PaperlayerView exact revision on arXiv

Source-linked AI summary

Critical reflections on asymptotically safe gravity

Alfio Bonanno, Astrid Eichhorn, Holger Gies, Jan M. Pawlowski, Roberto Percacci, Martin Reuter, Frank Saueressig, Gian Paolo Vacca

arXiv:2004.06810v1gr-qchep-th

TL;DR

The paper addresses open technical and conceptual questions in asymptotically safe gravity and the need to clarify its current status. It critically reviews existing results and identifies systematic pathways for testing them, while highlighting unresolved limitations including EFT matching and RG improvement.

  • Problem

    Further progress in understanding quantum spacetime requires resolving open technical and conceptual questions surrounding asymptotically safe gravity.

  • Method

    The paper critically reviews the program’s state of the art, examines its open questions, and identifies systematic research pathways using multiple methods and observable analyses.

  • Results

    The review clarifies the current status of asymptotically safe gravity while identifying unresolved issues concerning vacuum structure, Planckian scattering, spectral representations, and unitarity.

  • Takeaways & Limitations

    The paper concludes that cross-checks between quantum-gravity approaches and concerted use of more than one method are called for.

  • Takeaways & Limitations

    Matching asymptotic safety to low-energy effective field theory and assessing the limitations of RG improvement remain necessary parts of the analysis.

Abstract

from arXiv · show

Asymptotic safety is a theoretical proposal for the ultraviolet completion of quantum field theories, in particular for quantum gravity. Significant progress on this program has led to a first characterization of the Reuter fixed point. Further advancement in our understanding of the nature of quantum spacetime requires addressing a number of open questions and challenges. Here, we aim at providing a critical reflection on the state of the art in the asymptotic safety program, specifying and elaborating on open questions of both technical and conceptual nature. We also point out systematic pathways, in various stages of practical implementation, towards answering them. Finally, we also take the opportunity to clarify some common misunderstandings regarding the program.

I. INTRODUCTION AND CONCLUSIONS

Asymptotic Safety is presented as a candidate quantum theory of gravity within relativistic QFT, potentially including matter fields. The paper critically reviews the program’s achievements, open technical and conceptual questions, and pathways for future progress.

  • Asymptotic Safety seeks a quantum theory of gravitational interactions using relativistic QFT and metric degrees of freedom, with matter inclusion conceptually straightforward.
  • The paper provides a critical assessment of the field’s achievements, shortcomings, open questions, and future research directions rather than an introductory exposition.
  • The review examines functional-RG uncertainties, including uncontrollable approximations and dependence on the background-field method.
  • It discusses cross-checking asymptotic-safety results with different methods because of theoretical uncertainties.
  • Open problems include the physical meaning of running couplings, observable calculations, EFT matching, scale versus conformal symmetry, unitarity, and Lorentzian-signature calculations.

A. The main idea

Asymptotic Safety links ultraviolet completion, renormalizability, and predictive power to RG fixed points. The section reviews the fixed-point mechanism and evidence from non-gravitational models, including complementary Euclidean, Lorentzian, analytical, simulational, and bootstrap approaches.

  • A. The main idea: Asymptotic Safety identifies quantum scale invariance in the UV with non-perturbative renormalizability and a finite number of free parameters.
  • A. The main idea: RG trajectories describe physical theories, while trajectories emanating from fixed points as k decreases can define asymptotically safe theories.
  • A. The main idea: The fixed point’s stability matrix determines attractive and repulsive directions, providing the basis for asymptotic-safety predictivity.
  • A. The main idea: Fermionic non-Gaussian fixed points connect to quantum phase transitions whose universal critical exponents are studied by simulations and conformal bootstrap.
  • A. The main idea: Quantitative agreement with Quantum Monte Carlo shows asymptotic safety in these systems is visible in both Euclidean and Lorentzian formulations.
  • A. The main idea: At non-Gaussian fixed points, anomalous dimensions can make perturbatively marginal operators relevant and produce power-law rather than logarithmic coupling scaling.

III. FUNCTIONAL RENORMALIZATION GROUP

The Functional Renormalization Group is the primary tool used to investigate Asymptotic Safety through the scale-dependent effective average action. Its flow integrates fluctuations shell by shell and enables searches for interacting fixed points.

  • The FRG equation for the effective average action Γk is the primary tool for investigating Asymptotic Safety.
  • Γk contains metric, ghost, and possibly matter fields, while k suppresses quantum fluctuations below the infrared cutoff.
  • The regulator Rk suppresses modes with p^2 ≲ k^2, decays for p^2 ≳ k^2, and vanishes at k^2 = 0, making the flow finite at large momenta.
  • The Wetterich equation implements Wilsonian running by integrating quantum fluctuations shell by shell as k is lowered.
  • The FRG flow approaches the standard one-loop effective action at k = 0, so FRG approximations naturally contain one-loop results.

B. FRG approach to quantum gravity

In quantum gravity, the FRG uses background-field techniques and organizes the effective action through vertex and derivative expansions. These constructions introduce background dependence and face unresolved parameterization and convergence challenges.

  • B. FRG approach to quantum gravity: The gravitational background-field method decomposes the metric into an arbitrary background metric and fluctuations.
  • B. FRG approach to quantum gravity: Different parameterizations represent the same quantization only when they cover the same configuration space and include the path-integral Jacobian.
  • B. FRG approach to quantum gravity: The background enables momentum classification and gauge fixing but makes Γk depend on both fluctuation and background fields.
  • B. FRG approach to quantum gravity: The asymptotic-safety mechanism is not tied exclusively to metric variables; vielbein and Palatini formulations may also lead to asymptotic safety.
  • B. FRG approach to quantum gravity: The vertex expansion uses proper vertices as theory-space coordinates, while the derivative expansion organizes diffeomorphism-invariant operators by derivative order.
  • B. FRG approach to quantum gravity: The Einstein-Hilbert action is only the leading derivative-expansion terms and must be supplemented by gauge-fixing, ghost, and potentially higher-derivative contributions.

C. Results for asymptotically safe gravity

Studies of asymptotically safe gravity use FRG truncations, background and vertex expansions, and complete-flow calculations to characterize the Reuter fixed point and connect it to infrared physics. Results are encouraging, but systematic convergence, operator completeness, and full UV–IR trajectories remain open challenges.

  • Functional renormalization group: FRG studies of pure gravity explore increasingly broad truncations, including curvature invariants, polynomial f(R) actions, and effective actions with many couplings.Closed flow equations are also available for some infinite-dimensional functional truncations.
  • Open technical challenges: Systematic convergence remains difficult because commonly used backgrounds cannot distinguish all curvature invariants and vertex calculations use special kinematical configurations.Non-local operators also threaten predictivity if included explicitly, whereas quasi-local theory spaces avoid generating them under the flow.
  • Vertex expansion: Vertex expansions around flat space retain momentum dependence and have produced a full f(R)-potential beyond the background approximation.Higher-order momentum-dependent correlation functions provide evidence for the Reuter fixed point.
  • UV–IR trajectories: Complete trajectories connecting the Reuter fixed point to k = 0 are less advanced than fixed-point characterization and have been computed only in selected cases.The expected flow contains several regimes, including a UV regime near the Reuter fixed point and a simpler gravitational regime below the Planck scale.
  • Convergence question: Near-canonical scaling spectra in f(R) truncations support the near-perturbative assumption underlying canonical-power-counting choices, while apparent fixed-point convergence encourages the expectation that the Reuter fixed point exists in full theory space.A complete set of curvature-cube operators remains outstanding.

E. Do backgrounds matter?

Background fields are essential to the FRG formulation but create a central background-dependence problem. The effective average action can nevertheless implement background independence dynamically, with restoration at k = 0.

  • Background dependence: The FRG uses a background metric to implement gauge fixing and local coarse graining, but the regulator and gauge fixing break split symmetry.Consequently, Γk genuinely depends on both fluctuation and background metrics.
  • Background independence: The effective average action remains background independent because the background is a freely variable metric-type argument determined by its own equations of motion.The Ward identity relates background and quantum equations of motion, allowing the background to be fixed dynamically.
  • Background independence: At finite k, regulator contributions introduce genuine background dependence, so background independence is restored only in the physical limit k = 0.The self-consistent background metric is therefore a dynamical prediction rather than an input.
  • Open challenges: A major technical challenge is disentangling the separate hµν and ¯gµν dependence of Γk across a broad class of background metrics.Current calculations mainly use highly symmetric geometries or local Seeley-DeWitt heat-kernel information, and setting h = 0 can deform or remove fixed points.
  • Open challenges: Strong k dependence may imply multifractal effective spacetimes, but it remains unclear whether this behavior leaves an echo in the physical limit k → 0.The full momentum dependence of Γk→0 is needed to answer this question.
  • Open challenges: Background dependence is identified as a main conceptual and technical obstacle to applying the FRG to quantum gravity.This challenge motivates systematic future work on the separate background and fluctuation dependence.

IV. ADDITIONAL METHODS FOR ASYMPTOTIC SAFETY

The paper surveys complementary methods for testing asymptotic safety and clarifies how Wilsonian and perturbative renormalization-group meanings differ. These methods can reduce systematic uncertainties and broaden access to phase structure and phenomenology.

  • Motivation: Complementary methods are needed because FRG calculations face technical challenges and different approaches have different systematic errors.Regge calculus, random lattices, and tensor models can also probe phase diagrams and possible pre-geometric phases.
  • ε expansion: The ε expansion connects the four-dimensional Reuter fixed point continuously to the perturbative fixed point in 2 + ε dimensions.The connection provides a perturbative route for studying the fixed point, alongside links to Liouville gravity in two dimensions.
  • Lattice approaches: Lattice approaches test asymptotic safety through statistical theories of random geometries, using fixed triangulations with variable edge lengths or variable triangulations with fixed edge lengths.Regge simulations provide indications for asymptotic safety, but curvature-squared operators and conformal-factor instability remain important caveats.
  • Numerical simulations: Evidence for second-order phase-transition lines or points has been found numerically in higher-dimensional lattice studies, although systematic uncertainties remain.Further evidence requires suitable truncation extensions and improved understanding of emergent geometries.
  • Other approaches: Asymptotic-safety ideas are gaining traction in group field theories and spin-foam models, extending the search for universal continuum limits beyond FRG and lattice methods.The paper presents asymptotic safety as potentially relevant across several distinct quantum-gravity approaches.
  • Interpretation and prospects: A broad toolkit is needed to establish fixed-point existence, gain quantitative control, and develop robust phenomenological links while controlling systematic errors.The paper also clarifies that RG terminology, running couplings, and regularization procedures differ between Wilsonian and particle-physics settings.

B. Remarks on dimensional regularization

Dimensional regularization can reproduce universal logarithmic information but is insufficient for much of the power-law running relevant to asymptotic safety. The paper therefore emphasizes careful analytic-continuation procedures and momentum-dependent observables.

  • Regularization: FRG uses explicit momentum or spectral regulators, whereas perturbative particle physics usually uses dimensional regularization; scheme independence makes their relation an important question.The paper asks whether dimensional regularization can be consistently incorporated into the FRG treatment of asymptotically safe gravity.
  • Universality and divergences: One-loop curvature-quadratic operators with dimensionless couplings show the expected universality, but relevant and irrelevant operators do not share that degree of universality.Dimensional regularization is blind to power divergences and projects onto logarithmic divergences represented by 1/ε poles.
  • Analytic continuation: Dimensional regularization requires attention to analytic structure, especially for non-relativistic scattering and computations in large background fields.Proper-time or ζ-function formulations connected to heat-kernel methods can access information about power divergences.
  • Application to gravity: In large-background gravity computations, proper-time-based definitions provide a route for applying dimensional-regularization ideas, and propertime RG applications support the Reuter fixed point.The paper treats this as a setting where naive dimensional regularization is not sufficient.
  • Physical interpretation: Asymptotically safe gravity is characterized through momentum-dependent correlation functions, form factors, and running couplings rather than RG-scale dependence alone.These objects can identify ultraviolet, transition, and infrared scaling regimes and support the construction of observables.
  • Open limitations: Existing approximations identify asymptotically safe, transition, and classical infrared regimes, but they do not yet sustain large curvatures and require significant upgrades.Correlation functions in the scaling regime can exhibit indications of quantum scale invariance at large momenta.

VI. OBSERVABLES

Meaningful observables are essential for testing asymptotic safety, but diffeomorphism invariance makes local observables difficult to define in quantum gravity. The paper therefore focuses on phenomenologically relevant observables supported by dynamically generated backgrounds.

  • Phenomenological observables: Testing quantum gravity requires moving from correlation functions and effective quantities toward observables with direct phenomenological relevance.The paper emphasizes observables as the route to making asymptotic safety testable.
  • Defining observables: Diffeomorphism invariance makes spacetime points unphysical, so quantum gravity generally lacks local gauge-invariant observables.Meaningful observables must instead be constructed as integrals of scalar densities over spacetime.
  • Phenomenological observables: The relevant observables depend strongly on the intended observations and often require a dynamically generated background to provide a notion of locality.The paper distinguishes three classes of observations for testing asymptotic safety.

A. Particle physics at the Planck scale

The paper examines how asymptotic safety might be tested through Planck-scale, low-energy, and cosmological observables, while emphasizing substantial theoretical and experimental limitations. It also discusses linking asymptotic safety to effective field theory through complementary perturbative and functional-renormalization-group descriptions.

  • Particle-physics observables: Planck-scale scattering and decay observables require an asymptotically flat background, but neither the theoretical nor experimental comparison is currently available.Such observables are also unlikely to be measured with current technology, and the background assumption excludes situations relevant to quantum gravity.
  • Low-energy imprints: Low-energy observables may retain asymptotic-safety imprints through higher-order operators or constraints on matter-sector parameters.Higher-order effects are typically unmeasurably small across the Planck-to-infrared scale separation, whereas matter predictions include the Abelian gauge coupling and bottom mass, subject to small truncations and Euclidean-to-Lorentzian assumptions.
  • Cosmological observables: Cosmological observables are formulated as integrated correlators on backgrounds such as Friedmann-Robertson-Walker spacetime, with curvature fluctuations carrying scale-dependent physics information.The paper discusses possible influences on inflation, scalar-potential flatness, and early-universe configurations, including suppression of initial singularities, anisotropies, and inhomogeneities in one higher-derivative action.
  • Cross-checks and geometrical observables: Lattice studies could cross-check continuum results, but quantum-gravity observables are difficult to define and implement outside asymptotically flat settings.Geometrical observables such as lengths, areas, volumes, and curvatures can also be investigated through composite-operator flow equations, although their measurement remains unclear.
  • Effective-field-theory connection: A realistic renormalization-group trajectory should connect the ultraviolet fixed point to Einstein-gravity effective field theory near the free-theory fixed point.The paper proposes matching functional-renormalization-group and perturbative descriptions at a scale where perturbation theory becomes applicable, reconstructing the loop expansion systematically.

B. Effective vs. fundamental Asymptotic Safety

The paper considers asymptotic-safety fixed points both as possible fundamental ultraviolet descriptions and as intermediate scaling regimes within a theory completed differently at shorter distances. It argues that this broader role requires determining the theory’s self-consistent ground state, which remains poorly understood beyond severe approximations.

  • Effective versus fundamental roles: An asymptotically safe fixed point could matter even if a different theory describes quantum gravity in the deep ultraviolet.The fixed point may provide an effective description below a finite ultraviolet cutoff and extend the validity of perturbative gravitational EFT.
  • Effective versus fundamental roles: The fixed point can serve as a UV starting point, an IR endpoint, or an intermediate finite-scale scaling regime.Its multiple infrared-attractive directions allow these distinct roles in different scenarios.
  • Vacuum structure: Connecting asymptotic safety to gravitational EFT requires establishing whether flat spacetime is the theory’s self-consistent ground state.The bridge is potentially intricate because the true ground state may not be flat, and this issue has so far been studied only in a severe approximation.
  • Vacuum structure: The only explicit Γk=0 vacuum investigation found a layered “lasagna vacuum” within a conformally reduced R + R2 truncation.That truncation includes only conformal-factor fluctuations, so the proposed structure remains tied to a restricted treatment of dynamics and degrees of freedom.
  • Vacuum structure: Identifying the correct k-dependent ground state is important for interpreting the cosmological-constant problem and the physical implications of high-scale curvature.The paper leaves open whether the curvature implied by Λk = λ∗k2 becomes physical as k approaches zero, with implications for singularity resolution and phenomenology.

D. RG improvement

RG improvement uses scale-dependent couplings with a geometrical or momentum identification to estimate quantum-gravity effects, but its results are generally qualitative and potentially ambiguous. Physical effects must instead be derived from the fully integrated effective action, while scale invariance is recovered at fixed points under the Wilsonian realization.

  • RG-improvement procedure: RG improvement retains running couplings such as G_k and Λ_k and identifies the RG scale with a geometrical quantity or momentum.Applications include black holes, gravitational collapse, and cosmological scenarios.
  • RG-improvement procedure: Applying the improvement at the action, field-equation, or solution level introduces freedom that can produce ambiguous results.The ambiguity arises because these stages are not equivalent by construction.
  • RG-improvement procedure: At a fixed point, G_k = g_*k^-2 and Λ_k = λ_*k^2; identifying k^2 with the Ricci scalar generates an R^2 action.The generated interaction has a natural place in the effective action, but the procedure yields at most qualitative insights.
  • Examples and reliability: In QED and selected gravitational examples, RG improvement can reproduce qualitative behavior and, for ξ = 0 or ξ = 1/6, correct numerical coefficients.The gravitational comparison concerns quantum corrections to the Newtonian potential from scalar-field loops.
  • Examples and reliability: Physical quantum effects must be calculated from Γ_k→0, whereas RG-improved black-hole and cosmological models are only quantum-gravity-inspired.They can provide qualitatively sensible but not necessarily precise answers when one clearly identifiable scale dominates.
  • Scale anomaly: At a Wilsonian fixed point, quantum scale invariance is realized through a rescaling implementation different from the standard particle-physics one.The alternative realization transforms the cutoff together with the fields, and the anomaly vanishes at the fixed point.

B. Black hole entropy

The paper critically examines arguments that black-hole entropy conflicts with asymptotic safety. Those arguments rely on assumptions about high-energy states, semiclassical entropy scaling, quantum corrections, and dimensionality that remain insufficiently established in quantum gravity.

  • Argument and assumptions: The incompatibility argument assumes that black holes dominate the high-energy density of states, an assumption called asymptotic darkness.The chain then compares black-hole and conformal-field-theory entropy scaling.
  • Argument and assumptions: In four dimensions, the asymptotic-darkness argument gives S_BH ∝ E^2, whereas CFT degrees of freedom follow a different dimension-dependent scaling law.This mismatch is central to the proposed tension with a QFT description of gravity.
  • Critical review: The quantum analysis lacks a corresponding study of black-hole formation in Planckian scattering, leaving the relevance of the classical expectation unresolved.Effective-action form factors or 1PI vertices are expected to be important for trans-Planckian scattering.
  • Critical review: Semiclassical entropy scaling may fail near the Planck scale because quantum corrections become increasingly important for small black holes.Therefore, applying the simple area-law scaling in the quantum-gravity regime is not established.
  • Critical review: The correct dimensionality controlling scaling laws in fluctuating spacetimes is uncertain because quantum gravity can exhibit multiple notions of dimension.This complicates direct extensions of flat-space scaling arguments.
  • Conclusion: Combining semiclassical asymptotic-darkness arguments with flat-space CFT arguments produces tension with asymptotic safety, but their applicability to quantum gravity remains unshown.The paper concludes that substantially more work is needed before the conflict can be established.

IX. UNITARITY

Unitarity in asymptotically safe gravity is harder to assess than in flat-space QFT because the relevant S-matrix and background structure are nonstandard. Ghost-like modes, positivity violations, and Euclidean-to-Lorentzian issues require evaluating the fully integrated effective action and developing Lorentzian methods.

  • Conceptual status: A unitary S matrix is standard in flat Lorentzian QFT, but generalizing it to arbitrary backgrounds and gravitational interactions is highly non-trivial.Euclidean reflection positivity guarantees analytic continuation only in the conventional flat-space setting described here.
  • Conceptual status: Unphysical modes in a chosen background do not automatically imply inconsistency because they may signal instability of that background.The paper notes that non-standard backgrounds can remove the Euclidean conformal-mode problem.
  • Assessment criterion: CDT provides an indicator of possible unitarity through a self-adjoint, bounded transfer matrix and a well-defined Lorentzian continuation.This evidence comes from a related quantum-gravity program rather than the full asymptotic-safety framework.
  • Conclusion: The paper emphasizes that unitarity limitations remain intrinsic to background-independent quantum gravity.These cautions constrain interpretation of positivity violations and candidate ghost modes.
  • Ghost modes: Higher-derivative propagators can contain negative-residue ghost modes, violating reflection positivity and signaling Lorentzian non-unitarity for physical asymptotic states.This violation is already present classically as an instability of the theory.
  • Ghost modes: Proposed ways around Ostrogradsky instability include non-local ghost-free propagators, Lorentz-violating higher spatial derivatives, or accepting microscopic causality violation.These routes trade or modify different structural principles.
  • Assessment criterion: Unitarity must be assessed from Γ_k=0 propagators because intermediate-scale Γ_k propagators can contain artificial poles.A ghost mass may diverge along the flow, causing the associated degrees of freedom to decouple.

C. Spectral function of the graviton

Negative spectral weights and the uncertain existence of gauge-field spectral representations complicate unitarity tests for the graviton. The paper therefore stresses that asymptotic safety's unitarity remains unresolved and that Lorentzian, non-perturbative analyses are needed.

  • Spectral weights: A general graviton propagator can produce spectral functions with negative parts, extending beyond isolated ghost contributions.Negative spectral weights are therefore a broader issue than the single ghost mode discussed earlier.
  • Spectral weights: Yang-Mills theory shows that negative gluon spectral weights can occur without directly corresponding to asymptotic physical states.The paper uses this analogy to qualify the interpretation of positivity violations in gravity.
  • Spectral representations: The existence of spectral representations for gauge fields with nonlinear gauge symmetries remains an open issue.In gravity, states associated with large-mass negative residues need not correspond to asymptotic states.
  • Interpretation: Future work may find physical ghost modes, yet an asymptotically safe fixed point could still extend the gravitational EFT regime and inherit unitarity from a deeper description.This is presented as a possible effective-asymptotic-safety setting, not an established result.
  • Conclusion: The current status of asymptotically safe gravity's unitarity is unclear, requiring conceptual analysis of correlators and non-perturbative Lorentzian computations.The paper calls for cross-checks between quantum-gravity approaches and multiple methods.
  • Lorentzian signature: Euclidean-to-Lorentzian continuation is obstructed by absent global Killing vectors, complicated propagator analytic structure, and structural differences between Euclidean and Lorentzian effective actions.Momentum regularization in the FRG can introduce additional symmetry breaking, poles, or cuts.
  • Lorentzian signature: ADM-based studies provide initial indications that asymptotic safety persists for Lorentzian metrics within very small truncations, but systematic Lorentzian FRG remains under development.Current calculations do not yet resolve the structural differences between signatures.
Loading 2004.06810v1…