Consistent completion of the conformal-factor-first formulation of canonical quantum gravity: resolving the divergent induced Weyl-squared effective action

't Hooft (2010) proposed that in the functional integral of canonical quantum gravity one should perform the integration over the local conformal factor $\omega(x)$ of the metric $g_{\mu\nu}=\omega^2\hat g_{\mu\nu}$ (with $\det\hat g=-1$) before all other functional integrations, expecting a locally scale-invariant and hence finite effective theory for the unit-determinant metric $\hat g_{\mu\nu}$ and matter. The Gaussian $\omega$-integral instead generates a one-loop divergent effective action equal to the Weyl-squared action with an effectively infinite coefficient, to which every conformally coupled matter species contributes with the same sign, so cancellation by matter is impossible. The source enumerates five candidate resolutions (matter cancellation; a Weyl-squared counterterm; including $\hat g_{\mu\nu}$ loops; accepting no counterterm so that $\hat g_{\mu\nu}$ becomes effectively classical; modifying the theory at the Planck scale) and finds each one problematic or incomplete. The source's own sequels (2010-2015) postpone the $\hat g_{\mu\nu}$ sector and transform the difficulty into the equally unresolved negative-metric-state ('gravitello') problem; adjacent lines (PT-symmetric, i.e. invariant under combined parity and time reflection, treatments of fourth-order ghosts; asymptotic safety, i.e. ultraviolet completion at a renormalization-group fixed point, of conformally reduced or quadratic gravity; BRST (Becchi-Rouet-Stora-Tyutin) quantization of conformal gravity; perturbative cures of the Euclidean conformal instability, the wrong-sign conformal-mode problem of the gravitational path integral) address components or neighbors but not this completion. The problem of finding a physically consistent, unitary, low-energy-correct completion of the conformal-factor-first program - or proving none exists - remains open.

Background

A Hamiltonian or functional-integral quantization of general relativity ("canonical quantum gravity") integrates $e^{iS}$ over all components of the spacetime metric, with $S$ the Einstein-Hilbert action - the standard curvature integral of general relativity - plus matter. Since the classic one-loop calculation of 't Hooft and Veltman (1974), such integrals are known to generate curvature-squared ultraviolet divergences outside the Einstein-Hilbert form, making perturbative gravity non-renormalizable (its divergences cannot be absorbed into a finite number of additional terms in the action). 't Hooft and Veltman, One-loop divergencies in the theory of gravitation, Ann. Inst. Henri Poincare A 20 (1974) 69-94, DOI 10.1142/9789814539395_0001 (reprint), https://doi.org/10.1142/9789814539395_0001

In the source, 't Hooft splits the metric as $g_{\mu\nu}(x)=\omega^2(x)\hat g_{\mu\nu}(x)$ with the gauge-like constraint $\det(\hat g)=-1$, so that $\omega(x)$ carries the local overall scale (he calls $\omega$ a "metascalar" and $\hat g_{\mu\nu}$ a "metatensor", because they scale with unconventional powers of the coordinate-transformation Jacobian), and proposes to perform the functional integral over $\omega$ first. The motivation is twofold: (1) the matter Lagrangian is (at first) assumed conformally invariant - invariant under local scale transformations $\hat g_{\mu\nu}\to\hat g_{\mu\nu}+\lambda(x)\hat g_{\mu\nu}$ - so after integrating out $\omega$ the remaining effective action for $\hat g_{\mu\nu}$ should be locally scale invariant and plausibly finite; (2) an effective theory in which the conformal component is integrated out could realize black hole complementarity - the proposal that an infalling and a distant observer need not agree about the vacuum and the back-reaction of Hawking radiation (the thermal emission of black holes), the disagreement being absorbed into $\omega(x)$ - as argued in 't Hooft's earlier Erice lecture "Quantum gravity without space-time singularities or horizons" (arXiv:0909.3426).

The $\omega$-integral is Gaussian but has the wrong sign in Euclidean signature, so $\omega$ must be integrated along a complex contour $\omega=1+i\alpha$, $\alpha$ real. The resulting one-loop divergence is fixed by covariance and dimensionality to be the square of the Weyl tensor $W_{\mu\nu\rho\sigma}$ - the traceless part of the Riemann curvature tensor - whose square equals $2(R_{\mu\nu}R^{\mu\nu}-\frac13 R^2)$ up to a topological invariant (the Euler density): $\Gamma^{\rm div}=\frac{1}{960\pi^2(4-n)}\int d^nx\sqrt{-\hat g}(\hat R_{\mu\nu}\hat R^{\mu\nu}-\frac13\hat R^2)$, where $n$ is the spacetime dimension continued away from $4$ in dimensional regularization and $\hat R_{\mu\nu}$, $\hat R$ are the Ricci tensor and curvature scalar constructed from $\hat g_{\mu\nu}$. This is an "infinite coefficient" in front of the Weyl action that is not proportional to any small coupling (Newton's constant scales out of the $\omega$ integral), and it continues a divergence structure first noted by Capper and Duff (1975) in the conformal-anomaly problem - the quantum breaking of classical scale invariance. In the source's five-option analysis: (A) cancellation by matter is excluded because scalars, spinors and vectors all contribute the same sign to the coefficient $C=\frac{1}{120}(1+N_0)+\frac{N_{1/2}}{40}+\frac{N_1}{10}$, where $N_0$, $N_{1/2}$, $N_1$ count conformally coupled scalar, spinor, and vector species; (B) subtracting a Weyl-squared counterterm yields a propagator $(k^2+m^2-i\varepsilon)^{-2}\log(k^2/\mu^2)$ (with $m$ a mass parameter and $\mu$ the renormalization scale) with an extra pole at $k^2\approx\mu^2$ - a "Landau ghost" (a ghost-like excitation of indefinite norm, named after the Landau pole of quantum electrodynamics) - violating unitarity; (C) including the $\hat g_{\mu\nu}$ field's own loops would restore the usual perturbative renormalization group - in some supergravity theories (gravitational theories with supersymmetry) the conformal anomaly even cancels (Fradkin and Tseytlin, 1984) - but this abandons the $\omega$-first ordering and raises the unresolved question of how the $\hat g_{\mu\nu}$ functional measure scales under local scale transformations (the measure is only well-defined after gauge fixing, which the program wishes to avoid at this stage); (D) accepting the infinite coefficient with no counterterm drives the quantum fluctuations of $\hat g_{\mu\nu}$ to zero, so $\hat g_{\mu\nu}$ - and hence the gravitons it carries - becomes effectively classical, apparently violating the action-reaction principle and the possibility of graviton interference and entanglement (a gedanken graviton-photon converter made of strong transverse magnetic fields would test this); (E) cutting off the $\omega$ integration at momenta $|k|\approx M_{\rm Pl}$ (the Planck mass) breaks conformal invariance and reduces the construction to ordinary non-renormalizable perturbative gravity unless a specific Planck-scale modification and its low-energy signatures are supplied. Classically, the Weyl-squared action yields field equations $\Box R_{\mu\nu}-\frac13\partial_\mu\partial_\nu R-\cdots=0$ (with $\Box$ the wave operator) whose solutions are Einstein spacetimes sourced only by massless matter; fourth-order gravity theories of this type have been studied classically since Mannheim and Kazanas (1989), and Bender and Mannheim (2008) exhibited a no-ghost theorem for the prototypical fourth-order Pais-Uhlenbeck oscillator - the standard toy model whose equation of motion is fourth order in time - via PT-symmetric quantization (quantization of Hamiltonians invariant under combined parity and time reflection, which can define a positive inner product), a route the source mentions but does not adopt.

Problem Statement

Within the conformal-factor-first formulation of canonical quantum gravity, the spacetime metric is decomposed as $g_{\mu\nu}(x)=\omega^2(x)\hat g_{\mu\nu}(x)$ with the constraint $\det(\hat g)=-1$, so that $\omega(x)$ is the local overall scale of the geometry, and the functional integral over $\omega$ - taken along the complex contour $\omega=1+i\alpha$ with $\alpha$ real, because the Euclidean-signature integral diverges - is performed before all other functional integrations. The Gaussian $\omega$-integral generates the divergent one-loop effective action

$$\Gamma^{\rm div}=\frac{1}{960\pi^2(4-n)}\int d^nx\,\sqrt{-\hat g}\left(\hat R_{\mu\nu}\hat R^{\mu\nu}-\tfrac13\hat R^2\right),$$

where $n$ is the spacetime dimension continued away from $4$ in dimensional regularization, $\hat R_{\mu\nu}$ and $\hat R$ are the Ricci tensor and curvature scalar constructed from $\hat g_{\mu\nu}$, and the bracket equals the square of the Weyl tensor (the traceless part of the Riemann curvature tensor), which in four dimensions is $2(R_{\mu\nu}R^{\mu\nu}-\frac13R^2)$ up to the topological Euler density. The divergence is thus a Weyl-squared term with an effectively infinite coefficient, to which all conformally coupled matter species contribute with the same sign.

The problem is to find a physically consistent completion of the resulting effective theory for $\hat g_{\mu\nu}$ and matter - or to prove that none exists. A resolution must, within one coherent construction: (1) specify a definite prescription - one of the five options enumerated in the source or a new mechanism - for handling this divergence (cancellation by a symmetry or high-energy completion; a local Weyl-squared counterterm; inclusion of $\hat g_{\mu\nu}$ loops; acceptance of an infinite coefficient making $\hat g_{\mu\nu}$ effectively classical; or a specific modification of the integration domain or fundamental action at momenta $|k|$ of order the Planck mass $M_{\rm Pl}$); (2) demonstrate that the resulting theory is unitary (probabilities conserved, no negative-norm states) and causal - in particular resolving the fate of the Landau ghost, a ghost-like pole of indefinite norm, or of the negative-metric massive spin-2 state that any finite Weyl-squared term propagates, and of the 'partly classical' gravitons of the no-counterterm option, whose effective classicality appears to conflict with the action-reaction principle and with graviton interference and entanglement; (3) derive the low-energy limit, recovering Einstein gravity with quantized gravitons and standard matter couplings; and (4) settle the associated measure-and-ordering question: whether the $\omega$-first ordering is physically permissible in a nonperturbative, gauge-fixed functional measure, i.e. how the $\hat g_{\mu\nu}$ functional measure scales under local scale transformations. The motivating application - whether such a framework realizes black hole complementarity (the proposal that an infalling and a distant observer need not agree about the vacuum and the back-reaction of Hawking radiation, the disagreement being absorbed into $\omega(x)$) and information-preserving Hawking evaporation - completes the program but is downstream of (1)-(4) and is not required for the core resolution. Equivalently, a proof that no construction satisfying (1)-(4) exists within this framework would also resolve the problem.

References

1. Gerard 't Hooft, 'Probing the small distance structure of canonical quantum gravity using the conformal group', Subnuclear Series vol. 48 (2013) 155-183, DOI 10.1142/9789814522489_0007, https://doi.org/10.1142/9789814522489_0007; preprint arXiv:1009.0669 [gr-qc] (2010), https://arxiv.org/abs/1009.0669 (source paper)
2. Gerard 't Hooft, 'The Conformal Constraint in Canonical Quantum Gravity', arXiv:1011.0061 [gr-qc] (2010), https://arxiv.org/abs/1011.0061
3. Gerard 't Hooft, 'A class of elementary particle models without any adjustable real parameters', Foundations of Physics 41 (2011) 1829-1856, DOI 10.1007/s10701-011-9586-8, https://doi.org/10.1007/s10701-011-9586-8; preprint arXiv:1104.4543 [hep-th], https://arxiv.org/abs/1104.4543
4. Gerard 't Hooft, 'Local Conformal Symmetry: the Missing Symmetry Component for Space and Time', International Journal of Modern Physics D 24 (2015) 1543001, DOI 10.1142/S0218271815430014, https://doi.org/10.1142/S0218271815430014; preprint arXiv:1410.6675 [gr-qc] (2014), https://arxiv.org/abs/1410.6675
5. Gerard 't Hooft, 'Singularities, horizons, firewalls, and local conformal symmetry', Proceedings of the 2nd Karl Schwarzschild Meeting on Gravitational Physics, Springer Proceedings in Physics vol. 208 (2018) 1-12, DOI 10.1007/978-3-319-94256-8_1, https://doi.org/10.1007/978-3-319-94256-8_1; preprint arXiv:1511.04427 [gr-qc] (2015), https://arxiv.org/abs/1511.04427
6. Gerard 't Hooft, 'Quantum gravity without space-time singularities or horizons', Subnuclear Series 47 (2011) 251-265, DOI 10.1142/9789814374125_0010, https://doi.org/10.1142/9789814374125_0010; preprint arXiv:0909.3426 [gr-qc] (2009), https://arxiv.org/abs/0909.3426
7. Gerard 't Hooft, 'The Cellular Automaton Interpretation of Quantum Mechanics', Fundamental Theories of Physics vol. 185 (Springer, 2016), DOI 10.1007/978-3-319-41285-6, https://doi.org/10.1007/978-3-319-41285-6; preprint arXiv:1405.1548 [quant-ph] (2014), https://arxiv.org/abs/1405.1548
8. G. 't Hooft and M.J.G. Veltman, 'One-loop divergencies in the theory of gravitation', Annales de l'Institut Henri Poincare A 20 (1974) 69-94; reprinted in 'Euclidean Quantum Gravity' (World Scientific, 1993) 3-28, DOI 10.1142/9789814539395_0001 (reprint), https://doi.org/10.1142/9789814539395_0001
9. D.M. Capper and M.J. Duff, 'Conformal anomalies and the renormalizability problem in quantum gravity', Physics Letters A 53 (1975) 361-362, DOI 10.1016/0375-9601(75)90030-4, https://doi.org/10.1016/0375-9601(75)90030-4
10. E.S. Fradkin and A.A. Tseytlin, 'Conformal anomaly in Weyl theory and anomaly free superconformal theories', Physics Letters B 134 (1984) 187-193, DOI 10.1016/0370-2693(84)90668-3, https://doi.org/10.1016/0370-2693(84)90668-3
11. Philip D. Mannheim and Demosthenes Kazanas, 'Exact vacuum solution to conformal Weyl gravity and galactic rotation curves', Astrophysical Journal 342 (1989) 635-638, DOI 10.1086/167623, https://doi.org/10.1086/167623
12. Carl M. Bender and Philip D. Mannheim, 'No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model', Physical Review Letters 100 (2008) 110402, DOI 10.1103/PhysRevLett.100.110402, https://doi.org/10.1103/PhysRevLett.100.110402; preprint arXiv:0706.0207 [hep-th], https://arxiv.org/abs/0706.0207
13. Philip D. Mannheim, 'Ghost problems from Pauli-Villars to fourth-order quantum gravity and their resolution', International Journal of Modern Physics D 29 (2020) 2043009, DOI 10.1142/S0218271820430099, https://doi.org/10.1142/S0218271820430099; preprint arXiv:2004.00376 [hep-th], https://arxiv.org/abs/2004.00376
14. Philip D. Mannheim, 'Solution to the ghost problem in higher-derivative gravity', Il Nuovo Cimento C 45 (2022) 27, DOI 10.1393/ncc/i2022-22027-6, https://doi.org/10.1393/ncc/i2022-22027-6; preprint arXiv:2109.12743 [hep-th] (QGRAV 2021 proceedings, 8 pages), https://arxiv.org/abs/2109.12743
15. Carlos M. Nieto, Roberto Percacci and Vedran Skrinjar, 'Split Weyl transformations in quantum gravity', Physical Review D 96 (2017) 106019, DOI 10.1103/PhysRevD.96.106019, https://doi.org/10.1103/PhysRevD.96.106019; preprint arXiv:1708.09760 [gr-qc], https://arxiv.org/abs/1708.09760
16. Alfio Maurizio Bonanno, Maria Conti and Sergio Luigi Cacciatori, 'Ultraviolet behavior of conformally reduced quadratic gravity', Physical Review D 108 (2023) 026008, DOI 10.1103/PhysRevD.108.026008, https://doi.org/10.1103/PhysRevD.108.026008; preprint arXiv:2304.12011 [hep-th] (arXiv preprint title: 'On the ultraviolet behavior of conformally reduced quadratic gravity'), https://arxiv.org/abs/2304.12011
17. Ichiro Oda and Misaki Ohta, 'Quantum conformal gravity', Journal of High Energy Physics 02 (2024) 213, DOI 10.1007/JHEP02(2024)213, https://doi.org/10.1007/JHEP02(2024)213; preprint arXiv:2311.09582 [hep-th] (2023), https://arxiv.org/abs/2311.09582
18. Gianluca Calcagni and Leonardo Modesto, 'Path integral and conformal instability in nonlocal quantum gravity', Journal of High Energy Physics 07 (2024) 277, DOI 10.1007/JHEP07(2024)277, https://doi.org/10.1007/JHEP07(2024)277; preprint arXiv:2402.14785 [hep-th], https://arxiv.org/abs/2402.14785
19. Gerard 't Hooft, 'How studying black hole theory may help us to quantise gravity', arXiv:2211.10723 [gr-qc] (2022), https://arxiv.org/abs/2211.10723 (conference contribution)
20. Gerard 't Hooft, 'Alternative theory for the quantum black hole and the temperature of its quantum radiation', arXiv:2410.16891 [gr-qc] (2024), https://arxiv.org/abs/2410.16891