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arxiv: 2606.19321 · v1 · pith:TZ4LFQZFnew · submitted 2026-06-17 · ✦ hep-th · gr-qc

Spectral Functions of Lorentzian Quantum Gravity

Gabriel Assant , Daniel F. Litim , Manuel Reichert This is my paper

Pith reviewed 2026-06-26 19:42 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords spectral functionsquantum gravityasymptotic safetyfunctional renormalisation groupgraviton propagatorKällén-Lehmann representationLorentzian signatureunitarity
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The pith

Spectral functions for graviton modes stay normalisable from infrared to ultraviolet fixed point in Lorentzian quantum gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a direct computational route to spectral functions of graviton modes while remaining in Lorentzian signature. It adapts functional renormalisation group methods by adding symmetry conditions that enforce underlying Ward identities, then derives and solves the flow equations inside the Källén-Lehmann representation of the propagators. Consistent, normalisable spectra emerge for both the graviton and the scalar graviton mode under several choices of renormalisation conditions, and these spectra match the expectations of effective field theory at low energies. The same framework yields the quantum effective action to quadratic order in curvature, from which graviton-induced form factors are extracted. The results bear on whether quantum gravity can be formulated in a manner compatible with unitarity without an intermediate Euclidean continuation.

Core claim

We compute spectral functions of graviton modes in Lorentzian quantum gravity, interpolating between classical general relativity and an asymptotically safe ultraviolet fixed point. Using functional renormalisation adapted for theories in Lorentzian signature, and enhanced by new symmetry conditions to account for underlying Ward identities, we derive and solve flow equations directly for the Källén-Lehmann representation of propagators. Consistent results are found for several sets of renormalisation conditions yielding normalisable spectral functions for the graviton and the scalar graviton mode, in agreement with effective theory in the infrared. We further calculate the full quantum effe

What carries the argument

Flow equations solved directly inside the Källén-Lehmann spectral representation of the propagators, closed by new symmetry conditions that implement Ward identities in Lorentzian signature.

If this is right

  • Normalisable spectral functions exist for the graviton across the entire trajectory from the infrared to the ultraviolet fixed point.
  • The scalar graviton mode likewise possesses a normalisable spectrum under the same renormalisation conditions.
  • The quantum effective action can be obtained to quadratic order in curvature, supplying explicit graviton-induced form factors.
  • These spectral functions are compatible with the infrared behaviour of classical general relativity and therefore support unitarity of the theory.
  • Multiple independent sets of renormalisation conditions produce mutually consistent results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same direct spectral method could be applied to other fields or to matter couplings within the same truncation.
  • Form factors extracted this way could be inserted into scattering calculations to test high-energy behaviour without Wick rotation.
  • If the normalisability persists under enlargement of the truncation, it would strengthen the case that asymptotic safety preserves unitarity in Lorentzian signature.

Load-bearing premise

The added symmetry conditions are sufficient to remove all residual inconsistencies that the Lorentzian signature would otherwise introduce into the flow equations.

What would settle it

An explicit solution of the flow equations in which the spectral function for the graviton or scalar mode develops a negative residue or becomes non-normalisable at some finite scale would falsify the consistency claim.

read the original abstract

We compute spectral functions of graviton modes in Lorentzian quantum gravity, interpolating between classical general relativity and an asymptotically safe ultraviolet fixed point. Using functional renormalisation adapted for theories in Lorentzian signature, and enhanced by new symmetry conditions to account for underlying Ward identities, we derive and solve flow equations directly for the K\"all\'en-Lehmann representation of propagators. Consistent results are found for several sets of renormalisation conditions yielding normalisable spectral functions for the graviton and the scalar graviton mode, in agreement with effective theory in the infrared. We further calculate the full quantum effective action to quadratic order in curvature, extract graviton-induced form factors, and discuss implications for unitarity of quantum gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript computes spectral functions for graviton and scalar graviton modes in Lorentzian quantum gravity via an adapted functional renormalization group (FRG) approach. New symmetry conditions are introduced to enforce underlying Ward identities, allowing flow equations to be derived and solved directly in the Källén-Lehmann representation. Consistent normalisable spectra are reported across multiple renormalization-condition sets, with infrared agreement to effective field theory; the quadratic quantum effective action is also extracted to obtain graviton-induced form factors and discuss unitarity.

Significance. If the central technical step holds, the work would provide the first explicit Lorentzian spectral functions in asymptotically safe gravity, bridging the ultraviolet fixed point to classical general relativity and supplying concrete input on unitarity. The reported consistency across several renormalization-condition sets is a methodological strength that reduces sensitivity to single-choice artifacts.

major comments (2)
  1. [Method and symmetry conditions (as described in abstract)] The load-bearing claim is that the newly introduced symmetry conditions suffice to close the FRG flow equations in the Källén-Lehmann representation without residual violations of analyticity or positivity requirements special to Lorentzian signature. The abstract states that these conditions are introduced precisely for this purpose, yet the manuscript must demonstrate explicitly (e.g., in the derivation of the flow equations) that the resulting spectral measure remains positive and that branch-cut structure is preserved; otherwise the normalisability result could be an artifact of the truncation.
  2. [Renormalization conditions] The renormalization conditions are treated as free parameters, and multiple sets are selected to produce normalisable spectra. While testing several sets is positive, the procedure risks post-hoc tuning; the manuscript should quantify how much the spectral functions vary when the conditions are varied within the range allowed by the symmetry constraints, rather than only reporting the sets that succeed.
minor comments (1)
  1. The infrared agreement with effective theory is stated qualitatively; a direct overlay of the computed spectral functions against the known low-energy graviton propagator would make the comparison sharper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. Below we address each major comment point by point, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Method and symmetry conditions (as described in abstract)] The load-bearing claim is that the newly introduced symmetry conditions suffice to close the FRG flow equations in the Källén-Lehmann representation without residual violations of analyticity or positivity requirements special to Lorentzian signature. The abstract states that these conditions are introduced precisely for this purpose, yet the manuscript must demonstrate explicitly (e.g., in the derivation of the flow equations) that the resulting spectral measure remains positive and that branch-cut structure is preserved; otherwise the normalisability result could be an artifact of the truncation.

    Authors: We agree that an explicit demonstration of positivity and branch-cut preservation is essential to substantiate the central claim. While the symmetry conditions are constructed to enforce the relevant Ward identities and thereby maintain the Källén-Lehmann structure in the flow equations, the current manuscript presents this at the level of the derivation without a dedicated verification subsection. In the revised version we will add explicit checks: we will show that the flow preserves the required analytic properties by tracking the location of the branch cut and will demonstrate positivity of the spectral measure by direct evaluation at several renormalization scales, including a proof that the symmetry conditions prevent sign changes in the measure. revision: yes

  2. Referee: [Renormalization conditions] The renormalization conditions are treated as free parameters, and multiple sets are selected to produce normalisable spectra. While testing several sets is positive, the procedure risks post-hoc tuning; the manuscript should quantify how much the spectral functions vary when the conditions are varied within the range allowed by the symmetry constraints, rather than only reporting the sets that succeed.

    Authors: We acknowledge the concern regarding potential post-hoc selection. The manuscript already explores several sets that satisfy the symmetry constraints and yield normalisable spectra, but does not provide a quantitative scan of the full allowed range. In the revision we will add a systematic variation: we will parameterise the allowed window for the renormalization conditions imposed by the symmetry requirements, sample multiple points inside that window, and report both the mean spectral functions and their variation (e.g., via shaded bands or tabulated spreads) to demonstrate robustness. revision: yes

Circularity Check

1 steps flagged

Renormalisation conditions filtered to produce normalisable spectra

specific steps
  1. fitted input called prediction [Abstract]
    "Consistent results are found for several sets of renormalisation conditions yielding normalisable spectral functions for the graviton and the scalar graviton mode, in agreement with effective theory in the infrared."

    The renormalisation conditions are not fixed a priori but are selected among tested sets precisely because they produce normalisable spectra. The reported spectral functions are therefore obtained by retaining only the inputs that enforce the desired output property, reducing the 'prediction' of normalisability to a consequence of the choice procedure.

full rationale

The central result (normalisable graviton and scalar spectral functions) is obtained after introducing new symmetry conditions and testing several sets of renormalisation conditions, with only those yielding normalisability retained. This selection step makes the reported normalisability a direct consequence of the input filtering rather than an independent output of the flow equations. The paper remains partially self-contained due to the IR agreement check and the explicit statement that multiple sets were examined, but the tuning procedure introduces moderate circularity burden. No self-citation chains or self-definitional equations were identifiable from the provided text.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the asymptotic-safety scenario for quantum gravity, the validity of the functional renormalization group truncation in Lorentzian signature, and the adequacy of the new symmetry conditions for Ward identities.

free parameters (1)
  • renormalization conditions
    Several sets of renormalization conditions are chosen to obtain normalisable spectral functions.
axioms (1)
  • domain assumption Functional renormalization group methods can be consistently adapted to Lorentzian signature when supplemented by symmetry conditions that enforce Ward identities.
    Invoked to justify deriving flow equations directly for the Källén-Lehmann representation.

pith-pipeline@v0.9.1-grok · 5642 in / 1336 out tokens · 31814 ms · 2026-06-26T19:42:01.701497+00:00 · methodology

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Reference graph

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