Lorentzian Asymptotic Safety
The newest active line in the program. Derives asymptotic-safety results directly in Lorentzian (physical) signature rather than the Euclidean signature used by almost all prior calculations. Answers a long-standing critique that the framework might be a Euclidean artifact.
Placeholder for a 3D visualisation of Asymptotic Safety. The interactive scene will land in Phase 3. Asymptotic Safety proposes that gravity is a sensible quantum theory on its own, without strings, supersymmetry, or any extra structure. Weinberg suggested in 1979 that gravity's troublesome ultraviolet behavior might be tamed not by extra physics but by a non-trivial fixed point: as you push to higher energies, the gravitational couplings approach a fixed, finite value rather than blowing up. Reuter 1998 made the proposal calculable using the functional renormalization group, an exact equation that tracks how couplings change with energy. Three decades of subsequent work have asked, in progressively more realistic approximations, whether the fixed point really exists, whether it survives the inclusion of Standard Model matter, whether higher-derivative gravitational operators preserve it, and whether results derived in Euclidean signature carry over to the Lorentzian signature of the spacetime we actually live in. The most striking quantitative claim is Shaposhnikov-Wetterich's 2010 prediction of the Higgs boson mass at 126 GeV, made two years before the LHC measured 125.1 GeV. Either the field's clearest empirical success or its most striking accident.
In one sentence
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.
The claim
Almost all renormalization-group calculations in asymptotic safety, and in the broader quantum-field-theory toolbox, are done in Euclidean signature. Time is replaced by imaginary time, the four-dimensional spacetime metric becomes positive-definite, oscillating quantum amplitudes become exponentially decaying integrands, and many integrals that diverge in Lorentzian signature converge cleanly in Euclidean signature. This is fine as long as you can Wick-rotate back to physical Lorentzian signature at the end. For gauge theories on flat backgrounds the Wick rotation is well-defined and the procedure is standard. For gravity, where the metric itself is being summed over, whether the Wick rotation makes sense at all has been a contested question for decades.
The critique that asymptotic safety might be a Euclidean artifact has been voiced inside the program and outside it. Several specific concerns: (1) the gravitational path integral has saddles in Euclidean signature with no Lorentzian counterpart; (2) the renormalization-group flow equations may depend on signature in subtle ways through the ghost and gauge-fixing sectors; (3) physical observables (light cones, causal structure) are only directly accessible in Lorentzian signature. The 2024-2025 work on foliated asymptotic safety, Saueressig and Wang's paper as the canonical reference, addresses these concerns by deriving the renormalization-group flow directly in Lorentzian signature via an Arnowitt-Deser-Misner decomposition that splits spacetime into space-like slices and tracks evolution explicitly.
Honest editorial framing: this is the newest active line in the program. The Saueressig-Wang 2025 paper has 19 citations at the time of writing because it is genuinely 2025 work, not because the topic is fringe; the Lorentzian critique has been live for decades but the constructive direct-derivation response is recent. Recent technical progress within an active program rather than a closed-case resolution. Whether the Lorentzian results agree with the Euclidean ones across the range of asymptotic-safety predictions is the question now being checked.
The family stance
Gravity needs no new physics beyond itself and the Standard Model to be a complete quantum theory. The Einstein-Hilbert action, treated as the leading approximation to a more complete quantum theory and run to high energies via the renormalization group, approaches a non-trivial fixed point where all couplings remain finite. Combined with the right matter content, the same framework yields a Higgs-mass prediction within experimental accuracy. After four decades, asymptotic safety has produced consistent fixed-point evidence in increasingly realistic truncations and one striking quantitative empirical success.
Predictions
- 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; this is in principle testable by computing causal observables directly in 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
Evidence
- Saueressig-Wang 2025 derived the foliated formulation explicitly and showed initial results consistent with the Euclidean-signature asymptotic-safety predictions; the work is technically detailed and was peer-reviewed in Phys. Rev. D
- The foliated framework is internally consistent and produces a Lorentzian renormalization-group flow that is non-trivial; the existence of such a derivation responds to a long-standing technical concern about the program
- Earlier work by the Saueressig group and others (e.g. on time-dependent backgrounds) had laid groundwork for the foliated approach; the 2025 paper is the consolidation of a multi-year program rather than a one-off result
- The agreement with Euclidean predictions, where it has been checked, is evidence that the original program's claims survive the move to physical signature in the regime of validity of the foliated framework
Counterpoints
- The variant is very new; the foliated framework has been published in 2025 and the full range of asymptotic-safety predictions has not yet been re-derived in Lorentzian signature. Most of the consistency checks remain to be done. The light citation count (19 on the founding paper) reflects this newness rather than fringe status, but it does mean 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
Variants in this family
▸Go deeperTechnical detail with proper terminology
Arnowitt-Deser-Misner (ADM) decomposition: a foliation of spacetime into space-like hypersurfaces parameterized by a time coordinate. The metric splits into a lapse function (clock-rate), a shift vector (relative motion of the foliation), and the induced spatial metric. ADM variables are the natural starting point for Hamiltonian formulations of general relativity and for derivations of canonical quantum gravity.
Foliated quantum gravity: the renormalization-group flow is derived for the ADM variables directly, treating the lapse, shift, and spatial metric as the quantum fields. The signature stays Lorentzian throughout, and the Wick rotation is performed explicitly on the time-component contour rather than assumed at the level of the full metric.
Wick rotation prescription: in Saueressig-Wang 2025, the rotation is performed on the lapse function (the time-time component of the metric) in a controlled way that preserves causal structure. The prescription gives a definite Lorentzian flow equation that can be solved or numerically integrated.
Comparison with Euclidean results: at the level of fixed-point existence and basic dimensionless ratios (cosmological constant times Newton's constant, anomalous dimensions of leading operators), the 2025 Lorentzian results agree with the Euclidean ones within calculational accuracy. Whether the agreement extends to more refined predictions (matter-coupled fixed points, higher-derivative operator scaling) is the active investigation.
References
Last reviewed May 18, 2026
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