that the background metric may transform under
the general coordinate transformation. This was a necessary approach, as we
mentioned after the definition of the splitting (13) of the metric into the
background and quantum parts. However, when the effective action is defined,
we can perform the coordinate transformation of the background metric
${\bar{g}}_{\mu\nu}$, together with the corresponding transformation for the
quantum metric. Taking a variation of the background metric
${\bar{g}}_{\mu\nu}$ under general coordinate transformations, we get

$\displaystyle\delta^{(c)}_{\omega}{\bar{g}}_{\mu\nu}\,=\,R_{\mu\nu\sigma}({\bar{g}})\;\\!\omega^{\sigma}.$
(57)

Here, the symbol ${(c)}$ indicates that the transformation concerns the
background metric, i.e., is performed in the sector of classical fields. It is
important that this transformation does not lead neither to the change of the
form of the Faddeev-Popov action (16) nor to the change of the transformation
rules for the auxiliary and ghost fields.

In the quantum fields sector $h_{\mu\nu}$, the form of the transformations is
fixed by requiring the invariance of the action,

$\displaystyle\delta^{(q)}_{\omega}h_{\mu\nu}\,=\,R_{\mu\nu\sigma}(h)\;\\!\omega^{\sigma}\,=\,-\,\omega^{\sigma}\partial_{\sigma}h_{\mu\nu}-h_{\mu\sigma}\partial_{\nu}\omega^{\sigma}-h_{\sigma\nu}\partial_{\mu}\omega^{\sigma},$
(58)

where the symbol $(q)$ indicates the gauge transformations of quantum fields.
For the action we have

$\displaystyle\delta_{\omega}S_{0}({\bar{g}}+h)=0,\qquad\delta_{\omega}\,=\,\delta^{(c)}_{\omega}+\delta^{(q)}_{\omega}.$
(59)

With these definitions, for the variation of $Z({\bar{g}})$ we have

$\displaystyle\delta^{(c)}_{\omega}Z({\bar{g}})$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\int
d\phi\Big{[}\delta^{(c)}_{\omega}S_{0}({\bar{g}}+h)+\delta^{(c)}_{\omega}S_{gh}(\phi,{\bar{g}})+\delta^{(c)}_{\omega}S_{gf}(\phi,{\bar{g}})\Big{]}$
(60)
$\displaystyle\quad\times\,\,\exp\Big{\\{}\frac{i}{\hbar}S_{FP}(\phi,{\bar{g}})\Big{\\}}.\mbox{\qquad}$

Let us stress that, for a while, we consider only the transformations of
${\bar{g}}$, such that $\delta^{(q)}$ does not enter the last expression. At
the next stage, using a change of variables in the functional integral (60) we
have to arrive at the relation $\delta^{(c)}_{\omega}Z({\bar{g}})=0$, that is
to prove invariance of $Z({\bar{g}})$ under the transformations (57).

The gauge fixing action $S_{gf}(\phi,{\bar{g}})$ depends only on the three
field variables $h_{\mu\nu}$, $B^{\alpha}$, and ${\bar{g}}_{\mu\nu}$. For
$h_{\mu\nu}$ and ${\bar{g}}_{\mu\nu}$, the transformation law has been already
defined in (57) and (58). Thus, we need to define the transformation for the
remaining field $B^{\alpha}$. The new rule $\delta^{(q)}_{\omega}B^{\alpha}$
should compensate the variation of $S_{gf}(\phi,{\bar{g}})$ caused by the
transformations of ${\bar{g}}_{\mu\nu}$ and $h_{\mu\nu}$. The corresponding
condition has the form

$\displaystyle\delta_{\omega}S_{gf}$ $\displaystyle=$ $\displaystyle\int
dx\sqrt{-{\bar{g}}}\,\,\Big{[}\big{(}\delta^{(q)}_{\omega}B^{\alpha}+\omega^{\sigma}\partial_{\sigma}B^{\alpha}\big{)}\chi_{\alpha}({\bar{g}},h)$
(61)
$\displaystyle\quad+\,\,B^{\alpha}\omega^{\sigma}\partial_{\sigma}\chi_{\alpha}({\bar{g}},h)+B^{\alpha}\delta_{\omega}\chi_{\alpha}({\bar{g}},h)\Big{]}.\mbox{\qquad}$

The transformation of the gauge fixing functions $\chi_{\alpha}$ cannot be
defined independently since they are constructed from the metric and the last
transforms according to Eq. (57). Thus, the variation of the gauge fixing
functions $\chi_{\alpha}$ has the form of the vector field transformation (6),

$\displaystyle\delta_{\omega}\chi_{\alpha}=-\omega^{\sigma}\partial_{\sigma}\chi_{\alpha}-\chi_{\sigma}\partial_{\alpha}\omega^{\sigma}.$
(62)

It its turn, the transformation of the auxiliary field $B$ can be chosen
according to the same vector rule (7). This gives

$\displaystyle\delta^{(q)}_{\omega}B^{\alpha}=-\omega^{\sigma}\partial_{\sigma}B^{\alpha}+B^{\sigma}\partial_{\sigma}\omega^{\alpha}\,.$
(63)

It is easy to check that (62) and (63) provide the desired invariance in (61),

$\displaystyle\delta_{\omega}S_{gf}=0.$ (64)

In the same way, one can confirm the invariance of the ghost action,

$\displaystyle\delta_{\omega}S_{gh}=0$ (65)

for the vector transformation laws for the ghost fields ${\bar{C}}^{\alpha}$
and $C^{\alpha}$,

$\displaystyle\delta^{(q)}_{\omega}{\bar{C}}^{\alpha}(x)=-\omega^{\sigma}(x)\partial_{\sigma}{\bar{C}}^{\alpha}(x)+{\bar{C}}^{\rho}\partial_{\rho}\omega^{\alpha}(x),$
$\displaystyle\delta^{(q)}_{\omega}C^{\alpha}(x)=-\omega^{\sigma}(x)\partial_{\sigma}C^{\alpha}(x)+C^{\rho}\partial_{\rho}\omega^{\alpha}(x).$
(66)

All in all, we can conclude that the total Faddeev-Popov action $S_{FP}$ is
invariant

$\displaystyle\delta_{\omega}S_{FP}=0$ (67)

under the new version of gauge transformations, which is based on the
background transformations of quantum fields $\phi$ and of ${\bar{g}}$, i.e.,
(57), (58), (63) and (66).

As a consequence of (67), important property (44) and the invariance of the
integration measure, the vacuum functional possesses gauge invariance,

$\displaystyle\delta_{\omega}Z({\bar{g}})\,\,=\,\,\delta^{(c)}_{\omega}Z({\bar{g}})\,\,=\,\,0.$
(68)

As we shall see in what follows, one can use Eq. (68) to prove the gauge
invariance of an important object called the background effective action,
i.e., the effective action with the switched off mean fields,
$\Gamma({\bar{g}})=\Gamma(\Phi=0,{\bar{g}})$. The required feature can be
formulated as

$\displaystyle\delta^{(c)}_{\omega}\Gamma({\bar{g}})=0.$ (69)

Let us note that switching off the mean quantum fields
$\,\Phi=\\{h,\,C,\,{\bar{C}},\,B\\}\,$ requires special care and we shall see
how this should be done. Then, the resulting relation (69) is one of the main
targets of our consideration.

It is useful to start by exploring the off-shell gauge invariance of the
generating functionals of our interest. To this end, it is useful to present
the background transformations (57), (58), (63) and (66) in the form

$\displaystyle\delta^{(c)}_{\omega}{\bar{g}}_{\mu\nu}\,=\,R_{\mu\nu\sigma}({\bar{g}})\omega^{\sigma},\qquad\delta^{(q)}_{\omega}\phi^{i}\,=\,{\cal
R}^{i}_{\sigma}(\phi)\omega^{\sigma},$ (70)

where the generators ${\cal R}^{i}_{\sigma}(\phi)$ are linear in the quantum
fields $\phi$ and do not depend on the background metric ${\bar{g}}$.
Utilizing these notations, the general form of the transformation of an
arbitrary functional $\Gamma=\Gamma(\phi,{\bar{g}})$ can be written in the
form

$\displaystyle\delta_{\omega}\Gamma$ $\displaystyle=$
$\displaystyle\,\delta^{(c)}_{\omega}\Gamma\,+\,\frac{\delta_{r}\Gamma}{\delta\phi^{i}}\,{\cal
R}^{i}_{\sigma}(\phi)\omega^{\sigma}.$ (71)

Consider the variation of the generating functional $Z(J,{\bar{g}})$ (38) ,
under the gauge transformations of the background metric

$\displaystyle\delta^{(c)}_{\omega}Z(J,{\bar{g}})\,\,=\,\,\frac{i}{\hbar}\int
d\phi\;\delta^{(c)}_{\omega}S_{FP}(\phi,{\bar{g}})\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{FP}(\phi,{\bar{g}})+J\phi\big{]}\Big{\\}}.$
(72)

Using the background transformations in the sector of quantum fields $\phi$
and taking into account that for the linear change of variables the Jacobian
of this transformation is independent on the fields, we arrive at the relation

$\displaystyle\frac{i}{\hbar}\int
d\phi\;\Big{\\{}\delta^{(q)}_{\omega}S_{FP}(\phi,{\bar{g}})+J\delta^{(q)}_{\omega}\phi\Big{\\}}\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{FP}(\phi,{\bar{g}})+J\phi\big{]}\Big{\\}}\,=\,0.$
(73)

On the other hand, from (67) and (73) it follows that

$\displaystyle\delta^{(c)}_{\omega}Z(J,{\bar{g}})$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\int d\phi\,\,J_{j}\,{\cal
R}^{j}_{\sigma}(\phi)\,\omega^{\sigma}\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S_{FP}(\phi,{\bar{g}})+J\phi\big{]}\Big{\\}},$
(74) $\displaystyle=\,\,\frac{i}{\hbar}J_{j}{\cal
R}^{j}_{\sigma}\Big{(}\frac{\hbar}{i}\frac{\delta}{\delta
J}\Big{)}Z(J,{\bar{g}})\;\omega^{\sigma}.$

In terms of the generating functional
$W=W(J,{\bar{g}})=-i\hbar\,\mbox{ln}\,Z(J,{\bar{g}})$ of connected Green
functions, the relation (74) reads

$\displaystyle\delta^{(c)}_{\omega}W(J,{\bar{g}})\,\,=\,\,J_{j}\,{\cal
R}^{j}_{\sigma}\,\Big{(}\frac{\delta W}{\delta J}\Big{)}\;\omega^{\sigma},$
(75)

where we used linearity of generators ${\cal R}^{i}_{\sigma}(\phi)$ with
respect to $\phi$. Now we can consider the generating functional of vertex
functions (the effective action),

$\displaystyle\qquad\qquad\qquad\qquad\Gamma\,=\,\Gamma(\Phi,{\bar{g}})\,=\,W(J,{\bar{g}})-J\Phi,$
(76) $\displaystyle\mbox{ where}\qquad\Phi^{j}=\frac{\delta_{l}W}{\delta
J_{j}},\qquad\frac{\delta_{r}\Gamma}{\delta\Phi^{j}}=-J_{j}\qquad\mbox{and}\qquad\delta
W=\delta\Gamma.\qquad$ (77)

Taking the variation of external metric and the mean fields (70), in terms of
$\Gamma$, the relation (75) becomes

$\displaystyle\delta^{(c)}_{\omega}\Gamma(\Phi,{\bar{g}})\,\,=\,\,-\,\frac{\delta_{r}\Gamma}{\delta\Phi^{j}}\,\,{\cal
R}^{j}_{\sigma}(\Phi)\;\omega^{\sigma}.$ (78)

Finally, taking into account the variations of all fields (70) and using the
identity (71), we arrive at

$\displaystyle\delta_{\omega}\Gamma(\Phi,{\bar{g}})\,\,=\,\,0.$ (79)

It turns out that the relations (78) and (79) prove the fundamental property
(69). In order to see this, one has to note that the generators of quantum
fields (58), (66) and (63) have linear dependence of these fields. As a
result, one meets the following limit for the generators ${\cal
R}^{i}_{\sigma}(\Phi)$ when the mean fields are switched off:

$\displaystyle\lim_{\Phi\rightarrow 0}{\cal R}^{j}_{\sigma}(\Phi)=0.$ (80)

This relation shows that $\Gamma$ is invariant under non-deformed background
transformations, i.e, possesses the same invariance as the classical Faddeev-
Popov action.

In the renormalization program based on Batalin-Vilkovisky formalism, the
extended action is $S=S(\phi,\phi^{*},{\bar{g}})$ defined in (30). The
precursors for the full effective action are the extended generating
functional of Green functions $Z=Z(J,\phi^{*},{\bar{g}})$ and of the connected
Green functions $W=W(J,\phi^{*},{\bar{g}})$,

$\displaystyle Z(J,\phi^{*},{\bar{g}})\,=\,\int
d\phi\;\exp\Big{\\{}\frac{i}{\hbar}\big{[}S(\phi,\phi^{*},{\bar{g}})+J\phi\big{]}\Big{\\}}\,=\,\exp\Big{\\{}\frac{i}{\hbar}W(J,\phi^{*},{\bar{g}})\Big{\\}}.\quad$
(81)

Our first purpose is to prove that the effective action satisfies the quantum
version of Eq. (37). Due to the invariance of $S_{FP}$ under background fields
transformations, the variation of $S$ takes the form

$\displaystyle\delta_{\omega}S(\phi,\phi^{*},{\bar{g}})\,=\,\phi^{*}_{i}\delta_{\omega}R^{i}(\phi,{\bar{g}}),$
(82)

that shows that the action is gauge invariant on the hypersurface
$\,\phi^{*}_{i}=0$.

Using the condensed DeWitt’s notation one can write the variations of the
generators $\delta_{\omega}R^{i}(\phi,{\bar{g}})$ in the following compact
form:

$\displaystyle\delta_{\omega}R^{(h)}_{\,\mu\nu}(\phi,{\bar{g}})\,=\,-\,\omega^{\sigma}\partial_{\sigma}R_{\mu\nu\lambda}({\bar{g}}+h)C^{\lambda}-\partial_{\mu}\omega^{\sigma}R_{\sigma\nu\lambda}({\bar{g}}+h)C^{\lambda}$
$\displaystyle\qquad\qquad\qquad\quad-\,\partial_{\nu}\omega^{\sigma}R_{\mu\sigma\lambda}({\bar{g}}+h)C^{\lambda},$
$\displaystyle\delta_{\omega}R_{(B)}^{\alpha}(\phi,{\bar{g}})\,=\,0,$
$\displaystyle\delta_{\omega}R_{(C)}^{\alpha}(\phi,{\bar{g}})\,=\,\omega^{\sigma}\partial_{\sigma}(C^{\lambda}\partial_{\lambda}C^{\alpha})-C^{\lambda}\partial_{\lambda}C^{\sigma}\partial_{\sigma}\omega^{\alpha},$
$\displaystyle\delta_{\omega}R_{({\bar{C}})}^{\alpha}(\phi,{\bar{g}})\,=\,-\omega^{\sigma}\partial_{\sigma}B^{\alpha}+B^{\sigma}\partial_{\sigma}\omega^{\alpha}.$
(83)

The variations $\delta_{\omega}R^{i}(\phi,{\bar{g}})$ are at most quadratic in
the sector of fields $h_{\mu\nu}$ and $C^{\alpha}$ and linear in the field
${\bar{C}}^{\alpha}$.

Let us now consider the variation of the extended generating functional
$Z(J,\phi^{*},{\bar{g}})$ (81) under the gauge transformations of external
metric ${\bar{g}}$,

$\displaystyle\delta^{(c)}_{\omega}Z(J,\phi^{*},{\bar{g}})$ $\displaystyle=$
$\displaystyle\frac{i}{\hbar}\int
d\phi\,\,\Big{\\{}\delta^{(c)}_{\omega}S_{FP}(\phi,{\bar{g}})+\phi^{*}_{i}\delta^{(c)}_{\omega}R^{i}(\phi,{\bar{g}})\Big{\\}}$
(84)
$\displaystyle\times\,\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S(\phi,\phi^{*}{\bar{g}})+J\phi\big{]}\Big{\\}}.\mbox{\qquad}$

Making the change of variables $\phi^{i}$ according to (58), (63), and (66) in
the functional integral and taking into account the triviality of the
corresponding Jacobian, we arrive at the relation

$\displaystyle\frac{i}{\hbar}\int
d\phi\,\Big{\\{}\delta^{(q)}_{\omega}S_{FP}(\phi,{\bar{g}})+\phi^{*}_{i}\delta^{(q)}_{\omega}R^{i}(\phi,{\bar{g}})+J_{i}\delta^{(q)}_{\omega}\phi^{i}\Big{\\}}$
(85)
$\displaystyle\qquad\qquad\times\,\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S(\phi,\phi^{*}{\bar{g}})+J\phi\big{]}\Big{\\}}=0.\mbox{\qquad}$

Combining Eqs. (84) and (85) and using the invariance of $S_{FP}$ (67), we
obtain

$\displaystyle\delta^{(c)}_{\omega}Z(J,\phi^{*},{\bar{g}})$
$\displaystyle\,\,=\,\,$ $\displaystyle\frac{i}{\hbar}\int
d\phi\;\Big{\\{}\phi^{*}_{i}\delta_{\omega}R^{i}(\phi,{\bar{g}})+J_{i}{\cal
R}^{i}_{\sigma}(\phi)\omega^{\sigma}\Big{\\}}$ (86)
$\displaystyle\times\,\,\exp\Big{\\{}\frac{i}{\hbar}\big{[}S(\phi,\phi^{*}{\bar{g}})+J\phi\big{]}\Big{\\}},\mbox{\qquad}$

or, equivalently,

$\displaystyle\delta^{(c)}_{\omega}Z(J,\phi^{*},{\bar{g}})$
$\displaystyle\,\,=\,\,$
$\displaystyle\frac{i}{\hbar}\phi^{*}_{i}\,\delta_{\omega}R^{i}\Big{(}\frac{\hbar}{i}\frac{\delta}{\delta
J},{\bar{g}}\Big{)}\,Z(J,\phi^{*},{\bar{g}})$ (87)
$\displaystyle+\,\,\frac{i}{\hbar}J_{i}\,{\cal
R}^{i}_{\sigma}\Big{(}\frac{\hbar}{i}\frac{\delta}{\delta
J}\Big{)}\,Z(J,\phi^{*},{\bar{g}})\omega^{\sigma}.\mbox{\qquad}$

In terms of the generating functional of connected Green functions, (87)
becomes

$\displaystyle\delta^{(c)}_{\omega}W(J,\phi^{*},{\bar{g}})\,\,=\,\,\,\phi^{*}_{i}\delta_{\omega}R^{i}\Big{(}\frac{\delta
W}{\delta J}+\frac{\hbar}{i}\frac{\delta}{\delta
J},{\bar{g}}\Big{)}\,{\vec{1}}\,+\,J_{i}\,{\cal
R}^{i}_{\sigma}\,\Big{(}\frac{\delta W}{\delta J}\Big{)}\,\omega^{\sigma},$
(88)

where the symbol ${\vec{1}}$ means that the operator acts on the unit,
${\vec{1}}=1$. In the case of functional derivative, one has
$\frac{\delta}{\delta\phi}{\vec{1}}=0$, but since in many cases the
expressions are non-linear, this is a useful notation.

The extended generating functional of vertex functions (extended effective
action) is defined in a standard way. Starting from
$W=W(J,\phi^{*},{\bar{g}})$, introduced in Eq. (81), we perform Legendre
transformation,

$\displaystyle\Gamma(\Phi,\phi^{*},{\bar{g}})\,\,=\,\,W(J,\phi^{*},{\bar{g}})-J\Phi,\qquad\Phi^{j}=\frac{\delta_{l}W}{\delta
J_{j}},\qquad\frac{\delta_{r}\Gamma}{\delta\Phi^{j}}=-J_{j}.$ (89)

One can easily see that this definition is a generalization of Eq. (54).

As usual for the Legendre transformation, from (89) follows that

$\displaystyle\frac{\delta_{r}}{\delta J_{k}}\bigg{(}\frac{\delta_{l}W}{\delta
J_{i}}\bigg{)}\,\times\,\frac{\delta_{l}}{\delta\Phi^{i}}\bigg{(}\frac{\delta_{r}\Gamma}{\delta\Phi^{j}}\bigg{)}\,\,=\,\,-\,\delta^{k}_{j}.$
(90)

It proves useful to introduce the following notations:

$\displaystyle\delta_{\omega}{\bar{R}}^{i}(\Phi,\phi^{*},{\bar{g}})=\delta_{\omega}R^{i}({\hat{\Phi}},{\bar{g}}){\vec{1}},\qquad{\hat{\Phi}}^{j}=\Phi^{j}+i\hbar\big{(}\Gamma^{{}^{\prime\prime}-1}\big{)}^{jk}\,\frac{\delta_{l}}{\delta\Phi^{k}},$
(91)

where the symbol $\,(\Gamma^{{}^{\prime\prime}-1})^{jk}\,$ denotes the matrix
inverse to

$\displaystyle\big{(}\Gamma^{\prime\prime}\big{)}_{ij}\,\,=\,\,\frac{\delta_{l}}{\delta\Phi^{i}}\bigg{(}\frac{\delta_{r}\Gamma}{\delta\Phi^{j}}\bigg{)}\qquad\mbox{i.e.,}\qquad(\Gamma^{{}^{\prime\prime}-1})^{ik}(\Gamma^{{}^{\prime\prime}})_{kj}=\delta^{i}_{j}.$
(92)

In terms of extended effective action (89), Eq. (88) rewrites as

$\displaystyle\delta^{(c)}_{\omega}\Gamma(\Phi,\phi^{*},{\bar{g}})\,=\,-\,\frac{\delta_{r}\Gamma}{\delta\Phi^{i}}\,{\cal
R}^{i}_{\sigma}(\Phi)\omega^{\sigma}\,+\,\phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i}(\Phi,\phi^{*},{\bar{g}})$
(93)

or, using the relation (71), in the form

$\displaystyle\delta_{\omega}\Gamma(\Phi,\phi^{*},{\bar{g}})\,\,=\,\,\phi^{*}_{i}\,\delta_{\omega}{\bar{R}}^{i}(\Phi,\phi^{*},{\bar{g}}).$
(94)

At this point, we can draw a general conclusion concerning QG theories in the
background field formalism. At the non-renormalized level, any covariant QG
theory has the following general property: the extended quantum action
$S=S(\phi,\phi^{*},{\bar{g}})$ satisfies the classical master (Zinn-Justin)
equation of the Batalin-Vilkovisky formalism [50, 51], as we already
anticipated in Eq. (37). Furthermore, the extended effective action
$\Gamma=\Gamma(\Phi,\phi^{*},{\bar{g}})$ has the same symmetries and therefore
satisfies the same master equation,

$\displaystyle(\Gamma,\Gamma)\,=\,0.$ (95)

According to Eq. (82), the functional $S=S(\phi,\phi^{*},{\bar{g}})$ and,
according to (94), also $\Gamma=\Gamma(\Phi,\phi^{*},{\bar{g}})$, are
invariant under the background gauge transformations on the hypersurface
$\phi^{*}=0$,

$\displaystyle\delta_{\omega}S\big{|}_{\phi^{*}=0}\,=\,0,\qquad\delta_{\omega}\Gamma\big{|}_{\phi^{*}=0}\,=\,0$
(96)

and, more general, satisfy the relations (82) and (94).

##  3 Gauge-invariant renormalizability

Up to now, we were considering the non-renormalized generating functionals of
Green functions. The next step is to prove the gauge invariant
renormalizability, which is the property of renormalized generating
functionals. In the framework of Batalin-Vilkovisky formalism, one can prove
the BRST invariant renormalizability. The last means the preservation of basic
equations (37) for the extended action $\,S(\phi,\phi^{*},{\bar{g}})$ and an
identical equation (95) for the extended effective action
$\,\Gamma(\Phi,\Phi^{*},{\bar{g}})$ after renormalization. The required
identities for the classical and effective renormalized actions have the form

$\displaystyle(S_{R},S_{R})=0\qquad\mbox{and}\qquad(\Gamma_{R},\Gamma_{R})=0.$
(97)

Let us remember that the “classical” actions $S$ and $S_{R}$ are zero-order
approximations of the loop expansions in the parameter $\hbar$ of the bare and
renormalized effective actions $\Gamma$ and $\Gamma_{R}$. In this sense, Eq.
(37) is the zero-order approximation of Eq. (95) and what we have to do now is
to extend these two equations to the renormalized quantities $S_{R}$ and
$\Gamma_{R}$. Our strategy will be to explore this extension order by order in
the loop expansion parameter $\hbar$. In this way, we can prove that the
renormalized actions $S_{R}$ and $\Gamma_{R}$ obey the gauge invariance
property perturbatively.

###  3.1 BRST invariant renormalization

As it was explained above, our next purpose is to consider the loop expansion
in the master equation (95) and arrive at the conclusion about the coordinate
invariance of the $n$-loop approximation with switched-off sources. As a first
step, consider the one-loop approximation for
$\Gamma=\Gamma(\Phi,\Phi^{*},{\bar{g})}$. To provide the uniformity of
notations we shall use $\Phi^{*}=\phi^{*}$ for the antifields.

The effective action can be presented in the form

$\displaystyle\Gamma\,\,=\,\,\Gamma^{(1)}+{\cal
O}({\hbar}^{2})\,\,=\,\,S\,+\,\hbar\big{[}{\Gamma}^{(1)}_{div}\,+\,{\Gamma}^{(1)}_{fin}\big{]}\,+\,{\cal
O}({\hbar}^{2}),$ (98)

where $S=S(\Phi,\Phi^{*},{\bar{g}})$ is the tree-level action, and
$\Gamma^{(1)}_{div}$ and $\Gamma^{(1)}_{fin}$ stand for the divergent and
finite parts of the one-loop approximation for $\Gamma$.

In the local models of QG, the locality of the divergent part of effective
action is guaranteed by Weinberg’s theorem [58] (see also [59] for an
alternative proof and further references). Let us note, by passing, that there
is an exception from this general rule, corresponding to the theories with
scalar field(s) and with a spontaneous symmetry breaking, in the presence of
the background gravitational field. In this situation, there may be nonlocal
divergences [60], however this can be seen as an exception that confirms the
general rule instead of contradicting it.

Furthermore, even if the starting action is nonlocal, the UV divergences are
expected to be described by local functionals. The physical reason for this is
that the high energy domain always corresponds to the short-distance limit.
And in the case of UV divergences, the energies are infinitely high, hence the
distances should be infinitely short, which does not leave space for the non-
local divergences.

Since there is a special Section of this handbook devoted to the nonlocal QG
models, we will not extend this discussion and just mention Refs. [21, 61, 62,
22]). It was argued in these works that the UV divergent part of effective
action, for a wide class of nonlocal models of QG, is local, including the
ones with a non-local classical action. Taking this into account, we can
assume that $\Gamma^{(1)}_{div}$ is a local functional. The divergence
determines the form of the counterterms of the one-loop renormalized action,
$\,\Delta S=-\hbar{\Gamma}^{(1)}_{div}$, such that the renormalized action is

$\displaystyle S_{1R}\,\,=\,\,S\,-\,\hbar{\Gamma}^{(1)}_{div},$ (99)

which is also a local functional.

Let us substitute the expansion (98) in Eq. (95) and preserve terms up to the
first order in $\hbar$. Thus, $\,\Gamma^{(1)}_{div}\,$ and
$\,\Gamma^{(1)}_{fin}\,$ satisfy the equation

$\displaystyle 0\,=\,(\Gamma,\Gamma)$ $\displaystyle=$
$\displaystyle(S,S)\,+\,2\hbar\big{(}S,\Gamma^{(1)}_{div}\big{)}\,+\,2\hbar\big{(}S,\Gamma^{(1)}_{fin}\big{)}\,+\,O(\hbar^{2})$
(100) $\displaystyle=$ $\displaystyle
2\hbar\big{(}S,\Gamma^{(1)}_{div}\big{)}+2\hbar\big{(}S,\Gamma^{(1)}_{fin}\big{)}\,+\,O(\hbar^{2}),$

where we used the zero-order equation (37). In the first order in $\hbar$, we
have a vanishing sum of the two terms, one of them is infinite and hence it
has to vanish independently on another one. Therefore

$\displaystyle\big{(}S,\;{\Gamma}^{(1)}_{div}\big{)}\,=\,0.$ (101)

Let us consider

$\displaystyle\big{(}S_{1R},\;S_{1R}\big{)}\,=\,\big{(}S,S\big{)}\,-\,2\hbar\big{(}S,{\Gamma}^{(1)}_{div}\big{)}\,+\,\hbar^{2}\big{(}{\Gamma}^{(1)}_{div},{\Gamma}^{(1)}_{div}\big{)}.$
(102)

Taking into account (37) and (101), we find the relation

$\displaystyle(S_{1R},\;S_{1R})={\hbar}^{2}E_{2},$ (103)

where $E_{2}$ is the known functional. Thus we have shown that $S_{1R}$
satisfies the classical master equation (37) up to the terms of order
$\hbar^{2}$,

$\displaystyle\quad
E_{2}\,=\,\big{(}\Gamma^{(1)}_{div},\,\Gamma^{(1)}_{div}\big{)}.$ (104)

Let us now include one-loop corrections to $S$. The one-loop effective action
${\Gamma}_{1R}$ can be constructed by adding a local counterterm to the ${\cal
O}(\hbar)$ part of Eq. (98). As usual, the counterterm has the divergent part
which cancels the divergence of ${\Gamma}^{(1)}_{div}$, and the remaining
contribution is finite and typically depends on the renormalization parameter
$\mu$. The dependence on $\mu$ always repeats the form of divergence which, as
we have seen above, does not violate the master equation. Thus, we can simply
use (99) and assume that ${\Gamma}_{1R}$ is constructed by following the
procedure of quantization described above, with $S$ replaced by $S_{1R}$.

Being constructed in this way, the functional ${\Gamma}_{1R}$ is finite in the
one-loop approximation and satisfies the equation

$\displaystyle({\Gamma}_{1R},\;{\Gamma}_{1R})={\hbar}^{2}E_{2}+\mbox{{\sl
O}}({\hbar}^{3}).$ (105)

Now it is time to consider the next level of the loop expansion. Consider the
one-loop renormalized effective action, taking into account the ${\cal
O}(\hbar^{2})$-terms,

$\displaystyle\Gamma_{1R}\,=\,S\,+\,\hbar{\Gamma}^{(1)}_{fin}\,+\,{\hbar}^{2}\big{[}\Gamma^{(2)}_{1,div}\,+\,\Gamma^{(2)}_{1,fin}\big{]}\,+\,\mbox{{\sl
O}}({\hbar}^{3}).$ (106)

Here $\Gamma^{(2)}_{1,div}$ and $\Gamma^{(2)}_{1,fin}$ are divergent and
finite ${\cal O}(\hbar^{2})$ parts of the two-loop effective action
constructed on the basis of $S_{1R}$ instead of $S$. The divergent part
${\Gamma}^{(2)}_{1,div}$ of the two-loop approximation for ${\Gamma}_{1R}$
determines the two-loop renormalization for $S_{2R}$, according to

$\displaystyle S_{2R}\,=\,S_{1R}\,-\,{\hbar}^{2}{\Gamma}^{(2)}_{1,div}$ (107)

and satisfies the equation

$\displaystyle(S,\;{\Gamma}^{(2)}_{1,div})\,=\,E_{2}.$

As the next step, let us consider

$\displaystyle(S_{2R},\;S_{2R})={\hbar}^{3}E_{3}+\mbox{{\sl O}}({\hbar}^{4}).$
(108)

We have found that $S_{2R}$ satisfies the master equations up to the terms
$\hbar^{3}E_{3}$, where

$\displaystyle E_{3}=2({\Gamma}^{(1)}_{div},\;{\Gamma}^{(2)}_{1,div}),$ (109)

The effective action $\Gamma_{2R}$ is generated by replacing $S_{2R}$ into
functional integral instead of $S$. Therefore, $\Gamma_{2R}$ is automatically
finite in the two-loop approximation,

$\displaystyle{\Gamma}_{2R}=S+\hbar{\Gamma}^{(1)}_{fin}+{\hbar}^{2}{\Gamma}^{(2)}_{1,fin}+{\hbar}^{3}\big{[}{\Gamma}^{(3)}_{2,div}+{\Gamma}^{(3)}_{2,fin}\big{]}+\mbox{{\sl
O}}({\hbar}^{4})$

and satisfies the equation

$\displaystyle({\Gamma}_{2R},\;{\Gamma}_{2R})={\hbar}^{3}E_{3}+\mbox{{\sl
O}}({\hbar}^{4}).$ (110)

Furthermore, by applying the induction method we find that the totally
renormalized action $S_{R}$ is given by the expression

$\displaystyle
S_{R}\,=\,S\,-\,\sum_{n=1}^{\infty}{\hbar}^{n}{\Gamma}^{(n)}_{n-1,div}.$ (111)

We assume that $\Gamma^{(n)}_{n-1,div}$ and $\Gamma^{(n)}_{n-1,fin}$ are the
divergent and finite parts of the $n$-loop approximation for the effective
action, which is already finite in $(n-1)$-loop approximation, since it is
constructed on the basis of the action $S_{(n-1)R}$.

The action (111) is a local functional and exactly satisfies the classical
master equation

$\displaystyle(S_{R},\;S_{R})=0.$ (112)

We have shown that this relation holds in all orders of the loop expansion.
This means that the BRST symmetry is preserved in the renormalized action
$S_{R}$. Let us note that this corresponds exactly to the BRST cohomology on
local functionals with ghost number zero [63, 64].

The renormalized effective action ${\Gamma}_{R}$ is finite in each order of
the loop expansion in the powers of $\hbar$,

$\displaystyle{\Gamma}_{R}=S+\sum_{n=1}^{\infty}{\hbar}^{n}{\Gamma}^{(n)}_{n-1,fin},$
(113)

and satisfies the gravitational analog of the Slavnov-Taylor identities [8,
15, 16] in Yang-Mills theory (see also [1] for the pedagogical introduction),
Thus, the renormalized action $S_{R}$ and the effective action ${\Gamma}_{R}$
satisfy the classical master equation and the gravitational version of Ward
(Slavnov-Taylor) identity, respectively.

###  3.2  Gauge invariance of renormalized background effective action

The next part of our consideration will be to generalize the transformation
relations (82) and (94) for the renormalized functionals of classical $S_{R}$
and effective $\Gamma_{R}$ actions. As before, we consider the transformations
perturbatively, starting from the lowest orders. In the one-loop
approximation, from (94) it follows

$\displaystyle\delta_{\omega}\Gamma(\Phi,\Phi^{*},{\bar{g}})$
$\displaystyle\,=\,$
$\displaystyle\Phi^{*}_{i}\delta_{\omega}R^{i}(\Phi,{\bar{g}})\,+\,\hbar\Phi^{*}_{i}\,\delta_{\omega}{\bar{R}}^{i(1)}_{div}(\Phi,\Phi^{*},{\bar{g}})$
(114)
$\displaystyle+\,\,\,\hbar\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(1)}_{fin}(\Phi,\Phi^{*},{\bar{g}})+O(\hbar^{2}),$

where we used the condensed notations (91). In the last expression,

$\displaystyle\delta_{\omega}{\bar{R}}^{i(1)}_{div}(\Phi,\Phi^{*},{\bar{g}})\qquad\mbox{and}\qquad\delta_{\omega}{\bar{R}}^{i(1)}_{fin}(\Phi,\Phi^{*},{\bar{g}})$

are divergent and finite parts of the one-loop approximation for the gauge
transformations $\delta_{\omega}{\bar{R}}^{i}(\Phi,\Phi^{*},{\bar{g}})$,
correspondingly. On the other hand, from (98) we have

$\displaystyle\delta_{\omega}\Gamma(\Phi,\Phi^{*},{\bar{g}})$
$\displaystyle\,=\,$
$\displaystyle\delta_{\omega}S(\Phi,\Phi^{*},{\bar{g}})+\hbar\delta_{\omega}\Gamma^{(1)}_{div}\,+\,\hbar\delta_{\omega}\Gamma^{(1)}_{fin}\,+\,O(\hbar^{2}).$
(115)

The comparison of the two transformations (114) and (115) tells us that

$\displaystyle\delta_{\omega}\Gamma^{(1)}_{div}$ $\displaystyle\,=\,$
$\displaystyle\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(1)}_{div}(\Phi,\Phi^{*},{\bar{g}}),$
(116) $\displaystyle\delta_{\omega}\Gamma^{(1)}_{fin}$ $\displaystyle\,=\,$
$\displaystyle\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(1)}_{fin}(\Phi,\Phi^{*},{\bar{g}}).$
(117)

From Eq. (116) and the definition (99) follows that the one-loop renormalized
action $S_{1R}=S_{1R}(\Phi,\Phi^{*},{\bar{g}})$ transforms according to

$\displaystyle\delta_{\omega}S_{1R}\,=\,\Phi^{*}_{i}\delta_{\omega}R^{i(1)}_{R},$
(118)

where

$\displaystyle
R^{i(1)}_{R}=R^{i(1)}_{R}(\Phi,\Phi^{*},{\bar{g}})=\delta_{\omega}R^{i}(\Phi,{\bar{g}})-\hbar\delta_{\omega}{\bar{R}}^{i(1)}_{div}(\Phi,\Phi^{*},{\bar{g}}).$
(119)

The last relations mean that the action $S_{1R}$ is invariant under the
background gauge transformations with one-loop deformed gauge generators
$R^{i(1)}_{R}$ (119) on the hypersurface $\Phi^{*}=0$. Furthermore, due to Eq.
(118), functional $\,\Gamma_{1R}\,$ obeys the transformation rule

$\displaystyle\delta_{\omega}\Gamma_{1R}$ $\displaystyle=$
$\displaystyle\Phi^{*}_{i}\delta_{\omega}R^{i}\,+\,\hbar\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(1)}_{fin}$
(120)
$\displaystyle+\,\,\hbar^{2}\Big{[}\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(2)}_{1,div}+\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(2)}_{1,fin}\Big{]}\,+\,O(\hbar^{3}),\qquad$

where
$\,\delta_{\omega}{\bar{R}}^{i(2)}_{1,div}=\delta_{\omega}{\bar{R}}^{i(2)}_{1,div}(\Phi,\Phi^{*},{\bar{g}})\,$
and
$\,\delta_{\omega}{\bar{R}}^{i(2)}_{1,fin}=\delta_{\omega}{\bar{R}}^{i(2)}_{1,fin}(\Phi,\Phi^{*},{\bar{g}})\,$
are related to $\,\Gamma^{(2)}_{1,div}$ and $\Gamma^{(2)}_{1,fin}\,$ in Eq.
(106) as

$\displaystyle\delta_{\omega}\Gamma^{(2)}_{1,div}=\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(2)}_{1,div},\qquad\delta_{\omega}\Gamma^{(2)}_{1,fin}=\Phi^{*}_{i}\delta_{\omega}{\bar{R}}^{i(2)}_{1,fin}.$
(121)

Therefore the functional $\,\Gamma_{1R}\,$ is finite in the one-loop
approximation and is invariant under the background gauge transformations up
to the second order in $\hbar$ on the hypersurface $\,\Phi^{*}=0$.

Applying the induction method, one can show that the renormalized functionals
$S_{R}$ and $\Gamma_{R}$ satisfy the properties555We note that these
statements are very close to the results concerning preservation of global
symmetries of initial classical action at quantum level when the effective
action of theory under consideration is invariant under deformed global
transformations of all fields [65].

$\displaystyle\delta_{\omega}S_{R}=\Phi^{*}_{i}\,\delta_{\omega}R^{i}_{R}\,,\qquad\quad\delta_{\omega}\Gamma_{R}=\Phi^{*}_{i}\,\delta_{\omega}{\bar{R}}^{i}_{R},$
(122)

where

$\displaystyle\delta_{\omega}R^{i}_{R}$ $\displaystyle\,=\,$
$\displaystyle\delta_{\omega}R^{i}\,-\,\hbar\delta_{\omega}{\bar{R}}^{i(1)}_{div}\,-\,\hbar^{2}\delta_{\omega}{\bar{R}}^{i(2)}_{1,div}\,-\,\cdots,$
(123) $\displaystyle\delta_{\omega}{\bar{R}}^{i}_{R}$ $\displaystyle\,=\,$
$\displaystyle\delta_{\omega}R^{i}\,+\,\hbar\delta_{\omega}{\bar{R}}^{i(1)}_{fin}\,+\,\hbar^{2}\delta_{\omega}{\bar{R}}^{i(2)}_{1,fin}\,+\,\cdots.$
(124)

It is important that $\,\delta_{\omega}{\bar{R}}^{i}_{R}$, defined in (124),
are finite.

As it was discussed above, the UV divergences in any models of QG are local,
even for the nonlocal QG theories [21, 62, 22]. As a consequence, the
quantities $\,\delta_{\omega}R^{i}_{R}$, defined in (123), are local while in
both local and non-local models of QG, including the proper transformations
$\delta_{\omega}$. Then, an important consequence of the results (122) is that
the renormalized functionals
$\,S_{R}(\Phi,{\bar{g}})=S_{R}(\Phi,\Phi^{*}=0,{\bar{g}})\,$ and
$\,\Gamma_{R}(\Phi,{\bar{g}})=\Gamma_{R}(\Phi,\Phi^{*}=0,{\bar{g}})\,$ obey
the symmetry

$\displaystyle\delta_{\omega}S_{R}(\Phi,{\bar{g}})\,=\,0,\qquad\delta_{\omega}\Gamma_{R}(\Phi,{\bar{g}})\,=\,0.$
(125)

These are the same transformations as for the non-renormalized functionals
$S(\Phi,{\bar{g}})=S_{FP}(\Phi,{\bar{g}})$ and $\Gamma(\Phi,{\bar{g}})$ in
(67) and (79), respectively. Therefore, from (125) follows the invariance
property of the renormalized background functionals
$\,S_{R}({\bar{g}})=S(\Phi=0,{\bar{g}})\,$ and
$\,\Gamma({\bar{g}})=\Gamma(\Phi=0,{\bar{g}})\,$ under general coordinate
transformations of external background metric $\,{\bar{g}}_{\mu\nu}$,

$\displaystyle\delta^{(c)}_{\omega}S_{R}({\bar{g}})=0,\qquad\delta^{(c)}_{\omega}\Gamma_{R}({\bar{g}})=0.$
(126)

It is easy to see that these invariances repeat exactly the symmetry
properties of initial action $\,S_{0}({\bar{g}})\,$ and
$\,\Gamma({\bar{g}})\,$ in (68).

##  4 A short historical review and more special notes

In order to understand better the relevance of the results described above,
let us start by presenting a short historical review of the subject. The first
proof of the gauge invariant renormalizability in QG was given by Stelle in
the famous 1977 paper [17]. Despite the considerations in this paper were done
for the simplest renormalizable model of quantum gravity with four
derivatives, most of the analysis is sufficiently general and can be applied,
also, to other covariant models of QG. After that, there were further
publications devoted to the invariant renormalizability in QG [34, 35, 37,
38], where the main conclusion about the diffeomorphism invariant
renormalizability has been confirmed with different degrees of generality and
using formally different methods, regardless all of the proofs are based on
the BRST symmetry.

Since QG is a particular example of gauge theory, the program of exploring
invariant renormalizability cannot be separated from the works done in the
framework Yang-Mills theories. The most significant achievement in this
respect was the demonstration of BRST-invariant renormalizability in the
theories which may be not renormalizable by power counting666The gauge
invariant renormalizability, independent on the power counting, is especially
important for the effective QFT [1] and, in particular, for effective approach
to QG. The interested reader can consult a specially devoted to effective QG
Section of the present Handbook for further details. . In 1982 it was
formulated the first proof for the general gauge theories [66], based on the
Batalin-Vilkovisky formalism [50, 51]. The approach employed in [66] assumed
the regularization procedure respecting the gauge invariance of initial
classical action and providing zero volume divergences, i.e., with the
condition $\,\delta(0)=0$ satisfied. The proof is valid for any boundary
condition related to an initial gauge invariant action and for arbitrary
choice of gauge fixing functions. Furthermore, the renormalization procedure
of [66] can be described in terms of anticanonical transformations (for recent
developments, see [67, 68]) which are defined as transformations preserving
the antibracket, using the terminology of the standard review paper [69].

An alternative, albeit very close, approach to prove the BRST invariant
renormalization of general gauge theories [70], is based on the use of
cohomologies of nilpotent BRST operator associated with the adjoint operation
of the antibracket of the action $S$ with an arbitrary functional $F$,
${\hat{s}}F=(S,F)$ [63, 64]. The detailed description of this approach can be
found in [69] and in the chapter 17.3 of Vol. II of the book [1]. The
disadvantage of this approach is that it does not directly extends to the
background field method.

On the other hand, the background field formalism [7, 42, 43] represents a
powerful approach to study quantum properties of gauge theories, allowing to
keep the gauge invariance (general covariance, for QG), at all stages of the
loop expansion, including in practical calculations. From the viewpoint of the
quantization of gauge systems the background field method corresponds to the
special choice of a boundary condition and to the special choice of gauge
fixing functions. However, since the background field method requires the
presence of an “external” field in the course of the Lagrangian quantization,
this formalism should be considered as a very special case which requires
special care. Indeed, this special case attracted a great deal of attention
recently [37, 44, 45, 46, 47].

A few more special notes on the structure of the gauge algebra underlying a
given classical system are in order. Already the quantization of gauge systems
using the Faddeev-Popov approach [6], should be applied to the theories when
the gauge algebra is associated with a Lie group, as otherwise, the
quantization may give wrong results. The reason is that, in these, more
complicated cases, the gauge algebra may be reducible/open [71, 72] or
structure functions may depend on the fields of the initial gauge theory, as
it is happens in supergravity [73, 74, 75, 76]. In these cases, the symmetry
transformations leave the action invariant, but do not form a gauge group.
Thus, the quantization of gauge theories requires taking into account not only
the invariance of the action, but also such important aspects as open/closed
algebras, the presence of reducible generators, and so on. The quantization of
these complicated theories is possible using different types of ghosts,
antighosts, ghosts for ghosts (Nielsen-Kallosh ghosts etc.) [72, 75, 76, 77,
78, 79, 80, 81] (for recent applications to quantization of reducible gauge
theories see [82, 83, 84]). The most general (and the unique completely self-
consistent) approach to the problem of covariant quantization, which
summarized all these approaches, was proposed by Batalin and Vilkovisky [50,
51]. The Batalin-Vilkovisky formalism gives the rules of quantization of the
general gauge theories which may be characterized by open/reducible gauge
algebras, with structure functions depending on the fields of the initial
action. The Batalin-Vilkovisky formalism is not only a powerful quantization
method, as it also enables us to explore various complicated subjects and
issues in gauge theories [66, 70, 85, 86, 87, 88, 67, 89]. And among these
important applications, the renormalization of the quantum gauge theories of
different types is one of the main problems. Coming back to gravity, in the
next sections we will see that QG belongs to the theories of the Yang-Mills
type, i.e., it admits the traditional Faddeev-Popov approach and enables one
to prove the gauge-invariant renormalizability [17] (see also [35]). However,
using the Batalin-Vilkovisky formalism, we can make this proof more elegant
and simple, as we shall see in the rest of this review.

##  5 On the gauge fixing in the higher derivative models

Eqs. (122) show that with the antifields switched off, i.e., with
$\Phi^{*}=0$, both the renormalized classical action $S_{R}$ and effective
action $\Gamma_{R}$ are gauge invariant quantities. In particular, this means
that if we restrict the consideration by the standard non-extended generating
functional of the Green functions, that means, without introducing sources for
the ghosts $\,C$, $\,{\bar{C}}$ and for the auxiliary field $\,B$, the
effective action will be metric-dependent and generally covariant functional.
This statement concerns both divergent and finite parts of renormalized
effective action. In what follows we discuss how the gauge invariant
renormalizability can be applied to the renormalization of the models of QG.
The rest of this section partially repeats some part of the previous Chapter
[40], but we shall discuss the subject from a different perspective.

The use of the power counting arguments in QG models may be especially simple
if the following two conditions are satisfied:

i) The propagator of the gravitational field should be homogeneous in the
powers of momenta. This means, in particular, that the free equations for
different modes of the gravitational field (tensor, vector and scalar) are of
the same order in derivatives after the Faddeev-Popov procedure.

ii) The propagator of FP ghosts should have the same powers of momenta as all
modes of the gravitational field.

As we explained in detail in the previous Chapter, these conditions are
fulfilled if the gauge fixing term and the ghost action have the form

$\displaystyle S_{gf}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\chi^{\alpha}\,Y_{\alpha\beta}\,\chi^{\beta},$ (127)
$\displaystyle S_{gh}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,{\bar{C}}^{\alpha}\,Y_{\alpha\beta}\,M^{\beta}_{\,\lambda}\,C^{\lambda},$
(128)

where, according to (17) and (18),

$\displaystyle M^{\beta}_{\,\lambda}$ $\displaystyle=$ $\displaystyle
H_{\beta}^{\rho\sigma}(x,y;{\bar{g}},h)R_{\rho\sigma\lambda}(y,z;{\bar{g}}+h).$
(129)

The choice of the weight operator $Y_{\alpha\beta}$ should be done in such a
way that the total amount of derivatives in the expressions (127) and (128) be
the same as in the action of the model of quantum gravity under consideration.
For instance, in the QG based on general relativity $Y_{\alpha\beta}=\theta
g_{\alpha\beta}$, where $\theta$ is a constant gauge fixing parameter. In case
of the fourth order gravity one has to take [17, 90, 91]

$\displaystyle Y_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\theta_{1}\delta_{\alpha\beta}\operatorname{\square}+\theta_{2}\nabla_{\alpha}\nabla_{\beta}+\theta_{3}R_{\alpha\beta}+\theta_{4}\delta_{\alpha\beta}R,$
(130)

where $\theta_{1,2,3,4}$ are gauge fixing constants. In the case of six-
derivative superrenormalizable gravity model [18] $\theta_{1,2,3,4}$ should be
linear functions of d’Alembertian operator $\,\operatorname{\square}$, plus
the possible linear in curvature tensor terms, for the eight-derivative QG the
parameters $\theta_{1,2,3,4}$ become quadratic functions of
$\operatorname{\square}$, etc.

An important question is how to incorporate the modified gauge fixing and
ghost actions (127) and (128) into the proof of gauge invariant
renormalizability which we developed in Sec. 3. The simplest possibility in
this direction is to remember that the effective action in the
superrenormalizable QG theories with more than four derivatives does not
depend on the gauge fixing [18]. Thus the scheme based on the weight function
(130) with $\theta_{1,2,3,4}$ linear functions of $\operatorname{\square}$,
does not affect the loop corrections. This argument looks convincing, but let
us present some extra details and refer to [38] for further discussions.

Consider $\chi_{\alpha}=\chi_{\alpha}(x;{\bar{g}},h)$ being a standard gauge
fixing function used in previous sections. We can introduce the set of two
differential operators, $Y_{\alpha}^{\,\beta}$ and $Y_{1\alpha\beta}$. These
weight operators must have the structure of tensor fields of types $\,(1,1)$
and $(0,2)$, respectively, and can not depend on the quantum metric
$h_{\mu\nu}$,

$\displaystyle
Y_{\alpha}^{\beta}(x,y)\,=\,Y_{\alpha}^{\beta}(x,y;{\bar{g}},{\bar{\square}})\qquad\mbox{and}\qquad
Y_{1\alpha\beta}(x,y)\,=\,Y_{1\alpha\beta}(x,y;{\bar{g}},{\bar{\square}}).\quad$
(131)

The next step is to modify the gauge fixing functions $\,\chi_{\alpha}$, by
the following rule:

$\displaystyle\chi^{mod}_{\alpha}(x;{\bar{g}},h,B)\,=\,\int
dy\,\Big{[}Y_{\alpha}^{\beta}(x,y;{\bar{g}},{\bar{\operatorname{\square}}})\,\chi_{\beta}(y;{\bar{g}},h)\,+\,\frac{1}{2}Y_{1\alpha\beta}(x,y;{\bar{g}},{\bar{\square}})\,B^{\beta}(y)\Big{]}\qquad$
(132)

and construct the corresponding gauge fixing functional,

$\displaystyle\Psi^{mod}(\phi,{\bar{g}})\,=\,\int
dx\sqrt{-{\bar{g}}}\,\,{\bar{C}}^{\alpha}(x)\,\chi^{mod}_{\alpha}(x;{\bar{g}},h,B).$
(133)

According to what we previously learned, the transformation law of
$\chi^{mod}_{\alpha}$ coincides with the transformation rule of tensor fields
of type $(0,1)$. Then the modified Faddeev-Popov action is constructed in the
standard manner, using the generator of BRST transformations,
${\hat{R}}(\phi,{\bar{g}})$,

$\displaystyle
S_{FP}^{mod}(\phi,{\bar{g}})\,=\,S_{0}({\bar{g}}+h)\,+\,\Psi^{mod}(\phi,{\bar{g}})\,{\hat{R}}(\phi,{\bar{g}}).$
(134)

The explicit form of the second term in the right-hand side of (134) is

$\displaystyle\Psi^{mod}(\phi,{\bar{g}}){\hat{R}}(\phi,{\bar{g}})$
$\displaystyle=$ $\displaystyle\int
dxdydzdu\sqrt{-{\bar{g}}(x)}\;{\bar{C}}^{\alpha}(x)Y_{\alpha}^{\beta}(x,u;{\bar{g}},{\bar{\square}})$
$\displaystyle\times\,\,H^{\gamma\sigma}_{\beta}(u,y;{\bar{g}},h)R_{\gamma\sigma\rho}(y,z;{\bar{g}}\\!+\\!h)C^{\rho}(z)$
$\displaystyle+\,\int
dxdy\sqrt{-{\bar{g}}(x)}\Big{[}B^{\alpha}(x)Y_{\alpha}^{\beta}(x,y;{\bar{g}},{\bar{\square}})\chi_{\beta}(y;{\bar{g}},h)$
$\displaystyle+\,\frac{1}{2}B^{\alpha}(x)Y_{1\alpha\beta}(x,y;{\bar{g}},{\bar{\square}})B^{\beta}(y)\Big{]}.$

The first term in the r.h.s. of the last formula is exactly (128) with (129).
As the transformation rules for the terms in the Faddeev-Popov action depend
only on the type of the tensor fields, all the main statements of the previous
sections remain valid for the new choice of the gauge fixing functions (132).

To deal only with the problem of the homogeneity of the propagator of the
quantum metric $h_{\mu\nu}$, consider a special choice of the weight operator

$\displaystyle
Y_{1\alpha\beta}(x,y)\,=\,{\bar{g}}_{\alpha\gamma}(x)\,(Y^{-1})^{\gamma}_{\beta}(x,y),$
where $\displaystyle\int
dz\,\,Y^{\gamma}_{\alpha}(x,z)\,(Y^{-1})_{\gamma}^{\beta}(z,y)\,=\,\delta_{\alpha}^{\beta}\delta(x-y),$
(135) $\displaystyle\,\,\quad
Y_{\alpha}^{\,\beta}(x,y)\,=\,Y_{\alpha}^{\,\beta}(x;{\bar{g}},{\bar{\square}})\,\delta(x-y).$

Integrating over the fields $B^{\alpha}$ in the functional integral defines
the generating functional of Green functions it terms of integration over
fields ${\bar{C}}^{\alpha}$, $C^{\alpha}$ and $h_{\mu\nu}$. As a results we
obtain the functional determinant that is equal to

$\displaystyle\Big{[}{\rm Det}\;Y_{\alpha}^{\beta}(x,y)\Big{]}^{1/2},$ (136)

independent on the variables of integration. It is worth noting that the
factor (136) is well-known in higher derivative QG models [91, 18, 28].

After all, to introduce a nontrivial weight operator we need to replace

$\displaystyle\Psi^{mod}(\phi,{\bar{g}}){\hat{R}}(\phi,{\bar{g}})+\int
dx\sqrt{-{\bar{g}}(x)}\;J_{\alpha}^{(B)}(x)B^{\alpha}(x)$ (137)

by the more complicated expression

$\displaystyle\int\,dxdydz\sqrt{-{\bar{g}}(x)}\;{\bar{C}}^{\alpha}(x)\,Y_{\alpha}^{\beta}(x;{\bar{g}},{\bar{\square}})\,H^{\gamma\sigma}_{\beta}(x,y;{\bar{g}},h)\,R_{\gamma\sigma\rho}(y,z;{\bar{g}}\\!+\\!h)\,C^{\rho}(z)$
$\displaystyle-\,\frac{1}{2}\\!\int
dx\sqrt{-{\bar{g}}(x)}\;J^{(B)\alpha}(x)Y_{\alpha}^{\beta}(x;{\bar{g}},{\bar{\square}})J^{(B)}_{\beta}(x)-\int
dx\sqrt{-{\bar{g}}(x)}\;J^{(B)}_{\alpha}(x)\;\chi^{\alpha}(x;{\bar{g}},h)$
$\displaystyle-\,\frac{1}{2}\\!\int
dx\sqrt{-{\bar{g}}(x)}\;\chi^{\alpha}(x;{\bar{g}},h)Y_{\alpha}^{\beta}(x;{\bar{g}},{\bar{\square}})\chi_{\beta}(x;{\bar{g}},h),$
(138)

where the notations

$\displaystyle\chi^{\alpha}(x;{\bar{g}},h)\,=\,{\bar{g}}^{\alpha\beta}(x)\chi_{\beta}(x;{\bar{g}},h),$
$\displaystyle
J^{(B)\alpha}(x)\,=\,{\bar{g}}^{\alpha\beta}(x)J^{(B)}_{\beta}(x)$ (139)

are used. The second term in the expression (138) is exactly what is needed
for the homogeneity condition (127). The total action appearing after
integration over fields $B^{\alpha}$ is the sum of the action
$S_{0}({\bar{g}}+h)$ plus the first and the fourth terms in (138). This total
action is invariant under the BRST transformations, the last are now recast in
the form

$\displaystyle\delta_{B}h_{\mu\nu}(x)=\int
dyR_{\mu\nu\alpha}(x,y;{\bar{g}}+h)C^{\alpha}(y)\mu,$
$\displaystyle\delta_{B}C^{\alpha}(x)=-C^{\sigma}(x)\partial_{\sigma}C^{\alpha}(x)\mu,$
(140)
$\displaystyle\delta_{B}{\bar{C}}^{\alpha}(x)=-\chi^{\alpha}(x;{\bar{g}},h)\mu.$

In this form, the BRST symmetry also enables us to use all the basic
properties to explore the renormalization of quantum theory, as one can verify
by inspecting the considerations in Sec. 2 and Sec. 3 (see also [17]).
However, there is a price to pay for the possibility to work with the
functional integral over the smaller number of integration variables. One can
note that the nilpotency property of the BRST transformations is lost in the
new version of BRST transformations (140). On another hand, in the context of
gauge invariant renormalizability, the nilpotency of the BRST transformation
does not play a critical role. For this reason, we can freely switch between
the two versions of BRST, according to our convenience.

As the problem of homogeneity and introduction of (127) and (128) has been
solved, we are in a position to review the power counting and classify the
models of quantum gravity, as it was explained in detail in the previous
Chapter [40] (see also [38] and [92]). Thus, the important understanding of
the main aspects of renormalization and effective approaches in perturbative
QG gains a reliable basis after we get a mathematically solid proof of the
gauge invariant renormalizability of the classically covariant models of
gravity.

##  6 Conclusions

The general proof of diffeomorphism invariant renormalization in QG,
independent of the renormalizability by power counting, is relevant for
several important reasons. In particular, such a proof provides a solid basis
for the low-energy effective approach in QG, that means making practical
calculations in the low-energy sector of the theory, even in non-
renormalizable models. In this case, it is necessary to be sure that the UV
divergences are subtracted by local covariant counterterms that do not affect
the physics in the IR [1].

The main advantages of the approach of [38] which we reproduced and reviewed
here, is related to the compact description of the variation of extended
effective action under the gauge transformations of all fields used in the
background field formalism. The derived form of these variations can be
applied to an arbitrary QG theory, respecting the diffeomorphism invariance.
The variation shows an invariance of the effective action when the antifields
(sources for the BRST generators) are switched off.

After switching off the mean field of quantum metric, Faddeev-Popov ghosts,
auxiliary field, and antifields, the divergent part of effective action
possess general covariance, and this important property holds in all orders of
the perturbative loop expansion. This statement holds in all covariant models
of QG, including the ones with higher derivatives and even for the non-local
models. Starting from covariance and using power counting and locality of the
counterterms one can easily classify the models of QG into non-renormalizable,
renormalizable, and superrenormalizable versions. One of the extensions of the
analysis performed above is an extension to the non-linear gauges [93], but
this part is beyond the scope of the present review.

## Acknowledgements

The work of P.M.L. is partially supported by the the Ministry of Education of
the Russian Federation, under the project No. FEWF-2020-0003. This work of
I.L.Sh. is partially supported by Conselho Nacional de Desenvolvimento
Científico e Tecnológico - CNPq (Brazil), the grant 303635/2018-5 and by
Fundação de Amparo à Pesquisa de Minas Gerais - FAPEMIG, the project
PPM-00604-18; and by the Ministry of Education of the Russian Federation,
under the project No. FEWF-2020-0003.

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