Quantum Conformal Gravity

We present the manifestly covariant canonical operator formalism of a Weyl invariant (or equivalently, a locally scale invariant) gravity whose classical action consists of the well-known conformal gravity and Weyl invariant scalar-tensor gravity, on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism. It is shown that there exists a Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(8|8)$ global symmetry as in Einstein's general relativity, which should be contrasted to the case of only the Weyl invariant scalar-tensor gravity where we have a more extended Poincar${\rm{\acute{e}}}$-like $\mathit{IOSp}(10|10)$ global symmetry. This reduction of the global symmetry is attributed to the presence of the St\"{u}ckelberg symmetry.


Introduction
General relativity (GR) by Einstein is a mathematically beautiful and physically successful classical theory.It is constructed by two fundamental principles, general coordinate invariance and equivalence principle, and is described in terms of Riemann geometry where the metric tensor is regarded as fundamental dynamical variables.Since general relativity can account for many astrophysical and cosmological phenomena without any conflict with observations and experiments done so far, there is no need for modifying it at least at large distance scales.
On the other hand, it is a well-established fact that the physics must be described by quantum field theory (QFT).Unfortunately, it seems to be difficult to construct a quantum field theory of general relativity owing to its nonrenormalizability although the perturbative non-renormalizability has nothing to do with the consistency of the theory.In the interest of renormalizability, it is natural to alter the Einstein-Hilbert Lagrangian by adding to it the most general quadratic Lagrangian L of dimension four at most: which is known as a renormalizable gravitational theory [1].However, a notorious problem happens and it is associated with the last term involving conformal tensor C µνρσ : As far as this term exists in the Lagrangian, we have a spin-2 massive ghost which makes not only the classical theory be unstable because of unbounded energy from below but also the quantum theory be non-unitary owing to the ghost with negative norm.At extremely high energies, it is expected that the kinetic term dominates the mass term and as a result all particles can be effectively regarded as massless particles.In such a situation, a global or local scale symmetry naturally appears in addition to general coordinate invariance.Since the global scale symmetry could be broken by the no-hair theorem of black holes in a curved space-time [2], it is plausible to suppose that the local scale symmetry, which we call Weyl symmetry, plays a role at high energies.If we impose the Weyl symmetry on the Lagrangian (1.1) and assume that Einstein's general relativity is restored at low energies, we would have the following Lagrangian: where in the unitary gauge φ = 3 4πG the terms except for conformal gravity on the right-hand side (RHS) produce the Einstein-Hilbert term.
In this article, as the first step for understanding the problem of the massive ghost, we wish to construct the manifestly covariant canonical operator formalism of the Weyl invariant gravity as defined in the Lagrangian (1.2) on the basis of the Becchi-Rouet-Stora-Tyupin (BRST) formalism [3].We will see that this construction is very subtle since we have to carefully pick up gauge fixing conditions in order to introduce as many independent BRST transformations as possible.
The paper is organized as follows.In Section 2, we review the classical theory where the Lagrangian is constructed out of the well-known conformal gravity and the Weyl invariant scalar-tensor gravity.In Section 3, we shed light on quantum aspects of our theory.We will find that the existence of the Stückelberg symmetry makes it impossible to construct three independent BRST transformations and it allows us to make only two independent BRST transformations.In Section 4, we perform the canonical quantization.In Section 5, on the basis of the canonical formalism in Section 4, we derive various equaltime (anti)commutation relations.In Section 6, we analyze asymptotic fields by expanding not only the metric around a flat Minkowski metric but also the scalar field around a constant background.In Section 7, we derive the four-dimensional (anti)commutation relations from the equal-time (anti)commutation relations and clarify that the physical modes of our theory are composed of a massive ghost with indefinite norm and a massless graviton, and the other modes belong to the BRST quartets which appear in the physical subspace only as zero norm states.The final section is devoted to discussion.
Two appendices are put for technical details.In Appendix A, a derivation of the equal-time commutation relation between Ȧµ and b µ is given, and in Appendix B we present various equal-time (anti)commutation relations which are necessary in deriving the four-dimensional (anti)commutation relations in Section 7.

Classical theory
In this section, we consider a classical gravitational theory which is invariant under both general coordinate transformation (GCT) and Weyl transformation (or equivalently, a local scale transformation) in four dimensional Riemann geometry.Our classical Lagrangian consists of Weyl invariant scalar-tensor gravity [4] and conformal gravity 1 where Here φ is a real scalar field with a ghost-like kinetic term, R the scalar curvature, α c a dimensionless positive coupling constant (α c > 0) and C µνρσ is conformal tensor defined as In order to perform the canonical quantization, it is more convenient to introduce an auxiliary symmetric tensor K µν = K νµ and a Stückelberg-like vector field A µ , 2 and rewrite L CG , which is the Lagrangian of conformal gravity, into a form [5][6][7][8]: 1 We follow the notation and conventions of Misner-Thorne-Wheeler (MTW) textbook [2].Lowercase Greek letters µ, ν, . . .and Latin ones i, j, . . .are used for spacetime and spatial indices, respectively; for instance, µ = 0, 1, 2, 3 and i = 1, 2, 3.The Riemann curvature tensor and the Ricci tensor are, respectively, defined by where G µν ≡ R µν − 1 2 g µν R denotes the Einstein tensor, and γ and α are dimensionless coupling constants which obey a relation where α > 0. It is easy to see that carrying out the path integral over CG produces the Lagrangian of conformal gravity L CG .Actually, taking the variation of K µν leads to the equation Moreover, taking the trace of this equation yields Inserting (2.7) to (2.6) gives us the expression of K µν Finally, substituting Eqs.(2.7) and (2.8) into the Lagrangian (2.4) and using the relation (2.5), we can arrive at the Lagrangian of conformal gravity, L CG in (2.2) up to surface terms.This can be achieved by use of the identity where I is defined as which is locally a total derivative in four dimensions.From now on, as a classical Lagrangian L c we take a linear combination of The classical Lagrangian L c is invariant under three local transformations, those are, infinitesimal general coordinate transformation (GCT) δ (1) , Weyl transformation δ (2) and Stückelberg transformation δ (3) .Concretely, the GCT takes the form δ (1) (2.12) As for the Weyl transformation, we have δ (2) g µν = 2Λg µν , δ (2) Note that δ (2) K µν has been obtained via Eq.(2.8).Finally, the Stückelberg transformation is given by In the above, ξ µ , Λ and ε µ are infinitesimal transformation parameters.
To close this section, let us count the number of phyical degrees of freedom since it is known that this counting is more subtle in higher derivative theories than in conventional second-order derivative theories [9,10].In the formalism at hand, however, the introduction of the auxiliary field K µν makes it possible to rewrite conformal gravity with fourth-order derivatives to a second-order derivative theory, so we can apply the usual counting method.The fields g µν , φ, K µν and A µ have 10, 1, 10 and 4 degrees of freedom, respectively.We have three kinds of local symmetries, those are, the GCT, Weyl and Stückelberg symmetries with 4, 1 and 4 degrees of freedom, respectively.Thus, we have totally (10 + 1 + 10 + 4) − (4 + 1 + 4) × 2 = 7 physical degrees of freedom, which will turn out to be the massless graviton of 2 physical degrees with positive-definite norm and the massive ghost of spin-2 of 5 degrees with indefinite norm.

Quantum theory
To fix three local symmetries and obtain a BRST invariant quantum Lagrangian, we have to introduce three kinds of gauge fixing conditions and the corresponding Faddeev-Popov (FP) ghost terms in the classical Lagrangian (2.11).In our previous papers [11][12][13], we have constructed two independent BRST transformations corresponding to general coordinate transformation (GCT) and Weyl transformation in the sense that the two nilpotent BRST charges anticommute with each other.To do so, it has been emphasized that a gauge condition for one local symmetry must respect the other symmetry [12].Concretely speaking, a gauge condition for the GCT must be invariant under the Weyl transformation while a gauge condition for the Weyl transformation must be so under the GCT.We would like to stress that the existence of independent BRST transformations makes it easy to derive many equal-time (anti)commutation relations with the help of the canonical (anti)commutation relations and field equations as can be seen in Section 5, However, it will turn out that we cannot find such suitable gauge fixing conditions in the present formalism since the gauge fixing condition for the Stückelberg gauge transformation necessarily breaks the Weyl symmetry.Thus, in this article, instead of making three independent BRST charges we will construct only two independent BRST charges, by which physical states and observables are defined consistently.
The suitable gauge condition for the GCT, which preserves the maximal global symmetry as will be seen later, is given by "the extended de Donder gauge condition" [12]: where we have defined gµν ≡ √ −gg µν .This gauge condition breaks the GCT (2.12) but is invariant under both the Weyl transformation (2.13) and the Stückelberg transformation (2.14).
As the gauge fixing condition for the Weyl transformation, we shall choose, what we call, "the traceless gauge condition": Let us note that the traceless gauge condition is invariant under the GCT (2.12) and the Stückelberg transformation (2.14).
Incidentally, what is called, "the scalar gauge": which, together with the extended de Donder gauge condition (3.1), assures the masslessness of the dilaton, turns out be inappropriate since it does not fix any dynamical degree of freedom associated with the gauge field A µ .Finally, let us consider the gauge fixing condition for the Stückelberg transformation.It is here that we cannot find the gauge fixing condition which breaks the Stückelberg transformation but is invariant under both the GCT and the Weyl transformation.Let us argue this issue in detail since this problem is interesting in its own right.For instance, the gauge fixing condition for the Stückelberg transformation which is invariant under the GCT and the Weyl transformation would be where F µν is the field strength of the Stückelberg vector field A µ defined as However, the existence of the identity ∂ µ ∂ ν ( √ −gF µν ) = 0 implies that the gauge condition gives us only three independent equations.To supplement one more equation, we further impose a gauge-fixing condition 3 Let us note that this gauge condition breaks the general coordinate invariance, but it is invariant under the general linear transformation GL(4).It is also straightforward to show that gauge conditions for the other local symmetries, which will be discussed later, do not violate the GL(4) symmetry.Thus, the quantum Lagrangian is also invariant under the GL(4). 4As seen in Eq. (2.7), this gauge condition is equivalent to the condition of the vanishing scalar curvature, R = 0 at the classical level [14][15][16].
At first sight, the gauge conditions (3.4) and (3.6) might be a suitable choice as the gauge condition for the Stückelberg transformation but after integrating over the auxiliary symmetric tensor K µν , we can restore the term C 2 µνρσ .This fact implies that since the Stückelberg vector field A µ plays no role in removing the second-class constraint, the gauge conditions (3.4) and (3.6) do not do the job for quantizing conformal gravity properly, and we are therefore led to imposing the gauge condition on K µν .Then, a natural gauge fixing condition reads Since this gauge condition is manifestly invariant under the GCT but is not so under the Weyl transformation, we cannot define three independent BRST charges, but only two independent BRST charges.We will call this gauge condition (3.7) "the K-gauge".
The BRST transformation corresponding to the GCT, which is called GCT BRST transformation δ (1) B , can be obtained from (2.12) by replacing the transformation parameter ξ µ with the Faddeev-Popov (FP) ghost c µ δ (1) where cµ and B µ are respectively an antighost and a Nakanishi-Lautrup (NL) field, and a new NL field b µ is defined as which will be used in place of B µ in what follows.
On the other hand, because of the K-gauge condition (3.7), in order to construct another BRST transformation which is independent of the GCT BRST transformation (3.8), we make a BRST transformation in a such way that it involves both the Weyl and the Stückelberg transformations simultaneously.This new BRST transformation δ B , which we call "WS BRST transformation", can be made by replacing Λ and ε µ with the FP ghosts c and ζ µ , respectively, as follows: where c and ζµ are antighosts, and B and β µ are NL fields.In place of ζ µ , it is more convenient to introduce a new FP ghost ζµ , which is defined as In addition to it, we introduce a new NL field b which is defined as Using the new FP ghost ζµ and the new b field, the WS BRST transformation for K µν , A µ , ζµ and b can be written as In order to make the two nilpotent BRST transformations be anticommutative, i.e., {δ B } = 0, we must determine the remaining BRST transformations: As for the GCT BRST transformation, the BRST transformations on fields, which do not appear in (3.8) but appear in (3.10) are determined in such a way that they coincide with their tensor structure, for instance, δ On the other hand, in cases of the WS BRST transformations, one simply defines the vanishing BRST transformations, e.g., δ B cµ = 0. ( Now that we have chosen gauge fixing conditions and established BRST transformations, we can construct a gauge fixed and BRST invariant quantum Lagrangian by following the standard recipe: where surface terms are dropped.
From the Lagrangian L q , it is straightforward to derive the field equations by taking the variation with respect to each fundamental field in order.All the field equations are summarized as follows: where we have defined the following quantities: Moreover, we have introduced symmetrization with weight one by round brackets, e.g., Based on these field equations, we can write down the simpler type of equations for several fields.First of all, using Eqs.(3.21) and (3.24), it is easy to see that (3.30) Furthermore, taking the GCT BRST transformation of the field equation for cρ in (3.30) enables us to derive the field equation for b ρ [13]: In other words, setting X M = {x µ , b µ , c µ , cµ }, X M turns out to obey the very simple equation: This equation, together with the gauge condition ∂ µ (g µν φ 2 ) = 0, produces the two kinds of conserved currents: where we have defined Using these currents, we can show that there is a Poincaré-like IOSp(8|8) symmetry in the present theory as in Einstein's gravity [17,18], which should be contrasted to the IOSp(10|10) symmetry in both Weyl invariant scalar-tensor gravity in Riemann geometry [12] and Weyl conformal gravity in Weyl geometry [13].
Here it is worth mentioning that this reduction of the global symmetry is relevant to the fact that the Einstein's gravity is in a sense similar to the quantum electrodynamics (QED) while the quantum conformal gravity under consideration is similar to the quantum chromodynamics (QCD).For instance, as a representative of the global symmetries, let us consider the BRST charges.As is well known, in the QED, the BRST charge takes the simple form where B and c are the NL field and antighost for the U(1) gauge symmetry, respectively.On the other hand, in the QCD, the BRST charge has nonlinear and interacting terms as well as quadratic ones where B a and c a are the NL field and antighost for the nonabelian gauge symmetry, respectively, D µ the covariant derivative, g the coupling constant, and f abc the structure constant.Analogously, in the Weyl invariant scalar-tensor gravity, the Weyl BRST charge is of the form [13] whereas in the quantum conformal gravity, it turns out that the WS BRST charge has a very complicated nonlinear structure. 5In this sense, the quantum conformal gravity is similar to QCD rather than the QED.Lastly, let us note that such nonlinear global symmetries cannot be described by the generators of the Poincaré-like IOSp(8|8) symmetry.

Canonical commutation relations
In this section, we derive the concrete expressions of canonical conjugate momenta and set up the canonical (anti)commutation relations (CCRs), which will be used in evaluating various equal-time (anti)commutation relations (ETCRs) among fundamental variables in the next section.To simplify various expressions, we obey the following abbreviations adopted in the textbook of Nakanishi and Ojima [18]: where we assume that g00 is invertible.
To remove second order derivatives of the metric involved in R and G µν , and regard b µ as a non-canonical variable, we perform the integration by parts and rewrite the Lagrangian (3.16) as where Kµν is defined as and a surface term V µ is given by Since the NL fields b µ , b and β µ have no derivatives in L q , we can regard them as non-canonical variables.Using this Lagrangian (4.2), it is straightforward to derive the concrete ex-pressions of canonical conjugate momenta.The result is given by where we have defined the time derivative such as ġµν ≡ ∂gµν ∂t ≡ ∂gµν ∂x 0 ≡ ∂ 0 g µν , and differentiation of ghosts is taken from the right.
Next let us set up the canonical (anti)commutation relations: where the other (anti)commutation relations vanish.In setting up these CCRs, it is valuable to distinguish non-canonical variables from canonical ones.Recall again that in our formalism, the NL fields b µ , b and β µ are not canonical variables.

Equal-time commutation relations
Since we have presented the canonical (anti)commutation relations (CCRs) in the previous section, we would like to evaluate various nontrivial equal-time (anti)commutation relations (ETCRs) which are necessary for the algebra of symmetries and computations in later sections.In what follows, we will derive various important equal-time (anti)commutation relations (ETCRs) on the basis of the canonical (anti)commutation relations, field equations and BRST transformations.In deriving ETCRs, we often use a useful identity for generic variables Φ and Ψ which holds for the anticommutation relation as well.
To begin with, we wish to derive the ETCR between g µν and b µ , which is one of the important ETCRs and plays a role in proving the algebra of symmetries.For this purpose, let us first consider the antiCCR, {c µ , π ′ cν } = iδ µ ν δ 3 , which gives us Next, we find that the CCR, [g µν , π ′ cρ ] = 0 leads to where we have used the CCR, [g µν , c′ ρ ] = 0 and the formula (5.1).It then turns out that the GCT BRST transformation (3.8) of the CCR, [g µν , c′ ρ ] = 0 yields where we have used Eqs.(3.9), (5.2) and (5.3).
From this ETCR, we can easily derive ETCRs: (5.5) Here we have used the following fact; since a commutator works as a derivation, we can have formulae: where Φ is a generic field.Now we would like to derive another important ETCR, [ ġρσ , g ′ µν ].To this aim, let us focus on the canonical conjugate momentum π µν K , from which we can describe ġij as This expression immediately gives us the ETCR [ ġij , g ′ µν ] = 0. ( Here we have used the ETCR [β ρ , g ′ µν ] = 0, (5.9) which can be easily shown by taking the WS BRST transformation (3.10) of the CCR, [ ζρ , g ′ µν ] = 0.In order to calculate the remaining ETCR, [ ġ0ρ , g ′ µν ], we utilize the extended de Donder gauge condition (3.1), from which we can obtain the equation: This equation makes it possible to express ġ0ρ in terms of ġkl and φ as follows: where the ellipsis denotes terms without time-derivatives.Then, using Eq.(5.8), we find that [ ġ0i , g ′ µν ] = 0 and (5.12) To evaluate the right-hand side (RHS), we need to use the canonical conjugate momentum π φ , from which we can express φ in terms of π φ , b ρ and ġij as (5.13) Then, with the help of Eqs.(5.4) and (5.8), Eq. (5.13) enables us to evaluate the ETCR, [ φ, (5.14) From Eqs. (5.12) and (5.14), we have Hence, we can arrive at the result Incidentally, we can also offer a different proof of Eq. (5.16) on the basis of symmetry of this ETCR.The ETCR, [ ġρσ , g ′ µν ] has in general a symmetry under the simultaneous exchange of (µν) ↔ (ρσ) and primed ↔ unprimed in addition to the usual symmetry µ ↔ ν and ρ ↔ σ.We can therefore write down its general expression [ ġρσ , g ′ µν ] = c 1 g ρσ g µν + c 2 (g ρµ g σν + g ρν g σµ ) where c i (i = 1, • • • , 5) are some coefficients.To fix the coefficients c i , let us make use of the extended de Donder gauge condition (3.1), which can be rewritten as (g 0λ g ρσ − 2g λρ g 0σ ) ġρσ + 4φ −1 g λρ ∂ ρ φ = (2g λρ g σi − g ρσ g λi )∂ i g ρσ .
(5.18) Using (5.14), Eq. (5.18) yields It then turns out that this equation provides us with relations among the coefficients: (5.20) To fix the coefficients completely, we further take account of the CCR, [π αβ K , g ′ µν ] = 0, which can be cast to the form [g 0α (g βσ g 0ρ − g 0β g ρσ ) + g αρ (g 0β g 0σ − g 00 g βσ )][ ġρσ , g ′ µν ] = 0, (5.21)where Eq. (5.9) was used.Substituting (5.17 we can obtain that (5.28) Taking the GCT BRST transformation of the former equation in (5.27), we have (5.29) Using Eqs.(5.2), (5.28) and [ ġµν , c′ ρ ] = 0, which is easily proved, we reach the ETCR between K µν and b ρ : (5.30) Along the same line of argument, we can prove that ρ ] has been already calculated by using the method developed in our previous article [11].Only the result is written out as Here note that the second term on the LHS can be simplified to (5.38) Furthermore, following the derivation in Appendix A, we find that In a similar manner, we can also show that (5.40) Following our previous calculation [11], we can prove B of this CCR reads where we have used the ETCR { ζρ , ċ′ } = 0, ( which can be easily obtained from the CCR, { ζρ , π ′ c} = 0. To calculate the right-hand side (RHS) of Eq. (5.42), let us first calculate the ETCR, [g µν , ζ′ ρ ].The CCR, [g µν , π ρ′ ζ ] = 0 leads to the equation Since from Eq. (5.16) we can derive the ETCR Eq. ( 5.44) provides us with This ETCR enables us to evaluate the ETCR Next, let us consider the CCR, { ζµ , π ν′ ζ } = iδ ν µ δ 3 , which can be cast to the form This equation, together with Eq. (5.47), yields the ETCR (5.49) Then, using Eqs.(5.47) and (5.49), (5.42) is calculated to With the help of Eqs.(5.7), (5.11), (5.13) and ( 5.50), it turns out that To close this section, it is worthwhile to mention that any ETCRs can be in principle calculated by using the CCRs, field equations, the BRST transformations and the ETCRs presented thus far although we have not given all the ETCRs in this article.

Linearized field equations
In this section, we analyze asymptotic fields under the assumption that all fields have their own asymptotic fields and there is no bound state.We also assume that all asymptotic fields are governed by the quadratic part of the quantum Lagrangian apart from possible renormalization.
We define the gravitational field ϕ µν on a flat Minkowski metric η µν and the scalar fluctuation φ on a nonzero fixed scalar field φ 0 : For sake of simplicity, we use the same notation for the other asymptotic fields as that for the interacting fields.Then, up to surface terms the quadratic part of the quantum Lagrangian (3.16) reads In this and next sections, the spacetime indices µ, ν, . . .are raised or lowered by the Minkowski metric η µν = η µν = diag(−1, 1, 1, 1), and we define ≡ η µν ∂ µ ∂ ν , ϕ ≡ η µν ϕ µν and Kµν ≡ K µν − 1 2 η µν K. Based on this Lagrangian, it is straightforward to derive the linearized field equations: ∂ µ K µν = 0. (6.9) Now we are ready to simplify the field equations obtained above.Before doing so, it is more convenient to make use of the linearized BRST transformations in order to seek for the linearized field equations for the NL fields b µ , b and β µ .Taking the linearized GCT BRST transformation δ Of course, Eqs.(6.12), (6.13) and (6.14) can be also derived by solving the linearized field equations directly.
Next, operating ∂ µ on Eq. (6.14) leads to Moreover, acting on Eq. (6.14) and using Eq.(6.15), we have which implies that β µ is a dipole field.In a perfectly similar manner, Eq. (6.11) gives us Now it is easy to see that with the help of Eqs.(6.8) and (6.9), Eq. (6.6) provides7 Given Eq. (6.13), this equation shows that the gauge field A µ is a dipole field obeying By use of Eq. (6.8), this equation means that K is a simple field: Next, to exhibit that the scalar field φ is also a dipole field, let us take the trace of Eq. (6.5) whose result can be written as where Eq. (6.8) was utilized.Substituting this equation into Eq.(6.4) yields .22)Operating on this equation produces the desired result that φ is a dipole field: where we used Eqs.(6.12), (6.13) and (6.15).The divergence of Eq. (6.7) takes the form Using three equations (6.21), (6.22) and ( 6.24), we can describe ϕ and ∂ µ ∂ ν ϕ µν as which imply two equations: Here it is useful to express K µν in terms of the other fields by starting with Eq. (6.5) and utilizing some equations obtained thus far, whose result is described as Finally, let us focus on the linearized Einstein equation (6.3).After some calculations using several equations, it turns out Eq. ( 6.3) can be rewritten into a more compact form: where we have defined mass squared, m 2 ≡ 3γ 2 , which demands us to take the positive α c as assumed before.Furthermore, operating on (6.28), we can obtain the gravitational equation for ϕ µν : Eq. (6.29) implies that there are both massless and massive modes in ϕ µν .In order to disentangle these two modes, let us act on Eq. (6.27): This RHS can be further rewritten by using Eqs.(6.27) and (6.28) as Provided that we take a linear combination of fields given as we find that ψ µν corresponds to an infamous massive ghost of spin-2 of 5 physical degrees of freedom since it satisfies the equations of motion On the other hand, if we choose the following linear combination we find that h µν obeys the field equation Then, Eq. (6.35) implies that h µν is a dipole field satisfying 2 h µν = 0. (6.36) Later we will show that two transverse components of h µν is nothing but a massless spin-2 graviton.

Analysis of physical states
Following the standard technique, we can calculate the four-dimensional (anti)commutation relations (4D CRs) between asymptotic fields.The point is that the simple pole fields, for instance, the Nakanishi-Lautrup field b µ (x) obeying b µ = 0, can be expressed in terms of the invariant delta function D(x) as Here the invariant delta function D(x) for massless simple pole fields and its properties are described as where ǫ(k 0 ) ≡ k0 |k0| .With these properties, it is easy to see that the right-hand side (RHS) of Eq. (7.1) is independent of z 0 , and this fact will be used in evaluating 4D CRs via the ETCRs shortly.
To illustrate the detail of the calculation, let us evaluate a 4D CR, [h µν (x), b ρ (y)] explicitly.Using Eq. ( 7.1), it can be described as As mentioned above, since the RHS of Eq. (7.1) is independent of z 0 , we put z 0 = x 0 in (7.3) and use relevant ETCRs to obtain In a similar manner, we can calculate the four-dimensional (anti)commutation relations among ψ µν , h µν and b µ etc.To do that, let us note that since ψ µν obeys a massive simple pole equation (6.33), it can be expressed in terms of the invariant delta function ∆(x; m 2 ) for massive simple pole fields as where ∆(x; m 2 ) is defined as

.7)
As for h µν , since it is a massless dipole field as can be seen in Eq. ( 6.36), it can be described as where we have introduced the invariant delta function E(x) for massless dipole fields and its properties are given by As in Eq. ( 7.1), we can also show that the RHS of both (7.6) and (7.8) is independent of z 0 .By using the ETCRs summarized in Appendix B, after a lengthy but straightforward calculation, we find the following 4D CRs among ψ µν , h µν , φ, b µ , b, β µ , c µ , cµ , c and c: [h µν (x), ψ στ (y)] = 0. (7.12) a Weyl invariant scalar-tensor gravity.Once the unitary gauge, φ = 3 4πG for the Weyl symmetry is taken, the classical theory becomes equivalent to general relativity plus conformal gravity, so at low energies our theory properly reduces to Einstein's general relativity while at high eneries it reduces to conformal gravity where a local scale symmetry, or equivalently the Weyl symmetry, emerges in addition to the general coordinate invariance.This fact would give us some distict phenomenological consequences from those obtained through only Einstein's general relativity for inflation and the scale invariant spectrum of the Cosmic Microwave Background (CMB) radiation etc.
One of the important ingredients in the present formalism lies in the choice of gauge conditions for three local symmetries, those are, the general coordinate invariance, the Weyl symmetry and the Stückelberg symmetry.We have required that the proper gauge conditions should not only fix the gauge symmetries completely but also give us the maximal global symmetry.As a result, we are led to selecting the extended de Donder gauge condition, the traceless gauge condition and the K-gauge condition.We think that these gauge conditions are almost unique up to terms involving the NL fields multiplied by the gauge parameters.
A question often asked in gravitational theories is that global symmetries such as the Poincaré-like IOSp(8|8) symmetry are effective symmetries existing only at low energies or exact symmetries holding even at high energies.To address this question, for instance, one has to construct a renormalizable quantum gravity and show that such global symmetires still exist in such a ultraviolet (UV) complete quantum gravity.Since quantum conformal gravity under consideration is a renormalizable theory as long as the Weyl symmetry is free from Weyl anomaly, the Poincaré-like IOSp(8|8) symmetry is not an effective but an exact global symmetry.Moreover, this symmetry is closely related to purely quantum fields such as ghosts and the Nakanishi-Lautrup fields, so it is not violated by black hole's no-hair theorem [2].
As future's works, we wish to comment on two important issues.One of them is of course related to the issue of the massive ghost which violates the unitarity of the quantum theory.Recently it has been clearly shown that the Lee-Wick's prescription [19,20] dealing with the ghost fields does not work well at least within the standard framework of quantum field theories [21].Thus, if our theory makes sense as a quantum field theory, we cannot rely on the Lee-Wick's prescription any longer and should develop a new dynamical mechanism.Regarding this problem, it might be useful to recall that as mentioned in Section 3 the quantum conformal gravity is in a sense similar to the QCD while the quantum Einstein's gravity is similar to the QED.It is known that in the QCD, gluons and quarks are confined to the unphysical sector.Thus, we could conjecture that the global symmeties existing in the quantum conformal gravity might play a role to make the massive ghost be confined to the unphysical sector.
The other important issue is relevant to Weyl anomaly.More recently, this issue has been considered in Ref. [22] where it is mentioned that there is no Weyl anomaly in the unbroken phase ( φ = 0) but Weyl anomaly appears in the broken phase ( φ = 0) in Weyl geometry which is a generalization of Riemann geometry.We wish to understand whether the similar results hold even in our theory formulated in Riemann geometry.Actually, the fact that the Weyl symmetry could be maintained and manifest even at the quantum level in Riemann geometry has been already discussed in Refs.[23][24][25][26].We would like to return these two issues in future.