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Conformally Reduced Gravity vs Lorentzian Asymptotic Safety

Asymptotic Safety· within family

	Conformally Reduced Gravity Frontier	Lorentzian Asymptotic Safety Frontier
Proposed	2009	2025
Key figures	Martin Reuter, Holger Weyer	Frank Saueressig, Jian Wang
In one sentence	Conformally reduced gravity restricts the full quantum spacetime metric to its conformal mode, the single overall scale factor that says how big each region of spacetime is. The simplification permits analytic calculations and lets researchers verify whether asymptotic safety's basic mechanisms work as the full theory claims. Reuter-Weyer 2009 is the canonical reference; the framework now functions more as a pedagogical tool and verification testbed than a frontier research line.	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	The conformal mode flows to a non-trivial fixed point in the renormalization-group equation, consistent with the full theory's fixed-point claims Specific dimensionless ratios in the conformally reduced fixed-point structure should match those in the full theory's same sector; mismatch would indicate truncation artifacts The mechanism of asymptotic safety, fixed-point approach plus finite anomalous dimensions, should be visible already at the level of the conformal mode alone, not requiring the full geometric content to manifest	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 variant cannot be a candidate description of nature: freezing nine of ten metric components is a calculational convenience, not a physical claim. Results are only suggestive for the full theory, not conclusive If the full theory's fixed-point evidence is an artifact of how the calculations are organized, conformally reduced gravity is unlikely to reveal that because it is part of the same calculational framework The mode-by-mode reduction depends on a choice of conformal frame; different choices can give different intermediate results, complicating the interpretation Most active asymptotic-safety research has moved to matter-coupled and Lorentzian extensions; the conformally reduced approach now appears primarily in textbook treatments, similar to how toy models in other quantum-gravity programs (e.g. simple AdS/CFT examples) live in textbook treatments without driving frontier results	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 simplified-model problem: the encouraging results come from a stripped-down version of gravity, the conformally reduced sector, and no one knows whether the full ten-component theory of spacetime behaves the same way.	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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Conformally Reduced Gravity

2009 · Frontier

Lorentzian Asymptotic Safety

2025 · Frontier

Proposed

2009

2025

Key figures

Martin Reuter, Holger Weyer

Frank Saueressig, Jian Wang

In one sentence

Conformally reduced gravity restricts the full quantum spacetime metric to its conformal mode, the single overall scale factor that says how big each region of spacetime is. The simplification permits analytic calculations and lets researchers verify whether asymptotic safety's basic mechanisms work as the full theory claims. Reuter-Weyer 2009 is the canonical reference; the framework now functions more as a pedagogical tool and verification testbed than a frontier research line.

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

The conformal mode flows to a non-trivial fixed point in the renormalization-group equation, consistent with the full theory's fixed-point claims
Specific dimensionless ratios in the conformally reduced fixed-point structure should match those in the full theory's same sector; mismatch would indicate truncation artifacts
The mechanism of asymptotic safety, fixed-point approach plus finite anomalous dimensions, should be visible already at the level of the conformal mode alone, not requiring the full geometric content to manifest

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 variant cannot be a candidate description of nature: freezing nine of ten metric components is a calculational convenience, not a physical claim. Results are only suggestive for the full theory, not conclusive
If the full theory's fixed-point evidence is an artifact of how the calculations are organized, conformally reduced gravity is unlikely to reveal that because it is part of the same calculational framework
The mode-by-mode reduction depends on a choice of conformal frame; different choices can give different intermediate results, complicating the interpretation
Most active asymptotic-safety research has moved to matter-coupled and Lorentzian extensions; the conformally reduced approach now appears primarily in textbook treatments, similar to how toy models in other quantum-gravity programs (e.g. simple AdS/CFT examples) live in textbook treatments without driving frontier results

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 simplified-model problem: the encouraging results come from a stripped-down version of gravity, the conformally reduced sector, and no one knows whether the full ten-component theory of spacetime behaves the same way.

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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