###### Abstract

A method for consistent quantization of conformal gravity treating conformal
symmetry in a very controllable way is presented. First, we discuss local
conformal symmetry in the framework of gravitational interactions, where we
view it as an example of a general gauge theory. We also present some early
attempts at quantization of conformal gravity and use the generalized
framework of covariant quantization due to Faddeev and Popov. Some salient
issues such as the need for conformal gauge-fixing, an issue with conformal
third ghosts, and discontinuities in conformal gravity are studied as well.
Finally, we provide some explanations of the original ad hoc methods of
computation valid at the first quantum loop level in conformal gravity.

###### keywords:

quantum gravity; quantum field theory; higher-derivative gravity; conformal
symmetry; conformal gravity; covariant quantization

1 1 0 Academic Editors: Giulio Francesco Aldi, Luca Buoninfante, Giuseppe
Gaetano Luciano, Luciano Petruzziello and Luca Smaldone Received: 16 February
2022; Accepted: 02 April 2022; Published: 06 April 2022 Introduction to
Quantization of Conformal Gravity Lesław Rachwał ${}^{1,\,{{\dagger}}}$ Leslaw
Rachwal

##  1 Introduction and Motivation

Conformal gravity is a very promising model of relativistic gravitational
dynamics. It embodies not only symmetries of general relativity
(diffeomorphisms) but also is invariant under conformal transformations of the
metric tensor. In this class of theories we view the metric tensor
$g_{\mu\nu}$ as the fundamental variable completely describing the
gravitational field. In $d=4$ spacetime dimensions, conformal gravity, (also
known as Weyl gravity) is naturally a four-derivative model. This means that
from the beginning we have to deal with higher-derivative theories
Stelle:1976gc ; stelle2 , contrary to the original Einstein–Hilbert theory
which was described just by two-derivative dynamics. Hence, Weyl gravity is
the first model of modified gravitational dynamics, and this was introduced by
Hermann Weyl in 1918, just a few years later after the original Einstein
construction. When describing the dynamics of the gravitational field in a
conformal manner we have to be very careful and pay some special attention to
this higher-derivative nature of quantum conformal gravity shapiroconf1 ;
shapiroconf2 . Generally, the higher-derivative nature of gravity is quite
inevitable when one considers the effects of quantum matter fields
Utiyama:1962sn . Conformal gravity is a special case of such a generated (or
induced) gravitational theory when all the matter fields are massless.

Conformal gravity is the simplest gravitational model in which we have only
dimensionless gravitational couplings. This is why one of the prerequisites
for it is scale invariance on the classical level. By placing further
restrictions, this latter is constrained more and results in conformal
invariance, where the metric tensor is effectively charged under conformal
group. This being said, in conformal gravity, we cannot have any standard
dimensionful gravitational coupling such as famous gravitational Newton’s
constant $G_{N}$ or the cosmological constant. On the classical level, this
theory is completely described by the action which is a $4$-dimensional volume
integral of the conformal Lagrangian. The last one is given by the square of
the Weyl tensor (tensor of conformal curvature) according to the formula with
contractions,
$C^{2}=C_{\mu\nu\rho\sigma}^{2}=C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$,
where, in the last tensor $C^{\mu\nu\rho\sigma}$, the indices are raised using
the contravariant metric tensor $g^{\mu\nu}$. This is the unique four-
dimensional Lagrangian, which is conformally covariant (transforms with a
conformal weight $w=-4$) in such a way that together with the proper metric
density $\sqrt{|g|}$ they make the Lagrangian density $\sqrt{|g|}C^{2}$ a
conformal invariant exclusively in $d=4$ spacetime dimensions.

There are various motivations for considering classical conformal gravity as a
viable theory of the gravitational field mannheim1 ; mannheim2 ; mannheim3 .
First, the relativistic dynamics of such a system are constrained even more
due to the presence of additional symmetry besides the diffeomorphism
symmetry. Local conformal symmetry leads classically to new conservation laws
and to new integrals of motion. For example, the trace of the energy-momentum
tensor $T=g_{\mu\nu}T^{\mu\nu}$ must vanish all the time. This conformal
symmetry is also essential for constructing new, exact gravitational
solutions. It is also essential for the issue of black holes dynamics, high-
energy particle physics hooft1 ; hooft2 ; hooft3 , and also for the issues of
singularities hooft4 . Conformal gravity is a first scale-invariant model;
hence, it is consistent with what is generated from quantum loops of matter
when the last ones are integrated out on the level of path integral. The
gravitational action of conformal gravity can absorb all UV divergences which
are produced on non-trivial curved spacetime when massless matter fields exist
there. This also implies that conformal gravity in its minimal framework gives
rise to a consistent renormalizable theory of gravitational interactions when
it is considered in the quantum field theoretical (QFT) framework. On the
classical vacuum level, one proves that all Ricci-flat solutions of Einstein
gravity are automatically solutions in conformal gravity in four dimensions as
well exactsol . Finally, the presence of additional symmetry of gravitational
interactions is instrumental in solving the issue with spacetime
singularities, which are otherwise quite ubiquitous Bambi:2016wdn ;
Modesto:2016max ; singproc . With the power of conformal symmetry, one can
show that all curvature singularities are eradicated Modesto:2016max  and that
they are not a problem anymore for classical gravitational theory.

Above, we have given mainly theoretical reasons for studies of conformal
gravity. There exists also successful phenomenological predictions of
conformal gravity which makes this gravitational model completely falsifiable
using future gravitational experiments and also observations from astrophysics
and cosmology. As a model of modified gravitational dynamics (but still
preserving Lorentz symmetry), conformal gravity explains, for example, flat
rotation curves in galaxies without the need of local dark matter components.
Moreover, when picking the solutions of conformal gravity, one has to pay some
special attention to the global (conformal) aspects of spacetime, since the
conformal gravitational dynamics fully realize Machian ideas. Additionally, in
conformal gravity, one has naturally classical cosmological solutions which
describe both early inflationary phase as well as late-time accelerated
expansion of our Universe. This is achieved without explicitly adding the
cosmological constant term to the action or without a specially designed
inflaton scalar field. In the latter case, for exponential evolution, we do
not need any dark energy ingredients in the contents of the Universe. To
summarize conformal gravity solves the current important problems of present-
day cosmology without explicitly invoking dark matter, dark energy, and
inflaton fields. For a review of these experimental and observational issues,
one can consult expjizba ; exp1 ; exp2 . One can also study interactions of
conformal gravity with particle physics models, especially with its impact on
the Higgs potential, Stueckelberg fields, and also on spontaneous breaking of
conformal symmetry generated from matter fields such as in Ghilencea:2018dqd ;
Ghilencea:2019jux . For phenomenological applications, it is interesting, for
example, to study evaporation of black holes in conformal gravity evaporation
; evaporationconf , and their computation of entropy entanglement , the RG
flow towards the IR regime Jizba:2019oaf ; Rachwal:2021fhj  in one-loop
conformal quantum gravity, and also its implications on the dark sectors of
the universe as studied in Jizba:2020hre .

###  1.1 Motivations and Important Issues on the Quantum Level

With the above theoretical and phenomenological successes of conformal gravity
on the classical level, one is tempted to use it also to describe the quantum
dynamics of the gravitational field. Here, one enters into the domain of
quantum gravity (QG), where the things become much more complicated. For the
moment, there is not any experimental proof or evidence that the conformal
gravity is a correct theory of quantum gravity. In our framework, adopted in
this paper, we want to quantize the dynamics of the gravitational field in a
covariant relativistic manner and in a way that attempts to preserve all
classical symmetries of the model. The most successful and suitable approach
here is to consider the quantized theory of gravitational interactions in the
setup of old good quantum field theories. In this minimal and conservative
approach, we study the quantum theory of the gravitational field and its
interactions with matter fields. In particular, we focus on the propagation of
quanta of gravitational radiation (gravitons), their consistent interactions
with other matter particles, and also their self-interactions. The way quantum
particles interact is dictated and is very strongly constrained by the
symmetries of the quantum theory in question, and here, the fact that we also
use conformal symmetry should help in finding a unique and consistent theory
of quantum gravitational interactions.

The first results regarding quantum conformal gravity are promising. This
model in $d=4$ spacetime dimensions is renormalizable so there is a control
over UV divergences. This already improves over the situation existing in
quantum gravity described by the Einstein–Hilbert model in $d=4$. On the
quantum level, one can ensure that gravitons’ interactions are consistently
described with the diffeomorphism symmetry of any gravitational theory. One
generally knows quite well how to quantize the system in such a way that this
last local symmetry is completely preserved on the level of quantum dynamics
and in the general-relativistic (GR) setup. There exist well-known procedures
of covariant quantization due to Faddeev and Popov, and ways to deal with
gauge-fixings, gauge conditions, and additional fields of ghostly nature
needed only on the quantum level for diffeomorphism local invariance. All
these developments can be understood in the framework of quantization of
general gauge theories, where the symmetries are local. However, the situation
with local conformal symmetry understood in the framework of GR (so
equivalently on curved spacetime background) is more intricate.

The quantization of gravitational theories with conformal symmetry faces the
problem of the fate of this symmetry on the true quantum level. If one works
in the framework of conformal field theories (CFT), then on the quantum level,
in order to preserve conformal invariance, very special conditions must be
satisfied DiFrancesco:1997nk . The presence of unbroken conformal invariance
means that, in particular, all correlation functions must show a scale-
invariant behavior and they cannot depend on any mass or energy parameter
Rachwal:2018gwu . This implies that there cannot be any UV divergences in the
quantum model. The theory must be completely UV-finite. There are only a few
known examples of such theories in the gravitational framework Modesto:2014lga
; Modesto:2015lna ; Modesto:2017xha ; fin2 , and in particular around
maximally symmetric spacetime backgrounds Koshelev:2017ebj . More simple,
however, are models with only gauge symmetries on flat spacetime
Modesto:2015foa . All the results of loop computations in such models must
provide finite convergent results. There should not be any reason for
renormalization of the theory and for hiding infinities, hence the scale of
arbitrary renormalization $\mu_{0}$ should be absent in the formulation of the
theory on the quantum level. No need for $\mu_{0}$, no need for
renormalization, and only finite redefinitions of couplings are possible in
such a model with true quantum conformality. From the point of view of the
renormalization group (RG) flow, such a model sits always at the fixed point
(FP) so there is actually no RG flow. All the beta functions of couplings must
be zero at these circumstances. Due to the absence of any dependence on the
energy scale, the quantum fluctuations, corrections, and phenomena, they all
look the same, no matter at which scale one looks at them. The situation at
the FP of RG, where the scale invariance is automatically realized, allows one
to consider full upgrade to the quantum conformal symmetry. Then, one can
speak about true quantum CFT, where the conformal symmetry is so powerful to
severely constrain correlation functions (for example, $3$\- and $4$-point
functions are completely determined), and holding conformal Ward identities
places strong conditions on the resulting dynamics of the quantum system. All
this is happening obviously when the conformal symmetry is not violated on the
quantum level.

When one tries to quantize the pure conformal gravity model $C^{2}$ in $d=4$
spacetime dimensions in a natural covariant way, one unfortunately finds that
the above conditions do not hold. The theory, for example, at the one-loop
level, is not UV-finite; there are non-vanishing beta functions of
dimensionless coupling constants of the theory. There is, however a hope,
since not all is lost regarding the conformal symmetry in such circumstances.
The counterterms that one needs in this quantum model are also conformally
invariant when understood as terms in the gravitational action. Therefore,
although the UV divergences are present, then the divergent form of the
effective action also _formally_ preserves the conformal symmetry. One could
say that at this level the conformal invariance is hardly violated

In actuality, there are various ways that the conformal symmetry can be broken
on the quantum level. They include violations due to the conformal anomaly
(CA) confan ; confan2 ; confan3 ; confreview , a non-trivial RG flow of some
dimensionless coupling parameters, a presence of some dimensionful parameter
in the quantum theory and its running Dou:1997fg , an addition of some non-
conformal deformation parameter, or, finally, via vacuum expectation value of
some conformally charged scalar field present in the theory Ghilencea:2018dqd
. We would like to strongly emphasize here that despite that in most generic
situations the conformal symmetry is present on the classical level, but then
disappears on the first quantum level, etc., the quantization procedure is
never a formal reason for the violation of conformality. After all,
quantization is a mathematical procedure invented by humans, and nature does
not have to follow this, nor did it ever , since we know that the quantum
Planck constant $\hslash$ is already non-zero. We stress that if the quantum
conformal symmetry is broken, then this is because of the effects of some
quantum physical phenomena. As remarked above, there are various physical
means due to which conformal symmetry may not be fully realized on the quantum
level, but we will not discuss them here, leaving this interesting, though
difficult, future research.

We stated that the generic situation with Weyl gravity on the quantum level is
that at the quantum level the conformal symmetry is only partially realized.
At the classical level this symmetry is fully present, then on the first loop
level it is violated by the RG running of the dimensionless coupling constant
of the theory, i.e., $\beta_{\alpha_{C}}\neq 0$, where $\alpha_{C}$ is a
scale-dependent coupling in front of the $C^{2}$ term in the action of the
Weyl theory. Still, at the one-loop level, counterterms are conformally
invariant and of the form $\sqrt{|g|}C^{2}$ and $\sqrt{|g|}{\rm GB}$, where
the last one is the famous Gauss–Bonnet topological density. Then, due to the
presence of CA already at the one-loop level, the worse situation is expected
at the two-loop level. The anomaly heralds the presence of the $R^{2}$ term in
the effective action, which is not generally conformally invariant—it is only
invariant with respect to the so-called restricted conformal transformations
satisfying the condition $\square\Omega(x)=0$. However, in pure Weyl gravity,
the situation at the second and higher loops requires more detailed studies.
Generally after the quantization, in Weyl gravity, conformal symmetry is not
fully visible on the quantum level, but for this drawback, the formal
quantization procedure cannot be blamed.

###  1.2 A Need for New Conformal Quantization Method

From the purely theoretical view, one should still have a reliable
quantization method that allows for a clean distinction between spurious
artifacts and true physical effects which are responsible for violation of
quantum conformality. For example, such a method when applied to the FT
conformal supergravity model ought to provide results and rules which do not
break conformal symmetry, except when this is really needed (but then it is
fully under control), such as in the case of performing conformal gauge-
fixing. The same method, when used in the case of ordinary conformal gravity
(due to Weyl), would give everyone a theoretical advantage of clear separation
of sources for violation of conformal invariance on the quantum level. Then,
one could better understand the fate of conformal symmetry there and whether
this survives quantum corrections. Additionally, such a framework with
properly quantized conformal gravitational theory will be a good starting
point for writing all covariant Feynman rules of the theory. This, in a long
perspective, will allow us to perform perturbative computation of various
theoretical processes in quantum conformal gravity to higher loop orders too.
It is known, for example, that the covariant technique of traces of
differential operators is not easily applicable to the two-loop level and
beyond, while the computation using Feynman diagrams, although very tedious,
could be still attempted using some computer software programs that now can
deal very effectively with tensor algebra.

In particular, such two-loop explicit results could shed some light on the
fate of consistency of pure quantum Weyl gravity since, as we remarked above,
the two-loop level is the first one where we should see, perhaps, some
disastrous influence of conformal anomaly. Therefore, this computation there
could be crucial, and for this we need a very secure way of obtaining Feynman
rules of the theory. We need to include the propagators of all fields in the
spectrum (also those additional, such as Faddeev–Popov ghost and third ghost
fields) and also pay some special attention to perturbative vertices of
interactions and the role of proper gauge-fixing of all local symmetries
present in the model. The same also regards, for example, the case of the two-
loop level accuracy of computation in the FT supergravity theory in $d=4$. It
is expected that at this level one could explicitly derive the relation that
must exist between the Yang–Mills and conformal gravitational couplings in
this model. The one-loop condition for UV-finiteness of the full quantum
coupled model is not sensitive to such relation; however, the presence of it
is needed for a consistent absence of perturbative divergences at higher loop
orders and also for support of some claims of even more extended supersymmetry
there, which would be, presumably, here realized nonlinearly on fields of the
theory.

In order to write with confidence such Feynman rules and perturbative spectrum
of all propagating modes, one needs to have a very clear and methodological
means of quantization of general gravitational theories with conformal
symmetry present in the local version. Generally, we think that such a scheme
of quantization which treats the conformal symmetry very gently is naturally
needed to write and deal with perturbative rules of the models, in which it is
known that conformal symmetry is present also in the quantum domain of the
theory. Then, by achieving this, one could provide a necessary tool of the
preferred quantization method which is consistent with all the symmetries of
the model, present both on the classical and quantum level. Such a
quantization method does not violate conformal symmetry accidentally, and if
it does so, this is performed in a very controllable way. The consistent
formulation of such a quantization method within the framework of general
gauge theories (with local symmetries) and with various other constraints is
possible and we devote the main part of this paper to providing an
introduction to such a topic.

We think that, above, we have sufficiently motivated the need for a new
quantization method when dealing with local conformal symmetry in the
gravitational framework. The quantization method which conserves conformal
symmetry is a natural generalization of previous methods which were suitably
constructed in such ways to preserve gauge symmetries (in YM theories),
diffeomorphisms (in general gravitational theories), and also local
supersymmetry (in supergravitational models). We think that this
generalization to treat the conformal symmetry consistently is a natural, and
probably an ultimate, generalization of all these above methods. This claim
could be supported by the belief that the local conformal symmetry in the
framework of gravitational interactions is probably the last gauge symmetry to
be discovered since it requires a limiting attempt to reach formally
infinitely high energies. As we explained above, the true quantum conformal
symmetry should probably reside there in the deep UV regime, so then it could
be very difficult to be proven or rejected by experiments in the QG domain.

###  1.3 Contents of the Article

The structure of this paper is as follows. First, we discuss the conformal
symmetry as a gauge symmetry. This we study in the framework of flat
spacetime, curved spacetime (GR), and also in the setup of general gauge
theories. In the next section, we present the main details of the construction
of the suitable quantization method that preserves conformal symmetry. For
this purpose, we initially analyze all the necessary elements of the
construction present already when the theories with diffeomorphisms are
quantized using generalized Faddeev–Popov trick. Then, we motivate the need
for conformal gauge-fixing and conformal third-ghosts fields. To end this
section, we also present a small account of Veltman-like discontinuities in
conformal gravity on the classical level and also on the problems with
counting perturbative degrees of freedom in the spectrum of Weyl gravity in
$d=4$. Finally, we discuss the pioneering, although a bit ad hoc, method of
Fradkin and Tseytlin to arrive at the correct one-loop quantum results in
conformal gravity. At the end, we draw our conclusions and sketch some
possible future directions of developments of these interesting conformal
topics.

##  2 Conformal Symmetry as a Gauge Symmetry in Gravity

###  2.1 Conformal Symmetry in Flat Spacetime

First, one can consider conformal symmetry in the flat spacetime framework. In
this setting, one can also generate gravity as a gauge theory of small
tensorial fluctuations around flat spacetime. Moreover, conformal
transformations may also have their origin here. Originally, we start with
scale transformations or global conformal transformations. When one picks up a
Cartesian coordinate system $(x^{\mu})$ to describe events on flat Minkowski
spacetime, the following transformations,

$x^{\mu}\to\left(x^{\mu}\right)^{\prime}=x^{\prime\mu}=\lambda x^{\mu}$ (1)

are called scale transformations for $\lambda>0$ and $\lambda={\rm const}$.
They change how the scales (units) of lengths and distances are measured and
defined. By rules of quantum physics, the scale of length corresponds to
inverse of the energy scale (the transcription is performed using the Planck
constant $\hbar)$, so we also effectively change characteristic scales of the
energy of the physical system under consideration. The transformation (1)
leads to

$ds^{2}\to ds^{\prime 2}=\lambda^{2}ds^{2},$ (2)

where the Minkowskian infinitesimal length element (or proper time $d\tau^{2}$
along geodesics) $ds^{2}$ is shown _not_ to be invariant under scale
transformations. Instead, it rescales with the positive factor $\lambda^{2}$,
so its sign is untouched and this also implies invariance of the character of
the world-lines in relativistic physics (time-like, light-like, or space-
like). This informs us that the causal structure is not modified by scale
transformations, at least on flat spacetime background. It is obvious that for
any non-zero $\lambda$, the square $\lambda^{2}$ is positive in (2), while for
definiteness we have chosen $\lambda>0$ in (1). The infinitesimal distance
$ds^{2}$ being only covariant under scale transformations is fully consistent
with finite lengths and energies, also transforming with some scale factors
under changes of the scale.

Already in the flat spacetime field theory, a requirement of scale invariance
(so, invariance of the physical observables of a model under scale
transformations (1)) is quite constraining. Namely, in the construction of
such a classical field theory we cannot obtain help from any dimensionful
quantity such as an energy scale or a length scale (in high energy units,
where $[E]=[L]^{-1}={\rm GeV}$). In addition, all coupling constants of the
model describing interactions have to be properly dimensionless when the
dimension of the underlying Minkowski spacetime is fixed. In particular, this
condition excludes any presence of mass parameters for fields in the model.
The fields themselves, of course, may carry energy dimensions according to
their canonical (engineering) dimensionalities. As a result, we end up with a
model which, for its definition, does not rely upon any particular choice of
physical units of mass, time, length, energy, etc. This is contrary to other
models that, for their definitions, have to use ratios of some scales with
some other reference scales. Additionally, one could say that in such a theory
there is no scale (scale-less or scale-free models). On the quantum level,
full quantum scale invariance implies that there is no renormalization group
(RG) flow of all the necessarily dimensionless couplings of the theory, so the
theory sits all the time at the fixed point (FP) of RG flow.

Still, on flat spacetime, which in four dimensions comes with the full
Poincaré symmetry group $SO(1,3)\ltimes T^{4}$, scale symmetry can be viewed
just as a one-dimensional group of global transformations isomorphic to a
positive ray of the real line: $\mathbb{R}_{+}$. The subgroup of rescalings
(also known as conformal dilatations) directly multiplies the full Poincaré
group since it is known that a commutator of two conformal rescalings does not
additionally boost, rotate, or translate the physical system. This is not the
case with the relation between Lorentz subgroup $SO(1,3)$ and translation
subgroup $T^{4}$, and they enter into Poincaré group being multiplied semi-
directly. If one focuses on the 10-element Poincaré group, then one can map it
into $SO(2,3)$ group with translations $T^{4}$ included as “rotations” in
extended five-dimensional spacetime. This results in a non-trivial structure
of the algebra of commutators between generators (both boosts and rotations)
from the full Poincaré group.

In almost all classical field theory examples usually considered on flat
spacetime, the scale invariance can be promoted to a bigger symmetry group,
which includes also “vectorial”-like transformations (or “translations”-like
transformations) Migdal:1971fof ; DiFrancesco:1997nk . This group, together
with Poincaré factor, constitutes a 15-dimensional conformal group ${\rm
Conf}$ on four-dimensional flat spacetime. The five additional generators
compared to the group containing standard spacetime symmetries (Poincaré
algebra has 10 generators) are one dilatation (for scale symmetry) and a four-
dimensional vector of special conformal transformations (SCTs). One can see
that the full conformal group ${\rm Conf}$ is isomorphic to $SO(2,4)$, so to
the group related to Lorentz transformations but only in a six-dimensional
spacetime with signature $(+,+,-,-,-,-)$. This could be understood by
following the so called “embedding formalism” for conformal symmetry due to
Dirac. The structure of the full 15-element conformal group ${\rm Conf}$ is
more complicated (than just of the underlying Poincaré group $SO(1,3)\ltimes
T^{4}$) due to various semi-direct products used in its definition. Moreover,
it is known that two subsequent SCTs also generate resulting Poincaré
transformations on a physical system, and that their generators behave similar
to Lorentz vectors under boosts and rotations. Therefore, the structure of
commutators in the ${\rm Conf}$ algebra is more intricate around flat
spacetime, but we will not need an explicit form for it here.

Finally, we mention that while dilatations can be presented as a global part
of the full conformal group ${\rm Conf}$ and they can be performed with
constant transformation parameters $\lambda$, the intrinsic nature of SCTs
prevents the same identification for them. The character of SCTs is that they
depend on some special Lorentz vector (usually denoted by $b_{\mu}$), and in
an explicit transformation law in four-dimensional Cartesian coordinates, one
can see also a non-trivial dependence on the spacetime location point and the
Minkowskian distance from the origin $x^{2}=x^{\mu}x_{\mu}$. This is not so
surprising when one recalls that the full conformal group ${\rm Conf}$ of flat
spacetime contains conformal inversion transformations, that is
$x^{2}\to\frac{1}{x^{2}}$. To complicate this case, such dependence is found
in the denominators of the nonlinear transformation laws for Cartesian four-
dimensional standard coordinates111The situation looks much more simple when
one uses Dirac formalism and works in extended six-dimensional spacetime,
although such a spacetime with the signature $(2,4)$ does not have a clear
physical meaning. Moreover, here it can be understood as a useful, but only a
theoretical, tool.. The conclusion is that the SCTs cannot be performed with
constant parameters and their action actually depends on the spacetime point
characterized by Cartesian coordinates $x^{\mu}$. This means that, for
example, two SCTs with the same vector $b_{\mu}$ will act differently in two
different spacetime points $x_{1}^{\mu}$ and $x_{2}^{\mu}$ with $x_{1}^{2}\neq
x_{2}^{2}$. We can say without too much rigor that part of the full conformal
group which generates SCTs is already in the “local” version, contrary to the
other parts which are global (as dilatations are). This implies that it is
actually incorrect to try gauging the generators of SCTs by force Lord:1985fdw
; Ferrara:1977ij . One can also convince oneself about this fact by comparing
a count of number of generators in the full 15-dimensional conformal group
with the number of associated vector gauge bosons. In the conformal part of
this group (consisting of dilatations and SCTs) after gauging and making the
gauge potentials dynamical, one finds only one gauge boson (Weyl vector field,
originally associated incorrectly by Weyl to a photon) corresponding to the
generator of dilatations. The SCTs do not add new fields in the gauge theory
framework.

One remark is in order here. When we consider matter field theories around
flat spacetime background, which are without dimensionful coupling parameters,
thus, scale-invariant and later easily promoted to conformally invariant in
the sense of transformations given in (1), and from the full ${\rm Conf}$
group, then we mean global scale transformations, with parameters of conformal
transformations which are _not_ spacetime-dependent functions of coordinates
$x^{\mu}$. As explained in the introductory sections, this is the story on the
classical level. On quantum level to require full conformality, one must have
vanishing beta functions for all dimensionless couplings of the theory, or in
other disguised scale invariance of all Green functions, when the fields on
external legs are not rescaled. The set of examples for such a behavior is, of
course, more restricting, the ${\cal N}=4$ super-Yang–Mills (SYM) theory being
the standard one. However, we remark that, despite that ${\cal N}=4$ SYM on
flat background is completely UV-divergence free and scale-invariant, this
model of QFT is not invariant under gauged conformal transformations; for
example, it is not invariant under conformal dilatation transformations with
$\lambda=\lambda(x)$, where the latter is an arbitrary function of spacetime
points. Hence, gauging of the conformal group ${\rm Conf}$ (only parts related
to dilatation generator) brings a new kind of symmetry to the system and
constrains the dynamics even more. This symmetry with new added values can be
fully understood only in the proper gravitational context.

The class of theories (matter models) which are symmetric under
transformations from the global and full conformal group ${\rm Conf}$ (in
particular, with respect to dilatations with constant parameters $\lambda={\rm
const}$) is already very special and the theories in it are called conformal
field theories (CFTs in short). Such CFTs describe quite constrained quantum
dynamical matter systems different than ordinary QFTs because here there is no
RG flow. The CFT is located at the quantum FP of the RG. The requirement of
global (rigid) conformal symmetry places severe constraints on the dynamics,
and, for example, quantum (to all orders) two- and three-point functions are
completely determined. Higher $n$-point functions satisfy some recurrence
relations, but all the story so far has been without any influence of gravity
and for global scale (or conformal) transformations with $\lambda={\rm
const}$. When one performs the gauging of dilatations and the final theory is
brought to the form invariant under local scale transformations with
$\lambda=\lambda(x)$, then this framework is already more constraining than
the original CFTs were. Similarly, the original CFTs are not global conformal
anomaly-free when they are coupled to external background metric of spacetime
different than Minkowski metric $\eta_{\mu\nu}$222One notices that the
Minkowski metric $\eta_{\mu\nu}$ is completely unchanged under transformations
from (1) (conformal rescalings), even with $\lambda(x)$ spacetime-dependent.
The same is true also for SCTs acting on the metric of flat spacetime..
Therefore, coupling to external metric field (to gravity) or gauging the
conformal group ${\rm Conf}$ takes us out from the domain of CFTs analyzed
around flat background and we must necessarily consider conformal symmetry in
the gravitational setup. Here, there are various ways possible to perform the
mentioned above gauging Iorio:1996ad , such as Ricci gauging, Weyl gauging,
etc. For the moment, only very special coupled theories (matter+gravity)
satisfy demands of local quantum conformal symmetry, and the participation of
quantum conformal gravity in this dynamics is essential.

The analysis of dimensions of various subgroups and the structure of the full
conformal group ${\rm Conf}$ can be extended to be valid in any dimension of
spacetime $d$. For this purpose, one only notes that if there is one time
dimension (to have a consistent physical description) and $(d-1)$ spatial
dimensions, then the Lorentz group is $SO(1,d-1)$ with $\frac{d(d-1)}{2}$
generators, and the Poincaré group is $SO(1,d-1)\ltimes T^{d}$ isomorphic to
$SO(2,d-1)$ with $\frac{d(d+1)}{2}$ generators. Finally, adding one
$d$-dimensional vector of SCTs and one generator of dilatations, we obtain the
full conformal group ${\rm Conf}$ isomorphic to $SO(2,d)$ with
$\frac{(d+1)(d+2)}{2}$ generators. Of course, these general considerations
make sense only for dimensions $d>2$ since it is well known that in $d=2$
special dimensions of spacetime (or two-dimensional Euclidean space), the
conformal group and algebra are infinite. This infinitely enhanced power of
two-dimensional conformal symmetry has its roots in powerful complex analysis
on Argand plane, as it is well known and used in mathematics. Moreover, this
powerful infinite-dimensional conformal algebra is the reason for various
successes of $d=2$ worldsheet formulation (both Euclidean and Minkowskian) of
string theory. Eventually, this gauged conformal symmetry in string
worldsheets is so omnipotent that it cancels target spacetime ultraviolet
divergences otherwise ubiquitous and usually problematic in other quantum
gravity models.

In order to obtain a gauged theory of gravity, one can perform gauging of all
10 Poincaré generators as well as of the dilatation generator from the full
conformal groups Cabral:2020fax . The SCTs do not need to be gauged. When all
symmetries are made local from global and a special care is exerted to
construct physical actions, which are invariant under these new local
transformation symmetries, then one can conclude that an extended gauge theory
of conformal gravity is accomplished. This is in the framework of gauged
Poincaré gravity (with curvature and torsion as two independent field
strengths) and with an addition of conformal symmetry Hobson:2020bms . For
this end, one only gauges dilatations from the conformal part of the full
conformal group ${\rm Conf}$. In what follows, we will concentrate only on
this conformal part of the whole story. Actually, one can show that first, it
is consistent to consider only gauging of the conformal part, leaving
untouched the gravitational part of the full 15-element ${\rm Conf}$ group,
and second, for the last end, only dilatation generator and scale
transformations (1) need to be changed from global to local (with spacetime
dependent parameters $\lambda\to\lambda(x)$ in full generality in (1)).

###  2.2 Conformal Symmetry in Curved Spacetime

Instead of discussing the explicit gauging of the dilatation generator, we
will follow below a different, in a sense more geometrical, route. Namely, we
will consider conformal transformations and conformal symmetry in the
framework brought about by differential geometry of curved manifold
backgrounds. That is, in the spacetime setting, we discuss conformal symmetry
in a fully general-relativistic (GR) framework Fulton:1962bu ; Kaku . By this
we do not mean a full dynamical setup coming with Einstein’s gravitational
theory. For what matters here we will consider only the kinematical aspects of
gravitational theory while the specification of precise dynamical content in
some models to be considered later will be clear when we will discuss the
quantization of dynamical gravitational systems also enjoying conformal
symmetry. Therefore, by GR we mean here a theory describing spacetime physics
independently of the particular coordinate system chosen and all ensuing
consequences of such a view on physical systems.

In actuality, the origin of the construction of the word “conformal” for
conformal transformations, maps, images, and geometry is related to the way to
preserve a specific geometric form of some objects studied in geometry. In a
rough translation from Italian, “con forme” means in English “with the forms”
(to be understood with appendix “preserved”), so conformal means “with the
forms preserved”. The “forms” in question are angles between vectors which are
unchanged in this kind of geometry333One can note another interesting
linguistic coincidence here, but this time in Polish language. The second
morpheme “formal” in the word “conformal” (translated as Polish “konforemny”)
is related in Polish to the way that very symmetric regular polygons in plane
geometry are named, originally known as “wielokąty foremne”. Of course, such
regular flat polygons have all the internal angles identical..

In conformal geometry of spacetime, (hyperbolic) angles between four vectors
are preserved, and the causal structure (also named because of this conformal
structure) remains untouched when one performs conformal geometric
transformations. Such a transformation does modify lengths or magnitudes of
vectors and scalar products between them. This means that the differential
structure on the manifold is not invariant in conformal geometry and depends
on the choice of conformal gauge. In addition, the distances (in the sense of
four-dimensional proper Minkowskian intervals) are now not absolute, not
invariant, and become relative. They all depend on the conformal gauge. The
last one is realized by a selection of the conformal factor of the metric
(such as $\lambda^{2}(x)$ in (2) or $\Omega^{2}$ in (3) below) since,
typically, the conformal symmetry does not come with its own set of
coordinates, and borrows the ones also used standardly in differential
geometry, nor does it often come with its own set of gauge potential fields to
be fixed by this choice (the notable exception here is the gauge theory
proposed by Weyl with Weyl gauge connections). In the framework of conformal
geometry and field theories enjoying conformal symmetry, only angles are still
absolute, but they always constitute very beautiful geometrical structures
(exemplified in Escher’s pictures of anti-de-Sitter-like geometries).

In the GR setup (on a general curved manifold), the scale transformation is
the following active operation on the covariant metric tensor field
$g_{\mu\nu}=g_{\mu\nu}(x)$ on the manifold

$g_{\mu\nu}\to
g^{\prime}_{\mu\nu}=\Omega^{2}g_{\mu\nu}=e^{2\ln\Omega}g_{\mu\nu},$ (3)

where $\Omega$ is a constant parameter of the transformation.

This very much resembles a global $U(1)$ transformation of QED for a charged
scalar field $\phi=\phi(x)$ given by

$\phi\to\phi^{\prime}=e^{i\ln\alpha}\phi,$ (4)

where we also require that $\alpha={\rm const}$. One notices that there is a
difference of imaginary unit $i$ between exponents in formulas (3) and (4),
respectively. Similarly to the QED case, the metric field $g_{\mu\nu}$ can be
treated as being effectively charged under conformal group. Or, even more, the
metric fields are matter fields from the point of view of conformal symmetry,
they transform similar to electrons but with the real, not a complex,
prefactor phase, and the module of the conformal coupling charge of the metric
is universal and equals two (by convention).

In general gauge theory, the fields can be divided into two groups: gauge
potentials or matter fields. The difference is only how they transform under
gauge transformations. Let us assume that these transformations are linear
(the fields in question must for this furnish a linear representation of a
gauge group) and for definiteness we only consider them in an infinitesimal
form. Then, if the transformation laws are homogeneous, then the fields upon
which the transformation is performed are matter fields, if instead they are
inhomogeneous but still linear, then these are gauge transformations of gauge
potentials. We remark that for different symmetries the same field can play
different roles, similar to the metric tensor for diffeomorphisms, and GCT is
a gauge potential but it behaves similar to matter field for conformal
symmetry.

However, there are also very important differences between these two cases,
giving rise to the dichotomy between gauge theories (in internal spaces) as
used extensively in non-gravitational field theory models and conformal
gravitational theories (in external spacetime) used sometimes to describe
relativistic gravitational fields. First, for the complex-valued scalar field
$\phi$ in (4), the invariant module square, that is
$|\phi|^{2}=\phi^{\dagger}\phi$, is QED-invariant, while for the real-valued
metric in (3), the Lorentz-invariant infinitesimal length element $ds^{2}$ is
not invariant under scale transformations (cf. (2))444But one sees a trivial
thing here, that $g^{\mu\nu}g_{\mu\nu}=d$ is invariant under conformal
transformations, stating obviously that the dimension of a spacetime is not
modified by conformal transformations in (3). This may resemble an invariance
in QED of $|\phi|^{2}=\phi^{\dagger}\phi$, although the sense of conjugation
is completely different here for real metrics..

Moreover, due to the fact that in the QED case, the complex phase factor
$e^{i\ln\alpha}$ is present, the $U(1)$ group is topologically identical to
the circle $S^{1}$ and hence is compact. In the case of the real and positive
scale factor $\Omega^{2}$ in (3), this is isomorphic to $\mathbb{R}_{+}$ and
hence this is a non-compact group of dilatations. This implies that some
mathematical aspects will be very different for the $\mathbb{R}_{+}$ group of
conformal dilatations and the $U(1)$ group of complex phase transformations in
QED. For example, when one tries to construct a gauge theory of a scalar field
$\phi$ which is invariant under real rescaling of the field, then one cannot
use the invariant $|\phi|^{2}=\phi^{\dagger}\phi$ or even simple square
$\phi^{2}$ (such as for the real-valued Higgs field) in construction of the
invariant action functional, and even the globally symmetric $\mathbb{R}_{+}$
theory on Minkowski background is not well-defined. To achieve this, one has
to leave the flat absolute Minkowski background and move to the framework of
general relativity, where these scale transformations show the structure of
spacetime under the influence of gravitational field embodied in the proper
metric tensor field $g_{\mu\nu}(x)$.

On the other hand, if one has overcome the above problems, and tries to build
a locally symmetric theory, then the differences are present, too. In the case
of QED, it is well established that the proper covariant derivative of the
scalar field is inherently complex-valued, when the $U(1)$ gauge connection
field is taken as real, but this part of the definition for the $U(1)$ gauge
covariant derivative must be with the imaginary unit $i$. On the contrary, in
the case of group of dilatations, the proper covariant derivative (Weyl
covariant derivative with a real Weyl gauge connection field) is entirely
real-valued and there is no room for any imaginary unit. Eventually, one
should recall the London brothers’ correspondence, thanks to which the
complex-valued electron fields $\psi$ (analogs of used here charged matter
scalar field $\phi$) give interpretation to complex-valued and normalizable
electron wave functions in quantum mechanics. Of course, such interpretation
is not possible when one deals with real-valued metric fields $g_{\mu\nu}$
which are charged under conformal symmetry group, but their transformation law
is with the real positive overall factor $\Omega^{2}$, so one stays all the
time in the real space formalism, which cannot be applied directly to quantum
theory. Additionally, the original identification by Weyl of the Weyl gauge
potential as the vector potential of the electromagnetic field suffers from
all the above problems and cannot be retained anymore in field theory models.
We can understand that in his time Weyl was seeking for a very economical
model combining (or even unifying based on some geometrical arguments)
together gravitation and electromagnetism. As we think nowadays, he was not
right and the road towards a full unification is still very long, because
gravitation (even if the models of conformal gravitation are being considered)
is quite different from internal space gauge theories (such as
electromagnetism and non-Abelian Yang–Mills (YM) theories). However, as we
will show below, the formal procedure of gauging the global symmetries in
these two models works very similarly and we do not see any big differences
here.

By exploiting the well-known formula

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu},$ (5)

one sees that (3) implies (2)

$ds^{2}\to ds^{\prime 2}=\Omega^{2}ds^{2},$ (6)

in accordance with our previous considerations of conformal transformations on
flat spacetime background described in Cartesian coordinates. For this, in the
GR framework, contrary to (1), one postulates that coordinates $x^{\mu}$, in
any coordinate system, are not changed by the transformations, the same for
their contravariant differentials $dx^{\mu}$.

Gauging means that we promote the constant parameters of transformations to
spacetime dependent parameters of the same kind of transformations, namely,

$\alpha\to\alpha=\alpha(x)$ (7)

in QED and

$\Omega\to\Omega=\Omega(x)$ (8)

in spacetime physics. This last operation produces full-fledged conformal
transformations within the meaning of GR:

$g_{\mu\nu}\to
g_{\mu\nu}^{\prime}=\Omega^{2}(x)g_{\mu\nu}=e^{2\ln\Omega(x)}g_{\mu\nu}$ (9)

with the local parameter of conformal transformations $\Omega=\Omega(x)$ and
there is no need of further gauging of these transformations. This is local
scale invariance or conformal invariance in GR. This is, in a sense, a gauging
of global scale invariance (3) understood in the GR framework. For general
conformal transformations, the parameters $\Omega=\Omega(x)$ are local
functions of the spacetime point $x$. We emphasize that within the GR, the
conformal transformations are already in the local version and the conformal
symmetry could be viewed here as a gauge symmetry of the system, provided that
the description of the configurations of the system is invariant under action
of these transformations. This, of course, depends on the dynamics in the
gravitational sector of the theory. We note that, not for all theories, the
conformal symmetry in the version from (9) is a symmetry of gravitational
dynamics. In actuality, the requirement for such a symmetry of the
gravitational action is very restrictive in gravitational models, also when
they are coupled to conformal matter models. Basically, when one fixes the
number of spacetime dimensions, the number of possible different theories
enjoying conformal symmetry, even on the classical level, is finite and is
usually quite small. When one moves to lower degenerate dimensions, such as
$d=2$, such theories may not exist at all. Therefore, the requirement for
consistent conformal symmetry of the gravitational system constrains quite
tightly the possible form of the dynamics in this system.

The local conformal symmetry finds its proper place in the context of
gravitational physics, where it is understood as a possible transformation of
the spacetime, thus of the gravitational background. Despite that it was
originally studied and constructed on flat absolute Minkowski spacetime and
promoted there from simple scale invariance to the full 15-element conformal
group (in $d=4$), its natural embedding is in the setup of differential
geometry. This is because a conformal transformation really touches the
measurement of distances, so crucial to the mathematical definition of
infinitesimal (differential) geometry. Since gravitational field has as one of
its manifestations the form of curved spacetime as differential manifold, then
the conformal symmetry, or the conformal group in general, will also primarily
be related to the dynamics of gravitational fields. At the end, we conclude
that the conformal symmetry is one of the possible symmetries of relativistic
gravitational fields. This symmetry in a gauged form can be an additional (and
last to be discovered) gauge symmetry of gravitation in addition to the local
Poincaré group of spacetime symmetries.

We also emphasize that, strictly speaking, the examples of CFTs that we
considered in the previous subsection on the flat background (such as ${\cal
N}=4$ SYM theory), which are conformal on the full quantum level, are not
invariant under scale transformations in GR, even when they are described by
constant parameters $\Omega={\rm const}$. In the last case, one could come
back easily to the form of Minkowski metric (absolute, not rescaled) by
performing a compensating rescaling of Cartesian coordinates according to (1)
with $\lambda=\Omega^{-1}$. On flat background, Minkowski metric
$\eta_{\mu\nu}$ is considered as an absolute element, which is in obvious
contradiction with GR, but even if we allow this compensating coordinate
transformation, then this extended setup cannot work on more general manifolds
than the flat one. Or, speaking physically, when we couple a flat spacetime
CFT to external gravitational field, then such a theory will not be
automatically invariant under local (gauged, so with $\Omega=\Omega(x)$ and
not rigid anymore) scale transformations with the sense from GR, from (3), and
(9) below. Moreover, we know that after such transformations the Minkowski
metric $\eta_{\mu\nu}$ changes into a general conformally flat metric of
spacetime and not into a necessarily flat one. We remind the reader that the
original CFT was not defined in such circumstances. This disappointing feature
is also related to the fact that most of flat spacetime CFTs are not free from
the conformal anomaly problem in external metric fields.

It must be noted that gauging of dilatation transformations is not the same as
appending SCTs to global scale transformation group (isomorphic to
$\mathbb{R}_{+}$) and subsequent promotion of global scale invariance to full
global conformal invariance under transformations from the full 15-dimensional
${\rm Conf}$ group. In the flat spacetime framework presented in the previous
subsection, the conformal group ${\rm Conf}$ was still a group of global
transformations, even if it contained somewhat local parts related to SCTs.
The situation with gauging of scale transformations (9) on the metric tensor
as understood in GR leads to a different group of conformal transformations.
These last transformations should be understood only in the framework of GR,
and after the gauging is finished, their general form is with precisely one
spacetime dependent parameter $\Omega(x)$. Thus, the resulting group of
transformations is different than a 15-dimensional global ${\rm Conf}$ group.
The relations between these groups and their flat spacetime global analogs is
the same as between a diffeomorphism group ${\rm Diff}$ and the global
Poincaré group $SO(1,3)\ltimes T^{4}$ on flat spacetime. The former is known
to be infinite-dimensional (because it contains local transformations), while
the latter has only 10 generators. Similarly, a local conformal group in GR is
infinite-dimensional, while there are one generator of dilatations, four of
SCTs, or altogether 15 generators in global flat spacetime ${\rm Conf}$ group.
Hence, these conformal groups are clearly different.

We also remark that although naturally placed in the gravitational and
differential geometry context, the conformal symmetry is not the same as a
symmetry pertaining to general coordinate transformations (GCTs). For them, we
have

$x\to x^{\prime}=x^{\prime}(x),$ (10)

where $x^{\prime}(x)$ are differentiable but arbitrary functions. Moreover,
general coordinate transformations preserve the value of the most fundamental
scalar invariant of differential geometry, namely of $ds^{2}$. In actuality,
this is not a consequence, but rather a postulate from which one can derive,
for example, the transformation law under GCTs of the covariant metric tensor
field $g_{\mu\nu}(x)$. Taking a quick look at (6), one easily understands that
general conformal transformations in GR are, most of the time, not general
coordinate transformations. This is, for example, clear from the fact that the
group of conformal transformations is different than the group of GCTs, known
as diffeomorphisms (and denoted below by ${\rm Diff}$). Therefore, in the most
general situation in the dynamics of gravitational fields, we can have two
symmetries of diffeomorphisms and of conformal transformations. These two
symmetries do not overlap each other and can be treated quite separately, for
example, for the quantization aims. In a rough sense, one can state that in
such a case, the total group is a direct product of subgroups of
diffeomorphisms ${\rm Diff}$ and of a local conformal symmetry (this is a
different group than 15-element ${\rm Conf}$ group, it is, rather, only a
group of symmetries under local conformal transformations, so it is, in a
sense, a one-dimensional group of gauged dilatations). The two groups do not
disturb each other, although they use the same field variables as the objects
upon which corresponding transformations act. This is a peculiar feature of
metric gravitational theories, where the metric tensor is the basis for
everything related to gravitation. These two groups both act on the metric;
their actions are, however, distinguished by the character of transformations
and by different parameter(s) used to characterize such transformations.

###  2.3 Conformal Symmetry in General Gauge Theories

As emphasized above, the local conformal symmetry in the GR framework can be
understood as a local gauge symmetry of the gravitational system, possibly
also coupled to some matter models. Therefore, for this symmetry one can apply
the general formalism of general gauge theories and their ensuing
Faddeev–Popov quantizations. Placing the conformal symmetry in this formalism
is particularly convenient for the sake of covariant quantization approach
shapiroconf1 ; shapiroconf2 ; Eli3  applied to the classical system of
gravitational field, enjoying full conformal symmetry with the hope of
obtaining consistent theory of conformal and gravitational interactions. To
this end, one first needs to recall some facts about conformal symmetry in
this framework. First, we will use a general metric background characterized
by the symmetric rank-2 tensor field $g_{\mu\nu}(x)$. Below, and in later
sections, we will use the condensed notation due to DeWitt. One can also
consult the textbooks Buchbinder:1992rb ; Buchbinder:2021wzv , where this
notation was employed for the first time for the case of general gauge
theories. One can consider the possibility of obtaining conformal gravity from
general gauge theory as in Johansson:2017srf .

The conformal symmetry transformations were defined in (9) but for the sake of
emphasizing the spacetime dependence, we write the last formula as

$g_{\mu\nu}(x)\to g_{\mu\nu}(x)^{\prime}=\Omega(x)^{2}g_{\mu\nu}(x),$ (11)

where $\Omega=\Omega(x)$ is a finite parameter of conformal transformations,
possibly also spacetime-dependent. In the infinitesimal version, this
transformation reads,

$\delta g_{\mu\nu}=\delta g_{\mu\nu}(x)=\xi(x)g_{\mu\nu}(x)=\xi g_{\mu\nu},$
(12)

where the infinitesimal parameter of transformation $\xi\ll 1$ is a general
spacetime-dependent function $\xi(x)$ and we have that $\Omega^{2}\approx
1+\xi$. Of course, as obvious from (12), the infinitesimal conformal
transformations are characterized by only one scalar transformation parameter
$\xi$, which is spacetime-dependent for local conformal symmetry (compared
with the case of diffeomorphisms, where a GCT is characterized by $d$
parameters collected in one GR-covariant vector $\xi_{\rho}$). For the purpose
of covariant quantization, it is very important to derive the form of an
infinitesimal generator ${\rm R}$ of conformal transformations555To avoid
clash of notation with various curvature tensors, all denoted by the same
letter “R” due to Riemann, Ricci, etc., we use the letter “R” in the Roman
font to denote various infinitesimal generators of transformations..

The general formula for ${\rm R}$ reads,

${\rm R}={\rm R}(x,y)=\frac{\delta\phi(x)}{\delta\xi(y)}.$ (13)

In our case, when we look for the generator of infinitesimal conformal
transformations, the “charged” field is the metric field, so we take
$\delta\phi(x)=\delta g_{\mu\nu}(x)$, and the parameter of this transformation
is $\xi=\xi(y)$. Therefore, we obtain that in our case, the corresponding
formula is

${\rm R}={\rm R}_{\mu\nu}(x,y)=g_{\mu\nu}\delta^{(d)}(x,y).$ (14)

As we see, the general form of this generator ${\rm R}$ inherits two covariant
indices from the metric tensor. Moreover, it also possesses ultra-local
dependence on two spacetime points $x$ (where the charged field is located)
and $y$ (where the parameter $\xi(y)$ resides), and this dependence is
expressed via the $d$-dimensional Dirac delta function. In general, one could
say that the generator in this case is a bitensor, or more precisely, a rank-2
symmetric tensor (such as the metric tensor $g_{\mu\nu}$) from the point of
view of the point $x$, and a scalar from the point of view of the point $y$
Buchbinder:1992rb ; Buchbinder:2021wzv .

Already, here, one can notice a few important things which will be important
in what follows later regarding the quantization in different choices of
conformal gauges. First, the conformal gauge transformations act on a metric
field in a linear way. This is true not only for the infinitesimal version of
these transformations, but also for finite local conformal rescalings. This is
a good feature simplifying the covariant quantization procedure. The
generators create a linear algebra of conformal gauge symmetries, which closes
off-shell (and then we do not have to employ the Batalin–Vilkovisky more
complicated quantization method). Secondly, the conformal transformations do
not involve any derivatives of the metric tensor which are charged here. This
is also a feature which is well received, since this will have some positive
consequences on the ghost sector of conformal symmetry. In actuality, in the
quantization, we can exploit both of them. With the explicit formula in (14),
we can conclude the part with understanding of conformal symmetry as a gauge
symmetry in the gravitational setup. All further developments depend on the
choice of the conformal gauge for quantization procedure, and hence they are
not universal pertaining to the knowledge of general gauge theories.

###  2.4 Fradkin–Tseytlin Conformal Supergravity

We must also remark here that there exists an extended model of conformal
gravity which is UV-finite on the full quantum level. This is realized when
additional matter fields are included but they are all united with gravity in
the same multiplet. Moreover, the symmetry which achieves this also mixes
bosonic and fermionic degrees of freedom, hence we have the right to call it
extended supersymmetry. In a model when this extension is maximal possible and
where in the pure bosonic spin-2 sector we deal with Weyl gravity in $d=4$, we
may have a special situation where all perturbative and also non-perturbative
divergences completely vanish. Such an extended supergravitational model was
first proposed by Fradkin and Tseytlin in 1985 Fradkin:1985am . These two
copies of ${\cal N}=4$ super-Yang–Mills theories are coupled to ${\cal N}=4$
conformal supergravity, which is a supersymmetric generalization of the Weyl
conformal gravity in the spin-2 sector of fluctuations.

In the ${\cal N}=4$ Fradkin–Tseytlin (FT) conformal supergravity model, the UV
divergences were checked for the complete absence at the first perturbative
loop level. However, the argument for UV-finiteness in this case is much
stronger since the presence of quantum conformal symmetry is guaranteed on the
quantum level from some algebraic considerations, and this holds to all loop
orders and also non-perturbatively. Basically, being in the same multiplet of
currents, the conservation of conformal (virial) current is related by the
extended ${\cal N}=4$ supersymmetry argument to the conservation of the matter
energy–momentum tensor $\nabla_{\mu}T^{\mu\nu}=0$ together with the
conservation of vectorial Yang–Mills (YM) current $D_{\mu}j^{\mu,a}=0$. From
the last statement about the conformal current, one derives the implication of
non-violated conformal symmetry in the model purely from algebraic reasons, so
it does not matter whether this is on the classical or on the quantum level.
Such conservation holds automatically in the theory.

The example of FT supergravity shows that it is possible to find a QFT theory
(actually a CFT) which also includes quantum gravitational interactions that
are completely free of any UV infinities and also of classical singularities
of spacetime, since the conformal symmetry and invariance are present both on
the classical and quantum levels of this model. From the point of view of RG,
this is a very special and highly symmetric theory which is defined at the FP
of RG, and the CFT which describes it is supersymmetric and with gravity
included. It was very difficult to find similar CFT models with gravitation
since other known examples considered only matter theories and on flat
spacetime backgrounds (such as mentioned before, ${\cal N}=4$ super-Yang–Mills
theory Namazie:1982br ). On the quantum level, the finiteness also implies
complete absence of conformal (trace) anomaly, and even the trace of the
quantum energy–momentum tensor is vanishing, i.e., $T=0$. The conformal
symmetry is fully realized on the quantum level and around any spacetime
background in this model. There is not any problem with its violation, and
moreover, the power of conformal symmetry is at use here since all Green
functions are heavily constrained, similar to in any CFT. This is properly an
anomaly-free gravitational theory FradkinT3 . Gravitational conformal Ward
identities hold and this is the true representation of the fact that conformal
symmetry is unbroken here.

One could also wonder whether in such symmetric theories, which are also
quantum conformal, there is a further enhancement of supersymmetries beyond
the case of ${\cal N}=4$. However, here, the counting of possible generators
(charges) of the superconformal algebra excludes such possibilities—at least
that such a hypothetical gravitational CFT with higher than 16 supercharges
cannot exist around flat spacetime and when all these symmetries are realized
linearly Cordova:2016emh . Maybe some nonlinear versions can exist with
superconformal algebras with ${\cal N}=8$ unifying together conformal
supergravity and also quantum conformal ${\cal N}=4$ super-Yang–Mills
theories. For such a model of quantum conformal gravitational interactions,
from the physical standpoint we shall view it emerging as true and UV-finite,
and thus a completely sensible theory of QG interactions at the very high
energies (in the deep UV regime) . Therefore, one expects that when one lowers
energy scale from the UV limit, the conformal invariance is somehow broken and
we end up with effective models not so symmetric and without full constraining
power of all symmetries of the gravitational theory. One should relate this,
in the future low-energy phenomena, with the breaking of conformal symmetry.
Moreover, one should be able to embed in this gravitational framework without
UV-divergences some UV-finite models of particle physics, as were proposed in
Meissner:2006zh ; Chankowski:2014fva .

##  3 An Example of Covariant Quantization of Diffeomorphisms

At the beginning, we remind the successful covariant quantization prescription
when applied to a generic higher-derivative gravitational theory. As
emphasized in Section 2, gravitational theories with conformal symmetry, away
from dimensionality of spacetime being two, $d>2$, are necessarily very
special types of higher-derivative gravitational theories, and with only all
couplings dimensionless. The general higher-derivative (HD) theories were
quantized in the covariant way by applying the general Faddeev–Popov (FP)
methods based on functional analysis (originally such covariant quantization
scheme, but only working up to one-loop level, was invented by Feynman). In
the general gravitational setup, FP analysis has to be exploited for
diffeomorphism symmetries, which are the gauge symmetries of gravitation (at
least in the minimal setting). The consideration of higher-derivative theories
of gravity, rather than Einsteinian gravity, is here beneficial, as we will
see shortly. The situation with higher derivatives is the most general
possible one, more general than the case of two-derivative gravitational
theories. This will also be important for quantization of conformal symmetry
in various gauges and in various dimensions of spacetime.

The expression for the true quantum partition function of a general
gravitational theory with diffeomorphisms reads,

$Z_{{\rm grav}}\left[J^{\mu\nu}\right]=\\!\int\\!{\cal D}h_{\mu\nu}{\cal
D}\bar{C}^{\alpha}{\cal D}C_{\beta}{\cal D}b_{\gamma}\exp\left[i\left(S_{{\rm
grav}}+S_{{\rm gfix}}+S_{{\rm gh}}+h_{\mu\nu}J^{\mu\nu}\right)\right],$ (15)

where the main quantum field is the metric fluctuation field $h_{\mu\nu}$
defined as a deviation from the background metric $g_{\mu\nu}$ such that the
total metric is $g_{\mu\nu}+h_{\mu\nu}$. Notice that we have decided that our
main quantum field will be a symmetric rank-2 tensor field with two covariant
indices—this is our choice of the parametrization of the quantum metric field.
As stated before, we work exclusively in metric theories of gravity and we do
not allow any dynamical or quantum variables with torsion or non-metricity of
spacetime. Therefore, the choice of the covariant fluctuations $h_{\mu\nu}$ as
the quantum variable is very natural. The covariant metric tensor field
$g_{\mu\nu}$ is the most standard choice since the coordinates are typically
chosen as contravariant objects $x^{\mu}$ and this metric tensor enters into
the expression (5) to construct a ${\rm Diff}$ invariant scalar, that is,
$ds^{2}$. Of course, other choices of the parametrization of quantum metric
fluctuation variable are also possible. Moreover, although not explicitly
mentioned, here we work in general background field method, keeping the
background classical metric $g_{\mu\nu}$ as fully general. In this way, we
realize all physical requirements related to hyped background independence of
quantum gravity. Because of this, the functional of quantum partition function
$Z_{{\rm grav}}$ in (15) also contains not-path-integrated background metric
$g_{\mu\nu}$.

In (15), we path-integrate (notation with ${\cal D}\phi$ for differentials of
fields under path integrals) not only over the quantum metric fluctuation
field $h_{\mu\nu}$, but also over a set of additional quantum fields needed
because of the diffeomorphism symmetry present in the model and because of a
general higher-derivative nature of gravitational theory. All fields in this
set are good ghost fields needed to keep the BRST remnants of diffeomorphism
symmetry also on the quantum level of the theory and to avoid potential
problems with anomalies. They are added to make the quantization procedure
covariant and to preserve diffeomorphism symmetry, such that the theory after
the procedure (quantum theory) is defined as with fully realized BRST symmetry
originating from the full classical ${\rm Diff}$ symmetry, the same as the
original classical theory defined by $S_{{\rm grav}}$. The quantization
performed in this way controls very well the local symmetries of the model,
and there is no any spurious or accidental breaking of diffeomorphism or any
other local symmetry. The framework fixes a coordinate gauge (of
diffeomorphisms) by adding the action $S_{{\rm gfix}}$, but in a very
controllable way. One can fully follow the dependence on gauge-fixing
parameters present in $S_{{\rm gfix}}$ and convince oneself that in the final
physical results the dependence on them completely drops out. In this way, the
quantization act does not violate the local symmetries, here diffeomorphism
symmetries. This is the essence of the meaning of covariant quantization
procedure. The quantum theory obtained in this way is gauge-fixed but with
gauge-fixing performed in a very clear and parametric way. The BRST symmetry
is what remains at the quantum level after gauge-fixing of all the local
symmetries. It controls the form of the total action $S_{{\rm tot}}$ appearing
in the exponent of the integrand of path integral in (15),

$S_{{\rm tot}}=S_{{\rm grav}}+S_{{\rm gfix}}+S_{{\rm gh}}.$ (16)

###  3.1 Ghost Fields

The ghost fields needed for the local gauge consistency of the model are of
two types here. The original FP ghosts are complex-valued quantum fields (the
first ghost $C_{\beta}$ and the second ghost $\bar{C}^{\alpha}$ being the
anti-ghost of $C_{\beta}$), and a third ghost real field $b_{\gamma}$. Since
the local symmetries here are diffeomorphisms, then each of these ghosts come
with one Lorentz index, so they have intrinsically vectorial nature from the
spacetime point of view. We emphasize that the canonical position of these
spacetime indices is covariant index on the FP ghost field $C_{\beta}$,
contravariant index position on the FP anti-ghost field $\bar{C}^{\alpha}$,
and covariant index on the third ghost $b_{\gamma}$. Since the covariant
metric tensor of fluctuations $h_{\mu\nu}$ is our quantum variable, then it is
also natural to choose a linear diffeomorphism gauge-fixing condition as a
covariant vector $\chi_{\mu}$. Then, the fact that a parameter of
infinitesimal ${\rm Diff}$ transformation is also taken as a covariant vector
$\xi_{\nu}$ (because $\delta
g_{\mu\nu}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$) implies that the
ghost field has to be with one covariant index $C_{\beta}$ in the canonical
position, since in the derivation of the FP determinant this ghost field takes
position of an arbitrary parameter $\xi_{\nu}$. The anti-ghost field has to
contract in a natural way with the covariant index on the gauge-fixing
condition $\chi_{\mu}$ from the left side, so it must carry a contravariant
index according to $\bar{C}^{\alpha}$, as we wrote above. Finally, the third
ghost field substitutes the diffeomorphism gauge-fixing $\chi_{\mu}$, hence
its index is also naturally covariant in $b_{\gamma}$.

All these ghosts are spurious fields appearing only on the quantum level since
they have anti-commuting character and mathematically are described using
Grassmannian variables (similar to fermions), although they do not obey the
usual spin-statistics theorems. They never appear on external legs of any
correlation functions and also never in the perturbative spectrum of the
theory, hence they are harmless for the unitarity issue. However, they are
crucial for the issue of gauge invariance of the resulting effective quantum
theory. Since they were not present in the classical theory and they show up
only on the quantum level when one uses the covariant quantization
prescription, then the reason for their names is obvious. They are classically
non-existing, anti-commuting fields, therefore they satisfy classical anti-
commutation relations,

$\left\\{\bar{C}^{\alpha},\bar{C}^{\kappa}\right\\}=\left\\{\bar{C}^{\alpha},C_{\beta}\right\\}=\left\\{\bar{C}^{\alpha},b_{\gamma}\right\\}=\left\\{C_{\beta},C_{\lambda}\right\\}=\left\\{C_{\beta},b_{\gamma}\right\\}=\left\\{b_{\gamma},b_{\varkappa}\right\\}=0$
(17)

even before these fields are treated as quantum operators in the second
quantization (not needed here). These “phantom” fields are present only inside
perturbative loops of the quantum covariant theory and they are needed there,
for example, to secure the gauge invariance of the final results of
computation of some Green functions or physical processes using Feynman
diagrams. The necessity to introduce them as virtual particles running in the
loops is the price to pay when one wants to deal with gauge symmetry and
redundancy of the formalism of general gauge theories on the quantum level in
a perturbative and covariant manner. These ghost particles never appear as
asymptotic states or on external legs (that is, on-shell, satisfying classical
equations of motion (EOM) of the theory) of any Feynman diagram. In actuality,
it is without any sense to speak about classical EOM for ghosts or on-shell
conditions for ghost fields since the classical action and ensuing EOM for
physical fields are defined entirely by the functional $S_{{\rm
grav}}\left[g_{\mu\nu}\right]$.

FP ghosts are not visible on the level of classical theory described by the
action $S_{{\rm grav}}$, where the gauge-fixing procedure is not necessary to
obtain classical equations of motion for the classical metric field
$g_{\mu\nu}$. They are not present in the perturbative spectrum of classical
theory (obtained from classical two-point functions when the background is
chosen and some gauge-fixing has to be performed); hence, these fields have
nothing to do with the violation of unitarity in higher-derivative
theories666In actuality, it is completely opposite: for example, in standard
Yang–Mills theories, perturbative FP ghosts are needed to save the unitarity
of the perturbative covariant quantum non-Abelian gauge theory, as explained
in Peskin:1995ev . The Feynman diagrams with them running inside loops are
necessary to include for the whole consistency of the quantum theory.. If one
moves to the level of classical higher $n$-point functions (again when the
background is specified and gauge-fixing is performed), one also does not see
any effect of these truly quantum virtual modes. Therefore, we draw a clear
distinction between these good ghosts and bad ghosts present in higher-
derivative theories and related to the famous Ostrogradsky theorem. The latter
ones arise at the perturbative spectrum of classical theory, and in principle
can appear on external legs before they are not consistently forbidden to
appear there by various prescriptions of how to save perturbative unitarity in
classical higher-derivative and non-local theories tomb2 ; unita ; fakeunit .

Let us here discuss in more detail this issue with Ostrogradsky instabilities
on the classical level or equivalently with the presence of “bad ghosts” on
the tree level of the spectrum of the quantum theory. Generally, in any class
of higher-derivative theories (including not only gravity but also other
fundamental interactions), perturbative unitarity of $S$-matrix is in danger
because there are perturbative states with negative kinetic energy or
equivalently with negative mass square parameter—so-called ghosts or tachyonic
states, respectively Boulware:1972yco . In fact, the optical theorem implies
that the $S$-matrix based on the perturbation theory in such cases is non-
unitary. Obviously, both ghosts and tachyons are undesirable, and various
approaches have been invoked to remove them (or their effects) from the
observable predictions of the theory.

One of such approaches was based on the consideration of higher-than-quadratic
polynomials (in derivatives or curvatures) in the gravitational actions.
However, soon it was realized that with every polynomial theory, the ghost
states must inevitably appear in the physical spectrum of particles asorey .
Besides this, various cures have been proposed in the literature for dealing
with the ghosts–tachyon issue: Lee–Wick prescription lw ; lw2 , fakeons
anselmi ; anselmi2 ; anselmi3 , non-perturbative numerical methods tkach ;
smilga ; tomb ; Kaku , ghost instabilities Cusin:2015oza ; donoghue , non-
Hermitian $PT$-symmetric quantum mechanics bender ; bender2 , when applied to
conformal gravity mannheim1 ; mannheim2 ; mannheim3 ; mannheim4 , etc. (see
also other discussions in christ ; fakeunit ). One might even entertain the
idea that unitarity in quantum gravity is not a fundamental concept hartle ;
politzer ; loyd . We think that so far, none of the proposed solutions
conclusively solves the problem. However, we also express the hope that these
various research directions on the issue of unitarity in conformal gravity
will be revealed to have a lot in common. Maybe in the future they will merge
into one, conclusively and definitely solving this problem since in each of
them one finds a bunch of good theoretical ideas.

These last bad ghosts show up also in other disguises on the level of exact
and perturbative classical solutions salles1 ; salles2 , where they signal
instabilities of the theory in a form of various possible runaway solutions
CastardellidosReis:2019fgu ; Cusin:2015oza . Some progress with resolution of
these important problems can be seen in deOSalles:2018eon . The clearest way
to see the presence of Ostrogradsky ghosts (also called Boulware–Deser ghosts
in higher-derivative gravitational theory, or simply Weyl ghost in Weyl
conformal four-dimensional gravity) is to study the form of the perturbative
tree-level propagator of the theory, for example, around flat spacetime
background and to perform a simple fraction decomposition there. Bad
perturbative ghosts are defined as modes whose residues are negative in this
last decomposition of the propagator of all fluctuating modes of the theory.
They are dangerous for the unitarity of the resulting quantum theory but
various ingenious proposals were invented to circumvent these dangerous (or
even disastrous for the whole consistency of the theory) effects of them. The
FP ghosts are truly different.

Since FP ghosts are complex-valued, in order to give a real total action
$S_{{\rm tot}}$, they always have to show up in a complex pair, which we take
as $\bar{C}C$, as is usually performed with complex fields (such as fermions
or charged scalar fields). The remaining third ghost field $b_{\gamma}$ is
real and this condition does not apply; however, as one sees, the essence of
the FP construction of introducing anti-commuting ghost fields to the quantum
theory is to rewrite Faddeev–Popov determinants as expressions quadratic in
the corresponding ghost fields, so we also necessarily have a pair of third
ghost fields in the ghost actions below. Namely, we have

$S_{{\rm
gh}}=\\!\int\\!d^{d}x\sqrt{|g|}\left(\bar{C}^{\alpha}M_{\alpha}{}^{\beta}C_{\beta}+\frac{1}{2}b_{\gamma}G^{\gamma\delta}b_{\delta}\right).$
(18)

As is clear from the construction of this action $S_{{\rm gh}}$, the ghost
fields do not have higher self-interactions, and they do not interact between
themselves either. They do interact only with the background metric
$g_{\mu\nu}$, actually to infinite order in the last field. The requirement of
this construction for quadratic and diagonal action (between FP sector and the
sector of the third ghost) originates from the desire to rewrite various
positive powers of determinants of various differential operators as
integrations over some quantum fields under path integrals. Since these powers
are positive, one has to use Grassmannian field variables to achieve such
expansion. As is well known for this purpose, one can consistently concentrate
on the expressions which are quadratic in these new ghost variables. For
simplicity, higher terms with self-interactions of ghosts are not included in
this minimal FP construction. In actuality, the Grassmannian nature of ghosts
forces them to appear every time in the action with even powers (similar to
standard fermions), and the power two is here the minimal possible choice.

The numerical factors in (18) correspond to standard normalization of the
Lagrangians for canonically normalized scalar fields which are complex
(similar to FP ghosts) and real (similar to the third ghost). The two matrix-
differential operators $M_{\alpha}{}^{\beta}$ and $G^{\gamma\delta}$ (with the
canonical positions of spacetime indices as indicated) can be understood as
kinetic operators (defined on a general background) between respective ghost
fields. They give rise to the dynamics of the ghosts on the quantum level
(inside perturbative loops) and that is why the quantum propagator for these
ghost fields exists and is invertible. In conclusion, one can judiciously
write Feynman rules for the ghosts, in particular, the rules for ghost lines
(ghost propagators), and use them in construction of more complicated Feynman
diagrams of the theory. One notices that the two propagators are completely
independent and there is no mixing between FP ghosts and the third ghost
field. Moreover, the precise forms of these matrix-differential operators
$M_{\alpha}{}^{\beta}$ and $G^{\gamma\delta}$ can be found in general higher-
derivative gravitational theories, but they are not relevant in the present
discussion.

Simply speaking, the requirement to preserve the BRST remnant of gauge
invariance in the quantum theory forces, due to the presence of the FP
determinant, the introduction of FP two ghosts fields $\bar{C}^{\alpha}$ and
$C_{\beta}$, while the third ghost field $b_{\gamma}$ is introduced only when
one deals with higher-derivative theories (possessing higher than two-
derivative classical EOM). In actuality, in the case of two-derivative
theories, one can still deal with the third ghosts and their kinetic operator
$G^{\gamma\delta}$ (also called gauge-weighting operator), but this is a
superfluous construction. The simpler one is just to take $G^{\gamma\delta}$
in the form proportional to the contravariant metric function
$g^{\gamma\delta}$ multiplied by ultralocal $d$-dimensional Dirac delta
function in two spacetime points $x$ and $y$, without any differential
operator character. Then, in such a situation, the sector of the third ghost
is completely algebraic, since their action $S_{{\rm 3gh}}$ reduces to
$b^{2}=b^{\gamma}b_{\gamma}$ and then they do not have any dynamics, their
propagator does not exist, and they do not show up in perturbative loops.
Then, simply, one can safely forget about them on the quantum level and for
all covariant quantization purposes.

The third ghosts are instead helpful and are a convenient trick in higher-
derivative theories, especially when considered on general background or in
background field method formalism (so also in pure non-Abelian gauge
theories). Their role is related to the third and last part of addition in
$S_{{\rm tot}}$ in (16), namely, to the action functional denoted as $S_{{\rm
gfix}}$.

###  3.2 Gauge-Fixing

Now, we come to the discussion of this last element of the total covariant
action used in the Faddeev–Popov procedure for quantization. The last part of
the action in (16) is entirely responsible for adding the gauge-fixing in a
covariant manner to the gauge-invariant action, which for gravity was denoted
as $S_{{\rm grav}}$. Fixing of the gauge is a necessary step in the covariant
quantization procedure due to Faddeev and Popov. Typically, one achieves this
by exploiting some gauge-fixing conditions $\chi$ particularized to various
local symmetry groups present in the model. In the case of quantum gravity,
these conditions are standardly taken as linear local conditions in metric
fluctuation fields $h_{\mu\nu}$. Due to the character of a coordinate choice
related to a specific gauge choice, these gauge-fixing conditions should be
covariant vectors from the spacetime point of view $\chi_{\gamma}$ and they
should carry precisely one covariant index. Their Lorentz form is exactly the
same as for the third ghost fields $b_{\gamma}$. The precise form of the
linear gauge-fixing conditions $\chi_{\gamma}$ is not important for the
discussion that follows. We just remark that because of Lorentz covariance,
they should contain odd number of derivatives acting on the metric fluctuation
field $h_{\mu\nu}$ and, of course, in the smallest version, this means
precisely one covariant derivative (constructed with the usage of the
background metric $g_{\mu\nu}$) acting on one fluctuation field $h_{\mu\nu}$
(the last is due to the assumed linearity of the gauge-fixing condition
$\chi_{\gamma}$ in $h_{\mu\nu}$). However, of course, some forms of
$\chi_{\gamma}$ can be selected from some broad families of gauge-fixings and
then these conditions $\chi_{\gamma}$ contain a dependence on various gauge-
fixing parameters, usually called $\beta$, $\gamma$, etc.

In the minimal version, applicable with full success to two-derivative
theories, the gauge-fixing action $S_{{\rm gfix}}$ is enough to be constructed
from the square of the gauge fixing condition
$\chi^{2}=\chi^{\gamma}\chi_{\gamma}$ properly weighted by another gauge-
fixing parameter (which is often called $\alpha$), hence the name of the
gauge-weighting functional $G^{\gamma\delta}$; here, it is equal to
$\frac{1}{2}\alpha^{-1}g^{\gamma\delta}$. We need to remark that, in general,
the gauge-fixing functional $G^{\gamma\delta}$ should not be confused with the
gauge-fixing condition $\chi_{\gamma}$. In actuality, on the classical level,
to solve EOM one typically chooses a coordinate gauge for diffeomorphisms by a
condition $\chi_{\gamma}=\ell_{\gamma}$, where $\ell_{\gamma}$ is some
covariant vector field on the manifold; in particular, this last one can be
selected to be zero. What we do instead, on the level of action, is fix the
gauge by conditions which are quadratic in $\chi_{\gamma}$, which, of course,
are not gauge-invariant conditions. By varying these conditions on the level
of action with respect to the metric fluctuation field $h_{\mu\nu}$, one
obtains related linear gauge-fixing conditions, but still different than
original $\chi_{\gamma}$ (because they often contain more derivatives), to be
used during solving EOM or finding higher $n$-point functions of classical
theory. Moreover, by employing the gauge-fixing functional $G^{\gamma\delta}$,
we effectively perform an average over all conditions $\ell_{\gamma}$
according to ’t Hooft ideas, when, again, $G^{\gamma\delta}$ plays the role of
the weighting functional during this mathematical averaging process.

In a more general case, of theories with higher derivatives, the general form
of the gauge-fixing action is given by

$S_{{\rm
gfix}}=\\!\int\\!d^{d}x\sqrt{|g|}\frac{1}{2\alpha}\chi_{\gamma}G^{\gamma\delta}\chi_{\delta},$
(19)

where the matrix-differential operator $G^{\gamma\delta}$ is quite arbitrary
(besides concurring with Lorentz symmetry in its two contravariant indices
$\gamma$ and $\delta$, but it does not even have to be symmetric as an
operator in these two indices). The choice of the particular form of this
operator $G^{\gamma\delta}$ is dictated solely by practical purposes. Namely,
one also wants to be able to invert the kinetic operator between
(gravitational) fluctuations on a general background, in particular around
flat spacetime. This choice of $G^{\gamma\delta}$ is parameterized by various
gauge-fixing parameters, such as $\alpha$, $\beta,$ etc.

The situation around flat spacetime for metric fluctuations is that in two-
derivative Einsteinian theory, gauge-fixing achieved by addition of
$\frac{1}{2\alpha}\chi^{2}$ is completely sufficient, and the Hessian (kinetic
operator between fluctuations) is completely invertible as an operator. This
happens because, typically, the gauge conditions $\chi_{\gamma}$ already
contain one derivative acting on metric fluctuations $h_{\mu\nu}$ and, after
integrating by parts, one sees that from the term $\frac{1}{2\alpha}\chi^{2}$
one obtains precisely an addition to the kinetic term of graviton with exactly
two derivatives. This is sufficient to completely remove the degeneracy of the
kinetic operator between small gravitational perturbations and to find the
propagator (being the inverse of the Hessian) by algebraic inversion
procedure.

The general feature of theories with local symmetry is that the kinetic
operator (Hessian, or second variational derivative with respect to
fluctuating gauge modes) is degenerate as a consequence of local symmetries
present. To remove this problem and make the operator invertible, one needs to
add some gauge-fixing terms which explicitly break gauge symmetries, but are
still under full control due to the presence of controlling parameters, such
as $\alpha$, $\beta,$ etc. Only then can the perturbative propagator of gauge
modes be found, even around flat and flat gauge connection backgrounds.
Typically, such propagator then explicitly depends on gauge-fixing parameters
$\alpha$, $\beta,$ etc., but here this is not a problem since the propagator
by itself (or even when embodied as a two-point function in quantum theory) is
not an observable, physical quantity. The gauge-fixing functional $S_{{\rm
gfix}}$ is then crucial even for finding flat spacetime propagator.

Now, one can easily understand the nuisances of higher-derivative theories
(both gravitational or pure gauge theories). In such circumstances, the
kinetic operator is with higher derivatives, and addition of just a term
$\frac{1}{2\alpha}\chi^{2}$ containing always only two derivatives, does not
modify the degeneracy due to local gauge symmetry on the level of the Hessian
and on the level with the highest number of derivatives, which is then, here,
higher than two. One solves the problem by adding the full non-minimal form of
the gauge-fixing functional as presented in (19). One also has to require that
the matrix operator $G^{\gamma\delta}$ now contains some derivatives and
cannot be purely algebraic and proportional to $g^{\gamma\delta}$. Of course,
the precise number of derivatives in it depend on the number of higher
derivatives above two present in the construction of classical EOM of the
theory.

In general, the matrix-differential operator $G^{\gamma\delta}$ is not
algebraic and must be chosen as a self-adjoint operator for a consistent
construction of the quantum theory, or theory of linear perturbations to a
quadratic level, so on the level of the effective action to the accuracy of
one loop. Exactly the same operator $G^{\gamma\delta}$ appears in the kinetic
action for the third ghost fields. This is not a mere coincidence but a fact
stemming from the consistency of the whole covariant quantization formalism in
gauge theories. Hence, exactly the same considerations can be repeated for the
dynamics in the sector of third ghost fields. One now understands that they
are crucial for perturbative covariant treatment of gauge theories with higher
derivatives in the classical action.

As clearly seen from the action in (18) and resulting from the perturbative
vertices, the third ghosts interact only with background metric (or gauge)
fields. This is already a great simplification for their treatment in the
method of Feynman diagrams. For example, when restricted to the flat spacetime
perturbation theory, third ghosts may not couple to metric fluctuations
$h_{\mu\nu}$, and the background is trivial $\eta_{\mu\nu}$, so despite that
they have their own propagator and internal dynamics in the sector, one can
completely forget about their effects in all perturbative loops. For this, one
chooses the form of matrix-differential operator $G^{\gamma\delta}$ as
proportional to the metric $\eta^{\gamma\delta}$ and constructed out of
exclusively flat spacetime d’Alembertian operator $\partial^{2}$. Of course,
such a choice does remove the degeneracy in kinetic operator for gravitons
around flat spacetime and allows the perturbative propagator to be computed.
However, from some esthetical reasons, one can object that the form of the
operator $G^{\gamma\delta}$ when proportional to powers of $\partial^{2}$ is
in disagreement with general covariance of the framework used for
gravitational theories. Of course, this is the price for not having effects of
third ghost fields around flat spacetime.

One could instead use a generally covariant operator $G^{\gamma\delta}$
constructed now out of
$\square=\nabla^{2}=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, but then the
interactions between perturbative gravitons $h_{\mu\nu}$ and third ghost
fields $b_{\gamma}$ in the second part of (18) would be resurrected, even on
flat spacetime background. However, even on flat background when using a non-
covariantly-looking form of the matrix-differential operator
$G^{\gamma\delta}$ built with $\partial^{2}$, one can forget about the third
ghosts in Feynman diagrams, but one cannot forget about adding the gauge-
fixing term $\chi_{\gamma}G^{\gamma\delta}\chi_{\delta}$ to the action of the
theory to fix the degeneracy and to be able to find the perturbative
propagator.

When using the theory in other backgrounds than flat, the interactions of
third ghost fields $b_{\gamma}$ with gravitons $h_{\mu\nu}$ (perturbations
around this chosen background) are naturally generated. Therefore, the
presence of the third ghost field is very natural in higher-derivative gauge
theories when analyzed using background field method on general curved
backgrounds and in covariant manner. Moreover, one notices that the similar
arguments as raised above about the simplification happening on flat spacetime
background for the third ghost field cannot simplify the sector of complex
normal FP ghosts. This is because the matrix-differential operator
$M_{\alpha}{}^{\beta}$ cannot be constructed only using partial derivatives on
flat spacetime $\partial_{\mu}$ or their square in the form of the
d’Alembertian operator $\partial^{2}$. The construction of the FP ghosts
kinetic operator $M_{\alpha}{}^{\beta}$ requires that at least one derivative
there must be fully gravitationally (or gauge) covariant. That is, in the
framework of perturbative quantum gravity, this must be a generally covariant
derivative $\nabla_{\mu}$ since such is used in the infinitesimal generator
for local symmetry of diffeomorphisms. All this is a simple consequence of the
fact that the FP operator $M_{\alpha}{}^{\beta}$ is the _covariant_ change of
the gauge-fixing condition $\chi_{\gamma}$ under infinitesimal coordinate
transformations (diffeomorphisms in gravity or gauge transformations in full
generality). Of course, now the formalism looks much better when all
derivatives used in the operator $M_{\alpha}{}^{\beta}$ are generally
covariant, but this, of course, implies non-trivial form of interaction
between FP ghost fields and perturbative gravitons around any background and
up to infinite order in graviton fields.

###  3.3 Gravitational Action and Correlation Functions

Having discussed at length the issues related to gauge-fixing and ghost
actions and to their construction, now we come to the last ingredients present
in the expression for the general partition function in (15) in general
higher-derivative gravitational theories. This simplest element is the
gravitational action with higher derivatives $S_{{\rm grav}}$. In a general
setting, when we work within background field method, this functional

$S_{{\rm grav}}=S_{{\rm grav}}\left[\hat{g}_{\mu\nu}\right]=S_{{\rm
grav}}\left[g_{\mu\nu},h_{\mu\nu}\right]$ (20)

depends on both the fluctuation (graviton) fields $h_{\mu\nu}$ (over which we
path-integrate) as well as on the background metric $g_{\mu\nu}$. Only in the
special case of flat spacetime is this entirely a functional of only graviton
fields: $S_{{\rm grav}}\left[h_{\mu\nu}\right]$. In the general situation,
this action, when understood classically, is with all local symmetries
realized on the level of full metric tensor
$\hat{g}_{\mu\nu}=g_{\mu\nu}+h_{\mu\nu}$, hence it must be written generally
covariantly with respect to the full metric $\hat{g}_{\mu\nu}$. The
construction of such actions with higher derivatives is a standard topic that
we will not repeat here. As we emphasized in previous sections, we will deal
with actions that, on top of local symmetries related to diffeomorphism group,
also possess the invariance under local conformal transformations understood
in the proper GR framework. As explained earlier, the construction of such
actions is more tricky and the results are more constrained.

Above, we have discussed the issues of how to build parts of the total action
in (16) responsible for gauge-fixing, for fully consistent ghost sector(s),
and also for the main gravitational part. This discussion will be very useful
and very beneficial when we will deal in a strikingly similar manner with
quantization of local conformal symmetry in the next section. The form of
analysis of presence or algebraic triviality of various sectors, various
fields, or various operators for local conformal symmetry will parallel much
of what we have just said in this section. There are many common similarities
between quantization of diffeomorphism and local conformal symmetry, since
both of them fit in the broad framework of general gauge theories. We will see
that the situation with all additional quantum sectors present in $S_{{\rm
tot}}$ in (16), as it is in general HD gravitational theories, for local
conformal symmetry will find its counterparts in some models of conformal
gravity in some dimensions and with some specific conformal gauge choices.

Finally, we remark that in the form of the general partition function in (15),
we added in the exponent of the integrand a coupling of graviton fields
$h_{\mu\nu}$ to an external current (could be viewed as an external
energy–momentum tensor (EMT) of some external matter fields) $J^{\mu\nu}$
preserving both locality and general covariance of the extended action. This
is a known trick in quantum field theory since all Green functions are now, in
principle, obtainable by respective variational differentiation of the full
partition function $Z_{{\rm true}}\left[J^{\mu\nu}\right]$ with respect to the
current field $J^{\mu\nu}(x)$ and then setting its vanishing value. We add
that in general background field method, the partition function in (15)
depends also functionally on the background metric $g_{\mu\nu}$ which was not
path-integrated out in this formula, hence it is obvious that the full non-
perturbative partition function $Z_{{\rm grav}}$ is, in a sense, dependent on
the choice of the classical background.

Moreover, another emanation of ghostly nature of fields $\bar{C}^{\alpha}$,
$C_{\beta}$, and $b_{\gamma}$ is that they do not come with any classical
background values (although some generalizations in this direction are
possible Buchbinder:1992rb ) and we path-integrate only over their, in a
sense, quantum fluctuation fields. These fields possess only quantum loop
nature, as explained above. This means that from our form of the partition
function in (15) we cannot, even in principle, obtain correlation functions
with any of these ghost fields on external lines, because we can differentiate
only with respect to the current $J^{\mu\nu}$ naturally coupled to the
graviton field $h_{\mu\nu}$. Therefore, within this view it is senseless to
speak about Green functions with good ghosts on external legs, or even about
scattering amplitudes with ghosts on external lines, since it is without sense
to place them on-shell since they do not come with any well-defined classical
EOM. However, some generalizations in these directions are also possible, and
a quantum theory with correlators between ghosts and gauge fields can be also
formulated, and for some aspects these are even fruitful extensions.

Additionally, the Green functions one obtains from (15) when the external
current $J^{\mu\nu}$ is turned on are, as it is well known, not properly
gauge-invariant. They would be in the absence of this current, but then from
the partition function we could obtain only one sensible amplitude, namely,
the vacuum persistence amplitude or the transition of perturbative vacuum into
vacuum. Of course, such an amplitude has its meaning in gravitational theory,
_per se_ , although it is typical in usual quantum field theory around flat
spacetime or in statistical mechanics to treat this one number as just a
convenient normalization of other Green functions. Along with standard
knowledge here, we know that the presence of external current violates gauge
invariance of the full system of quantum correlators. However, the resolution
comes from the fact that the physical observables are typically taken not as
rough correlators, but as $S$-matrix elements between dressed and on-shell
fields derived from general $n$-point functions. For such objects, not for
mere quantum correlation functions, Weinberg’s theorems guarantee that
physical observables are gauge-invariant and independent of various gauge-
fixing parameters used at any intermediate step of computations. We add that
truly physical observables, such as cross sections, are obtained from
invariant squares (complex absolute moduli) of the $S$-matrix elements
(amplitudes) to remove complex phases that are often too difficult to measure.
This corresponds to measuring of probabilities and densities of them, but not
of mere scattering or transition amplitudes.

One can prove, using standard tools of general gauge theories, that the
partition function in the absence of the coupling to the external current
$J^{\mu\nu}$ is completely independent of the choice of the gauge-fixing
conditions $\chi_{\gamma}$ used to define it. The form of $\chi_{\gamma}$ and
the matrix-differential operator $G^{\gamma\delta}$ can be arbitrarily changed
and the resulting $Z_{{\rm grav}}$ remains the same. We only remark that for
different, rather than standard and minimal, choices of these auxiliary
elements needed for the definition of quantum theory, the explicit computation
may be very difficult, and a much more tedious task than for minimal cases.
However, the general proof holds for any local symmetries. This is a gauge-
fixing independence of the true representation for the partition function $Z$
without coupling to an external current. When one adds this coupling of the
form $h_{\mu\nu}J^{\mu\nu}$, then all Green functions (such as propagators or
effective higher-order vertices) are gauge-dependent objects. This happens
entirely due to the term $h_{\mu\nu}J^{\mu\nu}$ added to the total action
$S_{{\rm tot}}$. At the end, we indeed compute higher $n$-point functions
using the functional $Z\left[J^{\mu\nu}\right]$ with the current $J^{\mu\nu}$
turned on and by consecutive variational differentiations with respect to it
(and only typically after having performed the variations, setting it to zero
value). Finally, due to the mentioned equivalence theorems due to Weinberg, on
the level of single elements of the $S$-matrix, we see full gauge-independence
and also full gauge-fixing-independence. This last one signifies that nothing
there depends on arbitrary elements $\chi_{\gamma}$ and $G^{\gamma\delta}$ of
the FP construction related to fixation of gauges for all local symmetries
present in the models under study.

##  4 Early Attempts of Quantization of Conformal Symmetry

###  4.1 A Need for Conformal Gauge-Fixing

Now, we discuss the issues related to conformal gauge-fixing. This parallels
the discussion of fixing the gauge for diffeomorphism symmetry in general HD
gravitational theories. This issue is at the heart of covariant and
perturbative quantization of conformal gravity in any even dimension $d>2$. We
just emphasize that since local conformal symmetry is a different symmetry
than diffeomorphisms, then on the level of FP quantization it will come with a
different set of gauge-fixing conditions, different set of ghosts, different
weighting functionals, and different kinetic operators in the ghost sectors.
We will, in what follows, distinguish these new objects related to conformal
symmetry from ones related to diffeomorphism symmetries by appending the
subscript “c” on related quantities in the formalism of covariant FP
quantization for conformal symmetry. Besides this, there is an obvious fact
that the action for such a conformal gravitational theory is different than a
generic action of globally scale-invariant HD gravitational theory in any even
dimension $d>2$ of spacetime. We also emphasize that here it is not enough to
constrain and gauge-fix only diffeomorphism symmetry of an HD gravity, since
the conformal symmetry is in the local version too, and this local conformal
gauge invariance has its usual consequences as any local symmetry has on the
formalism of quantization and also on the structure of field configurations
and physical representatives of fields there. The only similarity and common
overlap between these two local symmetries is that they will use the same
field as their gauge potential—namely, the covariant tensor of metric
fluctuation field $h_{\mu\nu}(x)$, on which transformations act and which will
become a proper quantum variable. This is because we do not introduce here the
Weyl gauge connection field (an analogue of photon field for conformal
symmetry).

First and foremost, however, we need to fix the conformal gauge in quantum
conformal gravity, because the graviton kinetic operator in conformally
invariant theory is degenerate, and some steps with its processing are not
well-defined. This degeneracy is again the consequence of conformal symmetry
present in the local version. Moreover, we have to properly extract the factor
of the volume of the conformal group for the start of the successful FP
quantization procedure. As it is well known in general gauge theories, the
full space of configuration fields comes with a big over-baggage. This
redundancy comes from the fact that gauge fields are a very redundant
description of the physical system with a lot of symmetries. In actuality, in
the covariant framework of quantization, the more symmetries present, the more
fields need to be added, and the description is more redundant. This is in
clear opposition to the case of canonical treatment of gauge theories, when
more symmetric theories come with more constraints and the space of physical
field configurations is effectively smaller. For example, quantum conformal
gravity is a system with very particular and highly constrained relativistic
dynamics. In two special dimensions, $d=2$, the number of constraints is so
big that the quantum gravitational system is practically over-constrained and
it becomes a completely integrable and even solvable system of quantum
(conformal) gravitational field theory.

The field space in conformal gravity, besides the redundancies related to the
diffeomorphisms, has the structure of orbits related to conformal
transformations. We have to find proper conformal representatives of field
configurations, since in the covariant formalism, different field
configurations, but related by some conformal transformations, physically
describe the same configurations. Therefore, one has to remove this multiple
counting of the same physical configurations in a true representation for the
partition function of the quantum conformal gravitational system. In other
words, one has to properly extract the volume of the conformal group, the
five-dimensional part (or local part with GR scale transformations from (3))
of the full 15-element conformal group ${\rm Conf}$ in $d=4$ spacetime
dimensions. As usual in the FP procedure, this restriction of the path-
integration to a physical subspace spanned only by physical representatives of
gauge configurations, is achieved by a choice of the particular conformal
gauge. For the consistency of the whole procedure, we must ensure that we
integrate precisely once over each physical configurations, which must be
different from the point of diffeomorphism symmetries, and also they cannot be
conformally related.

The geometric structure of the full conformal gauge configuration space is an
interesting topic. However, as often performed in physics, rather than
applying very formal methods of computation, we use heuristic ones due to
Faddeev and Popov. Instead of describing the structure of conformal orbits in
the space of all metric fluctuation fields $h_{\mu\nu}(x)$ by picking some
suitable coordinate system for this infinite-dimensional linear space, and
instead of restricting the path integral only to a subspace which is physical
and crosses each conformal orbit precisely once, we prefer to add a conformal
gauge-fixing to the theory given by a suitable functional. We could restrict
the integration to this, usually a nonlinear curved subspace of lower but
still infinite dimensions and of conformal field representatives, and we could
introduce geometric coordinates there. However, due to explicit and intrinsic
nonlinearities present in such a natural geometric framework, we adopt a
different heuristic method. We still want to integrate over the full vastly
redundant configuration space, but we need then to modify the integrand of the
partition function and make it different from the naive original integrand
which was simply $\exp\left(iS_{{\rm grav}}\right)$. The modification should
take into account that we want to restrict the integration to some “physical”
subspace which completely solves the gauge-fixing condition with special
attention given to the fact that this submanifold may be extrinsically curved
in the embedding configuration space. All these geometrically sound conditions
and demands are taken fully by the addition of the FP determinant to the
integrand under path integral. By properly exponentiating this FP determinant,
one sees that the ensuing total action of the theory is modified and it does
not consist only of $S_{{\rm grav}}$: there are necessary quantum additions to
it with new quantum dynamical fields, new ensuing actions for them, and a
gauge-fixing part for local conformal symmetry.

On the other hand, the geometric condition that each conformal orbit crosses
the space of conformal representatives precisely once is typically difficult
to check explicitly. One can quite easily ensure this in a small neighborhood
of the first crossing point, but for further neighborhoods, this becomes more
challenging. This problem was first described by Gribov in gauge theories and
the resulting ambiguity brings his name, together with the possibility of
having different theories with different non-perturbative vacua, so-called
Gribov copies, coming from the same quantum partition function defined by the
same classical theory. We will discuss the tentative solution to the Gribov
ambiguity issue of local conformal symmetry in conformal gravity in one of the
later publications.

As a simple choice of the conformal gauge we may choose to fix the value of
the trace of the metric fluctuations

$h=g^{\mu\nu}h_{\mu\nu}$ (21)

for various quantum computations. This is one of the possible conformal gauge-
fixing conditions. In principle, there could be many of such gauge conditions.
We restrict our attention only to gauge conditions enforced on the metric
fluctuation field $h_{\mu\nu}(x)$, since this is the gauge potential in our
gravitational context available at our disposal. Moreover, we focus on linear
gauge-fixing conditions which do not contain higher than the first power of
the fluctuation $h_{\mu\nu}(x)$. This restricts quite a lot class of possible
conformal gauges. The trace of the metric is a scalar from the point of view
of flat spacetime Lorentz symmetry, hence this condition is quite simple.

It is important to note the following obvious thing: the gauge-fixing
condition has to be non-invariant under transformations of symmetries that it
is supposed to fix. By using a conformal invariant constructed out of the
metric fluctuation field $h_{\mu\nu}$, one would not sufficiently specify any
conformal gauge since arbitrary conformal transformations would still keep
this invariant untouched, hence there would still be a freedom in performing
them. One has to check that indeed the trace of the metric fluctuations $h$,
contrary to the expressions as built in footnote 4 , transforms non-trivially
under conformal transformations, and hence fixing it to some value constrains
the freedom of conformal transformations to the level that none of them can be
additionally performed anymore. This is detailed in the section where there is
a discussion of the solution to the conformal Gribov ambiguity problem.

In general, we can form a family of gauges by requiring that

$\chi_{c}=\chi_{c}\left[h_{\mu\nu}\right]=h-\ell_{c}=0$ (22)

for arbitrary (also spacetime-dependent) (non-dynamical) field
$\ell_{c}=\ell_{c}(x)$. The choice in (22) we will call the simple conformal
gauge. The choice we made for the scalar was not a vector, as a conformal
gauge condition is dictated by the fact that the parameter $\Omega=\Omega(x)$
of conformal transformation is also a scalar, so it does not carry any
spacetime or Lorentz index. This is an important property of the conformal
transformations, in distinction to diffeomorphisms when such a choice must be
made for a vector $\chi_{\mu}$. One can see that the vector (or, in general,
any tensor of odd rank) will not work well with the FP quantization procedure
since then, for local conformal symmetry, the FP operator would be with odd
number of spacetime indices (two from the infinitesimal generator in (14) and
three from the variational derivative of a hypothetical vector conformal
gauge-fixing condition $\chi_{c,\mu}$ with respect to the metric field which
is here a charged potential field for conformal symmetry; at the end we
contract two pairs of indices, but the parity remains unchanged). Of course,
then it is impossible to take a determinant of such an operator and then
naively the whole FP procedure fails777The same arguments correctly show that
in the case of diffeomorphism symmetry, due to the fact that the infinitesimal
generator of GCT comes with three indices, and we use a vector gauge-fixing
$\chi_{\mu}$, the contraction in two pairs of indices results in only two
indices left on the FP operator whose determinant can be easily defined and
taken.. One must also admit that the scalar conformal gauge-fixing condition
$\chi_{c}$ is much simpler than the next theoretically possible tensorial one
$\chi_{c,\mu\nu}$. For simplicity, here, we will restrict our considerations
only to this first choice.

The conditions in family (22) are linear admissible gauges, but they do not
contain any derivatives acting on the potential field $h_{\mu\nu}$. Of course,
this choice is also dictated by simplicity, however one can easily complicate
the simple choice from (22) by inserting in the middle some differential
operator constructed with the background metric and being covariant with
respect to it. For example, we can use GR-box operator
$\square=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ or simply two derivatives
$\nabla_{\mu}\nabla_{\nu}$ acting on the field of metric fluctuations
$h_{\mu\nu}$. By the arguments stipulated above, it is obvious that we must
use an even number of derivatives. These are more complicated gauges; however,
they may have some advantages over simple conformal gauges.We will also show
later that the physical results properly do not depend on this choice of the
conformal gauge-fixing conditions.

###  4.2 Hidden Local Conformal Symmetry and Veltman-like Discontinuities

According to the general philosophy, one must be very careful with all local
symmetries present in a model that is being prepared for quantization, both in
the visible (manifest) form and also in invisible forms. For example, if one
superficially looks at the theory

$S=\\!\int\\!d^{4}x\sqrt{|g|}\left(R_{\mu\nu}^{2}-\frac{1}{3}R^{2}\right),$
(23)

then one sees only diffeomorphic symmetries of general relativity and a theory
with HD gravitational action. One does not recognize that this action is
actually conformally invariant since the usage of Gauss–Bonnet identity in
$d=4$ dimensions valid under spacetime volume integral reduces manifestly
conformally-invariant-looking action with the square of the Weyl tensor
$C^{2}$ to this form above in (23). However, the conformal symmetry is still
present there but in a hidden form. It is still there in a local version and
it should be always properly treated and gauge-fixed for the quantization
purposes. One sees the problems with a naive interpretation of the action in
(23), if one attempts to compute a propagator for gravitational fluctuations
in such a theory. The standard tricks with minimal gauge-fixing for (only)
diffeomorphisms do not work here since after them the kinetic operator for
small fluctuations is still degenerate. One may rediscover in this way that
the theory is with enhancement of symmetries and that these symmetries need a
special treatment—basically, the hidden conformal symmetry should be revealed.
In conclusion, no matter what our knowledge is of the symmetries of the
theory, we always have to be prepared that for every not-manifest symmetry
that we find enhanced and hidden, we must perform a proper separate FP
quantization procedure (a special treatment of local gauge symmetries) if we
want to move to a quantum level with such a theory.

All gauge symmetries of a given model should be treated with care, gauge-
fixed, and their violations by inclusion of explicit gauge-fixing conditions
should be controlled in order not to produce any unwanted anomaly of gauge
symmetries on the quantum level. The latter ones are signaling that something
very bad is happening with the quantum version of the theory and that the
theory is sick and sustains itself only on the classical level and does not
survive the quantization procedure. This is our general philosophy in this
paper to treat all local gauge symmetries with their due respect, and they
should not be neglected (or overlooked) for the quantization process, even if
one knows _a posteriori_ that these local symmetries cease to exist on the
quantum level (for example, due to anomalies or explicit breaking), but this
is caused by some other physical circumstances related to other physical
mechanisms or phenomena. Basically, we do not want the quantization act by
itself to produce some fatal violations of local gauge symmetries that cannot
be recovered on the quantum level. We hope that all local symmetries are here
with adequate respect and that their breaking is performed in a fully
controllable manner. This is similar to what is performed, for example, with
the covariant treatment of gauge symmetries in standard YM theories, where one
has full control (due to the introduction of gauge-fixing parameters). One
introduces gauge-fixing conditions, properly parametrized, and as a result,
the gluon propagator in some gauges is able to be derived, allowing for
computation of more complicated Green functions at loop levels and off-shell.
However, at the end, one recovers on-shell quantum physical results as gauge-
invariant, gauge-independent, and gauge-fixing-independent. We want to apply
the same attitude, following from a successful FP quantization when applied to
YM theories, here to other symmetries in the local form, in particular to
conformal symmetry present in the gravitational context.

The action of conformal gravity in $d=4$ spacetime dimensions is typically
taken as $S=\\!\int\\!d^{4}x\sqrt{|g|}C^{2}$, since in this form conformal
invariance is manifest. One can perform integration by parts and neglect (or
add—depending on the point of view) some boundary terms. They are important
for the canonical approach to quantization of gravitational gauge theories. By
neglecting the Gauss–Bonnet (GB) density term (see also (31)), which
integrates to the boundary term, one can rewrite the original action in the
form

$S=2\\!\int\\!d^{4}x\sqrt{|g|}\left(R_{\mu\nu}^{2}-\frac{1}{3}R^{2}\right).$
(24)

Its properties under conformal transformations are the same (up to the
boundary terms, so one neglects the derivatives of the parameter $\Omega(x)$
on the boundary). The action as written above has simplicity and one notices
the fact that now one removes the terms with Riemann tensor. Since it is a
completely equivalent action on the classical level, then one can study its
exact solutions. For example, in gravitational vacuum situations, one finds
immediately that all Ricci-flat solutions are also exact vacuum solutions of
this higher-derivative theory exactsol .

Besides this nice feature of conformal gravity in $d=4$, the reason for using
the action of conformal gravity in the form (24) is also of the more
theoretical nature. One keeps neglecting the GB terms presently hidden in the
original action with Weyl tensors. The reason for doing this is evident, when
the canonical charges Henneaux:1992ig  subject to appropriate boundary
conditions are constructed in this theory. It is worthwhile to mention that
the surface terms as present implicitly in (24) render the conformal gravity
action (24) to have well-defined functional derivatives, under some suitable
boundary conditions (such as, for example, on AdS asymptotics
Irakleidou:2014vla ; Irakleidou:2016xot ). Not surprisingly, the resulting
total Hamiltonian derived from the form of the action in (24) is already an
improved gauge generator under these boundary conditions, so then it does not
need an improvement by surface integrals. Additionally, due to the specific
form of one of the secondary constraints, the extended Hamiltonian arising
from (24) also has well-defined functional derivatives under these boundary
conditions. Therefore, the inclusion of boundary terms and usage of the action
as in (24) is more favored theoretically, since asymptotic charges of
conformal gravity are then clearly extracted. We remark that the Hamiltonian
here is the generator of all local gauge transformations present in the system
(diffeomorphisms and local conformal (Weyl) rescalings).

According to what we said before, we would like to have a quantum theory
resulting from the covariant quantization procedure such that it treats well
(as much as possible) the local conformal symmetry present in some
gravitational models. The gauge symmetry has to be fixed, but at the same time
its violation has to be fully controlled by gauge-fixing parameters, which
later drop out from physical observables. This is the essence of the covariant
quantization act, but the quantization procedure is not the end of the story
on the quantum level. It is well known that one also has to regularize the
theory on the quantum level due to UV-divergences arising due to perturbative
loop diagrams. Such regularization procedures also have to satisfy some
stringent conditions. In particular, they have to preserve as much symmetry of
the original classical theory as possible. Therefore, they have to be selected
from a set of covariant regularization procedures. They must not touch the
gauge symmetries of the model; for example, they cannot be based on sharp
ultraviolet momentum cutoff. The preferred regularization for ordinary gauge
theories is dimensional regularization or covariant $\zeta$-function
regularization.

However, in the setup of a gravitational theory also enjoying conformal
symmetry, one cannot play with the dimensionality of regularized spacetime,
since the dimension is a very delicate feature there and conformal symmetry
depending on the form of a gravitational action is realized only in a fixed
number of dimensions. They cannot be even perturbed by a small
$\varepsilon>0$. Other covariant regularization schemes for UV-divergences
have to be employed in quantum conformal gravity to regularize these
divergences in a fully conformally invariant manner. Moreover, one requires
that the counterterms should be written in a conformally covariant way, so the
divergent part of the effective action is also a conformally invariant
functional in a fixed number of dimensions. Then, in such a setup of quantum
conformal gravity, one could say naively that the conformal symmetry is
preserved by the UV-regularization procedure and the effective action in
quantum theory also seems to look as with full conformal symmetry
realized888As introduced in Section 2, for the full quantum conformality, one
must not have any UV-divergences. Hence, the developments above are only
formal and the quantum effective action in theory with non-vanishing
divergences is only superficially conformally invariant.. In conclusion, both
the quantization procedure and the regularization of ultraviolet physics in
quantum field theory approaches to conformal gravity have to treat the
delicate local conformal symmetry gently.