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Higher-Derivative Gravity Extensions vs Lorentzian Asymptotic Safety

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	Higher-Derivative Gravity Extensions Frontier	Lorentzian Asymptotic Safety Frontier
Proposed	1977	2025
Key figures	Kellogg Stelle, Alessandro Codello, Roberto Percacci, Benjamin Knorr, Frank Saueressig	Frank Saueressig, Jian Wang
In one sentence	Stelle showed in 1977 that gravitational theories with curvature-squared terms (R-squared, Ricci-squared, Weyl-squared) added to the Einstein-Hilbert action are perturbatively renormalizable but contain ghosts: negative-norm states that ruin probability conservation. Codello-Percacci 2006 showed that within asymptotic safety's non-perturbative framework, fixed points exist for the higher-derivative couplings too, potentially resolving the ghost problem. Modern work on form factors by Knorr, Ripken, and Saueressig is the current state of the art.	Almost all asymptotic-safety calculations are performed in Euclidean signature (imaginary time), which makes the renormalization-group machinery tractable. Whether the results carry over to the Lorentzian signature of actual physical spacetime is a long-standing open question. Saueressig and Wang's 2025 'foliated' approach derives asymptotic safety directly in Lorentzian signature using an Arnowitt-Deser-Misner decomposition and a controlled Wick rotation.
Predictions	Fixed points in the renormalization-group flow exist for higher-derivative gravitational operators (R-squared, Ricci-squared, Weyl-squared couplings), not just for Newton's constant and the [[cosmological constant]] Curvature-dependent form factors, the functions parameterizing the full quantum corrections to the gravitational action, approach scaling forms at the fixed point that can be computed within truncation The classical ghost states of perturbative higher-derivative gravity are absent (or rendered harmless) in the non-perturbative asymptotic-safety completion Specific dimensionless ratios involving the higher-derivative couplings at the fixed point should match between different truncation schemes; agreement is consistency evidence, not a proof of the underlying theory	The asymptotic-safety fixed point exists in Lorentzian signature with structure consistent with the Euclidean-signature results; this is the central testable claim of the foliated approach The Wick rotation between Lorentzian and Euclidean asymptotic-safety calculations is controlled and well-defined, at least within the foliated framework; specific saddles and integration contours are tracked explicitly rather than assumed to be benign Causal structure (light cones, the timelike-spacelike distinction) is preserved by the renormalization-group flow in the foliated framework; the concrete check is whether the light-cone structure at the fixed-point couplings matches the low-energy light-cone geometry recovered from the infrared limit of the flow Specific dimensionless ratios at the fixed point in Lorentzian signature should match those from Euclidean calculations within calculational uncertainty; discrepancies would indicate signature dependence that the Euclidean program has missed
Where it breaks	The truncation convergence problem hits this variant especially hard: higher-derivative truncations are larger than the Einstein-Hilbert case, but the operator space is also bigger, so it is not obvious the convergence picture improves rather than just becoming more complicated The claim that ghosts are resolved by the non-perturbative completion is a hopeful interpretation of fixed-point evidence rather than a proof; a constructive demonstration that the non-perturbative theory is unitary is still missing Higher-derivative gravity theories are notoriously difficult to formulate causally; standard Cauchy-problem analyses produce instabilities that the perturbative framework cannot resolve The form-factor program produces consistent fixed-point structures, but the physical interpretation of those structures (what kind of theory they actually describe at high energies) is less developed than the technical calculations themselves	The foliated framework was published in 2025; most consistency checks against Euclidean predictions remain in progress. The full range of asymptotic-safety predictions has not yet been re-derived in Lorentzian signature, so caution is appropriate The foliated approach relies on specific choices: an Arnowitt-Deser-Misner-style decomposition of spacetime, a particular Wick rotation prescription, a choice of foliation surface. Whether the results are independent of these choices, the analog of background independence in the Lorentzian setting, is a technical question still being investigated Some authors have argued that the Lorentzian path integral for gravity is mathematically ill-defined in a deeper sense than the Wick rotation can resolve; if so, even a successful foliated derivation may be sitting on top of a structural problem The variant inherits all the truncation-convergence and BRST-symmetry concerns of the broader asymptotic-safety program; the move to Lorentzian signature does not address those questions
Key unresolved problem	The bad-probabilities problem: older versions of this kind of gravity produce ghost states that imply negative probabilities, and no one has yet proven the asymptotic-safety version is free of them, that it stays unitary.	The time-slicing problem: the real-time results may hinge on an arbitrary choice of how spacetime is sliced into moments, the ADM foliation, rather than reflecting physics that holds no matter how you slice it.
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Higher-Derivative Gravity Extensions

1977 · Frontier

Lorentzian Asymptotic Safety

2025 · Frontier

Proposed

1977

2025

Key figures

Kellogg Stelle, Alessandro Codello, Roberto Percacci, Benjamin Knorr, Frank Saueressig

Frank Saueressig, Jian Wang

In one sentence

Stelle showed in 1977 that gravitational theories with curvature-squared terms (R-squared, Ricci-squared, Weyl-squared) added to the Einstein-Hilbert action are perturbatively renormalizable but contain ghosts: negative-norm states that ruin probability conservation. Codello-Percacci 2006 showed that within asymptotic safety's non-perturbative framework, fixed points exist for the higher-derivative couplings too, potentially resolving the ghost problem. Modern work on form factors by Knorr, Ripken, and Saueressig is the current state of the art.

Almost all asymptotic-safety calculations are performed in Euclidean signature (imaginary time), which makes the renormalization-group machinery tractable. Whether the results carry over to the Lorentzian signature of actual physical spacetime is a long-standing open question. Saueressig and Wang's 2025 'foliated' approach derives asymptotic safety directly in Lorentzian signature using an Arnowitt-Deser-Misner decomposition and a controlled Wick rotation.

Predictions

Fixed points in the renormalization-group flow exist for higher-derivative gravitational operators (R-squared, Ricci-squared, Weyl-squared couplings), not just for Newton's constant and the [[cosmological constant]]
Curvature-dependent form factors, the functions parameterizing the full quantum corrections to the gravitational action, approach scaling forms at the fixed point that can be computed within truncation
The classical ghost states of perturbative higher-derivative gravity are absent (or rendered harmless) in the non-perturbative asymptotic-safety completion
Specific dimensionless ratios involving the higher-derivative couplings at the fixed point should match between different truncation schemes; agreement is consistency evidence, not a proof of the underlying theory

The asymptotic-safety fixed point exists in Lorentzian signature with structure consistent with the Euclidean-signature results; this is the central testable claim of the foliated approach
The Wick rotation between Lorentzian and Euclidean asymptotic-safety calculations is controlled and well-defined, at least within the foliated framework; specific saddles and integration contours are tracked explicitly rather than assumed to be benign
Causal structure (light cones, the timelike-spacelike distinction) is preserved by the renormalization-group flow in the foliated framework; the concrete check is whether the light-cone structure at the fixed-point couplings matches the low-energy light-cone geometry recovered from the infrared limit of the flow
Specific dimensionless ratios at the fixed point in Lorentzian signature should match those from Euclidean calculations within calculational uncertainty; discrepancies would indicate signature dependence that the Euclidean program has missed

Where it breaks

The truncation convergence problem hits this variant especially hard: higher-derivative truncations are larger than the Einstein-Hilbert case, but the operator space is also bigger, so it is not obvious the convergence picture improves rather than just becoming more complicated
The claim that ghosts are resolved by the non-perturbative completion is a hopeful interpretation of fixed-point evidence rather than a proof; a constructive demonstration that the non-perturbative theory is unitary is still missing
Higher-derivative gravity theories are notoriously difficult to formulate causally; standard Cauchy-problem analyses produce instabilities that the perturbative framework cannot resolve
The form-factor program produces consistent fixed-point structures, but the physical interpretation of those structures (what kind of theory they actually describe at high energies) is less developed than the technical calculations themselves

The foliated framework was published in 2025; most consistency checks against Euclidean predictions remain in progress. The full range of asymptotic-safety predictions has not yet been re-derived in Lorentzian signature, so caution is appropriate
The foliated approach relies on specific choices: an Arnowitt-Deser-Misner-style decomposition of spacetime, a particular Wick rotation prescription, a choice of foliation surface. Whether the results are independent of these choices, the analog of background independence in the Lorentzian setting, is a technical question still being investigated
Some authors have argued that the Lorentzian path integral for gravity is mathematically ill-defined in a deeper sense than the Wick rotation can resolve; if so, even a successful foliated derivation may be sitting on top of a structural problem
The variant inherits all the truncation-convergence and BRST-symmetry concerns of the broader asymptotic-safety program; the move to Lorentzian signature does not address those questions

Key unresolved problem

The bad-probabilities problem: older versions of this kind of gravity produce ghost states that imply negative probabilities, and no one has yet proven the asymptotic-safety version is free of them, that it stays unitary.

The time-slicing problem: the real-time results may hinge on an arbitrary choice of how spacetime is sliced into moments, the ADM foliation, rather than reflecting physics that holds no matter how you slice it.

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