Periodic Reporting for period 1 - NLQG (Understanding NonLocality in Quantum Gravity)
Période du rapport: 2023-12-01 au 2025-11-30
A vast amount of observational data has confirmed the theoretical predictions of Einstein's General Relativity (GR), making it the currently best known theory to describe classical aspects of the gravitational interaction, from cosmological to sub-millimeter scales. Despite this phenomenal success, there are still fundamental questions that remain unanswered. At small scales, GR predicts the existence of singularities in black holes and at the big bang, where the theory breaks down. At the quantum level, Einstein's theory lacks predictivity in the ultraviolet regime, i.e. at high energies, as it is perturbatively non-renormalizable. It is generally believed that a consistent theory of quantum gravity is needed to address these challenges.
In the last decades, new theoretical progress has been made toward developing quantum gravity approaches and gaining new insights into quantum aspects of gravity. Despite their intrinsic differences, most of these approaches describe new physics in the high-energy regime via additional higher-curvature invariants containing finite-order (i.e. local) and/or non-polynomial (i.e. non-local) differential operators, which can appear in the gravitational Lagrangian at effective and fundamental levels. These operators are known as form factors.
Higher derivative field theories often introduce "ghost" degrees of freedom which are sometimes considered responsible for classical instabilities (due to negative energies) and violation of S-matrix unitarity at the quantum level. However, in recent years the field of higher-derivative gravity has had a renaissance as new ideas for instability-free, unitary quantizations of ghost fields have been proposed in the case of local gravitational Lagrangians, while the existence of ghost-free form factors has been demonstrated for non-local Lagrangians. This means that it is now possible to formulate quantum field theories (QFTs) of gravity which are perturbatively stable and unitary despite the presence of higher-derivative form factors.
A full derivation of the form factors in the quantum-gravity Lagrangian is still pending in the known approaches and scenarios mentioned above. Moreover, when the locality principle is given up, it becomes difficult to uniquely select the Lagrangian because there exists an infinite class of ghost-free gravity models whose non-polynomial form factors are such that no ghost appears. This means that from a bottom-up point of view additional constraints are needed to reduce the degeneracy.
The overall goal of this project has been twofold: (i) to derive and enforce fundamental consistency requirements of causality, stability, unitarity, and healthy high-energy behavior in order to constrain the space of allowed gravitational QFTs; (ii) to use experimental constraints inferred from astrophysical and cosmological observations to further test and constrain the quantum-gravity Lagrangian.
In the last decades, new theoretical progress has been made toward developing quantum gravity approaches and gaining new insights into quantum aspects of gravity. Despite their intrinsic differences, most of these approaches describe new physics in the high-energy regime via additional higher-curvature invariants containing finite-order (i.e. local) and/or non-polynomial (i.e. non-local) differential operators, which can appear in the gravitational Lagrangian at effective and fundamental levels. These operators are known as form factors.
Higher derivative field theories often introduce "ghost" degrees of freedom which are sometimes considered responsible for classical instabilities (due to negative energies) and violation of S-matrix unitarity at the quantum level. However, in recent years the field of higher-derivative gravity has had a renaissance as new ideas for instability-free, unitary quantizations of ghost fields have been proposed in the case of local gravitational Lagrangians, while the existence of ghost-free form factors has been demonstrated for non-local Lagrangians. This means that it is now possible to formulate quantum field theories (QFTs) of gravity which are perturbatively stable and unitary despite the presence of higher-derivative form factors.
A full derivation of the form factors in the quantum-gravity Lagrangian is still pending in the known approaches and scenarios mentioned above. Moreover, when the locality principle is given up, it becomes difficult to uniquely select the Lagrangian because there exists an infinite class of ghost-free gravity models whose non-polynomial form factors are such that no ghost appears. This means that from a bottom-up point of view additional constraints are needed to reduce the degeneracy.
The overall goal of this project has been twofold: (i) to derive and enforce fundamental consistency requirements of causality, stability, unitarity, and healthy high-energy behavior in order to constrain the space of allowed gravitational QFTs; (ii) to use experimental constraints inferred from astrophysical and cosmological observations to further test and constrain the quantum-gravity Lagrangian.
Objective 1: Causality and unitarity in nonlocal field theories
Although the issue of S-matrix unitarity in nonlocal quantum field theories has been addressed in detail by several groups at the perturbative level, the question of what happens non-perturbatively is still not settled. Moreover, the questions of whether nonlocality violates causality and how to quantify such an effect are still open. The aim of this objective was to undertake a new non-perturbative study of unitarity and causality in generic nonlocal field theories by using methods of scattering amplitudes, dispersion relations and effective field theory constraints.
The requirements of unitarity and causality lead to significant constraints on the Wilson coefficients of an effective-field-theory expansion, known as positivity bounds. Their standard derivation relies on the crucial assumption of polynomial boundedness on the growth of scattering amplitudes in the complex energy plane, which is a property satisfied by local quantum field theories, and by weakly coupled string theory in the Regge regime. Together with my collaborators, I clarified the role of locality by deriving generalized positivity bounds under the assumption of exponential boundedness, typical of non-local quantum field theories where the Froissart-Martin bound is usually not satisfied. Using appropriately modified dispersion relations, I derived new constraints and found regions in the effective field theory parameter space that do not admit a local ultraviolet completion. At the same time, I provided an explicit example of an exponentially bounded amplitude that satisfies partial-wave unitarity and asymptotic causality. This study showed that it is possible to have nonlocal ultraviolet completions that satisfy unitarity and causality.
Publication (1): L. Buoninfante, L. Q. Shao and A. Tokareva, “Nonlocal positivity bounds: Islands in terra incognita”, Phys. Rev. D 112 (2025) no.2 L021904.
Objective 2: Degrees of freedom and stability in higher-derivative field theories
The counting of degrees of freedom of a higher-derivative or nonlocal Lagrangian is typically performed considering linear perturbations around some background and checking the propagator poles. This procedure is useful to verify stability and unitarity (e.g. absence of tachyons and ghosts) in quantum field theory, but it does not help analyse stability of the classical field equations at the non-linear level and independently of the background. The goal was to pioneer a nonperturbative and background-independent analysis of higher-derivative Lagrangians and use nonlinear stability conditions to constrain the theories.
I addressed part of this objectives in two ways:
(i) I considered non-perturbative resummations of the propagator in a higher-derivative field theory to check what happens to ghost-like degrees of freedom and whether non-perturbative effects can shed new light on their nature. In particular, I conducted a systematic study of ghost-like resonances, including the resummation leading to the the dressed propagator, pole structure, analytic continuation to the second Riemann sheet, and spectral representations. Notably, I demonstrated that, unlike ordinary unstable resonances, ghost propagators can exhibit complex conjugate poles in the first sheet for real masses above the multiparticle threshold. Moreover, the imaginary part of the dressed propagator is positive, unlike that of the bare propagator, suggesting that the ghost-like minus sign may disappear on-shell nonperturbatively.
Publication (2): L. Buoninfante, “Remarks on ghost resonances”, JHEP 02 (2025), 186.
(ii) I studied a case of nonlinear evolution in a higher-derivative theory of gravity by focusing on a gravitational collapse configuration. In particular, I carried out the first study of gravitational collapse in quadratic gravity, demonstrating that the evolution can be stable and that a horizon can form for a homogeneous and isotropic interior spacetime. This result challenges earlier static analyses that suggested horizonless solutions, marking a significant shift in the field. By pioneering a dynamical approach to compact-object formation, this work opened the door to further studies with relaxed symmetry assumptions and potential new phenomenology.
Publication (3): L. Buoninfante, F. Di Filippo, I. Kolar and F. Saueressig, “Dust collapse and horizon formation in quadratic gravity,” JCAP 01 (2025), 114
Objective 3: High-energy behaviour of higher-derivative and nonlocal field theories
Some higher-derivative and nonlocal theories of gravity are claimed to be super-renormalizable at the perturbative level, and ultraviolet divergences are only expected at one-loop. The initial goal of this objective was to determine the one-loop effective action for a generic higher-derivative nonlocal theory of gravity and discriminate among different theories by looking at the high-energy behaviour of the beta functions and consequently of the amplitudes. During the first year of MSCA I understood that by requiring the criterion of “strict renormalizability” (instead of superrenormalizability), it is possible to obtain a more unique and so predictive higher-derivative theory of gravity, i.e. quadratic gravity. I focused a lot on this theory, in particular on its motivation and physical motivations.
In particular, I showed that the historical success of the quantum field theory framework for the formulation of the Standard Model and the current cosmic microwave background observations point to quadratic gravity as a consistent, conservative renormalizable quantum field theory extension of general relativity in the high-energy regime. In particular, I demonstrated that quadratic gravity provides testable predictions for key cosmological parameters that can be verified by upcoming satellite missions in the next decade.
Publication (4): L. Buoninfante, “Strict renormalizability as a paradigm for fundamental physics”, JHEP 07 (2025), 175.
In addition to the above works, during the two years as MSCA fellow I also worked and published a book chapter as invited contributor, a report on black holes, a report on quantum gravity, and a set of lecture notes on quantum gravity.
Publication (5): L. Buoninfante and F. Di Filippo, “Is the information loss problem a paradox?”, Springer Series in Astrophysics and Cosmology, doi:10.1007/978-981-96-6170-1_3.
Publication (6): L. Buoninfante, et al., “Black Holes Inside and Out 2024: visions for the future of black hole physics”, [arXiv:2410.14414 [gr-qc]].
Publication (7): L. Buoninfante, et al., “Visions in quantum gravity,” SciPost Phys. Comm. Rep. 11 (2025).
Publication (8): I. Basile, L. Buoninfante, F. Di Filippo, B. Knorr, A. Platania and A. Tokareva, “Lectures in quantum gravity,” SciPost Phys. Lect. Notes 98 (2025), 1
Although the issue of S-matrix unitarity in nonlocal quantum field theories has been addressed in detail by several groups at the perturbative level, the question of what happens non-perturbatively is still not settled. Moreover, the questions of whether nonlocality violates causality and how to quantify such an effect are still open. The aim of this objective was to undertake a new non-perturbative study of unitarity and causality in generic nonlocal field theories by using methods of scattering amplitudes, dispersion relations and effective field theory constraints.
The requirements of unitarity and causality lead to significant constraints on the Wilson coefficients of an effective-field-theory expansion, known as positivity bounds. Their standard derivation relies on the crucial assumption of polynomial boundedness on the growth of scattering amplitudes in the complex energy plane, which is a property satisfied by local quantum field theories, and by weakly coupled string theory in the Regge regime. Together with my collaborators, I clarified the role of locality by deriving generalized positivity bounds under the assumption of exponential boundedness, typical of non-local quantum field theories where the Froissart-Martin bound is usually not satisfied. Using appropriately modified dispersion relations, I derived new constraints and found regions in the effective field theory parameter space that do not admit a local ultraviolet completion. At the same time, I provided an explicit example of an exponentially bounded amplitude that satisfies partial-wave unitarity and asymptotic causality. This study showed that it is possible to have nonlocal ultraviolet completions that satisfy unitarity and causality.
Publication (1): L. Buoninfante, L. Q. Shao and A. Tokareva, “Nonlocal positivity bounds: Islands in terra incognita”, Phys. Rev. D 112 (2025) no.2 L021904.
Objective 2: Degrees of freedom and stability in higher-derivative field theories
The counting of degrees of freedom of a higher-derivative or nonlocal Lagrangian is typically performed considering linear perturbations around some background and checking the propagator poles. This procedure is useful to verify stability and unitarity (e.g. absence of tachyons and ghosts) in quantum field theory, but it does not help analyse stability of the classical field equations at the non-linear level and independently of the background. The goal was to pioneer a nonperturbative and background-independent analysis of higher-derivative Lagrangians and use nonlinear stability conditions to constrain the theories.
I addressed part of this objectives in two ways:
(i) I considered non-perturbative resummations of the propagator in a higher-derivative field theory to check what happens to ghost-like degrees of freedom and whether non-perturbative effects can shed new light on their nature. In particular, I conducted a systematic study of ghost-like resonances, including the resummation leading to the the dressed propagator, pole structure, analytic continuation to the second Riemann sheet, and spectral representations. Notably, I demonstrated that, unlike ordinary unstable resonances, ghost propagators can exhibit complex conjugate poles in the first sheet for real masses above the multiparticle threshold. Moreover, the imaginary part of the dressed propagator is positive, unlike that of the bare propagator, suggesting that the ghost-like minus sign may disappear on-shell nonperturbatively.
Publication (2): L. Buoninfante, “Remarks on ghost resonances”, JHEP 02 (2025), 186.
(ii) I studied a case of nonlinear evolution in a higher-derivative theory of gravity by focusing on a gravitational collapse configuration. In particular, I carried out the first study of gravitational collapse in quadratic gravity, demonstrating that the evolution can be stable and that a horizon can form for a homogeneous and isotropic interior spacetime. This result challenges earlier static analyses that suggested horizonless solutions, marking a significant shift in the field. By pioneering a dynamical approach to compact-object formation, this work opened the door to further studies with relaxed symmetry assumptions and potential new phenomenology.
Publication (3): L. Buoninfante, F. Di Filippo, I. Kolar and F. Saueressig, “Dust collapse and horizon formation in quadratic gravity,” JCAP 01 (2025), 114
Objective 3: High-energy behaviour of higher-derivative and nonlocal field theories
Some higher-derivative and nonlocal theories of gravity are claimed to be super-renormalizable at the perturbative level, and ultraviolet divergences are only expected at one-loop. The initial goal of this objective was to determine the one-loop effective action for a generic higher-derivative nonlocal theory of gravity and discriminate among different theories by looking at the high-energy behaviour of the beta functions and consequently of the amplitudes. During the first year of MSCA I understood that by requiring the criterion of “strict renormalizability” (instead of superrenormalizability), it is possible to obtain a more unique and so predictive higher-derivative theory of gravity, i.e. quadratic gravity. I focused a lot on this theory, in particular on its motivation and physical motivations.
In particular, I showed that the historical success of the quantum field theory framework for the formulation of the Standard Model and the current cosmic microwave background observations point to quadratic gravity as a consistent, conservative renormalizable quantum field theory extension of general relativity in the high-energy regime. In particular, I demonstrated that quadratic gravity provides testable predictions for key cosmological parameters that can be verified by upcoming satellite missions in the next decade.
Publication (4): L. Buoninfante, “Strict renormalizability as a paradigm for fundamental physics”, JHEP 07 (2025), 175.
In addition to the above works, during the two years as MSCA fellow I also worked and published a book chapter as invited contributor, a report on black holes, a report on quantum gravity, and a set of lecture notes on quantum gravity.
Publication (5): L. Buoninfante and F. Di Filippo, “Is the information loss problem a paradox?”, Springer Series in Astrophysics and Cosmology, doi:10.1007/978-981-96-6170-1_3.
Publication (6): L. Buoninfante, et al., “Black Holes Inside and Out 2024: visions for the future of black hole physics”, [arXiv:2410.14414 [gr-qc]].
Publication (7): L. Buoninfante, et al., “Visions in quantum gravity,” SciPost Phys. Comm. Rep. 11 (2025).
Publication (8): I. Basile, L. Buoninfante, F. Di Filippo, B. Knorr, A. Platania and A. Tokareva, “Lectures in quantum gravity,” SciPost Phys. Lect. Notes 98 (2025), 1
The impact that my research results had during the two years of fellowship is confirmed by a significant number of citations that my publications received and by the several invitations I received for talks and interviews. In particular, I will now describe the impact for some of the publications listed above.
Publication (2): I showed that, unlike ordinary unstable resonances, ghost resonances are a new type of stable quantum objects that deserve further investigations. Indeed, understanding their true nature may help shed new light on quantum gravity in the sub-Planckian regime.
Publication (4): Rather than directly aiming at unification or a full quantum theory of spacetime, through this work I proposed an ambitious approach: developing a high-energy extension of general relativity, analogous to the transition from Fermi’s theory of weak interaction to the electroweak theory, and placing gravity on equal footing with the renormalizable interactions of the Standard Model. While progress in quantum gravity is often seen as relying solely on theoretical consistency, this work offered a novel, falsifiable perspective: extra degrees of freedom from quadratic curvatures generate new, testable physics at accessible energies below the Planck scale. This represents a rare and promising case of an approach to quantum gravity that may actually be confirmed or falsified in the near future.
Publication (7): This report on quantum gravity included the summaries of all the intensive panel discussions held during the three-week workshop organized as part of the NORDITA program in August 2024, along with individual reflections from all speakers and panelists, offering a clear snapshot of the current state of quantum-gravity research. This combined efforts, led by me together with the other three organizers, has been highly appreciated by the quantum-gravity community.
Publication (2): I showed that, unlike ordinary unstable resonances, ghost resonances are a new type of stable quantum objects that deserve further investigations. Indeed, understanding their true nature may help shed new light on quantum gravity in the sub-Planckian regime.
Publication (4): Rather than directly aiming at unification or a full quantum theory of spacetime, through this work I proposed an ambitious approach: developing a high-energy extension of general relativity, analogous to the transition from Fermi’s theory of weak interaction to the electroweak theory, and placing gravity on equal footing with the renormalizable interactions of the Standard Model. While progress in quantum gravity is often seen as relying solely on theoretical consistency, this work offered a novel, falsifiable perspective: extra degrees of freedom from quadratic curvatures generate new, testable physics at accessible energies below the Planck scale. This represents a rare and promising case of an approach to quantum gravity that may actually be confirmed or falsified in the near future.
Publication (7): This report on quantum gravity included the summaries of all the intensive panel discussions held during the three-week workshop organized as part of the NORDITA program in August 2024, along with individual reflections from all speakers and panelists, offering a clear snapshot of the current state of quantum-gravity research. This combined efforts, led by me together with the other three organizers, has been highly appreciated by the quantum-gravity community.