∗
Abstract
Recently in Eur. Phys. J. C 76 , 341 (2016) Myung has suggested that the problem with unphysical states in massive conformal gravity, which was pointed in Eur. Phys. J. C 76 , 188 (2016), is solved if we quantize the dilaton field in a classical background metric. Here we show that this solution is inconsistent and suggest a way to fix it.
* fff@uespi.br

Before we address the content of Myung’s paper [1], let us take a brief look on the results of Ref. [2]. For this purpose, we consider the massive
conformal gravity (MCG) action, which is given by 1
[3]
S
MCG
=
Z d 4 x
√
− g
1 m 2
C αβµν C
αβµν
− α ϕ 2 R + 6 ∂
µ ϕ∂ µ ϕ , (1) where m is a constant with dimension of mass, α is a dimensionless constant,
C α
βµν is the Weyl tensor, R is the scalar curvature, and ϕ is a scalar field
called dilaton. Using the Lanczos identity, we can write (1) as
S
MCG
=
Z d 4 x
√
− g
2 m 2
R
µν
R
µν
−
1
3
R 2
− α ϕ 2 R + 6 ∂
µ ϕ∂
µ ϕ , (2) where R
µν is the Ricci tensor.
By performing a perturbative quantization of (2) about the background
field expansions g
µν
= η
µν
+ h
µν
, (3) ϕ = r
2
α
(1 + σ ) , (4) it can be shown, after a long but straightforward calculation, that the Feynman propagators for the quantum fields σ and Ψ
µν
by [2]
D
σ
( x − y ) = − i
Z d 4 p
(2 π ) 4
= h
µν e
− ip
·
( x
− y ) p 2 + m 2 − iε
,
− η
µν h/ 2 are given
(5)
D
µν,αβ
Ψ
= i
2
η µα η νβ + η µβ η να − η µν η αβ
Z d
(2
4
π p
) 4 ( p 2 − m iε
2 e
−
)( p ip
·
( x
− x ′ )
2 + m 2
,
− iε )
(6) where ε is an infinitesimal parameter. Since the residue at the pole for the
massless term in (6) is negative, its corresponding quantum state is taken
to have negative energy or negative norm. In either case quantum MCG is unphysical.
The solution proposed by Myung in Ref. [1] is to consider
ϕ as being a quantum field and g
µν as being a classical background field, which eliminates the unphysical massless pole. In this case, the computation of graphs with
1 Here we use “absolute units” in which c = ~ = 16 πG = 1.
1

internal lines of ϕ and external lines of both ϕ and g
µν
contribution to the effective action [4]
gives the one-loop
Γ
(1)
MCG
=
α
160 π 2 (4 − n )
Z d n x
√
− g R
µν
R
µν
−
1
3
R 2 .
T ϕ
µν
= α ϕ 2 G
µν
+ 6 ∂
µ ϕ∂
ν ϕ − 3 g
µν
∂
ρ ϕ∂ ρ ϕ + g
µν ϕ 2
− ∇
µ
∇
ν ϕ 2
(7)
This divergence has the same form of the first term of (2), and thus it is
of the renormalizable type. Hence, when we quantize the dilaton field ϕ in the classical background metric g
µν
, quantum MCG is both unitary and renormalizable.
However, there is an important problem with this proposal as well. The
variation of (2) with respect to
g µν gives
2 m 2
W
µν
= T ϕ
µν
, (8) where W
µν is the Bach tensor,
(9) is the energy-momentum tensor of ϕ , G
µν is the Einstein tensor, and ϕ is
the generally covariant dAlembertian for a scalar field. We can see from (8)
that a classical g
µν will not react upon the presence of a quantum T ϕ
µν source, which violates the action = reaction principle in physics. Not very physical!
We go further and fix Myung’s proposal by demanding that the classical background metric is flat ( g
µν
= η
µν
) at scales close to the Planck scale, but it gets renormalized by the dilaton field at much lower scales. In this way, dilation fluctuations can generate the non trivial g
µν that we experience at the classical level. This is possible because the causal structure, that is, the size and orientation of the light cones, is determined by both ϕ and g
µν
MCG, as we can see from the MCG geodesic equation 2
in d 2 x λ dτ 2
+ Γ
λ
µν dx µ dτ dx ν dτ
1 ∂ϕ
+ ϕ ∂x ρ g
λρ
+ dx λ dτ dx ρ dτ
= 0 , (10) where Γ λ
µν is the Levi-Civita connection. In this case, quantum MCG becomes a consistent quantum theory of gravity.
2
In fact, the causal structure is defined by the geodesic deviation. However, using (10),
we can show that the MCG geodesic deviation also depends on both g
µν and ϕ .
2

[1] Y.S. Myung, Eur. Phys. J. C 76 , 341 (2016).
[2] F.F. Faria, Eur. Phys. J. C 76 , 188 (2016).
[3] F.F. Faria, Adv. High Energy Phys.
2014 , 520259 (2014).
[4] G. t Hooft and M.J.G. Veltman, Ann. Poincare Phys. Theor. A 20 , 69
(1974).
3