11institutetext: Departamento de Física, Universidade Federal de Juiz de Fora,
MG, 36036-900, Brazil
E-mail address<EMAIL_ADDRESS>
On leave from Tomsk State Pedagogical University, Russia

# The background information about perturbative quantum gravity

Ilya L. Shapiro corresponding author

###### Abstract

The purpose of this Chapter is to give a general introduction and status
review on the perturbative approach to quantum gravity (QG). This text is a
modified version of the corresponding chapters of Part II of the recent
textbook on quantum field theory (QFT) and QG, co-authored with I.L.
Buchbinder and published in Oxford University Press. We discuss the choice of
the starting action in the QG models, degrees of freedom and propagator of
metric perturbations, power counting and renormalizability of these models,
the problems related to higher derivative theories and ghosts, such as quantum
unitarity and the stability of classical solutions in general relativity; and
the perspective to overcome these problems. The gauge fixing and
parametrization dependencies are discussed in detail using the corresponding
general QFT theorems developed in gauge theories. On top of that, we present a
basic example of deriving the one-loop divergences and discuss an important
example of the renormalization group in QG. The gauge invariant
renormalizability of QG is considered in another Chapter of the Handbook,
written together with P.M. Lavrov.

## Keywords

Quantum gravity; gravitational action; gauge fixing; propagator; power
counting; renormalizability; higher derivatives; ghosts; stability.

##  1 Introduction

Perturbative quantum gravity (QG) aims to construct quantum gravitational
theory in a manner close to how QFT describes other fundamental forces, such
as electroweak and high-energy strong interactions. One of the main purposes
of perturbative QG is to get the quantum corrections to the classical (tree-
level) action of gravity and the corresponding effective equations of motion
for the metric.

The importance of loop corrections to the gravitational equations, coming from
QG or from the quantum effects of matter fields (semiclassical approach), is
partially owing to the fact that, at some point, we hope to be able to compare
these corrections with experimental or observational data. Another reason to
study QG is that the consistency of quantum theory may be useful to establish
the restrictions on the modifications of general relativity (GR). It is
assumed we may be able to select those gravity theories that are consistent at
the quantum level and discard other models. Independent of all these reasons,
in the last six decades the perturbative QG became an important part of the
general QFT scenario and, especially, of the gauge fields theories.

When initiating the QG, we meet two main choices representing kinds of the
points of bifurcation, where one can choose the direction of how to construct
the theory. The first question is to decide what should be the object of
quantization. In the traditional perturbative quantum gravity, we choose to
quantize the spacetime metric, and apply to it the rules of quantization that
are common to all gauge theories, such as the Yang-Mills. It is worth
mentioning that one can choose other quantum variables to describe gravity,
e.g., the tetrad or consider the connection independent of the metric (first
order formalism). Owing to the size and contents limitations, we will not
discuss these possibilities in what follows.

On the other hand, it is possible to extend the set of the quantum fields,
that is include other fields along with the metric, or even replacing the
metric. One of the most interesting approaches to make such extension is to
provide an extended symmetry, as it is done in supergravity. We can refer the
reader to the corresponding Section of the present Handbook for the reviews of
supergravity, but will not discuss it here. Another possibility is to add
quantum matter fields. Let us note that there is a reduced, semiclassical
approach, when only the matter fields are assumed quantum and metric is
regarded as a classical background. This kind of theories share many technical
and conceptual features with perturbative QG, so the reader may find useful to
study the semiclassical approach first. There are many useful monographs on
this subject, let us mention just a few of them, DeWitt65 ; birdav ; book ;
MuWi ; ParToms ; Toms  and also recent textbook OUP , where the reader can
also find an introduction to QG111For the interested reader, there are good
books and reviews of semiclassical and quantum gravity theories, e.g.,
Alvarez-88 ; Fulling ; Kiefer ; Percacci . One can find many other useful
sources starting form these references.. In what follows, we shall address the
same subjects which are discussed in sections 18-21 of this textbook. Most of
the present Chapter can be seen as a modified and more review-style extract
from this book.

The second important choice appears after we agree to quantize the metric. One
has to choose the classical theory which should serve as a basis for the
perturbative quantum gravity. Obviously, such initial model may be the
Einstein’s GR, however there are strong reasons to try also other models of
gravity. The choice of a model defines many important features of the theory,
including its particle contents, renormalizability, unitarity, stability of
classical solutions, high energy (UV) and low-energy (IR) behaviors, as we
shall discuss below.

In what follows, we review the main elements of the perturbative QG, referring
to the recent textbook OUP  for most of the technical details. First of all,
we consider the choice of the action and the corresponding gauge-fixing
conditions, mostly, in the framework of the background field method DeWitt65 ;
Abbott82  (the reader can find more references starting from these two). After
that, we review the structure of propagator of the metric perturbations in
various models and the corresponding analysis of the degrees of freedom in QG.

The proof of the gauge-invariant renormalizability in QG is left for the
second chapter of this Section LavSha-HQG . This analysis possesses higher
level of complexity compared to the mentioned textbook, but, as a result, the
main statements are obtained in a more general and more concrete way. The main
two outputs of this consideration, based on the BRST symmetry and the Batalin-
Vilkovisky technique, are the following two statements:

i) The counterterms in the generally covariant theory of QG may have the same
symmetry, i.e., the diffeomorphism invariance, as the initial classical
action. This symmetry holds at the quantum level if we use a regularization
preserving this symmetry, such that the general covariance is not violated by
quantum corrections. For instance, this is the case for the perturbative QG in
even dimension $n=2m+2$ (where $m=1,2,,\dots$), including $n=4$, the last will
be used by default in the rest of this Chapter.

ii) The dependence on the choice of the gauge fixing and on the
parametrization of quantum field, at any order of loop expansion, is
proportional to the effective equations of motion. As an important consequence
of this rule, the respective dependencies in the one-loop divergences vanish
on the classical mass shell.

The first feature implies that the general structure of divergences in any
loop order may be defined on the basis of power counting of the Feynman
diagrams, that means it can be reduced to the use of the dimensional
arguments. In this way we can, in most of the cases, say that the given model
of QG is renormalizable, non-renormalizable, or super-renormalizable, and even
establish the structure of possible divergences in any loop order – without
making explicit calculations. The remaining questions concern only the
coefficients of the given terms in the divergences.

The second feature is also important for practical purposes, for the following
two reasons. From one side, one can choose the gauge fixing condition in such
a way that makes the calculations technically simpler. And, on another hand,
we can extract that part of the quantum corrections which are gauge-fixing and
parametrization invariant and, in principle, consider this as a physical
output of the loop calculations.

The Chapter is organized as follows. In Sec. 2 we formulate the most relevant
models of QG that may constitute a sound basis of the perturbative treatment.
Let us note that we only slightly touch the nonlocal models which represent,
nowadays, a popular object of studies. The reason is that there is a special
Section devoted to the nonlocal theories and we do not like to have
repetitions. Thus, we mainly consider the QG based on GR and on the different
kinds of local polynomial models. In Sec. 3, one can find the bilinear
expansions of all metric-dependent terms that define the propagator of the
quantum metric. In Sec. 5 we describe these propagators different models of
QG. The next Sec. 6 is briefly formulating the the main statements about
gauge-invariant renormalization in QG, while the detailed discussion is
postponed to the separate Chapter of this Section. Sec. 7 describes the power
counting and classification of the QG models to renormalizable, non-
renormalizable and super-renormalizable ones. Sec. 8 discusses the problem of
ghosts in higher derivative QG. This problem certainly represents the main
difficulty of all the QG program. We give a basic introduction and a brief
report on the existing results in this area. In Sec. 9 we describe the gauge-
fixing and parametrization dependencies of the one-loop effective action and
Sec. 10 shows the detailed derivation of the one-loop divergences in quantum
GR, in the simplest parametrization and gauge fixing. Sec. 11 discusses an
interesting example of the renormalization group applied to the quantum GR in
a manner that fits the effective approach to QG. There is a special Section of
the present Handbook, devoted to the subject of effective QG, so we do not go
too deep into the subject and only show this particular example. Sec. 12 gives
a short review of the one-loop calculations in other models of QG. Owing to
the size limitations, this part is very much incomplete and does not cover the
extensive literature on the subject, so it is included just as a starting
point for the interested readership. Finally, in Sec. 13 we draw our
conclusions and present final discussions of the current situation in
perturbative QG.

In the rest of the Chapter, we use DeWitt notations, when the covariant
integral $\int d^{4}x\sqrt{-g}=\int_{x}$ may be assumed but not written
explicitly when this is obvious. We denote functional trace Tr and determinant
Det , that includes the same covariant integration over spacetime coordinates,
and use pseudoeuclidean notations such as $\sqrt{-g}$, even in case of using
heat kernel technique, that requires Euclidean metric. In these cases, we
assume the Wick rotation, regardless it may be a nontrivial issue in some
models of QG. On top of this, the notations include the signature
$\eta_{\alpha\beta}=\,\mbox{diag}\,(+\,-\,-\,-)$, the definition of the
Riemann tensor

$\displaystyle
R^{\lambda}_{\,\,\,\tau\alpha\beta}=\partial_{\alpha}\,\Gamma^{\lambda}_{\,\,\tau\beta}-\partial_{\beta}\,\Gamma^{\lambda}_{\,\,\tau\alpha}+\Gamma^{\lambda}_{\,\,\gamma\alpha}\,\Gamma^{\gamma}_{\,\,\tau\beta}-\Gamma^{\lambda}_{\,\,\gamma\beta}\,\Gamma^{\gamma}_{\,\,\tau\alpha},$
(1)

Ricci tensor ${R^{\alpha}}_{\mu\alpha\nu}=R_{\mu\nu}$, and its trace
$R=R_{\mu\nu}g^{\mu\nu}$, that is Ricci scalar.

##  2 Models of QG. General classification and gauge fixing

Our purpose is to construct the quantum theory of the metric field
$g_{\mu\nu}$. In GR, the metric is subject to the gauge transformation, also
called diffeomorphism, corresponding to the infinitesimal coordinate
transformation

$\displaystyle x^{\mu}\to
x^{\prime\,\mu}=x^{\mu}+\xi^{\mu},\qquad\xi^{\mu}=\xi^{\mu}(x).$ (2)

Let us start by deriving the diffeomorphism transformation for the metric.
Keeping only the terms of the first order in $\xi^{\mu}$ and their
derivatives, we get

$\displaystyle g^{\prime}_{\alpha\beta}(x)$ $\displaystyle=$ $\displaystyle
g^{\prime}_{\alpha\beta}(x^{\prime})-\frac{\partial
g^{\prime}_{\alpha\beta}(x^{\prime})}{\partial
x^{\prime\lambda}}\,\xi^{\lambda}=g^{\prime}_{\alpha\beta}(x^{\prime})-\partial_{\lambda}g_{\alpha\beta}\,\xi^{\lambda}.$

Using the tensor transformation rule,

$\displaystyle g^{\prime}_{\alpha\beta}(x^{\prime})$ $\displaystyle=$
$\displaystyle\frac{\partial x^{\rho}}{\partial
x^{\prime\alpha}}\,\frac{\partial x^{\sigma}}{\partial
x^{\prime\beta}}\,g_{\rho\sigma}(x)\,=\,\big{(}\delta^{\rho}_{\alpha}-\partial_{\alpha}\xi^{\rho}\big{)}\big{(}\delta^{\sigma}_{\beta}-\partial_{\beta}\xi^{\sigma}\big{)}g_{\rho\sigma}(x)$
(3)
$\displaystyle\quad\,=\,\,g_{\alpha\beta}-g_{\rho\beta}\,\partial_{\alpha}\xi^{\rho}-g_{\alpha\rho}\,\partial_{\beta}\xi^{\rho}$

and taking the two expressions together, we get

$\displaystyle\delta g_{\alpha\beta}(x)$ $\displaystyle\,=\,$ $\displaystyle
g^{\prime}_{\alpha\beta}(x)\,-\,g_{\alpha\beta}(x)\,=\,-g_{\lambda\beta}(x)\,\partial_{\alpha}\xi^{\lambda}(x)-g_{\alpha\lambda}(x)\,\partial_{\beta}\xi^{\lambda}(x)$
(4)
$\displaystyle\quad-\,\,\partial_{\lambda}g_{\alpha\beta}(x)\,\xi^{\lambda}(x)\,\,=\,\,-\,\nabla_{\alpha}\xi_{\beta}\,-\,\nabla_{\beta}\xi_{\alpha},$

that defines the generators of the gauge transformations

$\displaystyle\delta
g_{\mu\nu}(x)\,=\,R_{\mu\nu\,\lambda}(g)\,\xi^{\lambda},\qquad
R_{\mu\nu\,\lambda}(g)\,=\,-\,g_{\mu\lambda}\nabla_{\nu}\,-\,g_{\nu\lambda}\nabla_{\mu}$
(5)

We assume that any sort of the QG candidate action $S=S(g)$ possesses the
symmetry under (5), i.e., satisfies the Noether identity,

$\displaystyle\frac{\delta S}{\delta
g_{\mu\nu}}\,R_{\mu\nu\,\lambda}(g)\,=\,0.$ (6)

In the next Chapter LavSha-HQG  (see also OUP ), it is shown that QG is the
gauge theory of the Yang-Mills type, that means the algebra of the generators
is closed off shell. This feature represents the basis of the two statements
i) and ii), mentioned in the Introduction.

In many situations, it is useful to parameterize the metric as a perturbation
over the Minkowski spacetime

$\displaystyle g_{\mu\nu}\,=\,\eta_{\mu\nu}+h_{\mu\nu},$ (7)

and use the following representation of $h_{\mu\nu}$ (see Percacci  and Sec. 5
for more details):

$\displaystyle
h_{\mu\nu}\,\,=\,\,{\bar{h}}^{\bot\bot}_{\mu\nu}+\partial_{\mu}\epsilon^{\bot}_{\nu}+\partial_{\nu}\epsilon^{\bot}_{\mu}+\partial_{\mu}\partial_{\nu}\epsilon+\frac{1}{4}\,h\,\eta_{\mu\nu}.$
(8)

In this expression, the tensor component (spin-2 mode) is traceless and
transverse, i.e., ${\bar{h}}^{\bot\bot}_{\mu\nu}\eta^{\mu\nu}=0$ and
$\partial^{\mu}{\bar{h}}^{\bot\bot}_{\mu\nu}=0$. The irreducible vector
component (spin-1 mode) satisfies the condition
$\partial_{\mu}\epsilon^{\bot\mu}=0$. There are also two scalar fields (or
modes) $\epsilon$ and $h$. The indices are raised and lowered with the flat
metric.

In the rest of this section, we describe the most popular actions which are
used for constructing the QG models. This description includes the proper
action and the details of the DeWitt-Fadeev-Popov (or Fadeev-Popov) procedure
required for the Lagrangian quantization of the model. For the sake of
generality, we perform the consideration of these procedures using the
background field method. The last means that, instead of (7), the expansion is
performed around an arbitrary background metric,

$\displaystyle
g_{\mu\nu}\,\longrightarrow\,g^{\prime}_{\mu\nu}\,=\,g_{\mu\nu}+h_{\mu\nu}.$
(9)

The interested reader can find more details of the expansion (9) and useful
exercises in the book OUP .

###  2.1 DeWitt-Faddeev-Popov method for QG

Let us briefly sketch the Faddeev-Popov (or DeWitt-Faddeev-Popov) method for
QG. There is no critical difference with other gauge models, and we shall use
the notations close to the general ones. Namely, we shall denote
$g^{i}=g_{\mu\nu}$ and keep in mind that, depending on our intentions, $g^{i}$
may be also used for $\,h_{\mu\nu}$, as defined in (9). Then the
transformation rule and the generator from Eq. (5) will be denoted as

$\displaystyle\delta g^{i}=R^{i}_{\,\alpha}\xi^{\alpha}\qquad\mbox{and}\qquad
R^{i}_{\,\alpha}=R_{\mu\nu\,\alpha}.$ (10)

The last note about notations is that $g^{i}$ may be also used for other
parameterizations of the quantum metric222An example is Eq. (134) below. , and
the formulas can be modified accordingly.

The starting point is the naive expression for the functional integral over
the quantum metric

$\displaystyle Z\,=\,\int dg\,e^{iS(g)},$ (11)

where we use $g\equiv g^{i}$ for the arguments of a functional $S$ and the
integration variable, to make formulas more readable. We can generalize the
last integral to the generating functional of Green functions by replacing
$S\to S+gJ$ and assuming that the source term $gJ=\int_{x}g^{i}J_{i}$ is
diffeomorphism invariant. As this replacement does not cause real changes, we
shall work with the formula (11).

The space of integration $g^{i}$ includes the orbits, i.e. the subspaces
defined by the gauge transformations of the metric field (10). Since the
action remains constant over any such subspace, it is cleat that each of these
subspaces contributes infinite to the integral, which is, therefore, badly
defined. Such a divergence is similar to taking logarithm of determinant of a
degenerate matrix. This divergence it is a direct consequence of the gauge
invariance and represents the problem solved by the DeWitt-Faddeev-Popov
method DeWitt-67 ; faddeev-popov  in gravity and in the Yang-Mills theory (see
also, e.g., Popov-FI  for QG and further references in the next Chapter
LavSha-HQG ).

The first step is to insert the factor of unity in the integrand of (11), in
the form

$\displaystyle 1\,=\,\Delta(g)\,\int
d\xi^{\alpha}\,\,\delta\big{(}\chi^{\alpha}(g)-l^{\alpha}\big{)}.$ (12)

Here $l^{\alpha}$ is an arbitrary vector field and $\chi^{\alpha}(g)$ is the
gauge fixing condition. There can be different choices of this condition but,
in principle, one can think that the surface $\chi^{\alpha}(g)=l^{\alpha}$
(or, simply, $\chi^{\alpha}(g)=0$) crosses each orbit in a unique “point”,
such that the degeneracy is removed. Indeed, this is not the way Faddeev-Popov
method works, as we will see.

The integral (12) can be easily taken by making the change of integration
variables to $\chi^{\alpha}$,

$\displaystyle\Delta^{-1}(g)\,=\,\,\int
d\chi^{\beta}\,\,\mbox{Det}\,\Big{(}\frac{\delta\xi^{\alpha}}{\delta\chi^{\beta}}\Big{)}\,\delta\big{(}\chi^{\alpha}(g)-l^{\alpha}\big{)}.$
(13)

The Jacobian of this transformation is inverse to the matrix

$\displaystyle\Big{(}\frac{\delta\xi^{\alpha}}{\delta\chi^{\beta}}\Big{)}^{-1}\,=\,\,\frac{\delta\chi_{\alpha}}{\delta\xi_{\beta}}\,=\,\frac{\delta\chi_{\alpha}}{\delta
g^{i}}\,R^{i\,\beta}\,=\,\frac{\delta\chi_{\alpha}}{\delta
g_{\mu\nu}}\,R_{\mu\nu}^{\quad\,\beta}\,\,=\,\,M_{\alpha}^{\,\,\beta}.$ (14)

It is important that the determinant of the matrix $M_{\alpha}^{\,\,\beta}$
does not depend on $\xi$, but only on $g^{i}$ and the form of the gauge fixing
condition $\chi(g)$. In this way, we arrive at the equivalent, albeit already
non-degenerate, form of (11),

$\displaystyle Z\,=\,\int
dg\,\,\,\delta\big{(}\chi^{\alpha}(g)-l^{\alpha}\big{)}\,\,\mbox{Det}\,\big{(}M_{\alpha}^{\,\,\beta}\big{)}\,\,e^{iS(g)}\,.$
(15)

To achieve a more useful form of this expression, we insert in the integrand a
unit in the form

$\displaystyle 1\,=\,\big{(}\,\mbox{Det}\,Y_{\alpha\beta}\big{)}^{1/2}\,\int
dl^{\alpha}\,\exp\Big{\\{}\frac{i}{2}\,l^{\alpha}Y_{\alpha\beta}l^{\beta}\Big{\\}}.$
(16)

Here $Y_{\alpha\beta}$ is a new object identified as weight operator. We will
discuss several forms of this operator, adapted to different models of QG, in
what follows.

Taking the integral with the delta function, we get

$\displaystyle Z\,=\,\int
dg\,\,\,\big{(}\,\mbox{Det}\,Y_{\alpha\beta}\big{)}^{1/2}\,\big{(}\,\mbox{Det}\,M_{\alpha}^{\,\,\beta}\big{)}\,\,e^{iS+iS_{gf}},$
(17)

where

$\displaystyle
S_{gf}\,=\,\,\frac{1}{2}\,\chi^{\alpha}Y_{\alpha\beta}\,\chi^{\beta}.$ (18)

In some situations, it is useful to deal with the modified form of the
operator,

$\displaystyle\tilde{M}_{\alpha\beta}\,\,=\,\,Y_{\alpha\lambda}\,M_{\lambda}^{\,\,\beta},$
(19)

providing an alternative form of (17), i.e.,

$\displaystyle Z$ $\displaystyle=$ $\displaystyle\int
dg\,\,\,\big{(}\,\mbox{Det}\,\,Y_{\alpha\beta}\big{)}^{-1/2}\,\big{(}\,\mbox{Det}\,\,\tilde{M}_{\alpha}^{\,\,\beta}\big{)}\,\,e^{iS+iS_{gf}}$
(20) $\displaystyle\quad=\,\,\int
dg\,d\bar{C}\,dC\,\,\,\big{(}\,\mbox{Det}\,Y_{\alpha\beta}\big{)}^{-1/2}\,\,\,e^{iS+iS_{gf}+iS_{gh}},$

where the action of the Faddeev-Popov ghosts has the form

$\displaystyle
S_{gh}\,=\,\bar{C}^{\alpha}\,\tilde{M}_{\alpha}^{\,\,\beta}\,C_{\beta}\,.$
(21)

It is clear that the ghost fields $\bar{C}^{\alpha}$ and $C_{\beta}$ should be
fermions (i.e., have odd Grassmann parity) to provide the positive power of
$\,\mbox{Det}\,\tilde{M}_{\alpha}^{\,\,\beta}$. One can trade the functional
determinant $\big{(}\,\mbox{Det}\,\,Y_{\alpha\beta}\big{)}^{-1/2}$ for another
(third) ghost. This ghost may have even or odd Grassmann parity, respectively,
for the versions with $\,\tilde{M}_{\alpha\beta}\,$ or
$\,M_{\lambda}^{\,\,\beta}$.

Thus, the total Faddeev-Popov action $S_{t}=S+S_{gf}+S_{gh}$ depends on the
choice of the gauge fixing condition $\chi^{\alpha}$ and the weight operator
(sometimes called weight function) $Y_{\alpha\beta}$. In what follows, we
consider a few important examples of the choice of these two objects in
different models of QG. As with the most of this review, the reader can find
more detailed discussion in OUP .

###  2.2 Quantum general relativity

The first and most obvious candidate for the starting action of QG is the
Einstein-Hilbert functional

$\displaystyle S_{EH}\,=\,-\,\frac{1}{\kappa^{2}}\,\int
d^{4}x\sqrt{-g}\,\big{(}R+2\Lambda\big{)}.$ (22)

Here we denoted $\kappa^{2}=16\pi G$, as it proves useful when we consider the
perturbative expansion. The reason is that, modifying the expansion (9) to

$\displaystyle g_{\mu\nu}\quad\longrightarrow\quad
g^{\prime}_{\mu\nu}\,=\,g_{\mu\nu}+\kappa h_{\mu\nu},$ (23)

the bilinear in $h_{\mu\nu}$ terms are free from the parameter $\kappa$, while
the higher orders in $h_{\mu\nu}$ terms have positive powers $\kappa$. One of
the consequences is that $\kappa$ plays the role of the interaction constant
in this QG model. In quantum theory, $\kappa$ turns out the loop expansion
parameter. Let us stress that this is, typically, not so for other QG models,
which may have other parameters of the loop expansion. Other two relevant
observations is that, since gravity is a non-polynomial theory, the
parametrization (23) results in that the action (22) has unbounded powers of
$\kappa$. Furthermore, things get more complicated in more general
parameterizations but, typically, $\kappa$ remains the parameter of the loop
expansion.

The flat limit requires a vanishing cosmological constant and we get

$\displaystyle g_{\mu\nu}\,=\,\eta_{\mu\nu}+\kappa h_{\mu\nu}\,.$ (24)

As we shall see below, this means that the propagator of quantum metric is
free of $\kappa$ by construction.

Instead of the expansion of the metric $g_{\mu\nu}$, one can start from the
expansion of the inverse metric $g^{\mu\nu}$, or even a more general
parametrization of the quantum metric. Later on, we consider the version which
is the most general for the one-loop calculations and has various arbitrary
parameters.

The Faddeev-Popov procedure requires introducing the gauge fixing term
$S_{gf}$. Such a term is called to make the highest derivative in $h_{\mu\nu}$
terms of the action $S+S_{gf}$ non-degenerate. Then one can obtain the
propagator of this field or, e.g., apply the heat-kernel method and Schwinger-
DeWitt technique for calculating the divergences in a covariant form. Since
the theory (22) has at most second derivatives, we can choose the weight
operator proportional to the background metric and arrive at the gauge-fixing
term in the form

$\displaystyle S_{gf}\,=\,\frac{1}{\alpha}\,\int
d^{4}x\sqrt{-g}\,\chi_{\mu}\,\chi^{\mu},\qquad\mbox{where}\qquad\chi_{\mu}=\nabla_{\nu}h^{\nu}_{\,\,\mu}-\beta\nabla_{\mu}h$
(25)

and $\alpha$ and $\beta$ are the gauge-fixing parameters. The dependence on
the choice of the gauge fixing and on the parametrization of quantum metric,
represents an important part of the QG development.

Why the action (22) requires the simplest weight functional in the case of QG?
The reason is that the Lagrangian of GR (22) has at most two derivatives of
the quantum metric $h_{\mu\nu}$ in the action. Thus, a “correct” (or, better
to say, appropriate) way of breaking down the degeneracy in the total action
requires that the gauge fixing term $S_{gf}$ also has two derivatives. Since
each of $\chi_{\mu}$ in Eq. (25) is linear in derivatives, we are forced to
implement the choice $Y_{\mu\nu}=\mbox{const}\times g_{\mu\nu}$ in this case.
With some adjustments, the same logic will be used in all models of QG which
will be discussed below (and even those which will not be).

The action of ghosts is constructed in a standard way, as

$\displaystyle S_{gh}\,=\,\int
d^{4}x\sqrt{-g}\,\,\,\bar{C}^{\alpha}\,M_{\alpha}^{\,\,\,\beta}\,C_{\beta},$
(26)

where the operator $M$ is the variation of gauge fixing condition with respect
to the transformation function,

$\displaystyle
M_{\alpha}^{\,\,\,\beta}\,=\,\frac{\delta\chi_{\alpha}}{\delta\xi_{\beta}}\,=\,\frac{\delta\chi_{\alpha}}{\delta
h_{\mu\nu}}\,R_{\mu\nu}^{\quad\,\beta}\,.$ (27)

It is important that the ghost fields in (26) satisfy the second-order
equations. This means, the propagators of both gravitational field
$h_{\mu\nu}$ and ghosts have the same type of the UV behavior $1/k^{2}$. In
the next models, we shall see that providing the homogeneity of the
propagators of different modes of the metric (8) and of the ghosts may require
some extra efforts. And such a homogeneity worth these efforts, as without it,
the quantum theory gains a lot of artificial complications.

###  2.3 Fourth derivative gravity

The next model of common interest is based on the fourth-derivatives action.
If the previous choice, namely, the Einstein-Hilbert action of GR, is strongly
motivated by the success of Einstein’s classical gravitational theory, the
fourth-derivative model is motivated by the consistency conditions of
semiclassical gravity. It is known from the early paper by Utiyama and DeWitt
UtDW  (see also the aforementioned books birdav ; book ; ParToms ; OUP ) that
if the matter fields are quantum, the action of vacuum (i.e., the
gravitational action) of renormalizable theory has to include both Einstein-
Hilbert action (22) and the covariant local fourth-derivative terms

$\displaystyle S_{HD}\,=\,\int
d^{4}x\sqrt{-g}\,\Big{\\{}a_{1}R_{\mu\nu\alpha\beta}^{2}+a_{2}R_{\mu\nu}^{2}+a_{3}R^{2}+a_{4}\operatorname{\square}R\Big{\\}}.$
(28)

In QG, it is more useful to write this action in another basis, including the
square of the Weyl tensor and the integrand of the Gauss-Bonnet topological
term

$\displaystyle
C^{2}\,=\,C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}\,=\,R^{2}_{\mu\nu\alpha\beta}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2},$
$\displaystyle E_{4}\,=\,R^{2}_{\mu\nu\alpha\beta}-4R_{\mu\nu}^{2}+R^{2}.$
(29)

In this basis, the fourth-derivative action has the form

$\displaystyle S_{HD}\,=\,-\,\int
d^{4}x\sqrt{-g}\,\Big{\\{}\frac{1}{2\lambda}C^{2}\,-\,\frac{1}{\rho}E_{4}\,+\,\frac{1}{\xi}R^{2}\,+\,\tau\square
R+\frac{1}{\kappa^{2}}(R-2\Lambda)\Big{\\}},$ (30)

Sometimes other notations for the couplings $\rho$ and $\xi$ are used, e.g.,
$\theta=\lambda/\rho$ and $\omega=-3\lambda/\xi$, or the ones used in the
special chapter Nobu-HD  about the one-loop calculations and renormalization
group flows in the theory (28). The important part is the positive sign of the
coupling $\lambda$, as it is required by the positivity of the energy of the
massless tensor mode (graviton) in the high energy region Stelle77 ; stelle78
(see also an alternative treatment of the same problem in OUP ).

Introducing the gauge-fixing term in the fourth-derivative theory of QG is a
more complicated task compared to the quantum GR. Since we are interested to
maintain homogeneity of the propagator, the gauge-fixing term should have four
derivatives of the quantum metric and hence we introduce the expression for
the gauge-fixing action

$\displaystyle S_{gf}\,=\,\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\,\chi^{\mu}\,Y_{\mu\nu}\,\chi^{\nu}\,,$ (31)

where the gauge condition $\chi^{\mu}$ is still defined by the formula (25),
but the new weight function $Y_{\mu\nu}$ should be a non-degenerate operator
of the second order in derivatives. In the framework of background field
method, its most general form is

$\displaystyle Y_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\,\Big{(}g_{\mu\nu}\Box+\gamma\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu}+p_{1}R_{\mu\nu}+p_{2}R\,g_{\mu\nu}\Big{)}\,,$
(32)

where the non-degeneracy requires $\gamma\neq 0$. Compared to the quantum GR,
the new gauge-fixing condition depends on a larger number of arbitrary gauge-
fixing parameters, i.e., on
$\,\alpha_{i}\,=\,(\alpha,\,\beta,\,\gamma,\,p_{1},\,p_{2})$, where $\beta$
comes from Eq. (25). In the case of the flat background metric, $Y_{\mu\nu}$
has only two arbitrary parameters $\alpha$ and $\gamma$.

With the definition (31) and (32), all the modes of the quantum metric in (8)
have the same leading UV behaviour of the propagator, $G^{-1}_{i}(k)\propto
k^{4}$. As we shall see in the forthcoming sections, the power counting is
greatly simplified if the ghost action has the same number of derivatives as
the action of the $h_{\mu\nu}$ field. It is clear that the problem can be
solved by introducing a modified ghost action (21), namely

$\displaystyle S_{gh}\,=\,\int
d^{4}x\sqrt{-g}\,\,{\bar{C}}^{\alpha}\,{\tilde{M}}_{\alpha}^{\,\,\,\beta}\,C_{\beta}\,,\qquad\mbox{where}\qquad{\tilde{M}}_{\alpha}^{\,\,\,\beta}\,=\,M_{\alpha}^{\,\,\,\lambda}(g)\,Y_{\lambda}^{\,\,\,\beta}(g).\mbox{\qquad}$
(33)

The functional integral is defined by the general expression (20) with the
specific choice (32) of the weight operator. Since both
$\,M_{\alpha}^{\,\,\lambda}\,$ and $\,Y_{\lambda}^{\,\,\beta}\,$ are second
order operators, the propagator of the Faddeev-Popov ghosts behaves like
$\,k^{-4}$ in the UV, exactly as the propagator of gravitational
perturbations. This feature proves useful for evaluating the power counting of
the Feynman diagrams in this theory.

###  2.4 Quantum gravity models polynomial in derivatives

The previous two examples of QG models are minimal versions. In particular, GR
fits all observational and experimental tests for a classical gravity Will-
book  and, in this sense, can be regarded as a reference theory. On the
contrary, there is not a single experimental test for the fourth-derivative
model of gravity but, from another side, fourth derivative terms are required
for the renormalizability of semiclassical gravity. And the same model
guarantees also the renormalizability of QG Stelle77 . On another hand, the
fourth derivative theory has serious problems related to nonphysical
ghosts333Which have nothing to do with the Faddeev-Popov ghosts, the
traditional use of the same word here is a mere coincidence. and stability, as
we shall discuss below. In this situation, one may look beyond the minimal
theories and it is natural to try the models with more than four derivatives.
Then, the number of derivatives may be finite or infinite. In the first case
we meet the polynomial in derivatives models of QG, suggested in highderi .

To construct the polynomial models, we impose the condition that the highest-
derivative terms in the action should be homogeneous in the derivatives. In
the next section, we shall see that this homogeneity may provide the
superrenormalizability of the theory. The action of the theory has the form

$\displaystyle S_{N}$ $\displaystyle=$
$\displaystyle\int\\!d^{4}x\sqrt{-g}\,\Big{\\{}\vartheta_{N,R}R\,\square^{N}R+\vartheta_{N,C}C\,\square^{N}C+\vartheta_{N,\mathrm{GB}}\mathrm{GB}_{N}$
(34)
$\displaystyle+\,\vartheta_{N-1,R}R\,\square^{N-1}R+\vartheta_{N-1,C}C\,\square^{N-1}C+\vartheta_{N-1,\mathrm{GB}}\mathrm{GB}_{N-1}\,+\,\,\,...\,\,\,$
$\displaystyle+\,\vartheta_{0,R}R^{2}+\vartheta_{0,C}C^{2}+\vartheta_{0,\mathrm{GB}}\mathrm{GB}_{0}+\vartheta_{\rm
EH}R+\vartheta_{\rm cc}+{\mathcal{O}}(R_{\dots}^{3})\Big{\\}}\,,\mbox{\qquad}$

where $N=1,2,\dots$ and all $\vartheta$’s are arbitrary parameters of the
action. It is assumed that both $\vartheta_{N,R}$ and $\vartheta_{N,C}$ are
non-zero and that the maximal power of metric derivatives in the terms
${\mathcal{O}}(R_{\dots}^{3})$ is $2N+4$, i.e., is not higher than of the
terms of the second order in curvatures, i.e., ${\mathcal{O}}(R_{\dots}^{2})$.

Furthermore, there are the squares of the Weyl tensor with extra factors of
$\,\operatorname{\square}$,

$\displaystyle
C\,\square^{n}\,C\,=\,C_{\mu\nu\alpha\beta}\,\square^{n}\,C^{\mu\nu\alpha\beta}\,=\,R_{\mu\nu\alpha\beta}\square^{n}R^{\mu\nu\alpha\beta}-2R_{\mu\nu}\square^{n}R^{\mu\nu}+\frac{1}{3}\,R\square^{n}R.\mbox{\qquad}\mbox{\quad},$
(35)

Similarly, using integrations by parts and the Bianchi identities, the
generalized Gauss-Bonnet invariants can be shown to have the property

$\displaystyle\mathrm{GB}_{n}\,=\,R_{\mu\nu\alpha\beta}\square^{n}R^{\mu\nu\alpha\beta}-4R_{\mu\nu}\square^{n}R^{\mu\nu}+R\square^{n}R\,=\,{\mathcal{O}}(R_{\dots}^{3}).$
(36)

This term is not topological for $\,n\geq 1$, but it contributes only to the
third- and higher-order terms in the curvature tensor, and to the total
derivatives. Thus, it may affect the vertices but not the propagators of QG,
as we shall explicitly check out in what follows. On a flat background, the
${\mathcal{O}}(R_{\dots}^{2})$ terms may contribute to the propagator of the
gravitational perturbation, while ${\mathcal{O}}(R_{\dots}^{3})$ terms affect
only the vertices. In the action (34), the ${\mathcal{O}}(R_{\dots}^{2})$’s
are given in the basis of Weyl-squared and $R$-squared terms. As will be shown
below, the Weyl-squared terms affect the propagation of the tensor mode
${\bar{h}}^{\bot\bot}_{\mu\nu}$, and the $R$-squared $R\cdot R$ terms affect
the propagation of the scalar modes. Using the higher-derivative actions in
the form (34), we separate the propagators of the tensor and scalar modes at
the level of the action.

To provide the homogeneity in derivatives for the propagator of all modes (8),
the gauge-fixing terms should have the same highest power in derivatives as
the main action. Since the gauge fixing conditions $\chi_{\mu}$ are always
chosen in the form (25), the $\mathcal{O}\big{(}p^{-4-2N}\big{)}$ propagator
of the quantum metric requires the weight function to be

$\displaystyle
Y_{\mu\nu}\,=\,-\,\frac{1}{\alpha}\big{(}g_{\mu\nu}\square+\gamma\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu}\big{)}\square^{N+1}.$
(37)

One can add here many terms with lower order of derivatives, but this does not
change the UV behaviour of the propagator.

To provide the same power of derivatives in the ghost sector, as in the
quadratic in curvature action, one can redefine the ghost action as (33), this
time with the weight operator (37).

Furthermore, one can write the action (34) in an alternative form,

$\displaystyle S_{N}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\Big{\\{}\frac{1}{2}\,C_{\mu\nu\alpha\beta}P_{1}(\square)C^{\mu\nu\alpha\beta}\,+\,\frac{1}{2}\,RP_{2}(\square)R$
(38) $\displaystyle\quad+\,\,\,\vartheta_{\rm EH}R\,+\,\vartheta_{\rm
cc}\,+\,{\mathcal{O}}(R_{\dots}^{3})\Big{\\}},\mbox{\qquad}\mbox{\quad}$

where $P_{1,2}(x)$ are polynomials of the same order $N$ and the terms
${\mathcal{O}}(R_{\dots}^{3})$ have at most $4+2N$ derivatives of the metric.
One can make further generalization of (38), by trading the polynomials to the
infinite series of $\operatorname{\square}$. The discussion of these theories
can be found in the corresponding Section of the present Handbook. Let us just
quote the expression for the general (i.e., polynomial or non-polynomial)
action

$\displaystyle S_{gen}$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\,\Big{\\{}-\frac{1}{\kappa^{2}}\,(R+2\Lambda)\,+\,\frac{1}{2}\,C_{\mu\nu\alpha\beta}\,\Phi(\Box)\,C_{\mu\nu\alpha\beta}$
(39)
$\displaystyle\quad+\,\,\frac{1}{2}\,R\,\Psi(\Box)\,R\,+\,{\mathcal{O}}(R_{\dots}^{3})\Big{\\}}\,.$

To complete the bilinear in curvature part of the action we can add the third
higher-derivative term, which boils down to the Gauss-Bonnet topological term
for a constant form factor $\Omega$,

$\displaystyle S_{GB}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int
d^{4}x\sqrt{-g}\,\Big{\\{}R_{\mu\nu\alpha\beta}\,\Omega(\Box)\,R_{\mu\nu\alpha\beta}\,-\,4R_{\mu\nu}\,\Omega(\Box)\,R_{\mu\nu}\,+\,R\,\Omega(\Box)\,R\Big{\\}}.\qquad$
(40)

Despite this term is equivalent to ${\mathcal{O}}(R_{\dots}^{3})$ in (38), it
makes sense to verify this feature, at least at the level of the propagator.
The homogeneity of the propagator of all modes of the metric perturbations
(8), requires the functions $\Phi(x)$ and $\Psi(x)$ to have analogous behavior
in the UV. This can be achieved by requiring that

$\displaystyle\lim\limits_{x\to\infty}\,\frac{\Psi(x)}{\Phi(x)}\,\,=\,\,C,$
(41)

with $\,C\,$ being a non-zero constant. For the polynomial functions
$P_{1,2}(x)$ of the same order, in (38), the last condition is guaranteed. In
the case of the non-polynomial functions, the simplest useful choice is

$\displaystyle\Phi(x)=-\frac{1}{\kappa^{2}\,x}\,\big{(}e^{\alpha_{1}x}-1\big{)}\quad\mbox{and}\quad\Psi(x)=-\frac{C}{\kappa^{2}\,x}\,e^{\alpha_{2}x},$
(42)

such that condition (41) reduces to $\alpha_{1}=\alpha_{2}$.

The gauge-fixing term in theory (39) that provides the homogeneity of the
propagators, is of the standard form (31), but requires the special weight
operator

$\displaystyle
Y_{\mu\nu}\,\,=\,\,\big{(}g_{\mu\nu}\square-\gamma\nabla_{\mu}\nabla_{\nu}+p_{1}R_{\mu\nu}+p_{2}Rg_{\mu\nu}\big{)}\,W(\operatorname{\square}).$
(43)

In many cases, it is sufficient to consider
$W(\operatorname{\square})\propto\Phi(\operatorname{\square})$, but we shall
keep this function arbitrary for generality, until some point.

Finally, the homogeneity in the momentum at the UV for the quantum metric and
for the Faddeev-Popov ghosts, can be achieved by the standard replacement
(33).

##  3 Bilinear forms and linear approximation

The analysis of propagators in different models of quantum gravity requires
the bilinear expansions of the relevant quantities depending on the curvature
tensor, on a flat metric background. On the other hand, similar expressions
with an arbitrary background metric are useful for the one-loop calculations
in the background field method. Thus, we consider a general case and assume
the expansion (9), i.e.
$g_{\mu\nu}\,\longrightarrow\,g^{\prime}_{\mu\nu}\,=\,g_{\mu\nu}+h_{\mu\nu}$.

Let us refer the interested reader to the book OUP  for technical details and
only give the following list of basic expansions:

$\displaystyle{g^{\prime}}^{\mu\nu}=g^{\mu\nu}-h^{\mu\nu}+h^{\mu}_{\,\,\lambda}h^{\nu\lambda}-h^{\mu}_{\,\,\lambda}h^{\lambda}_{\,\,\tau}h^{\nu\tau}+..$
$\displaystyle\sqrt{-g^{\prime}}=\sqrt{-g}\,\Big{(}1+\frac{1}{2}\,h+\frac{1}{8}\,h^{2}-\frac{1}{4}\,h_{\mu\nu}h^{\mu\nu}+\,...\Big{)},$
$\displaystyle{\Gamma^{\prime}}^{\alpha}_{\,\,\beta\gamma}\,=\,\Gamma^{\alpha}_{\,\,\beta\gamma}+\delta\Gamma^{\alpha}_{\,\,\beta\gamma}\,,$
(44)

where

$\displaystyle\delta\Gamma^{\alpha}_{\,\,\beta\gamma}\,=\,\frac{1}{2}\big{(}g^{\alpha\lambda}-h^{\alpha\lambda}+h^{\alpha}_{\kappa}h^{\lambda\kappa}-h^{\alpha}_{\kappa}h^{\kappa}_{\tau}h^{\tau\lambda}+...\big{)}\,\big{(}\nabla_{\beta}h_{\gamma\lambda}+\nabla_{\gamma}h_{\beta\lambda}-\nabla_{\lambda}h_{\gamma\beta}\big{)}.$

In these formulas, the Greek indices are lowered and raised with the
background metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. One has to
remember that the variation $\delta{\Gamma}^{\lambda}_{\mu\nu}$ is a tensor,
and, therefore, can be a subject to covariant differentiation.

The first two orders of expansion of the Riemann tensor have the form

$\displaystyle R^{\prime\alpha}_{\,\,\,\cdot\,\beta\mu\nu}$ $\displaystyle=$
$\displaystyle R^{\alpha}_{\,\cdot\,\beta\mu\nu}+\delta
R^{\alpha}_{\,\cdot\,\beta\mu\nu},\quad\mbox{where }\quad\delta
R^{\alpha}_{\,\cdot\,\beta\mu\nu}=R^{(1)\alpha}_{\quad\,\cdot\,\beta\mu\nu}+R^{(2)\alpha}_{\quad\,\cdot\,\beta\mu\nu},\mbox{\qquad}$
(45) $\displaystyle R^{(1)\alpha}_{\quad\,\cdot\,\beta\mu\nu}$
$\displaystyle=$
$\displaystyle\frac{1}{2}\Big{(}\nabla_{\mu}\nabla_{\beta}h^{\alpha}_{\,\nu}-\nabla_{\nu}\nabla_{\beta}h^{\alpha}_{\,\mu}+\nabla_{\nu}\nabla^{\alpha}h_{\mu\beta}-\,\nabla_{\mu}\nabla^{\alpha}h_{\nu\beta}$
$\displaystyle\,\,\,+\,R^{\alpha}_{\,\,\tau\mu\nu}h^{\tau}_{\,\beta}-R^{\tau}_{\,\,\beta\mu\nu}h_{\tau\alpha}\Big{)},\mbox{\qquad}\mbox{\,\,}$
$\displaystyle R^{(2)\alpha}_{\quad\,\cdot\,\beta\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}h^{\alpha\lambda}\big{\\{}\nabla_{\mu}\nabla_{\lambda}h_{\nu\beta}-\nabla_{\nu}\nabla_{\lambda}h_{\mu\beta}+\nabla_{\nu}\nabla_{\beta}h_{\mu\lambda}-\nabla_{\mu}\nabla_{\beta}h_{\nu\lambda}$
$\displaystyle\,\,\,+\,\big{[}\nabla_{\nu},\nabla_{\mu}\big{]}h_{\beta\lambda}\big{\\}}+\frac{1}{4}\Big{\\{}(\nabla_{\mu}h^{\alpha\lambda})(\nabla_{\lambda}h_{\nu\beta}-\nabla_{\beta}h_{\nu\lambda}-\nabla_{\nu}h_{\lambda\beta})$
$\displaystyle\,\,\,-\,(\nabla_{\beta}h^{\lambda}_{\,\mu}+\nabla_{\mu}h^{\lambda}_{\,\beta})(\nabla_{\lambda}h^{\alpha}_{\,\nu}-\nabla^{\alpha}h_{\lambda\nu})+(\nabla^{\lambda}h_{\nu\beta})(\nabla^{\alpha}h_{\mu\lambda}-\nabla_{\lambda}h^{\alpha}_{\,\mu})$
$\displaystyle\,\,\,-\,(\nabla_{\nu}h^{\alpha\lambda})(\nabla_{\lambda}h_{\mu\beta}-\nabla_{\beta}h_{\mu\lambda}-\nabla_{\mu}h_{\lambda\beta})\,-\,(\nabla^{\lambda}h_{\mu\beta})(\nabla^{\alpha}h_{\nu\lambda}-\nabla_{\lambda}h^{\alpha}_{\,\nu})$
$\displaystyle\,\,\,+\,(\nabla_{\beta}h^{\lambda}_{\,\nu}+\nabla_{\nu}h^{\lambda}_{\,\beta})(\nabla_{\lambda}h^{\alpha}_{\,\mu}-\nabla^{\alpha}h_{\lambda\mu})\Big{\\}}.$

Here and in what follows, points indicate the positions of the raised indices.
Similar formulas for the Ricci tensor and scalar curvature are

$\displaystyle{R^{\prime}}_{\beta\nu}=R_{\mu\nu}+\delta{R}_{\mu\nu},\quad\mbox{where
}\quad\delta{R}_{\mu\nu}=R^{(1)}_{\mu\nu}+R^{(2)}_{\mu\nu},$ $\displaystyle
R^{(1)}_{\mu\nu}=\frac{1}{2}\big{(}\nabla_{\lambda}\nabla_{\mu}h^{\lambda}_{\,\nu}+\nabla_{\lambda}\nabla_{\nu}h^{\lambda}_{\,\mu}-\nabla_{\mu}\nabla_{\mu}h-\operatorname{\square}h_{\mu\nu}\big{)},$
(46) $\displaystyle
R^{(2)}_{\mu\nu}=\,\frac{1}{2}h^{\alpha\beta}\big{(}\nabla_{\alpha}\nabla_{\beta}h_{\mu\nu}+\nabla_{\mu}\nabla_{\nu}h_{\alpha\beta}-\nabla_{\alpha}\nabla_{\mu}h_{\beta\nu}-\nabla_{\alpha}\nabla_{\nu}h_{\beta\mu}\big{)}$
$\displaystyle\quad+\,\frac{1}{2}\big{(}\nabla_{\alpha}h_{\mu\beta}\big{)}\big{(}\nabla^{\alpha}h_{\nu}^{\,\,\beta}-\nabla^{\beta}h_{\nu}^{\,\,\alpha}\big{)}+\frac{1}{4}(\nabla_{\mu}h_{\alpha\beta})(\nabla_{\nu}h^{\alpha\beta})$
$\displaystyle\quad+\,\frac{1}{4}\,\big{(}2\nabla_{\beta}h^{\alpha\beta}-\nabla^{\alpha}h\big{)}\big{(}\nabla_{\alpha}h^{\mu\nu}-\nabla_{\mu}h^{\alpha\nu}-\nabla_{\nu}h^{\alpha\mu}\big{)}$
$\displaystyle\mbox{and}\qquad R^{\prime}\,=\,R\,+\,\delta R,\quad\mbox{where
}\quad\delta R\,=\,R^{(1)}+R^{(2)},$ (47) $\displaystyle
R^{(1)}\,=\,\nabla_{\mu}\nabla_{\nu}h^{\mu\nu}-\operatorname{\square}h-R_{\mu\nu}h^{\mu\nu},$
$\displaystyle
R^{(2)}\,=\,h^{\alpha\beta}\Big{(}\nabla_{\alpha}\nabla_{\beta}h+\operatorname{\square}h_{\alpha\beta}-\nabla_{\alpha}\nabla_{\mu}h^{\mu}_{\,\,\beta}-\nabla_{\mu}\nabla_{\alpha}h^{\mu}_{\,\,\beta}\Big{)}-\,\frac{1}{4}(\nabla_{\alpha}h)(\nabla^{\alpha}h)$
$\displaystyle\quad+\frac{1}{4}\big{(}\nabla_{\mu}h_{\alpha\beta}\big{)}\big{(}3\nabla^{\mu}h^{\alpha\beta}-2\nabla^{\alpha}h^{\mu\beta}\big{)}\,+\,(\nabla_{\alpha}h^{\alpha\beta})(\nabla_{\beta}h-\nabla_{\mu}h^{\mu}_{\,\,\beta})+R_{\mu\nu}h^{\mu}_{\,\,\alpha}h^{\nu\alpha}.$

Using the expressions listed above, one can easily get the expansions of the
terms in the action, up the second order in $\,h_{\mu\nu}$. The results can be
written for the terms in the four derivative action (30), but it is easy to
show how, in the particular case of a flat metric, these expansions can be
mapped to the more general action (39) with an arbitrary (finite or even
infinite) number of derivatives.

The first expansion has the form

$\displaystyle\Big{(}\int
d^{4}x\sqrt{-g^{\prime}}\big{[}R^{\prime}+2\Lambda\big{]}\Big{)}^{(2)}\,=\,\frac{1}{4}\int
d^{4}x\sqrt{-g}\,h^{\mu\nu}\Big{[}\delta_{\mu\nu,\alpha\beta}\operatorname{\square}-g_{\mu\nu}g_{\alpha\beta}\operatorname{\square}$
$\displaystyle\quad-\,2g_{\mu\alpha}\nabla_{\nu}\nabla_{\beta}+\big{(}g_{\mu\nu}\nabla_{\alpha}\nabla_{\beta}-g_{\alpha\beta}\nabla_{\mu}\nabla_{\nu}\big{)}-\big{(}g_{\mu\nu}R_{\alpha\beta}-g_{\alpha\beta}R_{\mu\nu}\big{)}$
$\displaystyle\quad+\,2R_{\mu\alpha\nu\beta}\,-\,\big{(}R+2\Lambda\big{)}\Big{(}\delta_{\mu\nu,\alpha\beta}-\frac{1}{2}g_{\mu\nu}g_{\alpha\beta}\Big{)}\Big{]}h^{\alpha\beta}\,,$

where we used the DeWitt notation for the unit matrix in the symmetric tensors
space

$\displaystyle\delta_{\mu\nu,\alpha\beta}\,=\,\frac{1}{2}\,\big{(}g_{\mu\alpha}g_{\nu\beta}+g_{\nu\alpha}g_{\mu\beta}\big{)}$
(48)

and the short notations that assume the symmetrization, e.g.,

$\displaystyle g_{\mu\alpha}\nabla_{\nu}\nabla_{\beta}\,\,$
$\displaystyle\longrightarrow$
$\displaystyle\,\,\frac{1}{4}\big{(}g_{\mu\alpha}\nabla_{\nu}\nabla_{\beta}+g_{\nu\alpha}\nabla_{\mu}\nabla_{\beta}+g_{\mu\beta}\nabla_{\nu}\nabla_{\alpha}+g_{\nu\beta}\nabla_{\mu}\nabla_{\alpha}\big{)}.$
(49)

For the sake of brevity, the remaining expansions will be given only for a
flat background. The complete expressions can be found, e.g., in OUP . The
simplest of the remaining relevant bilinear expansions is

$\displaystyle\Big{(}\int
d^{4}x\sqrt{-g^{\prime}}{R^{\prime}}^{2}\Big{)}^{(2)}_{\rm flat}$
$\displaystyle=$ $\displaystyle\int
d^{4}x\,\,\,h^{\mu\nu}\Big{[}\eta_{\alpha\beta}\eta_{\mu\nu}\operatorname{\square}^{2}-\eta_{\mu\nu}\operatorname{\square}\partial_{\alpha}\partial_{\beta}$
(50)
$\displaystyle-\,\,\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\operatorname{\square}\,+\,\partial_{\mu}\partial_{\nu}\partial_{\alpha}\partial_{\beta}\Big{]}h^{\alpha\beta}.$

One can note the absence of the term
$\delta_{\mu\nu,\alpha\beta}\operatorname{\square}^{2}$ in this expression. As
a result, the $R^{2}$ term does not affect the propagation of the spin-2 mode
${\bar{h}}^{\bot\bot}_{\mu\nu}$ of the metric perturbation (8).

The next expansions are the one for the square of the Riemann tensor,

$\displaystyle\Big{(}\int
d^{4}x\sqrt{-g^{\prime}}{R^{\prime}}_{\mu\nu\alpha\beta}^{2}\Big{)}^{(2)}_{\rm
flat}$ $\displaystyle\,\,=\,\,$ $\displaystyle\int
d^{4}x\,\,\,h^{\mu\nu}\Big{[}\delta_{\mu\nu,\alpha\beta}\operatorname{\square}^{2}+\partial_{\alpha}\partial_{\beta}\partial_{\mu}\partial_{\nu}$
(51)
$\displaystyle-\,\,2\eta_{\nu\beta}\operatorname{\square}\partial_{\mu}\partial_{\alpha}\Big{]}h^{\alpha\beta}$

and for the square of the Ricci tensor,

$\displaystyle\Big{(}\int
d^{4}x\sqrt{-g^{\prime}}{R^{\prime}}_{\mu\nu}^{2}\Big{)}^{(2)}_{\rm
flat}\,=\,\frac{1}{2}\int
d^{4}x\,\,\,h^{\mu\nu}\Big{[}\frac{1}{2}\,\big{(}\delta_{\mu\nu,\alpha\beta}+\eta_{\mu\nu}\eta_{\alpha\beta}\big{)}\operatorname{\square}^{2}-\,\eta_{\nu\beta}\operatorname{\square}\nabla_{\alpha}\nabla_{\mu}$
$\displaystyle\qquad+\,\nabla_{\alpha}\nabla_{\mu}\nabla_{\beta}\nabla_{\nu}-\,\frac{1}{2}\eta_{\mu\nu}\operatorname{\square}\nabla_{\alpha}\nabla_{\beta}-\frac{1}{2}\,\eta_{\alpha\beta}\operatorname{\square}\nabla_{\mu}\nabla_{\nu}\Big{]}h^{\alpha\beta}\,.$
(52)

The last two expansions (51) and (52) possess the
$\delta_{\mu\nu,\alpha\beta}\operatorname{\square}^{2}$ terms. This means,
these two terms contribute to the flat-space propagator of the transverse and
traceless mode of the gravitational perturbation
${\bar{h}}^{\bot\bot}_{\mu\nu}$ in the representation (8).

One can note that it would be impossible to have only one of the terms (51)
and (52) contributing to the propagation of the spin-2 mode, because (50) does
not contribute to this mode and the linear combination of the three terms (29)
form a topological invariant $E_{4}$.

##  4 Gravitational waves, quantization, and gravitons

The gravitational wave is a dynamical classical solution of Einstein’s GR
without matter sources. This term can be also used for the solutions of the
same sort in modified gravity models. However, since in these models the
additional modes are typically massive and, therefore, do not propagate for a
long distances, it is most common to attribute the notion of a gravitational
wave to the solutions in GR444The topics related to the models of massive
gravity are left beyond the present Handbook. The main reason is that, in
these models, quantum aspects do not play an important role.. Nowadays,
gravitational waves represent one of the most successful parts of
gravitational physics, both experimental and theoretical. However, in this
short section we present only a brief survey of the basic facts concerning the
gravitational waves on a flat background in GR.

###  4.1 Gravitational waves in a weak-gravity regime

We start with an action of gravity with $\Lambda=0$ and use the bilinear
expansion (3) on the flat background metric $g_{\mu\nu}=\eta_{\mu\nu}$. To
discuss the emission of the gravitational wave (in the simplest case), we also
add the action of matter and consider it approximation that provides a linear
equation for $h_{\mu\nu}$. In this way, we obtain the action of GR in the
linearized regime,

$\displaystyle S_{total}^{(lin)}$ $\displaystyle=$
$\displaystyle-\frac{1}{32\pi G}\int
d^{4}x\,\,\,h^{\mu\nu}\Big{\\{}\frac{1}{2}\,\delta_{\mu\nu,\alpha\beta}\operatorname{\square}-\frac{1}{2}\,\eta_{\mu\nu}\eta_{\alpha\beta}\operatorname{\square}-\eta_{\mu\alpha}\partial_{\nu}\partial_{\beta}\mbox{\qquad}$
(53)
$\displaystyle\quad+\,\frac{1}{2}\big{(}\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}-\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\big{)}\Big{\\}}h^{\alpha\beta}\,\,-\,\,\frac{1}{2}\int
d^{4}x\,h^{\mu\nu}T_{\mu\nu},$

where $\operatorname{\square}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}$ and
$T_{\mu\nu}$ is the energy-momentum tensor of matter in flat spacetime
background. The equation for metric perturbations has the form

$\displaystyle\Big{\\{}\delta_{\mu\nu,\alpha\beta}\operatorname{\square}-\eta_{\mu\nu}\eta_{\alpha\beta}\operatorname{\square}-2\eta_{\mu\alpha}\partial_{\nu}\partial_{\beta}+(\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}-\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\Big{\\}}h^{\alpha\beta}\,=\,16\pi
GT_{\mu\nu}.$ (54)

Multiplying both sides of Eq. (54) by the matrix

$\displaystyle
K^{\mu\nu,\rho\sigma}\,=\,\delta^{\mu\nu,\rho\sigma}\,-\,\frac{1}{2}\,\eta^{\mu\nu}\eta^{\rho\sigma},$
(55)

we arrive at the equation for the modified stress tensor,
$S_{\mu\nu}=T_{\mu\nu}-\frac{1}{2}T^{\,\lambda}_{\lambda}\,g_{\mu\nu}$,

$\displaystyle\partial_{\lambda}\partial_{\nu}h^{\lambda}_{\,\mu}+\partial_{\lambda}\partial_{\nu}h^{\lambda}_{\,\mu}-\operatorname{\square}h_{\mu\nu}-\partial_{\mu}\partial_{\mu}h\,=\,16\pi
G\,S_{\mu\nu}\,.$ (56)

Here $h=h_{\mu\nu}\eta^{\mu\nu}$. The last equation describes both propagation
and emission of the gravitational waves in the linear approximation. This
equation has to be supplemented by the gauge transformation (4) $\delta
h_{\mu\nu}\,=\,-\,\partial_{\mu}\xi_{\nu}\,-\,\partial_{\nu}\xi_{\mu}$, and
requires imposing the gauge-fixing condition, e.g., using the de Donder (also
called Fock-de Donder) condition

$\displaystyle\partial_{\mu}h^{\mu}_{\,\,\nu}\,-\,\frac{1}{2}\,\partial_{\nu}h\,=\,0.$
(57)

Using condition (57) in Eq. (56), the last is cast in the form (see, e.g.,
Weinberg )

$\displaystyle\operatorname{\square}h_{\mu\nu}\,\,=\,-\,16\pi G\,S_{\mu\nu}.$
(58)

For the plane wave propagating along an arbitrary axis, the components of the
metric perturbation (8) which are gauge invariant and can be called physical,
are transverse ones, ${\bar{h}}^{\bot\bot}_{\mu\nu}$. Thus, the gravitational
wave in GR is a propagation of the spin-2 state.

The emission of the gravitational wave in the linear regime corresponds to the
solution in the standard form of retarded potential,

$\displaystyle h_{\mu\nu}({{\bf x}},t)\,\,=\,\,\frac{4}{M_{P}^{2}}\int
d^{3}x^{\prime}\,\,\frac{S_{\mu\nu}({{\bf x}}^{\prime},\,t-\left|{{\bf
x}}-{{\bf x}}^{\prime}\right|)}{\left|{{\bf x}}-{{\bf x}}^{\prime}\right|}\,.$
(59)

The factor $\,1/M^{2}_{P}=G\,$ in this expression shows that the emission of
the gravitational waves is suppressed by the square of the Planck mass. And
after the wave travels a very long distance, there is a similar Planck
suppression at the moment of its detection, that explains the difficulty of
detecting the gravitational wave. The remarkable detection by LIGO is
explained by the incredible quantity of energy emitted in the merger of two
black holes or other extremely compact and massive objects, such as neutron
stars.

###  4.2 Quantization and the notion of graviton

At quantum level, the physical degrees of freedom corresponding to the state
of a free gravitational field on a flat background correspond to the degrees
of freedom of the linear gravitational wave described above. The corresponding
particle with zero mass and spin-2 is called a graviton.

To derive the spin-2 part of the propagator of the gravitational perturbation
$h_{\mu\nu}$, we can use the tensor part of the propagator [see, e.g., Eq.
(89) below] and setting $\Phi=0$. For the sake of simplicity, we omit the
coefficient $\kappa^{2}$ and obtain the spin-2 part of the Euclidean
propagator in the form

$\displaystyle\langle
h_{\mu\nu}h_{\alpha\beta}\rangle^{(2)}\,=\,G^{(2)}_{\mu\nu,\alpha\beta}(k)\,=\,\frac{P^{(2)}_{\mu\nu,\alpha\beta}(k)}{k^{2}}\,,$
(60)

where the projector $P^{(2)}_{\mu\nu,\alpha\beta}(k)$ to the spin-2 states
will be defined below. This equation describes the propagation of the massless
degrees of freedom associated with the spin-2 states in GR, which is the
graviton.

In other models of QG, the propagator can be more complicated because $\Phi$
(and also $\Psi$, because there may be a relevant dynamics in the scalar
sector) are typically nonzero. For instance, including the fourth-derivative
terms, there may be the following two changes:

i) Instead of the unique massless pole in the propagator (60), there may be
additional massive poles.

ii) The scalar components of the metric perturbation (8) gain a massive,
gauge-independent sector in the propagator.

Adding more derivatives, which means using polynomial or even nonlocal models,
the modifications always concern the same two points. Namely, there will be
(in the polynomial models) a growing number of poles in the spin-2 sector and
the scalar sector. In contrast, some choices of nonlocal action may provide
that there would not be any massive poles in the tree-level propagator, in
both spin-2 and spin-0 sectors.

It is worth noting that the count of degrees of freedom, based on the simple
analysis of the gravitational propagator, was confirmed by the canonical
quantization of the gravitational theory in the cases of quantum GR and
fourth-derivative quantum gravity (see, e.g., book ).

##  5 Propagator of metric and the Barnes-Rivers projectors

At this point we note a common aspect of all mentioned models of QG, namely
the ones based on the actions (22), (30), (39). Since the propagator of
$h_{\mu\nu}$ in flat background is defined by the quadratic in curvatures
(Riemann, Ricci or scalar $R$) terms in the action, the form factors depending
on $\operatorname{\square}$ do not influence the tensor structure of the
bilinear form.

The propagator $G$ of the quantum metric, in any QG model, obeys the equation

$\displaystyle
H^{\mu\nu,\alpha\beta}(x)\,G_{\alpha\beta,\rho\sigma}(x,y)\,\,=\,\,\delta^{4}(x-y)\,\delta^{\mu\nu}_{\quad,\rho\sigma}\,,$
(61)

where

$\displaystyle
H^{\mu\nu,\alpha\beta}(x)\,\delta^{4}(x-y)\,\,=\,\,\frac{1}{2\sqrt{-g}}\,\frac{\delta^{2}S}{\delta
g_{\mu\nu}(x)\,\delta g_{\alpha\beta}(y)}.$ (62)

is the bilinear in quantum fields form of the classical action $S$ of a model
of quantum gravity. The action in (62) should include the gauge-fixing term,
as otherwise we meet a degeneracy. For the sake of definiteness, we start from
the gauge-invariant action and denote the corresponding degenerate bilinear
form coming from the initial action, as $H_{(0)}^{\mu\nu,\alpha\beta}$, while
the non-degenerate version, after the Faddeev-Popov procedure, will be denoted
as $H^{\mu\nu,\alpha\beta}$.

In this part of the Chapter, all spacetime indices are raised and lowered with
the flat background metric. Then, independently of the model and the choice of
the gauge fixing, the operator $H^{\mu\nu,\alpha\beta}(x)$ has the following
tensor structure:

$\displaystyle
H_{\mu\nu,\alpha\beta}(x;g)=a_{1}\delta_{\mu\nu,\alpha\beta}\square+a_{2}\eta_{\mu\nu}\eta_{\alpha\beta}\square+a_{3}\big{(}\eta_{\mu\nu}\partial_{\alpha}\partial_{\beta}+\eta_{\alpha\beta}\partial_{\mu}\partial_{\nu}\big{)}$
$\displaystyle\qquad+\,a_{4}\big{(}\eta_{\mu\alpha}\partial_{\beta}\partial_{\nu}+\eta_{\nu\alpha}\partial_{\beta}\partial_{\mu}+\eta_{\mu\beta}\partial_{\alpha}\partial_{\nu}+\eta_{\nu\beta}\partial_{\alpha}\partial_{\mu}\big{)}\,-\,a_{5}\partial_{\alpha}\partial_{\beta}\partial_{\mu}\partial_{\nu},\quad$
(63)

where $a_{k}=a_{k}(-\square)$ are five model-dependent functions of the
d’Alembert operator. In the higher-derivative cases, all of these functions
are proportional to the linear combinations of $\Phi(\square)$ and
$\Psi(\square)$ in Eq. (39). In particular, for the fourth-derivative model
(30), $a_{1,2,3,4}$ are linear functions of $\square$ and $a_{5}=const$. In
the case of quantum GR, there are the constant functions $a_{1,..,4}$ and
$a_{5}=0$. In what follows, we consider the general analysis of the
propagator, which is valid for all types of models.

Making a Fourier transform, we can rewrite the bilinear form in the momentum
representation,

$\displaystyle
H_{\mu\nu,\alpha\beta}(k;\eta)\,=\,-\big{[}a_{1}(k^{2})\delta_{\mu\nu,\alpha\beta}k^{2}+a_{2}(k^{2})\eta_{\mu\nu}\eta_{\alpha\beta}k^{2}$
$\displaystyle\quad+\,a_{3}(k^{2})\big{(}\eta_{\mu\nu}k_{\alpha}k_{\beta}+\eta_{\alpha\beta}k_{\mu}k_{\nu}\big{)}\mbox{\,\,}$
(64)
$\displaystyle\quad+\,a_{4}(k^{2})\big{(}\eta_{\mu\alpha}k_{\beta}k_{\nu}+\eta_{\nu\alpha}k_{\beta}k_{\mu}+\eta_{\mu\beta}k_{\alpha}k_{\nu}+\eta_{\nu\beta}k_{\alpha}k_{\mu}\big{)}+a_{5}(k^{2})k_{\alpha}k_{\beta}k_{\mu}k_{\nu}\big{]},$

where $k^{2}=k_{\mu}k^{\mu}$ is the square of the four-dimensional momentum
and $\delta_{\mu\nu,\alpha\beta}$ is similar to (48), but this time it is
constructed from the flat metric $\eta_{\mu\nu}$.

It is useful to present (64) in a slightly different form, providing more
generality by using the $n$-dimensional versions of the formulas,

$\displaystyle{\hat{H}}\,=\,s_{1}\,{\hat{T}}_{1}\,+\,s_{2}\,{\hat{T}}_{2}\,+\,s_{3}\,{\hat{T}}_{3}\,+\,s_{4}\,{\hat{T}}_{4}\,+\,s_{5}\,{\hat{T}}_{5},$
(65)

where ${\hat{T}}_{n}\,=\,T^{(n)}_{\mu\nu,\alpha\beta}$ and

$\displaystyle{\hat{T}}_{1}\,=\,\delta_{\mu\nu,\alpha\beta},\qquad{\hat{T}}_{2}\,=\,\eta_{\mu\nu}\eta_{\alpha\beta},\qquad{\hat{T}}_{3}\,=\,\frac{1}{k^{2}}\big{(}\eta_{\mu\nu}k_{\alpha}k_{\beta}+\eta_{\alpha\beta}k_{\mu}k_{\nu}\big{)},$
(66)
$\displaystyle{\hat{T}}_{4}\,=\,\frac{1}{4k^{2}}\big{(}\eta_{\mu\alpha}k_{\beta}k_{\nu}+\eta_{\nu\alpha}k_{\beta}k_{\mu}+\eta_{\mu\beta}k_{\alpha}k_{\nu}+\eta_{\nu\beta}k_{\alpha}k_{\mu}\big{)}\,,\quad{\hat{T}}_{5}\,=\,\frac{1}{k^{4}}\,k_{\alpha}k_{\beta}k_{\mu}k_{\nu}.$

The coefficients depend on momentum, $s_{l}=s_{l}(k^{2})$, and these
dependencies may be nontrivial, e.g., in the polynomial or nonlocal models.
However, the tensor structure of the expressions (65) is the same for all QG
models, i.e., for quantum GR, or for a higher-derivative polynomial, or
nonlocal models (39).

To invert the operator (64) and take care about its possible degeneracy,
consider the operators called Barnes-Rivers projectors Barnes ; Rivers . The
starting point is to formulate the projectors to the transverse and
longitudinal subspaces of the vector space. In the momentum representation we
have

$\displaystyle\omega_{\mu\nu}=\frac{k_{\mu}k_{\nu}}{k^{2}},\qquad\theta_{\mu\nu}=\eta_{\mu\nu}-\omega_{\mu\nu},$
(67)

with the standard properties

$\displaystyle\omega_{\mu\nu}\,\omega^{\nu}_{\,\,\,\,\lambda}=\omega_{\mu\lambda},\qquad\theta_{\mu\nu}\,\theta^{\nu}_{\,\,\,\,\lambda}=\theta_{\mu\lambda},\qquad\omega_{\mu\nu}\,\theta^{\nu}_{\,\,\,\,\lambda}=0.$
(68)

Then, the projectors to the spin-2, spin-1 and spin-0 states in the symmetric
tensors space are written in the form

$\displaystyle{\hat{P}}^{(2)}$ $\displaystyle=$ $\displaystyle
P^{(2)}_{\mu\nu\,,\,\alpha\beta}\,=\,\frac{1}{2}(\theta_{\mu\alpha}\theta_{\nu\beta}+\theta_{\mu\beta}\theta_{\nu\alpha})-\frac{1}{n-1}\,\theta_{\mu\nu}\theta_{\alpha\beta}\,,$
$\displaystyle{\hat{P}}^{(1)}$ $\displaystyle=$ $\displaystyle
P^{(1)}_{\mu\nu\,,\,\alpha\beta}\,=\,\frac{1}{2}\,(\theta_{\mu\alpha}\omega_{\nu\beta}+\theta_{\nu\alpha}\omega_{\mu\beta}+\theta_{\mu\beta}\omega_{\nu\alpha}+\theta_{\nu\beta}\omega_{\mu\alpha})\,,$
$\displaystyle{\hat{P}}^{(0-s)}$ $\displaystyle=$ $\displaystyle
P^{{(0-s)}}_{\mu\nu\,,\,\alpha\beta}\,=\,\frac{1}{n-1}\,\theta_{\mu\nu}\theta_{\alpha\beta},\quad{\hat{P}}^{(0-w)}\,=\,P^{{(0-w)}}_{\mu\nu\,,\,\alpha\beta}\,=\,\omega_{\mu\nu}\omega_{\alpha\beta}.\mbox{\qquad}$
(69)

On top of these, to have the closed algebra of projectors in the scalar
sector, one needs the additional transfer operators

$\displaystyle{\hat{P}}^{{(ws)}}=P^{(ws)}_{\mu\nu\,,\,\alpha\beta}=\frac{1}{\sqrt{n-1}}\,\theta_{\mu\nu}\omega_{\alpha\beta},\quad{\hat{P}}^{{(sw)}}=P^{(sw)}_{\mu\nu\,,\,\alpha\beta}=\frac{1}{\sqrt{n-1}}\,\omega_{\mu\nu}\theta_{\alpha\beta}.\mbox{\qquad}$
(70)

The algebra for the vector and tensor projectors is simple,

$\displaystyle{\hat{P}}^{(2)}\,{\hat{P}}^{(i)}\,=\,{\hat{P}}^{(2)}\delta_{i2}\qquad\mbox{and}\qquad{\hat{P}}^{(1)}\,{\hat{P}}^{(i)}\,=\,{\hat{P}}^{(1)}\delta_{i1}\,,$
(71)

where $i=(2,\,\,1,\,\,0-w,\,\,0-s,\,\,sw,\,\,ws)$. In the scalar sector, one
has to construct the matrix projector operator

$\displaystyle{\hat{P}_{0}}\,\,=\,\,\frac{1}{2}\,\left\|\begin{array}[]{cc}P^{{(0-s)}}&P^{{(sw)}}\\\
P^{{(ws)}}&P^{{(0-w)}}\end{array}\right\|$ (74)

satisfying the relation ${\hat{P}_{0}}^{2}\,=\,{\hat{P}_{0}}$. To end this
part, the last two terms, which represent the scalar sector of (8), can be
written, in momentum representation, as

$\displaystyle
h_{\mu\nu}^{scalar}\,\,=\,\,\frac{1}{4}\,h\,\theta_{\mu\nu}\,+\,\Big{(}\frac{1}{4}\,h-k^{2}\epsilon\,\Big{)}\omega_{\mu\nu},$
(75)

such that acting by each of the two projectors $P^{{(0-s)}}$ and
$P^{{(0-w)}}$, one of these terms remains invariant and another vanish.

Now we are in a position to find the propagator. Solving Eq. (61) requires the
inversion of the expression

$\displaystyle{\hat{H}}\,=\,b_{2}{\hat{P}}^{(2)}+b_{1}{\hat{P}}^{(1)}+b_{os}P^{{(0-s)}}+b_{ow}{\hat{P}}^{(0-w)}+b_{sw}\big{[}P^{{(ws)}}+{\hat{P}}^{(sw)}\big{]},\mbox{\qquad}\mbox{\,\,}$
(76)

that means one has to find such an operator

$\displaystyle{\hat{G}}\,=\,c_{2}{\hat{P}}^{(2)}+c_{1}{\hat{P}}^{(1)}+c_{os}P^{{(0-s)}}+c_{ow}{\hat{P}}^{(0-w)}+c_{sw}\big{[}P^{{(ws)}}+{\hat{P}}^{(sw)}\big{]},\mbox{\qquad}\mbox{\,\,}$
(77)

that the product with ${\hat{B}}$ is unity,

$\displaystyle{\hat{H}}\,{\hat{G}}\,=\,{\hat{1}}\,=\,{\hat{P}}^{(2)}+{\hat{P}}^{(1)}+P^{{(0-s)}}+{\hat{P}}^{(0-w)}.$
(78)

Using the aforementioned algebra of projectors, we get the solution to this
problem,

$\displaystyle c_{2}=\frac{1}{b_{2}},\quad c_{1}=\frac{1}{b_{1}},\quad
c_{os}=-\,\frac{b_{ow}}{\Delta},\quad c_{ow}=-\,\frac{b_{os}}{\Delta},\quad
c_{sw}=\frac{b_{sw}}{\Delta},$ (79)

where $\quad\Delta=b^{2}_{sw}-b_{os}b_{ow}$.

It is clear that the action of a generally covariant theory before adding the
gauge fixing term has either $b_{1}=0$ or $\Delta=0$. From this perspective,
the purpose of the Faddeev-Popov procedure is to remove this degeneracy.

It remains to present the projectors (69) and the transfer operators (70) as
the linear combinations of the expressions (66), and v.v.. The first set of
formulas is

$\displaystyle{\hat{P}}^{(2)}\,\,=\,\,{\hat{T}}_{1}\,-\,\frac{1}{n-1}\,{\hat{T}}_{2}\,+\,\frac{1}{n-1}\,{\hat{T}}_{3}\,-\,2{\hat{T}}_{4}\,+\,\frac{n-2}{n-1}\,{\hat{T}}_{5},\qquad\quad{\hat{P}}^{(1)}\,\,=\,\,\,2\big{(}{\hat{T}}_{4}\,-\,{\hat{T}}_{5}\big{)},$
$\displaystyle{\hat{P}}^{(0-w)}\,\,=\,\,{\hat{T}}_{5},\qquad{\hat{P}}^{(0-s)}\,\,=\,\,\frac{1}{n-1}\,\big{(}{\hat{T}}_{2}\,-\,{\hat{T}}_{3}\,+\,{\hat{T}}_{5}\big{)},$
$\displaystyle
P^{{(ws)}}+P^{{(sw)}}\,\,=\,\,\frac{1}{\sqrt{n-1}}\,\big{(}{\hat{T}}_{3}-2{\hat{T}}_{5}\big{)}.$
(80)

Finally, by inverting these relations, one can express the matrices in (66) as

$\displaystyle{\hat{T}}_{1}\,=\,{\hat{P}}^{(2)}\,+\,{\hat{P}}^{(1)}\,+\,P^{{(0-s)}}\,+\,{\hat{P}}^{(0-w)},$
$\displaystyle{\hat{T}}_{2}\,=\,(n-1)P^{{(0-s)}}\,+\,\sqrt{n-1}\big{[}P^{{(ws)}}\,+\,{\hat{P}}^{(sw)}\big{]}\,+\,{\hat{P}}^{(0-w)},$
$\displaystyle{\hat{T}}_{3}\,=\,\sqrt{n-1}\big{[}P^{{(ws)}}\,+\,{\hat{P}}^{(sw)}\big{]}\,+\,2{\hat{P}}^{(0-w)},$
$\displaystyle{\hat{T}}_{4}\,=\,\frac{1}{2}\,{\hat{P}}^{(1)}\,+\,{\hat{P}}^{(0-w)},\qquad{\hat{T}}_{5}\,=\,{\hat{P}}^{(0-w)}.$
(81)

Correspondingly, the transformations between the coefficients of (65) and (76)
are given by the inverse relations to the ones of “basic vectors”, i.e.,

$\displaystyle b_{2}\,=\,s_{1}\,+\,s_{2}\,+\,s_{3}\,+\,s_{4},\qquad
b_{1}\,=\,(n-1)s_{3}\,+\,\sqrt{n-1}s_{5}\,+\,s_{4},$ $\displaystyle
b_{0s}\,=\,\sqrt{n-1}s_{5}\,+\,2s_{4},\qquad
b_{0w}\,=\,\frac{1}{2}\,s_{2}\,+\,s_{4},\qquad b_{sw}\,=\,s_{4}.$ (82)

and

$\displaystyle
s_{1}\,=\,b_{2}\,-\,\frac{1}{n-1}\,b_{1}\,+\,\frac{1}{n-1}\,b_{0s}\,-\,2b_{0w}\,+\,\frac{n-2}{n-1}\,b_{sw},\qquad
s_{2}\,=\,\big{(}b_{0w}-b_{sw}\big{)},$ $\displaystyle
s_{3}\,=\,\frac{1}{n-1}\big{(}b_{1}-b_{0s}+b_{sw}\big{)},\quad
s_{4}\,=\,b_{sw},\quad
s_{5}\,=\,\frac{1}{\sqrt{n-1}}\big{(}b_{0s}-2b_{sw}\big{)}.$ (83)

The solution to Eq. (61) consists of casting the bilinear form in the standard
form (66), using the relations between ${\hat{T}}_{l}$ and projectors (81),
inverting the result using (79) and, finally, using the inverse relations
(80). In principle, this procedure works for the bilinear form of the total
action (63) for an arbitrary model of quantum gravity if the chosen gauge-
fixing term makes the bilinear form non-degenerate. Usually, the Faddeev-Popov
procedure is sufficient in this respect, but if the original theory had an
extra symmetry (e.g., the conformal one), one needs to apply an additional
gauge fixing, e.g., setting $h=h^{\mu}_{\,\,\mu}=0$ frts82 .

To put the described procedure in practise, one has to use definitions (35),
with $\operatorname{\square}^{n}\to\Phi$, and (40), and insert the form
factors $\Phi$, $\Psi$ and $\Omega$ into expansions (51), (52), (50), and (3).
The bilinear form of the general gauge-fixing term (31) with the weight (43),
can be easily calculated,

$\displaystyle H^{(GF)}_{\mu\nu,\alpha\beta}(k;\,\eta)\,=\,{\hat{H}}_{GF}$
$\displaystyle=$ $\displaystyle
W(-k^{2})\,k^{4}\bigg{\\{}\beta^{2}(\gamma-1){\hat{T}}_{2}\,+\,\beta(1-\gamma){\hat{T}}_{3}\,-\,\frac{1}{4}{\hat{T}}_{4}\,+\,\gamma\,{\hat{T}}_{5}\bigg{\\}}.$

Summing up all the terms, including the bilinear form of the original action
$H_{(0)\,\mu\nu,\alpha\beta}(k;\,\eta)$, contribution of (40) and of the
gauge-fixing term, we arrive at the expression (65)

$\displaystyle{\hat{H}}\,=\,{\hat{H}}_{(0)}\,+\,{\hat{H}}_{GB}\,+\,{\hat{H}}_{GF},$
(85)

with the coefficients

$\displaystyle s_{1}\,=\,\frac{1}{2}\,\Phi
k^{4}+\frac{1}{2\kappa^{2}}\,k^{2}+\frac{\Lambda}{\kappa^{2}}\,,$
$\displaystyle
s_{2}\,=\,\Big{(}\Psi-\frac{1}{6}\,\Phi\Big{)}k^{4}-\frac{1}{2\kappa^{2}}\,k^{2}-\frac{\Lambda}{2\kappa^{2}}+\beta^{2}(\gamma-1)Wk^{4},$
$\displaystyle
s_{3}\,=\,\Big{(}\frac{1}{6}\,\Phi-\Psi\Big{)}k^{4}+\frac{1}{2\kappa^{2}}\,k^{2}+\beta(1-\gamma)Wk^{4},$
$\displaystyle s_{4}\,=\,-\,\Phi k^{4}-\frac{1}{\kappa^{2}}\,k^{2}-Wk^{4},$
$\displaystyle s_{5}\,=\,\Big{(}\frac{1}{3}\,\Phi+\Psi\Big{)}k^{4}+\gamma
Wk^{4}.$ (86)

The remarkable detail is that there is no $\Omega(-k^{2})$ in these
expressions. This is certainly an expected result because this function comes
from the “generalized” topological invariant (40), but we observe that this
feature holds for any $\Omega(-k^{2})$, not only for a constant, when this
term in the action is really topological.

Another expected characteristic of expressions (86) is that all the
coefficients except $s_{1}$ depend on the gauge fixing parameters. Since (85)
has three linearly independent coefficients $\frac{1}{4\alpha}$,
$\frac{\beta\gamma}{\alpha}$ and $\frac{1-\gamma}{\alpha}$, by using the
choice of the three parameters $\alpha$, $\beta$ and $\gamma$, one can
eliminate terms with the coefficients $a_{3,4,5}$ in the bilinear form (85).
As a result, in any QG model, one can provide the minimal form of the total
bilinear operator,

$\displaystyle
H^{total\,,\,minimal}_{\mu\nu,\alpha\beta}(k;\,\eta)\,=\,-\,\Big{[}a_{1}(k^{2})\delta_{\mu\nu,\alpha\beta}\,+\,a^{\prime}_{2}(k^{2})\eta_{\mu\nu}\eta_{\alpha\beta}\Big{]}k^{2},$
(87)

where $a^{\prime}_{2}$ differs from $a_{2}$ in Eq. (64) because of the
contribution of the gauge-fixing term. This property of the bilinear form
holds also for an arbitrary background metric and is important for the one- or
higher-loop calculations in QG.

The main advantage of the minimal form (87) compared to the general one, is
that the minimal operators are directly suited for the use of the heat-kernel
technique. Indeed, it is possible to work with the nonminimal operators in
quantum gravity, e.g., using the generalized Schwinger-DeWitt technique bavi85
, but it is always simpler to work with the minimal bilinear forms.

In some QG models, the choice of parameters is more restricted. E.g., in
quantum GR, we meet only two gauge-fixing parameters, i.e., $\alpha$ and
$\beta$, but there are only four nonminimal terms because of $a_{5}=0$. As a
result, one can choose $\alpha$ and $\beta$ to achieve the minimal form (87).
One technical observation is that, if the initial action includes,
simultaneously, higher derivative ($\Phi$\- and $\Psi$-terms) and the
Einstein-Hilbert action, one can provide minimality only in the higher order
terms (typically, in all of them, greatly simplifying calculations in the
superrenormalizable models), but not in the second derivative sector of the
operator.

Let us now analyse the situation from another perspective. Expressions (86)
remain the same in any dimension $n$. Thus, we can use (82) to transform the
operator ${\hat{H}}$ into the form (76). The result of this transformation is

$\displaystyle b_{2}\,=\,\frac{1}{2}\,\Phi
k^{4}+\frac{1}{2\kappa^{2}}\,k^{2}+\frac{\Lambda}{\kappa^{2}}\,,$
$\displaystyle b_{1}\,=\,-\frac{1}{2}\,Wk^{4}+\frac{\Lambda}{\kappa^{2}},$
$\displaystyle
b_{0s}\,=\,-\frac{n-4}{6}\,\Phi\,k^{4}+(n-1)\big{[}\Psi+\beta^{2}(\gamma-1)W\big{]}k^{4}\,-\,\frac{n-2}{2\kappa^{2}}\,k^{2}\,-\,\frac{(n-3)\Lambda}{2\kappa^{2}}\,,$
$\displaystyle
b_{0w}\,=\,(1-\beta)^{2}(\gamma-1)Wk^{4}\,+\,\frac{k^{2}}{\kappa^{2}}\,+\,\frac{\Lambda}{2\kappa^{2}}\,,$
$\displaystyle
b_{sw}\,=\,\sqrt{n-1}\,\Big{[}\beta(\beta-1)(\gamma-1)k^{4}W-\frac{\Lambda}{2\kappa^{2}}\Big{]}\,.$
(88)

Once again, only the spin-2 coefficient is gauge-fixing independent.

The inversion formulas (79) are trivial in the spin-2 and spin-1 sectors. In
the tensor sector, we get, for the $\Lambda=0$ case, independent on the
dimension $\,n\,$ and on the gauge fixing,

$\displaystyle
c_{2}(\Lambda=0)\,\,\,=\,\,\,\frac{2\kappa^{2}}{k^{2}\big{(}1+\kappa^{2}k^{2}\Phi\big{)}}.$
(89)

The vector part depends on the gauge fixing and has no direct physical
interpretation. Thus, we concentrate on the results in the scalar sector. The
reader can easily obtain the complete formulas, but since these formulas are
cumbersome, we shall present only the qualitative results and the most
interesting expression. In the cases of $\Lambda\neq 0$ or $n\neq 4$, all the
scalar coefficients are gauge-fixing dependent. However, in case $\Lambda=0$
and $n=4$, one important coefficient is invariant555I am grateful to Dr.
Leslaw Rachwał for indicating to me this feature.,

$\displaystyle
c_{os}(\Lambda=0,\,n=4)\,\,\,=\,\,\,-\,\frac{\kappa^{2}}{k^{2}\big{(}1-3\kappa^{2}\Psi
k^{2}\big{)}}.$ (90)

It is easy to see from these formulas, that the propagator (89) of the spin-2
mode depends only on the function $\Phi$, while the spin-0 propagator (90)
depends only on the function $\Psi$. Let us note that this output was
anticipated already at the level of bilinear expansions of the classical
action. Furthermore, both spin-2 and spin-0 propagators are gauge-fixing
independent and contribute to the tree-level $S$-matrix of gravitational
perturbations. It is interesting that these features, which are looking quite
special, hold independent on the form of the form factors
$\Phi(\operatorname{\square})$ and $\Psi(\operatorname{\square})$ in the
action (39). The same concerns the irrelevance of the third form factor
$\Omega$ for the propagator. Of course, this property is not valid for the
vertices, if $\Omega(x)$ is not a constant function.

A peculiar situation occurs in the usual second-derivative gravity. Even if
one starts from the pure GR, the result is the same as if setting $\,\Psi\to
0\,$ in (90), i.e.,666The factor of $\kappa^{2}$ in this formula appears
because we used the expansion (9) with a flat background
$g_{\mu\nu}=\eta_{\mu\nu}$. If using the expansion (24), there is no such
coefficient.

$\displaystyle c_{os}(GR\mbox{ \ with \
}\Lambda=0)\,\,\,=\,\,\,-\,\frac{\kappa^{2}}{k^{2}}.$ (91)

This formula is in apparent contradiction with what we saw in the analysis of
the gravitational waves in GR, where the unique sort of the propagating
degrees of freedom are the tensor (transverse and traceless) modes. The
explanation of this apparent discrepancy is that the smooth $\,\Psi\to 0\,$
limit in (91) corresponds to the theory after the Faddeev-Popov procedure,
which extends the space of the propagating modes and make the whole propagator
non-degenerate. On the contrary, the result for the gravitational waves is
based on another procedure, i.e., removing all degrees of freedom by using
gauge transformation and its remnant (see, e.g., Weinberg ).

From the QG perspective, the smooth limit (91) of the general expression (90)
for the scalar sector is important, as it provides universal IR limit of the
propagator (for $n=4$ only!) in any QG model (39) at the tree level, i.e., in
both relevant sectors of the propagator.

##  6 Gauge-invariant renormalization in quantum gravity

In the next Chapter of this Section there is a detailed proof of the two main
statements concerning the gauge-invariant renormalization in quantum gravity
Lavrov-renQG  (see also pioneer work Stelle77 , VorTyu-QG  and more recent and
BarvSib-QG ; Ohta ) .

Both theorems were already mentioned in the Introduction. However, since these
two statements are relevant for the rest of this Chapter, let us formulate
them here in more detail.

1\. The renormalization preserves the diffeomorphism invariance (general
covariance) of the model of QG in four spacetime dimensions (i.e., in $n=4$),
if the initial classical theory possesses this symmetry. This means, one can
remove the divergences in all loop orders by adding covariant counterterms.
This statement applies literally only to the divergences that take place in
the framework of the background field method, which is especially designed to
avoid non-covariant counterterms. Let us stress that the background field
method is not necessary for the gauge-invariant renormalization in QG.
However, using the non-covariant parametrization of the metric, such as (24),
one has to go through a relatively complicated procedure or additional
renormalization transformations, as described in Stelle77 . The final output
is always the same in the sense of the same essential covariant counterterms.
For this reason, in what follows we shall assume (24) when evaluating the
power counting in different models of QG. However, we shall switch to the more
general parametrization with the general background metric (9) when making the
practical calculations.

2\. The dependence on the choice of the gauge condition (31), (e.g., on the
parameters $\beta$, $\gamma$ and the function $W(\operatorname{\square})$ in
the weight operator (43) is proportional to the effective equations of motion.
The same concerns the dependence of the parametrization of the quantum metric.
In particular, both ambiguities vanish on the classical mass shell for the
one-loop divergences of the effective action. In what follows (see more
details in OUP ) we shall demonstrate how this feature works in the practical
calculations in QG.

On top of that, there is the third statement, representing the general feature
of QFT and valid, in particular, for QG. The counterterms required to remove
UV divergences, in all loop orders, are local functionals of the fields. The
mathematically rigid proof of this statement (usually called Weinberg’s
theorem Weinberg1960 ) is complicated and can be found, e.g., in Collins . In
what follows, we shall apply these three statement to describe the
renormalization in QG.

##  7 Power counting, and classification of quantum gravity models

To estimate the power counting for the Feynman diagrams in QG is somehow
simpler than in the quantum theories of other fields. The reason is that the
metric is a dimensionless field. For this reason, the dimensions of the
counterterm that emerge for diagram $G$, with $L$ loops, is defined only by
the number of derivatives of the background metric or, in case of a non-
covariant parametrization such as (24), by the number of derivatives of
$h_{\mu\nu}$. The power counting of a diagram is essentially equivalent to the
count of dimensions and, therefore, it does not depend on the number of
external lines of $h_{\mu\nu}$. Let us use the last version of the expansion,
but with $\kappa\to 1$ for generality, as this enables us to include the
higher derivative theories into consideration.

The superficial degree of divergences will be denoted $\omega(G)$ (sometimes
it is also called the index of divergence) and $d(G)$ is the number of partial
derivatives of the external lines of the field $h_{\mu\nu}$ in the diagram.
Taking into account the powers of momenta in all elements of the diagrams
(see, e.g., OUP  for a detailed general treatment), the general expression is

$\displaystyle\omega(G)+d(G)\,=\,\sum\limits_{l_{int}}(4-r_{I})\,-\,4V\,+\,4\,+\,\sum\limits_{V}K_{V},$
(92)

where the first sum is over all $I$ internal lines of the diagram, $r_{I}$ is
the inverse power of momentum in the propagator of an internal line and $V$ is
the number of vertices. The last sum is taken over all the vertices, where
$K_{V}$ is the power of momenta, (or number of derivatives, in the coordinate
representation) of all the lines coming to the given vertex.

It is easy to see that formula (92) is insufficient to evaluate the
renormalizability of the given QFT or QG model. In addition to this formula,
there is the simple topological relation

$\displaystyle L=I-V+1$ (93)

valid for all the relevant diagrams.

Before going on to consider concrete models of QG, let us make the following
observation. The diagrams in quantum gravity, which we intend to analyze, have
external lines of the field $h_{\mu\nu}$ only, but there are internal lines of
both $h_{\mu\nu}$ and the Faddeev-Popov ghosts. However, with the modified
definitions of the ghost actions (33), the values of $r_{I}$ are the same for
both kinds of quantum fields. E.g., in quantum GR, in both cases we have
$r_{I}=2$, in fourth-derivative gravity in both cases $r_{I}=4$, in the
polynomial models $r_{I}=2N+4$. Finally, in the nonlocal QG models (39), both
$r_{I}$ and $K_{V}$ are infinite for both metric and ghosts. Then the use of
the combination of (92) and (93) is not possible. However, we shall see how to
deal with this special case using the topological relation (93) alone. For a
while, we assume that in all models of interest, $r_{I}$ are identical for the
quantum metric and the ghosts.

###  7.1 Power counting in quantum gravity based on GR

As the first step, consider power counting in quantum GR, where with $r_{I}=2$
for all internal lines. The vertices coming from the Einstein-Hilbert term
have $K_{EH}=2$. If we include the cosmological constant term, there are also
vertices $K_{\Lambda}=0$. However, looking only for strongest divergences, at
first we consider only the diagrams with $K_{V}=2$ vertices. Then (92),
together with the topological relation (93), yields

$\displaystyle\omega(G)+d(G)\,=\,2I-4V+4+2V\,=\,2I-2V+4\,=\,2+2L.\mbox{\qquad}$
(94)

The last result clearly shows that the QG based on GR is non-renormalizable.
At one-loop $L=1$ and the logarithmic divergences with $\,\omega(G)=0\,$ have
$\,d(G)=4$. Taking into account the diffeomorphism invariance, this indicates
towards the counterterms repeating the covariant structures included in the
fourth-derivative action (30). Indeed, at the one-loop order, there are
counterterms of the Einstein-Hilbert form $\sim\int\sqrt{-g}R$ with $d(G)=2$
but with quadratic divergences only, since $\omega(G)=2$.

The logarithmic divergences of this type are also possible, but only if we
introduce cosmological constant term. If there is one vertex with
$K_{\Lambda}=0$, the diagram produces the logarithmic divergence with two
derivatives. Assuming the covariance, this means an Einstein-Hilbert type
counterterm. With two such vertices, we meet a logarithmic divergence without
derivatives, i.e., with $d(G)=0$. In one of the next sections, we confirm
these conclusions by direct calculations and also analyze the gauge-fixing and
parametrization dependence of the one-loop counterterms.

One can rewrite the one-loop divergences using the relations

$\displaystyle C^{2}=E_{4}+2W\,\,\,\big{(}\mbox{where}\quad
W=R_{\mu\nu}^{2}-\textstyle{\frac{1}{3}}R^{2}\big{)},\quad E_{4},\quad
R^{2},\quad\square R.$ (95)

We know that $E_{4}$ and $\square R$ are surface terms, which do not affect
the dynamics of the theory, and the other two terms vanish on the classical
equations of motion, when $R_{\mu\nu}=0$. Thus, the one-loop $S$-matrix in the
pure quantum GR (pure means without matter contents) is finite. In the
presence of matter this feature doesn’t hold hove ; dene , but let us concern
only pure QG.

In the two-loop order $L=2$. According to Eq. (94), the logarithmic
divergences without $K_{\Lambda}$ vertices have dimension six. A complete list
of the corresponding terms has been elaborated in the works on the conformal
anomaly in six spacetime dimensions Bonora . This list includes

$\displaystyle\Sigma_{1}=R_{\mu\nu}R^{\mu\alpha}R_{\alpha}^{\nu}$
$\displaystyle\Sigma_{2}=(\nabla_{\lambda}{R}_{\mu\nu\alpha\beta})^{2}$
$\displaystyle\Sigma_{3}={R}_{\mu\alpha\nu\beta}\nabla^{\mu}\nabla^{\nu}{R}^{\alpha\beta}$
$\displaystyle\Sigma_{4}={R}_{\mu\nu}{R}^{\mu\lambda\alpha\beta}{R}^{\nu}\,_{\lambda\alpha\beta}$
$\displaystyle\Sigma_{5}=R^{\mu\nu}\,_{\alpha\beta}R^{\alpha\beta}\,_{\lambda\tau}R^{\lambda\tau}\,_{\mu\nu}$
$\displaystyle\Sigma_{6}=R^{\mu}\,_{\alpha}\,{}^{\nu}\,_{\beta}R^{\alpha}\,_{\lambda}\,{}^{\beta}\,_{\tau}R^{\lambda}\,_{\mu}\,{}^{\tau}\,_{\nu}$
$\displaystyle\Sigma_{7}=(\nabla_{\lambda}{R}_{\mu\nu})^{2}$
$\displaystyle\Sigma_{8}={R}_{\mu\nu}{\operatorname{\square}}{R}^{\mu\nu}$
$\displaystyle\Sigma_{9}=(\nabla_{\mu}{R})^{2}$
$\displaystyle\Sigma_{10}={R}{\operatorname{\square}}{R}$
$\displaystyle\Sigma_{11}=(\nabla_{\alpha}{R}_{\mu\nu})\nabla^{\mu}{R}^{\nu\alpha}$
$\displaystyle\Sigma_{12}=R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}R$
$\displaystyle\Sigma_{13}=R_{\mu\nu}R_{\alpha\beta}{R}^{\mu\alpha\nu\beta}$
(96)

as well as the set of surface terms,

$\displaystyle\Xi_{1}={\operatorname{\square}}^{2}R\quad\qquad\Xi_{2}=\Box
R^{2}_{\mu\nu\alpha\beta}\qquad\Xi_{3}=\Box
R^{2}_{\mu\nu}\qquad\quad\Xi_{4}=\Box R^{2}$
$\displaystyle\Xi_{5}=\nabla_{\mu}\nabla_{\nu}\big{(}R^{\mu}\,_{\lambda\alpha\beta}R^{\nu\lambda\alpha\beta}\big{)}\qquad\Xi_{6}=\nabla_{\mu}\nabla_{\nu}\big{(}R_{\alpha\beta}R^{\mu\alpha\nu\beta}\big{)}$
$\displaystyle\Xi_{7}=\nabla_{\mu}\nabla_{\nu}\big{(}R_{\alpha}^{\mu}R^{\nu\alpha}\big{)}\qquad\qquad\quad\quad\,\,\Xi_{8}=\nabla_{\mu}\nabla_{\nu}\big{(}RR^{\mu\nu}\big{)},$
(97)

satisfying the identity

$\displaystyle\Xi_{2}-4\Xi_{3}+\Xi_{4}-4\Xi_{5}+8\Xi_{6}+8\Xi_{7}-4\Xi_{8}\,=\,0.$
(98)

All these structures can show up in the two-loop divergences, but only two of
these terms, namely, $\Sigma_{5}$ and $\Sigma_{6}$ are critically important
because they do not vanish on shell. The two-loop calculation were done in
gosa  and confirmed in ven  by using another calculational approach. The
results confirmed the non-zero coefficient of $\Sigma_{5}$. The conclusion is
that there are no miracles and the theory of QG based on GR is non-
renormalizable.

Within the standard perturbative approach, the non-renormalizability means the
theory has no predictive power. With every new order of the loop expansion,
there are new types of local covariant divergences, with the growing number of
derivatives of the metric. And every time when a new type of counterterm is
introduced, it is necessary to fix the renormalization condition. Each of
these conditions requires making a measurement and using its result to fix the
value of the corresponding parameter. In quantum GR, this sequence of
operations is formally infinite. Thus, before making a single prediction, it
is necessary to use an infinite amount of experimental data.

What are the possible ways out of this situation? The main options are as
follows.

1\. Change standard perturbative approach to something else. The reader can
consult other Sections of our Handbook, to see how the problem is solved in
the framework of non-perturbative approaches, superstring theory, etc. Let us
say that there are many interesting options, but their consistency and the
relation to the QG program are not completely clear, in all cases.

2\. Restrict the area of application of QG to the low-energy domain. The
reader can read about this possibility in the Section about effective QG. The
main problem with this approach is that the QG is initially supposed to be a
concept describing extreme high-energy Physics, with the typical energy scale
of the Planck order of magnitude. It is certainly important to know what
remains from the QG effects in the IR, but this does not reduce the importance
of formulating QG that would be applicable at high energies.

3\. Change the theory, i.e., start from the model different from GR, to
construct QG. This option is the mainstream direction in perturbative QG. It
is important that the problems we meet in this way, such as the problem of
nonphysical ghosts coming from higher derivative terms, persist in many non-
traditional approaches which we mentioned in the first point (see, e.g., the
discussion in ABSh2 ).

###  7.2 Power counting in fourth-derivative gravity models

The next example is the power counting in the fourth-derivative quantum
gravity (30). We assume the Faddeev-Popov procedure with the second-order
weight operator (32) and the modified action of ghosts (33). In this case, for
all modes of the gravitational perturbation $h_{\mu\nu}$ and ghosts, we have
$r_{l}=4$, while the vertices $K_{V}$ include $K_{4d}$, $\,K_{EH}$, and
$\,K_{\Lambda}$.

Let us denote $n_{4d}$ the number of vertices with fourth power of momenta,
$n_{EH}$ the one with two, and $n_{\Lambda}$ – with zero power of momenta.
Then

$\displaystyle n_{4d}+n_{EH}+n_{\Lambda}=V\quad\mbox{and}\quad
n_{4d}K_{4d}+n_{EH}K_{EH}+n_{\Lambda}K_{\Lambda}=\sum_{V}K_{V}.\qquad$ (99)

The general expression (92), together with the topological relation (93), give
the following result:

$\displaystyle\omega(G)+d(G)\,=\,4-2n_{EH}-4n_{\Lambda}\,.$ (100)

As a starting point, consider the diagrams with the strongest divergences,
where all of the vertices are of the $K_{4d}$ type, i.e., $V=n_{4d}$. In this
case, (100) means that the logarithmic divergences have $d(G)=4$. Taking into
account locality and covariance arguments, the possible counterterms are of
the $C^{2}$, $R^{2}$, $E_{4}$ and $\square R$ types. This means, in all loop
orders the divergences have the same form as the fourth-derivative terms in
the classical action (30). Then, for $n_{4d}=V-1$ and $n_{EH}=1$, we obtain
$d(G)=2$, corresponding to the counterterm linear in $R$. Finally, for
$n_{EH}=2$ and $V-2=n_{4d}$, or for $n_{\Lambda}=1$ and $V-1=n_{4d}$, there is
a counterterm with $d(G)=0$, which is the cosmological constant. Thus, the
theory under consideration is multiplicatively renormalizable. In the next
sections, we shall see that this does not mean that the theory is completely
consistent, as there is a massive nonphysical ghost in the spectrum and the
subsequent problems with quantum unitarity and even with the stability of
classical solutions.

For the particular case of the general model (30) without dimensional
parameters, the classical action has a global conformal symmetry under the
transformation $g_{\mu\nu}\to g_{\mu\nu}e^{2\lambda}$, with $\lambda=const$.
The power counting (100) can be perfectly well applied in this case, yielding
$\omega(G)+d(G)=4$. This means that the theory is also multiplicatively
renormalizable. The disadvantage of this model of gravity is that there is no
automatic Einstein limit in the low-energy domain. Let us remember that such a
limit is one of the main conditions of consistency of any model which
generalizes or modifies GR, so this situation should be seen as a problem of
the model.

We note, by passing, that one can start from such a theory of globally
conformal theory, coupled to a massless scalar field. At the quantum level,
the loop corrections to the scalar potential may produce such effective
potential that the global conformal symmetry is dynamically broken, producing
GR in the low-energy limit. This idea resurrected several times in the
literature (see, e.g., BuchWeyl ; induce ; Agravity ) in different QG
frameworks and, in general, looks attractive and promising. Unfortunately, the
real deal is that, in the IR one has to break the global conformal symmetry.
And then, in the broken phase, we come back to the massive ghost problems and
to the related issue of instabilities, as it will be discussed in the
subsequent section 8.

Another particular case, which is instructive to consider, is the
$R+R^{2}$-gravity, which is model (30) without the $C^{2}$-term. As we saw in
the previous sections, in this model the traceless component of the metric
${\bar{h}}_{\mu\nu}$ has $1/k^{2}$ propagator, while the scalar mode has a
propagator behaving as $1/k^{4}$ in the UV. Furthermore, there are vertices
$K_{4d}$ connecting all these modes. It is easy to check that the power
counting in this model is dramatically different from that in the general
fourth-derivative model. The theory is non-renormalizable and the power
counting is even much worse than in quantum GR.

The next example is the model (30) without the $R^{2}$ term. This particular
model may be interesting since the fourth-derivative part of the action
possesses local conformal symmetry. This symmetry is softly broken by the
Einstein-Hilbert and cosmological terms. The expression “soft breaking” means
that the symmetry does not hold in the terms with dimensional parameters. Can
it be that the softly broken conformal symmetry “saves” the power counting in
this model? The answer is certainly negative. The propagator of the traceless
mode of the metric, ${\bar{h}}_{\mu\nu}$, in this case, has the UV behavior
$1/k^{4}$, and that of the scalar mode the different UV behavior, $\propto
1/k^{2}$, due to the presence of the $R$-term. At the same time, there are
$K_{4d}$ vertices that link all of the modes, and hence the power counting is
qualitatively the same as in the previous $R+R^{2}$ case. The theory is not
renormalizable.

The situation is much more complicated if the original theory has the pure
Weyl-squared term in the Lagrangian, i.e., possesses local conformal symmetry.
The theoretical proof of the renormalizability in conformal theories exists
only for the quantum theory in curved space (semiclassical gravity) tmf84
and, in the literature, there is no proof for conformal QG. On the other hand,
at the one-loop order this theory is renormalizable, as was shown by direct
calculations frts82 ; antmot  and confirmed in Weyl . However, it is expected
that this model is not renormalizable at higher loops because of the conformal
anomaly. But, in the situation when this expectation is not supported by
direct higher-loop calculations or the analysis similar to tmf84 , the
question should be regarded as open.

###  7.3 Power counting in the polynomial theory

Consider power counting in the polynomial model (34). As before, we assume
that the Faddeev-Popov quantization is done with the weight operator (37) and
the correspondingly modified ghost term (33). In the general case, both
coefficients of the highest derivative terms $\omega_{N,R}$ and $\omega_{N,C}$
are non-zero. Then the propagators of both metric perturbations $h_{\mu\nu}$
and ghosts have the UV behavior $\propto k^{-4-2N}$, i.e., we have
$r_{l}\equiv 4+2N$. For the vertices, the generalization of Eq. (99) is

$\displaystyle\sum_{V}K_{V}\,=\,n_{4+2N}K_{4+2N}+n_{2+2N}K_{2+2N}+n_{2N}K_{2N}+\,.\,.\,.\,+n_{EH}K_{EH}+n_{\Lambda}K_{\Lambda}\,,$
$\displaystyle V\,=\,n_{4+2N}+n_{2+2N}+n_{2N}+\,.\,.\,.\,+n_{EH}+n_{\Lambda},$
(101)

where $K_{4+2N}=4+2N$, $K_{2+2N}=2+2N$, . . . $K_{EH}=2$, $K_{\Lambda}=0$, and
$n_{4+2N}$, $n_{2+2N}$, $n_{2N}$, . . . , $n_{EH}$, and $n_{\Lambda}$ are the
numbers of the respective vertices.

Consider the diagrams with the strongest divergences. This means $V=n_{4+2N}$,
such that the other types of vertices are absent. Then the power counting
becomes quite simple, because of $\,\sum_{V}K_{V}=V(4+2N)$. The expression
(92) becomes

$\displaystyle\omega(G)+d(G)$ $\displaystyle=$
$\displaystyle(4-4-2N)I\,-\,4V\,+\,4\,+\,V(4+2N)$ (102)
$\displaystyle\,\,=\,\,4\,+\,2N(V-I)\,=\,4\,+\,2N(1-L),$

where we used the topological relation (93) in the form $V-I=1-L$. It is easy
to see that the power counting in the four-derivative model (100) is a
particular case, corresponding to $N=0$. Thus, we assume $N\geq 1$.

According to (102), the sum $\omega(G)+d(G)$ decreases with a growing number
of loops $L$. The strongest divergences occur for $L=1$, when the
aforementioned sum equals 4 and the logarithmic divergences correspond to the
one-loop counterterms with, at most, four derivatives. Taking the covariance
and locality, this means that the one-loop counterterms are of the $C^{2}$,
$R^{2}$, $E_{4}$ and $\square R$ types. In other words, all the terms with six
and more derivatives do not need to be renormalized at the one-loop order.
But, if there just one vertex with two less derivatives, i.e., $n_{2+2N}=1$
and $n_{4+2N}=V-1$, then we meet only the Einstein-Hilbert type divergence.
And finally, in the case $n_{2+2N}=2$ or $n_{2N}=1$, the unique divergence is
that of the cosmological constant.

The features of the $L=1$ approximation that we listed above, do not depend on
$N\geq 1$. Starting from $L\geq 2$, the structure of divergences starts to
depend on the value of $N$. In particular, for $N\geq 3$, according to (102),
the second- and higher-loop diagrams are all finite. This creates a situation
when the one-loop beta functions are the exact ones. In the case of $N=2$,
there are two-loop divergences, but only of the cosmological constant type.
Finally, for $N=1$, there are two-loop divergences of the Einstein-Hilbert and
the cosmological constant type and also three-loop divergences, but only of
the cosmological constant type.

All in all, a theory with both $\omega_{N,R}\neq 0$ and $\omega_{N,C}\neq 0$,
is superrenormalizable. In contrast, in the degenerate case, when only one of
these coefficients is zero, the theory is non-renormalizable. The situation is
essentially similar to what we discussed above for two similar four-derivative
models of QG.

Finally, the power counting in the QG models (39) with non-polynomial
(typically exponential) functions $\Phi$ and $\Psi$ cannot be performed on the
basis of the formula (92), because both the number of derivatives in the
vertices and the parameter $r_{l}$ are infinite. However, if the condition
(41) is satisfied, the evaluation can be done using only the topological
relation CountGhosts . Let us assume, for simplicity, that the functions are
those from (42). Then, in Euclidean signature, each propagator brings the
exponential of $-\alpha k^{2}$ and each vertex contributes with the
exponential of $+\alpha k^{2}$. This means, if the number of vertices is
different from the number of propagators, the last integral in the given
diagram will be either strongly divergent, or completely convergent. Taking
this into account, from the relation (93) we learn that the divergences are
present only in the one-loop diagrams and that these divergences have fourth
powers of momenta. Thus, the power counting in the theories of this class is
the same as in the polynomial models with $N\geq 3$.

##  8 Massive ghosts in higher-derivative models

Previously we saw that quantum GR, based on the Einstein-Hilbert action, is a
non-renormalizable theory. On the other hand, by adding the general fourth-
derivative terms, we arrive at the model providing multiplicative
renormalizability. Another strong argument for including fourth-derivative
terms is that they are required for the renormalizability of the semiclassical
theory, when gravity is an external field birdav ; book ; ParToms ; OUP . And
this is something one cannot disregard. The point is that the concepts of QFT
from one side and of the curved space from another, are well-established and,
to a great extent, verified by experiments and observations. Thus, if QFT in
curved space produces the four-derivative terms in the action, we have to
admit that these terms are there. The problem concerns not exactly UV
divergences and formal renormalizability. Together with the logarithmic
divergences there are always logarithmic nonlocal form factors. In the IR such
a form factor behaves, effectively, as a constant GWprT ; OUP . This means, if
we do not include fourth derivatives into the action, they will come there
anyway as legitimate corrections coming from quantum matter fields. Therefore,
independent on which approach to QG we choose, it makes sense to include
fourth derivatives terms into the gravitational action and check out the
physical consequences of this inclusion.

At the classical level and in the low-energy domain, the fourth derivative
terms look irrelevant because the coefficients of these terms in the action
(30) are just a numbers while the coefficient of the $R$-term is $1/G$, that
is $M_{P}^{2}\approx 10^{38}GeV$. As a consequence, until the metric
derivatives, in the momentum representation, do not have Planck-order
frequencies, the fourth-derivative terms cannot compete with the Einstein-
Hilbert term. This situation is usually called a Planck suppression. So, from
the first sight the deal is perfect: including the fourth derivatives terms we
get a renormalizable QG (and semiclassical too!), while all classical
solutions remain the same as in GR and one can enjoy the well-verified
gravitational theory.

Unfortunately, even if theory (30) is, formally, multiplicatively
renormalizable, it does not make it consistent at either the quantum or even
classical levels. As we shall see in brief, the spectrum of this model
includes states that have negative kinetic energy. These states, or particles,
are called massive nonphysical ghosts. The presence of ghosts violates the
unitarity of the theory at the quantum level. Worse than that, in the presence
of massive ghosts or, more generally, ghost-like states, classical solutions
of the theory can be unstable with respect to the metric perturbations.
Qualitatively, the same situation takes place not only in the fourth-
derivative theory but in all polynomial models of quantum gravity.

The problem of ghosts is certainly the main obstacle for building a consistent
QG theory. For this reason, we consider this problem here. We shall closely
follow OUP . The interested reader can address this book for more details, or
go directly to the original works, such as the reviews Woodard-review  on the
Ostrogradsky instabilities, or the papers HD-Stab  and PP ; Salvio-19 ;
LinNonlin  for the approach we follow to explore and interpret the
instabilities caused by massive ghosts.

###  8.1 What means a massive ghost

Consider the propagator of the transverse and traceless part of the metric
(89) or the one of the scalar, gauge-invariant mode (90). For simplicity, we
can set the cosmological constant to be zero, then the fourth-derivative
action (30) becomes

$\displaystyle S\,=\,-\,\int
d^{4}x\sqrt{-g}\,\Big{\\{}\,\,\frac{1}{\lambda}\,R_{\mu\nu}^{2}\,+\,\frac{\omega-1}{3\lambda}\,R^{2}+\frac{1}{\kappa^{2}}\,R\,\Big{\\}}.$
(103)

Formulas (89) and (90) change accordingly, with $\Phi$ and $\Psi$ becoming
constants.

We can consider at once both scalar and tensor modes, because the formulas are
similar. For the definiteness sake, consider the tensor mode
$\,h={\bar{h}}^{\bot\bot}_{\mu\nu}$, which is not affected by the $R^{2}$
term. Using the expansions (50) and (3), the action of this mode becomes

$\displaystyle S^{(2)}_{tensor}=\int
d^{4}x\Big{\\{}-\frac{1}{4\lambda}\big{(}\operatorname{\square}h\big{)}^{2}-\frac{1}{4\kappa^{2}}h\operatorname{\square}h\Big{\\}}=-\frac{1}{4\lambda}\,\int
d^{4}x\,h\big{(}\operatorname{\square}+m_{2}^{2}\big{)}\operatorname{\square}h,\mbox{\quad}\mbox{\quad}$
(104)

where $m_{2}^{2}=\lambda/\kappa^{2}$ is the mass of the mode that is called a
tensor ghost, a massive tensor ghost or higher-derivative ghost. The reason
for this exotic name is that the Euclidean propagator of the spin-2 mode in
this theory can be cast in the form

$\displaystyle G_{2}(k)$ $\displaystyle\,\,\propto\,\,$
$\displaystyle\frac{1}{m_{2}^{2}}\Big{(}\frac{1}{k^{2}}\,-\,\frac{1}{k^{2}+m_{2}^{2}}\Big{)}\,\,\hat{P}^{(2)}.$
(105)

The negative sign of the second term indicates that the corresponding mode is
not a usual particle. In fact, we have not one but two degrees of freedom of
the tensor field. One of these degrees of freedom has positive kinetic energy
and zero mass, and it corresponds to the first term in Eq. (105). The second
degree of freedom has the mass $m_{2}$ and corresponds to the second term in
(105). As we shall see in what follows, its kinetic energy is negative, and,
for this reason, it is called a ghost.

The separation of the two degrees of freedom can be most simply explored by
using an auxiliary field $\Phi$ (see, e.g., Cremin , a more detailed
discussion in E.Alvarez-QG  and more general formulations in fREddi ).
Consider the Lagrangian density

$\displaystyle{\mathcal{L}}^{\prime}$ $\displaystyle=$
$\displaystyle-\,\frac{m_{2}^{2}}{4\lambda}\,h\operatorname{\square}h\,+\,\lambda\phi^{2}-\phi\operatorname{\square}h.$
(106)

The Lagrange equation for $\phi$ can be solved as
$\,\phi=\frac{1}{2\lambda}\operatorname{\square}h$. Substituting this
expression back into (106), we arrive at Eq. (104), which shows the dynamical
equivalence of the models (104) and (106).

The two fields $\Psi$ and $h$ in (106) are not factorized. To improve on this
issue, we change to the new variables $\theta$ and $\psi$,

$\displaystyle
h\,=\,\frac{\sqrt{2\lambda}}{\,m_{2}}(a_{1}\theta+a_{2}\psi),\qquad\phi\,=\,\frac{\sqrt{2\lambda}}{\,m_{2}}a_{3}\psi,$
(107)

where the unknown coefficients $a_{1,2,3}$ should provide the separation of
the modes and also the standard coefficients $\frac{1}{2}$ or $-\frac{1}{2}$
in the kinetic terms. A small algebra shows that the condition $a_{2}+a_{3}=0$
is necessary to separate the variables and, also, the condition
$a_{1}=a_{2}=1$ is required to provide standard normalization of the kinetic
terms. Then, in the new variables, the Lagrangian (106) becomes

$\displaystyle{\mathcal{L}}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\eta^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta\,-\,\frac{1}{2}\,\Big{(}\eta^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi-
m_{2}^{2}\psi^{2}\Big{)}.$ (108)

We can conclude that the theory (30) has healthy tensor massless degrees of
freedom $\theta$ and, on top of this, tensor massive degrees of freedom $\psi$
with negative kinetic energy, called nonphysical massive ghost.

###  8.2 Classification of ghosts and tachyons

Consider a basic classification of ghosts and tachyons following GWprT ; OUP .
The general action of a free second-order field $\,h(x)=h(t,{{\bf r}})\,$ can
be written as

$\displaystyle S(h)$ $\displaystyle=$ $\displaystyle\frac{s_{1}}{2}\int
d^{4}x\,\big{\\{}\eta^{\mu\nu}\partial_{\mu}h\partial_{\nu}h-s_{2}m^{2}h^{2}\big{\\}}$
(109) $\displaystyle\quad=\,\,\frac{s_{1}}{2}\int
d^{4}x\,\big{\\{}{\dot{h}}^{2}-(\nabla h)^{2}-s_{2}m^{2}h^{2}\big{\\}}\,.$

Here $s_{1}$ and $s_{2}$ are sign factors $\pm 1$ for different types of
fields. In what follows, we consider all four combinations of these signs.

It is useful to perform the Fourier transform in the space variables,

$\displaystyle h(t,{{\bf r}})\,=\,\frac{1}{(2\pi)^{3}}\int d^{3}k\,e^{i{{\bf
k}}\cdot{{\bf r}}}\,h(t,{{\bf k}}).$ (110)

In a free theory, one can consider the dynamics of each component $\,h\equiv
h(t,{{\bf k}})\,$ separately. Such a dynamics is defined by the action

$\displaystyle S_{{\bf k}}(h)$ $\displaystyle\,=\,$
$\displaystyle\frac{s_{1}}{2}\int dt\,\big{\\{}{\dot{h}}^{2}-{{\bf
k}}^{2}h^{2}-s_{2}m^{2}h^{2}\big{\\}}\,=\,\frac{s_{1}}{2}\int
dt\,\big{\\{}{\dot{h}}^{2}-m_{k}^{2}h^{2}\big{\\}},\mbox{\quad}$ (112)
$\displaystyle\mbox{where}\qquad{{\bf k}}^{2}={{\bf k}}\cdot{{\bf
k}}\qquad\mbox{and}\qquad m_{k}^{2}=s_{2}m^{2}+{{\bf k}}^{2}\,.$

The properties of the field are defined by the signs of $s_{1}$ and $s_{2}$.
The possible options can be classified as follows:

i) Normal healthy field corresponds to $s_{1}=s_{2}=1$. The kinetic energy of
the field is positive and the equation of motion has the oscillatory form,

$\displaystyle{\ddot{h}}+m_{k}^{2}h\,=\,0\,,$ (113)

with the usual periodic solutions.

ii) A tachyon has $s_{1}=1$ and $s_{2}=-1$. The classical dynamics of tachyons
is described in the literature, e.g., in Sudarshan ; Terletsky , but, for our
purposes, it is sufficient to give only a basic survey. For a relatively small
momentum $k=|{{\bf k}}|$, there is $m_{k}^{2}<0$ in Eq. (112), and the
equation of motion is

$\displaystyle{\ddot{h}}-\omega^{2}h\,=\,0\,,\qquad\omega^{2}=\left|m_{k}^{2}\right|\,,$
(114)

with exponential solutions

$\displaystyle h\,=\,h_{1}e^{\omega t}\,+\,h_{2}e^{-\omega t}\,.$ (115)

However, if such a particle moves faster than light, the solution is of the
normal oscillatory kind, indicating that such a motion is “natural” for this
kind of particle.

iii) A massive ghost has $s_{1}=-1$ and $s_{2}=1$. It is not a tachyon,
because $m_{k}^{2}\geq 0$. In this case, the kinetic energy of the field is
negative, but the Lagrange equation (leaving aside its derivation from the
least action principle) has a normal oscillatory equation (113).

iv) A tachyonic ghost has $\,s_{1}=s_{2}=-1$. For relatively small ${{\bf
k}}^{2}$, one meets $m_{k}^{2}<0$. The kinetic energy is negative and the mass
is imaginary. So, along with the problems typical for the ghosts, the free
wave solutions are exponential, as in (115). One can find a discussion of the
implication of tachyonic ghosts in GWprT .

###  8.3 Massive ghosts in the fourth-order model

Let us come back to the fourth-order gravity model (103). Consider the free
tensor modes in the theory (104) and change to the momentum representation,

$\displaystyle\operatorname{\square}h\,\,\,=\,\,\,{\ddot{h}}-\Delta
h\,\,\,\longrightarrow\,\,\,{\ddot{h}}+{{\bf k}}^{2}h.$ (116)

Here ${{\bf k}}$ is the wave vector of an individual mode. It is important
that, owing to the presence of both massless and massive modes, the standard
massless dispersion relation between the frequency and the wave vector does
not hold in this case. The Lagrange function of the wave with fixed ${{\bf
k}}$ can be obtained from (104):

$\displaystyle L$ $\displaystyle=$
$\displaystyle-\,\frac{1}{4\lambda}\,\big{(}{\ddot{h}}+{{\bf
k}}^{2}h\big{)}^{2}\,-\,\frac{1}{4\kappa^{2}}\,h\big{(}{\ddot{h}}+{{\bf
k}}^{2}h\big{)}.$ (117)

The Lagrange equation for $L=L(q,\,{\dot{q}},\,{\ddot{q}})$ has the form

$\displaystyle\frac{\partial L}{\partial q}\,-\,\frac{d}{dt}\frac{\partial
L}{\partial{\dot{q}}}\,+\,\frac{d^{2}}{dt^{2}}\frac{\partial
L}{\partial{\ddot{q}}}\,=\,0$ (118)

and the energy can be easily obtained in the form

$\displaystyle E\,=\,{\dot{q}}\,\Big{(}\frac{\partial
L}{\partial{\dot{q}}}\,-\,\frac{d}{dt}\frac{\partial
L}{\partial{\ddot{q}}}\Big{)}\,+\,{\ddot{q}}\,\frac{\partial
L}{\partial{\ddot{q}}}\,-\,L\,.$ (119)

In our case (117), this formula gives the energy of the individual wave with
momentum ${{\bf k}}$,

$\displaystyle E\,=\,\frac{1}{4\lambda}\,\Big{(}2h^{\left(\textsc{\tiny
III}\right)}{\dot{h}}-{\ddot{h}}^{2}\Big{)}\,+\,\Big{(}\frac{1}{4\kappa^{2}}+\frac{{{\bf
k}}^{2}}{2\lambda}\Big{)}\,{\dot{h}}^{2}\,+\,\Big{(}\frac{{{\bf
k}}^{2}}{4\kappa^{2}}+\frac{{{\bf k}}^{4}}{4\lambda}\Big{)}\,h^{2}\,.$ (120)

This formula provides some information about the fourth-derivative theory. We
can separate it into the following points:

i) In the limit $\lambda\to\infty$, the remaining expression for the energy is
positively defined, as it should be for Einstein’s gravity.

ii) The fourth time derivative terms are given by the first summand in (120).
It is easy to see that this term is not positively defined. This sign
indefiniteness should be expected, as a direct consequence of the presence of
the massive nonphysical ghost.

iii) In the model under discussion, the low-energy limit (IR) means

$\displaystyle{\ddot{h}}^{2}\ll{{\bf
k}}^{2}{\dot{h}}^{2}\qquad\mbox{and}\qquad|hh^{\left(\textsc{\tiny
III}\right)}|\ll{{\bf k}}^{2}{\dot{h}}^{2}.$ (121)

In this case, the first indefinite term (with fourth derivatives) in (120) is
small, and the sign of the energy is defined by the second term, providing a
relevant constraint on the action (30). The positivity of the theory in this
limit does not depend on fourth time derivatives. However, the kinetic energy
can be still unbounded from below for the negative coupling $\lambda<0$. Owing
to the violated dispersion relation between the wave vector ${{\bf k}}$ and
the time derivatives, it is possible to have a large ${{\bf k}}^{2}$ with the
conditions (121) satisfied. Thus, the sign of the coupling $\lambda$ in the
action (30) should be positive, as it was always assumed in the literature,
e.g., in the classical works Stelle77 ; stelle78  and frts82 .

The equation for tensor perturbations can be derived from (118),777The more
general equation describing the dynamics of tensor perturbations on the
cosmological background, will be discussed below; see Eq. (130).

$\displaystyle h^{\left(\textsc{\tiny IV}\right)}+2{{\bf
k}}^{2}{\ddot{h}}+{{\bf
k}}^{4}h\,+\,\frac{\lambda}{16\pi\kappa^{2}}\,\big{(}{\ddot{h}}+{{\bf
k}}^{2}h\big{)}\,=\,0.$ (122)

One can introduce the new notation,

$\displaystyle\frac{\lambda}{16\pi\kappa^{2}}=s_{2}m^{2},$ (123)

where $\,s_{2}={\rm sign}\,\lambda\,$ and $\,m^{2}>0$. Then Eq. (122) becomes

$\displaystyle\Big{(}\frac{\partial^{2}}{\partial t^{2}}+{{\bf
k}}^{2}\Big{)}\,\Big{(}\frac{\partial^{2}}{\partial
t^{2}}+m_{k}^{2}\Big{)}h\,=\,0\,,$ (124)

where $m_{k}^{2}$ was defined in (112). The solutions of the last equation can
be different, depending on the sign of $\lambda$ and, hence, that of $s_{2}$.
The general formula for the frequencies in $h\sim\exp{\\{\pm\omega t\\}}$ has
the form

$\displaystyle\omega_{1,2}$ $\displaystyle\approx$
$\displaystyle\pm\,i\big{(}{{\bf
k}}^{2}\big{)}^{1/2}\qquad\mbox{and}\qquad\omega_{3,4}\approx\pm\big{(}-\,m_{k}^{2}\big{)}^{-1/2}\,.$
(125)

For a positive $\lambda$, there are only imaginary $\omega^{\prime}s$ and,
hence, oscillator-type solutions. In contrast, for $\lambda<0$, we have
$s_{2}=-1$ and the roots $\omega_{3,4}$ are real, since, in this case,
$-m_{k}^{2}>0$ for sufficiently small ${{\bf k}}^{2}$. Indeed, the first
couple of roots corresponds to the massless graviton, and the second couple to
the massive particle. According to the classification presented above, this
particle is a ghost for $\,\lambda>0$ and it is a tachyonic ghost for
$\lambda<0$.

Finally, we conclude that the model (30) has ghosts (and maybe tachyonic
ghosts) owing to the presence of fourth derivatives.

###  8.4 Massive ghosts in the six and higher-order models

If the number of derivatives in the polynomial model of six or greater, the
structure of the propagator is more complicated than in the fourth-order
theory. First of all, the massive poles can be either real or complex. In the
last case, they emerge in the complex conjugate pairs. There is an important
theorem about the structure of the propagator in the case of real poles
highderi . In this case, instead of (105), we meet

$\displaystyle
G_{2}(k)\,=\,\frac{A_{0}}{k^{2}}+\frac{A_{1}}{k^{2}+m_{1}^{2}}+\frac{A_{2}}{k^{2}+m_{2}^{2}}+\,.\,.\,.\,+\frac{A_{N+1}}{k^{2}+m_{N+1}^{2}},$
(126)

where the squares of the masses $m_{j}^{2}$ are real. Assuming that there is
the following hierarchy of the masses:

$\displaystyle 0<m_{1}^{2}<m_{2}^{2}<m_{3}^{2}<\,.\,.\,.\,<m_{N+1}^{2},$ (127)

one can prove that the signs of the coefficients $A_{j}$ alternate, i.e.,
${\rm sign}\ [A_{j}]=-\;{\rm sign}\ [A_{j+1}]$. This feature means one cannot
choose the theory in a such a way that all degrees of freedom instead of the
heaviest one are normal particles and the mass of the ghost is infinitely
large.

The complex poles are also possible, but their detailed discussion lies beyond
the scope of this review. Interesting hints about the role of real and complex
poles come from the analysis of the Newtonian limit and the bending of light
in the polynomial models Modesto2014 ; Giacchini2016 ; ABSh2 .

###  8.5 On the quantum consistency and stability of classical solutions

We have seen that the theory with fourth derivatives, typically, has a massive
ghost or a tachyonic ghost, and this conclusion can be extended to the
polynomial models with more than four derivatives. So, it is interesting to
understand what the ghost means, from the physical viewpoint. Let us give just
a brief qualitative description of the situation.

A particle with negative kinetic energy tends to the minimum of the action and
therefore tends to achieve a maximal speed. If such a particle is free, it
cannot accelerate, as this would violate energy conservation. Hence, a free
ghost does not produce any harm to the environment, being isolated from it.
However, in the case when there is an interaction of a ghost with healthy
fields, the argument about energy conservation in a closed system does not
work. Since any physical system tends to the state with minimal action, a
ghost tends to accelerate, transmitting the extra positive energy to the
healthy fields interacting with it, in the form of the quantum or classical
emission of the corresponding particles. A systematic study of this situation
at the quantum level has been given by Veltman in Veltman-63 .

Since gravitons are massless and the metric-dependent gravitational theories
have non-polynomial interactions, a massive ghost always couples to an
infinite amount of gravitons. This fact may lead to dramatic consequences. The
energy conservation does not forbid a spontaneous creation of a massive ghost
from the vacuum, even in the flat Minkowski space. It is clear that such a
spontaneous creation of a ghost also implies that the corresponding amount of
positive energy should be released with the creation of massless gravitons. As
the mass of the ghost has the Planck magnitude, these gravitons have to
accumulate with the Planck energy density.

Assuming the existence of even a single real ghost, such a particle should
accelerate, emitting and scattering gravitons. The magnitude of the energy of
the ghost would increase, and hence the energy of the created and scattered
gravitons would increase too, without an upper bound for the emitted
gravitational energy. After a while, the ghost would acquire an infinite
amount of negative energy and start to emit an infinite amount of positive
energy. It is clear that if some objects of this sort would be around, we
would certainly know about it or, rather, we would feel it.

Thus, the main theoretical problem is to explain why this dramatic scenario
does not work. We have to say that, at the moment, there is no solution to
this important problem. The solution could be, e.g., an explanation of why
gravitons cannot agglomerate with the Planck density, but no mechanism for
this has been formulated. Obviously, we have to assume that some kind of
solution exists. Probably, it is related to the large mass of the massive
ghost in higher-derivative gravity.

The simplest way to preserve the unitarity is to admit the existence of a
ghost. But, as we described above, this leads to the physically inconsistent
output. Thus, one avoids ghosts by forming the in states only with gravitons.
Owing to the interactions, ghost wakes up from the vacuum and emerge in the
out states – that means the scattering matrix is not unitary Stelle77  and we
arrive at the contradiction.

Historically, the main efforts in solving the problem of ghosts was related to
the quantum aspects. In this respect we can start the list from the mentioned
paper by Veltman Veltman-63 , regardless it has nothing specific about quantum
gravity. Soon after the seminal work of Stelle Stelle77  with the proof of
renormalizability of the fourth-derivative QG, there were first works about
solving the problem of ghosts Tomboulis-77  and salstr . The main common idea
of these works was that the loop corrections to the propagator (105) transform
the unique massive pole into a pair of the complex conjugate poles, with the
positions of these poles being gauge fixing dependent antomb . In this case,
one can prove the unitarity of the $S$-matrix, violated by the presence of
massive ghosts Stelle77 . Unfortunately, it was shown Johnston , that definite
conclusion on this issue can be taken only on the basis of exact knowledge of
the dressed propagator of $h_{\mu\nu}$. E.g., the one-loop corrections or the
$1/N$ approximation are insufficient for solving the problem. It is
interesting that starting from the six-derivative theory, one can provide the
desirable features (pair of complex conjugate poles, for instance) already at
the tree-level and, also, prove that the loop corrections do not change this
structure. In this situation, one can use the optical theorem and prove the
unitarity of the $S$-matrix in the Lee-Wick approach Modesto-complex .

The problem with this solution of the ghost problem is that the Lee-Wick
approach assumes that the scattering occurs between the asymptotical states,
where in and out states describe the free particles. However, in gravity (and
especially in its quantum version) the notion of free massive particle is not
perfectly well defined, because any such particle produces a gravitational
field, starts to interact with it and, therefore, is not completely free.
Therefore, the definite resolution of the ghost problem by means of the
$S$-matrix does not look really promising, at least at the fundamental level.

What we can learn from all the quantum considerations is that the issues with
ghosts described above would become impossible in the presence of a natural
cut-off on the energy density of gravitons, such that this density never
achieves the Planck order of magnitude. Then the gravitons cannot agglomerate
to create a ghost and the $S$-matrix remains unitary. The problem is that
there is no theoretical mechanism for such a cut-off. One can say that this is
the main reason of why the problem of ghosts does not have a solution.

###  8.6 Stable solutions in the presence of massive ghosts

Another aspect of the problem of ghosts is related to the stability of
classical solutions. The most important issue related to massive ghosts is
whether their presence can be compatible with the stability of the classical
solutions of GR. As we mentioned above, in higher derivative models of gravity
one typically meets the Planck suppression. As a result, classical solutions
of GR represent high-quality approximations to the corresponding solutions in
the presence of the higher-derivative terms. This logic can be successfully
applied to the fourth-derivative theory and extended to the polynomial
theories (34), if we assume that all massive parameters are of the Planck
order of magnitude ABSh2 .

The excellence in the approximation of the solution of GR does not necessarily
mean the stability of this solution. In general, providing the stability of a
gravitational solution under arbitrary small perturbations (that do not have
the symmetry of solutions themselves) may be not simple even in GR, for some
gravitational backgrounds. Let us mention, for instance, the study of the
stability of the Schwarzschild solution in GR RW ; Zer  (see also
FrolovNovikov ). In the presence of $C^{2}$ term, one can expect that the same
solution will not be stable. Regardless of existing contradictions in the
literature, in general, this expectation is confirmed whitt ; myung . Owing to
the high level of technical difficulty, this case will not be discussed here.

From the technical side, it is much simpler to consider the stability of
classical cosmological solutions in the presence of fourth-order terms. The
advantage of this simpler case is that the physical interpretation of the
results is relatively explicit. The analysis of HD-Stab  and after that in PP
; Salvio-19 ; LinNonlin , was done for the fourth-derivative action (30) with
anomaly-induced semiclassical corrections (see, e.g., OUP  for the review). It
turns out that these corrections do not change the main result. We shall
explain this result omitting all technical details except the basic formulas.

The stability we need to explore is related to the presence of massive spin-2
ghost degrees of freedom, which means the transverse and traceless modes of
the metric perturbation on the homogeneous and isotropic, cosmological
background. According to the theory of cosmological perturbations (see, e.g.,
Dodelson ; Mukhanov ; RubakovGorbunov ), the background cosmological metric
with tensor perturbations is

$\displaystyle\mathrm{d}s^{2}\,=\,a^{2}(\eta)\big{[}\mathrm{d}\eta^{2}-\left(\gamma_{ij}+h_{ij}\right)\mathrm{d}x^{i}\mathrm{d}x^{j}\big{]},$
(128)

where $\eta$ is the conformal time, $a(\eta)$ corresponds to a background
cosmological solution, and we imposed the synchronous coordinate condition
$h_{\mu 0}=0$. Furthermore, for the sake of simplicity we consider the flat
space geometry $k=0$ (hence, $\gamma_{ij}=\eta_{ij}$) and set the cosmological
constant to vanish, $\Lambda=0$.

Since we are interested in the gravitational wave dynamics, it is sufficient
to retain only the traceless and transverse parts of $h_{ij}$, which are the
purely tensor modes, by imposing

$\displaystyle\partial_{i}\,h^{ij}=0\,,\qquad h_{kk}=0\,.$ (129)

As before, we do not need to write indices and set $h=h^{ij}$.

The Lagrange equation for $h(t)$, in terms of physical time, has the form GW-
Stab

$\displaystyle\frac{1}{3}h^{\left(\textsc{\tiny
IV}\right)}+2Hh^{\left(\textsc{\tiny III}\right)}+\left(H^{2}-\frac{\lambda
M_{\mathrm{{}_{P}}}^{2}}{16\pi}\right)\ddot{h}+\frac{2}{3}\left(\frac{1}{4}\frac{\nabla^{4}h}{a^{4}}-\frac{\nabla^{2}\ddot{h}}{a^{2}}-H\frac{\nabla^{2}\dot{h}}{a^{2}}\right)$
$\displaystyle\quad-\left(H\dot{H}+\ddot{H}+6H^{3}+\frac{3\lambda
M_{\mathrm{{}_{P}}}^{2}H}{16\pi}\right)\stackrel{{\scriptstyle.}}{{h}}+\left[\frac{\lambda
M_{\mathrm{{}_{P}}}^{2}}{16\pi}+\frac{4}{3}\left(\dot{H}+2H^{2}\right)\right]\frac{\nabla^{2}h}{a^{2}}$
$\displaystyle\quad-\left[24\dot{H}H^{2}+12\dot{H}^{2}+16H\ddot{H}+\frac{8}{3}H^{\left(\textsc{\tiny
III}\right)}+\frac{\lambda
M_{\mathrm{{}_{P}}}^{2}}{8\pi}\left(2\dot{H}+3H^{2}\right)\right]h=0\,.\mbox{\qquad}\mbox{\quad}$
(130)

The contribution of the fourth-derivative terms depends on the unique
parameter $\lambda$ from the action (30). The reason is that the Gauss-Bonnet
combinations do not affect the equations of motion and another invariant is
$R^{2}$, which contributes to the equation for the conformal factor $a(t)$,
but does not affect the propagation of the tensor mode.

The next step is to make the Fourier transformation for the space coordinates

$\displaystyle h_{\mu\nu}({{\bf r}},t)$ $\displaystyle=$
$\displaystyle\int\,\frac{d^{3}k}{(2\pi)^{3}}\,\,h_{\mu\nu}({{\bf
k}},t)\,\,e^{i{{\bf k}}\cdot{{\bf r}}}.$ (131)

One can treat the wave vector ${{\bf k}}$ as a constant and hence will be
interested only in the time evolution of the perturbation $h_{\mu\nu}({{\bf
k}},t)$. The validity of such a treatment is restricted to the linear
perturbations, but this is what we need now. In this way, the complicated
partial differential equation (130) are reduced to the much simpler ordinary
differential equation for each of the individual modes.

Using the notation $h=h(t,{{\bf k}})=h(t,k)$, the equation has the form

$\displaystyle h^{\left(\textsc{\tiny IV}\right)}+6Hh^{\left(\textsc{\tiny
III}\right)}+\left(3H^{2}-\frac{3\lambda
M_{\mathrm{{}_{P}}}^{2}}{16\pi}\right)\ddot{h}+\left(\frac{1}{2}\frac{k^{4}}{a^{4}}h+\frac{2k^{2}}{a^{2}}\ddot{h}+\frac{2k^{2}}{a^{2}}\,H\dot{h}\right)$
$\displaystyle\quad-\,3\left(H\dot{H}+\ddot{H}+6H^{3}+\frac{3\lambda
M_{\mathrm{{}_{P}}}^{2}H}{16\pi}\right)\stackrel{{\scriptstyle.}}{{h}}-\left[\frac{3\lambda
M_{\mathrm{{}_{P}}}^{2}}{16\pi}+4\left(\dot{H}+2H^{2}\right)\right]\frac{k^{2}}{a^{2}}h$
$\displaystyle\quad-\,\left[72\dot{H}H^{2}+36\dot{H}^{2}+48H\ddot{H}+8H^{\left(\textsc{\tiny
III}\right)}+\frac{3\lambda
M_{\mathrm{{}_{P}}}^{2}}{8\pi}\left(2\dot{H}+3H^{2}\right)\right]h=0\,,\qquad\qquad$
(132)

where $k=|{\bf k}|$ is the frequency of the massless field. Finally, the
initial conditions for the perturbations will be chosen according to the
quantum fluctuations of free fields. The spectrum is identical to that of a
scalar quantum field in Minkowski space birdav ,

$\displaystyle h(x,\eta)=h(\eta)\,e^{\pm i{\bf k}.{\bf r}}\,\,,\,\,\qquad
h(\eta)\propto\frac{e^{\pm ik\eta}}{\sqrt{2k}}\,.$ (133)

As before, $\eta$ is conformal time. The transition to the physical time is
$a(\eta)d\eta=dt$. The normalization constant in (133) is irrelevant for the
linear perturbations.

The possible instabilities can be explored using Eq. (132). According to the
known mathematical theorems about the stability of the fixed points of