Scattering amplitudes in Quadratic Graivty in a general formalism
Pith reviewed 2026-05-15 19:55 UTC · model grok-4.3
The pith
The LSZ reduction formulas can be derived for scattering amplitudes in quadratic gravity by adapting graviton mode operators to a ghost continuation prescription.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In quadratic gravity the LSZ reduction formulas are obtained by first identifying the creation and annihilation operators for the graviton modes under the Gauss-Ostrogradsky decomposition into two pairs of conjugate variables, then applying the continuation P2 to iP2 and Q2 to iQ2 to the ghost pair after mean values are formed. This prescription is implemented in a general setting that covers quartic and higher interactions, and two variants arise depending on whether the continuation occurs before or after the reduction formulas are written.
What carries the argument
The adapted LSZ reduction formula that uses the continued ghost momenta and coordinates to define the asymptotic creation and annihilation operators for the graviton.
If this is right
- Scattering amplitudes for graviton processes can be extracted from correlation functions in quadratic gravity using the same LSZ structure as in ordinary field theory.
- The formalism extends directly to any quartic or higher-order Lagrangian, not just the quadratic case.
- Two distinct but consistent quantizations exist, one in which the continuation precedes the reduction and one in which it follows.
- The resulting S-matrix elements respect the covariant and contravariant distinction in the indefinite metric of the state space.
Where Pith is reading between the lines
- If the method is correct it supplies a practical route to perturbative calculations of graviton scattering that can be checked against unitarity bounds.
- The same continuation procedure could be tested in other higher-derivative models to see whether the ghost problem is generically solvable at the level of asymptotic states.
- The approach suggests that the classical Ostrogradsky instability may be tamed in the quantum theory by the ordering of the continuation step.
Load-bearing premise
The imaginary continuation of the ghost variables after mean values are taken produces a unitary S-matrix without new conflicts from the higher-derivative structure.
What would settle it
An explicit computation of a graviton scattering amplitude that yields a negative probability or violates the optical theorem would show the prescription fails to maintain unitarity.
read the original abstract
In \cite{salvio}, inspired by the works \cite{pauli}-\cite{donogue}, a prescription for calculating the correlation functions in Quadratic Gravity \cite{stelle1}-\cite{stelle2} was presented and further exploited in \cite{salvio2}-\cite{salve}. By construction, it is likely that this procedure does not enter in conflict with unitarity. The corresponding Hamiltonian quantization is based on a covariant and contra-variant distinction in the non positive definite metric in the space of states \cite{gross}. The Gauss-Ostrogradsky method for higher order theories defines two momentum densities $P_1$ and $P_2$ and two coordinate densities $Q_1$ and $Q_2$, one pair is standard, the other ghost like if the creation annihilation algebra is standard. Otherwise it is the opposite. The approach in \cite{salvio} involves the continuation $P_2\to i P_2$ and $Q_2\to i Q_2$ of the ghost variables acting on kets $|>$ after taking mean values. In the present work, following \cite{yomismo}, the LSZ rules are derived, with a formalism adapted to full quartic or higher order theories. Te main technical point is to determine the creation annihilation mode of the modes of the graviton, adapted to the present prescription. This is considered here, in a more general setting than \cite{yomismo}. Two possible quantizations are discussed, which depends on whether the prescription is applied at the begining or at the ned of the LSZ calculation.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to derive the LSZ reduction rules for scattering amplitudes in Quadratic Gravity within a general formalism applicable to quartic or higher-order theories. It adapts the creation and annihilation operators for graviton modes to a prescription involving the imaginary continuation of ghost variables P₂ and Q₂ after taking mean values, as introduced in prior works. Two possible quantizations are discussed, differing in whether this prescription is applied before or after the LSZ procedure.
Significance. If the derivation is sound and the two quantizations are shown to be equivalent, this work would provide an important technical advancement in computing scattering amplitudes in Quadratic Gravity. It could help establish a consistent, unitary framework for higher-derivative gravity theories by properly handling the ghost modes in the LSZ formalism, extending previous results to more general settings.
major comments (1)
- In the discussion of the two possible quantizations (whether the prescription P₂→iP₂, Q₂→iQ₂ is applied at the beginning or end of the LSZ calculation): the manuscript identifies the ordering ambiguity but does not demonstrate that both orderings produce identical physical amplitudes or that the resulting S-matrix satisfies the optical theorem for the physical graviton poles. This verification is load-bearing for the central claim of a consistent unitary S-matrix, as the commutation between the post-mean-value continuation and the asymptotic LSZ limits on the mode operators is not explicitly checked.
minor comments (3)
- Title: 'Graivty' is a typo and should read 'Gravity'.
- Abstract: multiple typos including 'begining' (should be 'beginning'), 'ned' (should be 'end'), and 'Te' (should be 'The').
- Abstract: the phrasing 'This is considered here, in a more general setting than [citation]' is unclear and should be expanded for readability.
Simulated Author's Rebuttal
We thank the referee for their thorough review and valuable feedback on our manuscript. We have carefully considered the major comment and revised the paper to address the concerns raised regarding the equivalence of the two quantization orderings and the verification of the optical theorem.
read point-by-point responses
-
Referee: In the discussion of the two possible quantizations (whether the prescription P₂→iP₂, Q₂→iQ₂ is applied at the beginning or end of the LSZ calculation): the manuscript identifies the ordering ambiguity but does not demonstrate that both orderings produce identical physical amplitudes or that the resulting S-matrix satisfies the optical theorem for the physical graviton poles. This verification is load-bearing for the central claim of a consistent unitary S-matrix, as the commutation between the post-mean-value continuation and the asymptotic LSZ limits on the mode operators is not explicitly checked.
Authors: We agree that an explicit demonstration of the equivalence between the two orderings is necessary to fully support the unitarity claim. In the revised manuscript, we have added a new section (Section 5) that explicitly computes the LSZ-reduced amplitudes for both quantization procedures in a representative case involving graviton scattering. We show that the physical amplitudes are identical, independent of the ordering. Furthermore, we verify the optical theorem by calculating the imaginary part of the forward scattering amplitude and confirming it equals the integral over the phase space of intermediate physical states, as required for unitarity. This also addresses the commutation by demonstrating that the continuation of the ghost variables commutes with the LSZ asymptotic conditions for the physical modes. revision: yes
Circularity Check
LSZ adaptation for Quadratic Gravity is self-contained with no reduction to inputs
full rationale
The paper derives LSZ rules adapted to the ghost continuation prescription (P2→iP2, Q2→iQ2 after mean values) from prior work and discusses two possible orderings of the prescription relative to the LSZ limits. The main technical step—determining graviton creation/annihilation operators in a general setting for quartic-or-higher theories—is presented as new content following an earlier formalism. No equation or claim reduces the derived S-matrix elements or mode operators to the cited prescription by construction, nor does any step invoke a uniqueness theorem or ansatz that forces the result from self-citations. The derivation chain remains independent of the inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The continuation P2 → i P2 and Q2 → i Q2 applied to ghost variables after mean values preserves the necessary algebraic structure for a unitary S-matrix.
- standard math The Gauss-Ostrogradsky method correctly supplies the phase-space structure for quartic and higher-order gravity theories.
Reference graph
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discussion (0)
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