--- title: Perturbative Conformal Gravity url: https://www.emergentmind.com/topics/perturbative-conformal-gravity type: topic --- # Perturbative Conformal Gravity Searching arXiv for recent and relevant papers on perturbative conformal gravity, cosmological perturbations, ghosts, and quantization. arxiv_search(query="perturbative conformal gravity", max_results=10, sort_by="relevance") arxiv_search(query="conformal gravity perturbations ghost cosmology quantum gravity", max_results=10, sort_by="relevance") Perturbative conformal gravity denotes several closely related perturbative programs in which conformal symmetry, the conformal factor of the metric, or the Weyl-squared gravitational action plays a central role. In one major usage, conformal gravity is the purely metric, fourth-order theory with action \[ I_W = -\alpha_g \int d^4x \sqrt{-g}\, C_{\lambda\mu\nu\kappa} C^{\lambda\mu\nu\kappa}, \] whose field equations are written as \(4\alpha_g W_{\mu\nu}=T_{\mu\nu}\), with \(W_{\mu\nu}\) the Bach tensor [1109.4119]. In another usage, perturbative conformal gravity refers to perturbative quantum-gravity constructions in which the conformal factor has a wrong-sign kinetic term and therefore alters the Wilsonian renormalization-group structure, the admissible Hilbert space of interactions, and the nature of the continuum limit [1802.04281]. More recent work also embeds conformalized Einstein dynamics into first-order gauge formulations and deformed topological field theories, where a cosmological scalar field controls the local scale of the Cartan model geometry [1601.00567], [2508.05683]. ## 1. Classical definitions and first-order realizations In the purely metric formulation, conformal gravity is locally conformally invariant under \[ g_{\mu\nu}(x)\to e^{2\alpha(x)} g_{\mu\nu}(x), \] and in four dimensions its action can also be written, using the Gauss-Bonnet theorem, as \[ I_W = -2\alpha_g \int d^4x\,(-g)^{1/2}\left(R_{\mu\nu}R^{\mu\nu}-\frac13 R^2\right) \] [1805.06807]. The theory is fourth order in derivatives, and this higher-derivative character governs both its perturbative field equations and its ghost problem. A distinct first-order route begins from an \(SU(2,2)\) gauge theory with gauge field \(A_\mu\) and adjoint Higgs field \(W\), where the symmetry is broken as \[ SU(2,2)\rightarrow SO(1,3)\times SO(1,1). \] The action is polynomial in \(\{A_\mu,W\}\), the field equations are first order in derivatives, and the new \(SO(1,1)\) symmetry is interpreted in terms of an emergent scale symmetry [1601.00567]. In that framework, the recovery of conformalized General Relativity and fourth-order Weyl conformal gravity as limits of the theory is demonstrated. In particular, if \(f^I=\pm \Omega^2(x)e^I\), the action reduces to \[ S[g,\Omega] = \int d^4x \sqrt{-g} \left( \frac{\Omega^2}{16\pi G_0} R + \frac{6}{16\pi G_0} g^{\mu\nu} \partial_\mu \Omega \partial_\nu \Omega - \frac{\Lambda_0}{8\pi G_0} \Omega^4 \right), \] while further constraints yield the standard fourth-order Weyl gravity action \[ S = \int \frac{\alpha^2}{4\beta} \epsilon_{IJKL} C^{IJ} C^{KL} \] [1601.00567]. A more recent first-order/topological formulation starts from the MacDowell-Mansouri connection \[ \omega = A + \frac{1}{l} e, \] with curvature \(F\) and auxiliary field \(B\). When the fundamental length scale is promoted to a dynamical variable, the coframe part becomes \[ \omega^{IJ} = \begin{bmatrix} A^{ij} & \sqrt{\frac{\Lambda}{3}}\,\phi\, e^i \\ -\epsilon \sqrt{\frac{\Lambda}{3}}\,\phi\, e^j & 0 \end{bmatrix}, \] and the resulting deformed BF action yields field equations equivalent on shell to the conformal Einstein equations [2508.05683]. This suggests a perturbative notion of conformal gravity in which the conformal factor is geometrized as a cosmological scalar field controlling the local radius of the tangent model homogeneous space. ## 2. Linear perturbations of Weyl conformal gravity For metric perturbations around a background \(g^{(0)}_{\mu\nu}\), one writes \[ g_{\mu\nu}=g^{(0)}_{\mu\nu}+h_{\mu\nu}. \] A central structural fact is that, due to conformal invariance, the trace of the metric perturbation decouples kinematically from the first-order fluctuation equations. One may therefore decompose \[ h_{\mu\nu}(x)=K_{\mu\nu}(x)+\frac14 g_{\mu\nu}(x) h(x), \] with \(K_{\mu\nu}\) traceless, and the field equations depend only on \(K_{\mu\nu}\) [1109.4119]. This trace decoupling is one of the defining perturbative simplifications of the theory. On a flat Minkowski background, in the conformal gauge, the linearized Bach tensor reduces to \[ \delta W_{\mu\nu}(K)=\frac12 \partial_\alpha \partial^\alpha \partial_\beta \partial^\beta K_{\mu\nu}, \] that is, \[ \Box\Box K_{\mu\nu}=0. \] The general solution contains both oscillatory plane-wave terms and linearly growing terms, \[ K_{\mu\nu}=A_{\mu\nu} e^{ik\cdot x} + (n\cdot x) B_{\mu\nu} e^{ik\cdot x} + \text{c.c.}, \qquad k^2=0, \] reflecting the fourth-order character of the theory [1805.06807]. Around de Sitter space, tensor perturbations satisfy \[ h_k'''' + 2k^2 h_k'' + k^4 h_k = 0, \] with general solution \[ h_k(\eta)=A\,e^{ik\eta}+B\,e^{-ik\eta}+\eta C\,e^{ik\eta}+\eta D\,e^{-ik\eta}. \] The first two terms correspond to standard massless graviton waves, while the terms linear in \(\eta\) represent additional ghost modes [2306.13714]. In the notation of the cosmological perturbation literature, the general transverse-traceless solution may also be written as \[ K_{\mu\nu}(\tau,\mathbf{x}) = A_{\mu\nu} e^{i\mathbf{q}\cdot \mathbf{x} - iq\tau} + B_{\mu\nu}\tau e^{i\mathbf{q}\cdot \mathbf{x} - iq\tau}, \] where the \(B_{\mu\nu}\) term is a purely fourth-order feature [1109.4119]. These perturbative solutions already exhibit the central tension of the subject: the same fourth-order structure that yields conformal invariance and renormalizability in some perturbative analyses also introduces extra modes that are usually identified as ghosts. ## 3. Cosmological perturbation theory A substantial perturbative literature studies conformal gravity on Robertson-Walker and de Sitter backgrounds. For backgrounds conformal to flat, \[ g^{(0)}_{\mu\nu}=\Omega^2(x)\eta_{\mu\nu}, \] the main fluctuation equation takes the remarkably simple form \[ \delta W_{\mu\nu}(K)=\frac12 \Omega^{-2}\eta^{\sigma\rho}\eta^{\alpha\beta} \partial_\sigma\partial_\rho\partial_\alpha\partial_\beta(\Omega^{-2}K_{\mu\nu}), \] and solutions on the curved background can be generated from flat-space solutions via \[ K_{\mu\nu}^{\text{curved}}(x)=\Omega^2(x)K_{\mu\nu}^{\text{flat}}(x) \] [1805.06807], [1109.4119]. It is not necessary that the perturbative geometry associated with the fluctuations itself be conformal to flat. The scalar-vector-tensor decomposition is central in this subject. In a fully gauge-invariant treatment, the decomposition theorem holds in conformal gravity, so scalar, vector, and tensor modes propagate independently at linear order [2009.06841]. In the notation used for conformal-to-flat backgrounds, \[ ds^2=\Omega^2(x)[ (1+2\phi)\,dt^2 - 2(B_i + \partial_i B)\, dt\, dx^i - [(1-2\psi)\delta_{ij} + 2\partial_i \partial_j E + 2\partial_{(i} E_{j)} + 2 E_{ij}]dx^i dx^j ], \] and the perturbation equations depend only on gauge-invariant combinations: one scalar, one transverse vector, and one transverse-traceless tensor [1805.06807]. In the recombination era, where the background universe is curvature-dominated with \(k<0\), the fluctuation equations simplify strongly and admit an exact solution. A master radial equation unifies the scalar, vector, and tensor sectors, \[ \left[ \frac{d^2}{dx^2} + (F-1)\coth x \frac{d}{dx} + \frac{l(l+F-2)}{\sinh^2 x} + v^2 \right] f(x)=0, \] with the vector fluctuations in three spatial dimensions obeying the same radial equation as scalar fluctuations in five dimensions, and tensor fluctuations in three spatial dimensions satisfying the same radial equation as scalar fluctuations in seven dimensions [2009.06841]. The corresponding spatial normalizations are \[ \int_0^\infty dx \; \sinh^2 x \; [\text{mode}_1(x)] [\text{mode}_2(x)] = \delta(v_1 - v_2), \] for scalars, \[ \int_0^\infty dx \; \sinh^4 x \; [\text{vector}_{1}(x)] [\text{vector}_{2}(x)] = \delta(v_1 - v_2), \] for vectors, and \[ \int_0^\infty dx \; \sinh^6 x \; [\text{tensor}_{1}(x)] [\text{tensor}_{2}(x)] = \delta(v_1 - v_2), \] for tensors [2009.06841]. A frequently emphasized cosmological consequence is rapid early-time growth. In a conformal Robertson-Walker cosmology, early-universe fluctuations grow as \(t^4\) [1805.06807]. Earlier work likewise found tensor fluctuations that grow far more rapidly than those that occur in the analogous standard case, and stated that in the \(K<0\) Robertson-Walker geometry the dominant tensor fluctuation can grow as fast as \(t^4\) or even \(t^6\), depending on the mode class [1109.4119]. These results are among the sharpest perturbative contrasts with Einstein gravity. ## 4. Ghosts, boundary conditions, and the status of extra modes The perturbative ghost problem is a persistent point of controversy. Around both de Sitter and Minkowski backgrounds, the linearized tensor sector contains four degrees of freedom: two healthy spin-2 modes and two ghost modes arising from the higher-derivative structure [2306.13714]. In de Sitter space, Maldacena boundary conditions consist of the Neumann boundary condition \[ \left.\partial_\eta h_{ij}\right|_{\eta=0}=0 \] together with the positive-frequency mode condition. Applied to conformal gravity, these select \[ h_k(\eta)=B_T(1+ik\eta)e^{-ik\eta}, \] which matches a particular Einstein-gravity graviton solution but does not recover the full general solution of the fourth-order system [2306.13714]. Alternative boundary conditions were derived to project onto the Einstein-like sector in full generality. In de Sitter space, combining the Neumann boundary condition with \[ \left. \partial_\eta^3 h_k \right|_{\eta=0} = k^2 \left. \partial_\eta h_k \right|_{\eta=0} \] yields the general Einstein-gravity-like solution \[ h_k(\eta)=A(1-ik\eta)e^{ik\eta}+B(1+ik\eta)e^{-ik\eta}, \] while in Minkowski space the conditions \[ \left. \partial_\eta^3 h_k \right|_{\eta=0} = -k^2 \left. \partial_\eta h_k \right|_{\eta=0}, \qquad \left. h_k \right|_{\eta=0} = -k^{-2} \left. \partial_\eta^2 h_k \right|_{\eta=0} \] project onto \[ h_k(\eta)=A\,e^{ik\eta}+B\,e^{-ik\eta} \] [2306.13714]. These constructions remove the ghost modes at the level of the allowed perturbative solutions by imposing boundary conditions that break conformal invariance and select the Einstein sector. The cosmological perturbation literature adds a further subtlety: the tensor modes in conformal gravity that are also solutions in standard gravity do not carry energy in the conformal theory, while only the pure fourth-order \(B\)-type modes can transport energy [1109.4119]. This does not resolve the ghost problem, but it changes the perturbative interpretation of which linearized modes are dynamically relevant. A distinct but related perturbative claim appears in nonlocal quantum gravity. There, the Euclidean conformal instability is described as a gauge artifact in perturbation theory: by suitable gauge choices one can ensure a sign choice that makes the path integral well defined for Einstein gravity, Stelle gravity, and nonlocal gravity [2402.14785]. This does not remove the higher-derivative ghost problem of Weyl gravity itself, but it clarifies that the conformal instability and the ghost issue are not identical questions. ## 5. Scattering amplitudes and perturbative spectra Tree-level amplitudes provide another perturbative probe of conformal gravity. In conformal supergravity and its truncations, amplitudes can be obtained from the double-copy construction using gauge-theory amplitudes and compared to direct calculations from conformal supergravity Lagrangians [1806.05124]. A basic bosonic mass-deformed Weyl-gravity Lagrangian is \[ e^{-1}L = -\frac{1}{2}(W_{\mu\nu\rho\sigma})^2 - m^2 R, \] where the two-derivative term acts as a mass deformation [1806.05124]. In the massless limit this returns pure Weyl gravity, while for \(m\to\infty\) the amplitudes interpolate to Einstein gravity. For minimal \(\mathcal N=4\) conformal supergravity, the equations of motion are implied by those of \(\mathcal N=4\) Einstein supergravity, and for planewave external states tree amplitudes vanish in the massless limit [1806.05124]. More generally, \[ A_n^{(4)}(1,2,\ldots,n)=m^2\,A_n^{(2)}(1,2,\ldots,n), \qquad A_n^{(4)}(\tilde{1},2,\ldots,n)=-A_n^{(2)}(1,2,\ldots,n), \] so planewave amplitudes vanish as \(m\to 0\), whereas amplitudes with at least one non-planewave state remain nontrivial [1806.05124]. This perturbative distinction between planewave and non-planewave states is a characteristic fingerprint of four-derivative dynamics. The same amplitude program identifies non-minimal theories, in particular the Berkovits-Witten theory with holomorphic function \(F=i\bar\tau\), for which tree amplitudes are non-vanishing [1806.05124]. The double-copy map uses a four-derivative \((DF)^2\) gauge theory together with supersymmetric Yang-Mills, and the resulting gravity amplitudes agree with direct Lagrangian calculations for the cases studied. This suggests that perturbative conformal gravity amplitudes admit a precise organization even when the underlying theory contains ghostlike states. First-order gauge approaches offer a contrasting perturbative spectrum. Around maximally symmetric backgrounds in the \(SU(2,2)\) theory, the quadratic action splits into a massless graviton, a massive scalar field, and a massless one-form field, all with second-order equations for the propagating modes despite the first-order fundamental action [1601.00567]. A plausible implication is that “perturbative conformal gravity” is not a single perturbative spectrum but a family of spectra associated with distinct off-shell completions. ## 6. Quantum perturbation theory, the conformal sector, and RG constructions A different research program studies the conformal factor of the metric itself as the decisive perturbative ingredient in quantum gravity. In Euclidean signature, decomposing \[ g_{\mu\nu}=\delta_{\mu\nu}+\kappa H_{\mu\nu}, \qquad H_{\mu\nu}=h_{\mu\nu}+\frac14 \delta_{\mu\nu}\varphi, \] the Einstein-Hilbert kinetic term takes the form \[ L^{\mathrm{kinetic}}_{EH}=\frac12 (\partial_\lambda h_{\mu\nu})^2-\frac12 (\partial_\lambda \varphi)^2, \] so the conformal factor has a wrong-sign kinetic term [1802.04281]. In this framework, the Wilsonian RG is profoundly altered: complete flows generically exist only in the reverse direction, from the infrared to the ultraviolet, and the Gaussian fixed point supports a discrete basis of evanescent eigenoperators \[ \delta_n(\varphi)=\frac{\partial^n}{\partial\varphi^n}\exp\left(-\frac{a}{\hbar\kappa^2}\varphi^2\right), \] or, with the scale-dependent notation, \[ \delta_\Lambda^{(n)}(\varphi)=\frac{\partial^n}{\partial\varphi^n}\left(\frac{1}{\sqrt{2\pi\Omega_\Lambda}}e^{-\frac{\varphi^2}{2\Omega_\Lambda}}\right) \] [1804.03834], [2004.06475]. These operators are square-integrable under a growing exponential measure and are non-perturbative in \(\hbar\). Every interaction monomial \(\sigma\) is then dressed with a coefficient function \(f_\Lambda^\sigma(\varphi)\) satisfying the diffusion-type flow equation \[ \partial_t f_\Lambda(\varphi)=\frac12 \dot\Omega_\Lambda f_\Lambda''(\varphi), \] with expansion \[ f_\Lambda^\sigma(\varphi)=\sum_{n=n_\sigma}^\infty g_n^\sigma \delta_\Lambda^{(n)}(\varphi) \] [2004.06475]. At second order in perturbation theory, one must sum over all melonic Feynman diagrams to obtain the particular integral, and the renormalized trajectory is well defined provided the solution is anchored at a scale \(0<\mu