Manifestly Covariant Canonical Formalism of Quadratic Gravity

Ichiro Oda

REVIEW 3 major objections 5 minor 1 cited by

Manifestly Covariant Canonical Formalism of Quadratic Gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read In quadratic gravity the metric and all its time derivatives have vanishing equal-time commutators, so the metric is not a dynamical operator; physical modes include a massless graviton, a massive scalar, and a negative-norm massive ghost.

desk verdict A serious, explicit BRST quantization of quadratic gravity whose striking vanishing-ETCR claim (Eq. 64) is asserted rather than proved; the linearized mode analysis and non-unitarity derivation are solid and worth engaging. read the letter →

arxiv 2505.09149 v1 pith:G5HPAYCA submitted 2025-05-14 hep-th gr-qc

Ichiro Oda This is my paper · ORCID

classification hep-thgr-qc PACS 04.60.-m11.15.-q04.50.Kd

keywords quadraticgravityhigher-derivativeBRSTquantizationequal-timecommutationrelationsKugo-OjimaconditionmassiveghostunitaritydeDondergauge

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper presents a manifestly covariant BRST quantization of quadratic gravity in the de Donder gauge and derives exact equal-time commutation relations from the canonical commutators. Its central result is Eq. (64): every equal-time commutator between the metric tensor and any number of its time derivatives vanishes identically. If this is correct, the metric behaves as a classical, non-dynamical object for timelike separated correlations, with gravitational dynamics carried by linear combinations of metric, auxiliary, and ghost fields. The paper further identifies the physical states as a positive-norm massless graviton, a positive-norm massive scalar, and a negative-norm spin-2 massive ghost, concluding that the physical S-matrix is not unitary unless the massive ghost is confined.

What carries the argument

The load-bearing device is the first-order rewriting of quadratic gravity through an auxiliary symmetric tensor $K_{\mu\nu}$ and a Stueckelberg vector field $A_\mu$ in Eq. (4), combined with BRST gauge-fixing in the de Donder gauge $\partial_\mu\tilde g^{\mu\nu}=0$. The de Donder condition lets time derivatives of the metric be expressed through spatial derivatives via Eq. (41), and from the canonical (anti)commutation relations for $g_{\mu\nu}$, $K_{\mu\nu}$, $A_\mu$, and the ghosts, the paper proves the vanishing ETCRs by showing that all coefficients in a generic ansatz for $[\dot g_{\rho\sigma},g'_{\mu\nu}]$ vanish. At the linearized level, field combinations $\varphi$, $\psi_{\mu\nu}$, and $h_{\mu\nu}$ isolate the massive scalar, massive ghost, and massless graviton, respectively, and the Kugo-Ojima subsidiary conditions determine which fields are physical.

What would settle it

Compute the full Dirac constraint algebra of the first-order Lagrangian in Eq. (11) with the de Donder gauge conditions; if a secondary second-class constraint appears beyond those solved by the CCRs, then Eq. (38) is incomplete and Eq. (64) can fail. Alternatively, directly evaluate $[\dot g_{00},g'_{00}]$ using the full conjugate momentum $\pi^g_{\mu\nu}$; a single nonzero value would falsify the central claim.

Watch

Extended reading notes

Core claim

The paper claims that in quadratic gravity, the manifestly covariant canonical formalism yields the identity $[\partial^m g_{\rho\sigma}/\partial t^m, \partial^n g'_{\mu\nu}/\partial t^n]=0$ for all $m,n=0,1,2,\dots$, stated as Eq. (64). Consequently, unlike general relativity, where $[\dot g_{\rho\sigma},g'_{\mu\nu}]$ is nonzero, the metric tensor has trivial equal-time commutation relations with itself and with all its time derivatives. The paper then shows, at the linearized level, that the physical Hilbert space defined by the Kugo-Ojima conditions contains a positive-norm massless graviton, a positive-norm massive scalar, and a spin-2 massive ghost with negative norm, so that the physical S-matrix is not unitary unless a confinement mechanism removes the ghost.

Load-bearing premise

Everything rests on the assumption that the first-order $K$ - $A$ formulation has no hidden second-class constraints, so that the canonical commutation relations in Eq. (38) are the complete quantization rules and the coefficient-ansatz proof of the vanishing ETCR is exhaustive.

Editorial extensions

If this is right

The metric cannot propagate physical degrees of freedom in quadratic gravity: all ETCRs between $g_{\mu\nu}$ and its time derivatives vanish, so the metric behaves as a classical object for timelike separated correlations.
The physical gravitational modes are carried by the linearized fields $h_{\mu\nu}$, $\psi_{\mu\nu}$, and $\varphi$ rather than by $g_{\mu\nu}$; at leading order $h_{\mu\nu}$ has the same four-dimensional commutator as the graviton in general relativity.
Physical states contain a massless graviton and a massive scalar with positive norm plus a spin-2 massive ghost with negative norm, so the physical S-matrix is not unitary unless the massive ghost is confined.
Because the massive ghost is a BRST singlet, the standard Kugo-Ojima quartet mechanism does not automatically remove it; a ghost-confinement mechanism based on a BRST-partner bound state is needed for unitarity.
The vanishing ETCRs appear only when $R^2$ and $C^2$ terms coexist in the action, suggesting that this peculiar feature is connected with the perturbative renormalizability of quadratic gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Eq. (64) is taken literally, gravitational correlation functions built from the bare metric should be effectively c-number-like, so observable gravitational scattering must be defined through dressed fields such as $h_{\mu\nu}$ rather than the metric itself.
The same first-order and Stueckelberg machinery could be applied to special limits of the coupling parameters, such as $\beta_1+4\beta_2\to 0$ or $\beta_1+\beta_2\to 0$, where the massive scalar or the ghost changes character; those limits would provide sharp tests of whether the vanishing ETCRs survive.
A natural check of the ghost-confinement proposal is whether the assumed bound state with mass equal to the massive-ghost mass can emerge from the BRST cohomology of the interacting theory, since the paper assumes asymptotic completeness and leaves the binding mechanism open.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline claim — Eq. (64), that all ETCRs of the metric with its time derivatives vanish — is not proved in this paper, and that is the one thing a referee should pin down. The rest of the package is more solid than the headline.

What is genuinely new: a complete manifestly covariant BRST canonical quantization of quadratic gravity in de Donder gauge, carried through to explicit 4D commutation relations and a linearized mode decomposition. The paper does the work it advertises: BRST action, canonical momenta, CCRs, a long list of ETCRs, then the decomposition into φ (massive scalar), ψ_μν (massive spin-2 ghost), and h_μν (massless graviton). Appendix C is a real, detailed derivation of [h,h]; Eq. (189) transparently gives the ghost negative norm. The identification of h_μν rather than the bare fluctuation as the graviton carrier is a clean result. The masses m² and M² are derived from the coupling constants, not fitted, and the non-unitarity sign follows from the computed (anti)commutators, so I see no circularity in the central physical claim.

Now the soft spot, and it is load-bearing. The induction from [∂_t g,g]=0 and [∂_t g,∂_t g]=0 to [∂_t g,∂_t² g]=0 and beyond is asserted, not shown. The base cases are plausible and carefully computed. But ∂_t² g, through the field equations, depends on ∂_t K and ∂_t A, and the paper never shows that [∂_t g,∂_t K] and [∂_t g,∂_t A] vanish — it explicitly declines to compute the [∂_t K,·] ETCRs. The symmetry-and-de Donder ansatz of Eqs. (53)/(58) cannot close without control of those commutators. So the claim that the metric loses its dynamical role in quadratic gravity rests on an unproven step. It may be true — the base cases are suggestive — but as written it is a conjecture with strong evidence at the first two derivative orders, not a theorem.

Smaller issues: the CCRs (38) assume no second-class constraints survive in the Stueckelberg formulation; that is imported from refs. [14-16], not re-derived. Several needed ETCRs come from the author's earlier papers or from Appendix B "without detailed proofs." Not circular, but the verification trail is long. The field-dependent coefficients in the ETCR ansätze silently treat operator ordering as c-number-like — standard for this genre, minor. And the abstract promises a ghost-confinement route to unitarity that the text explicitly defers to future work; the conclusion is candid about this, so the abstract oversells.

Who this is for: people working on covariant operator quantization of gravity. It deserves a serious referee — the framework is coherent, the calculations are extensive, and the gap in Eq. (64) is addressable in revision. Send it out with instructions to complete that proof, or to state explicitly what remains conjectural.

Referee Report

3 major / 5 minor

Summary. The paper develops a manifestly covariant BRST-invariant canonical operator formalism for quadratic gravity in the de Donder gauge. The classical action is rewritten in first-order form with an auxiliary symmetric tensor K_{μν} and a Stückelberg vector A_μ; after gauge fixing, canonical momenta are derived and canonical (anti)commutation relations are imposed. The main formal result is the set of equal-time commutation relations stated in Eq. (64): all ETCRs between the metric and arbitrary time derivatives vanish. The paper then linearizes around flat space, decomposes the fields into a massive scalar φ (mass m^2), a massive spin-2 field ψ_{μν} (mass M^2), and a massless dipole field h_{μν}, computes four-dimensional commutation relations and their Fourier transforms, and uses the Kugo-Ojima subsidiary conditions to identify physical modes. It concludes that the physical subspace contains a positive-norm massive scalar, a positive-norm massless graviton, and a negative-norm massive ghost, so the physical S-matrix is not unitary; a brief discussion of possible ghost confinement is included.

Significance. The paper is valuable for its explicit and detailed operator-formalism treatment of a renormalizable higher-derivative gravity. It derives the masses m^2 and M^2 directly from the action parameters rather than fitting them, and it computes a large set of ETCRs and 4D CRs, with appendices documenting several nontrivial derivations. If the central vanishing-ETCR claim (64) could be fully established, the result that the metric loses its dynamical role in quadratic gravity would be a striking structural difference from general relativity, conformal gravity, and f(R) gravity. The non-unitarity conclusion is consistent with the standard mode content, and the explicit BRST quartet analysis of the Stückelberg sector is a useful contribution. However, the proof of the most surprising claim is incomplete, and the physical-content conclusions are derived under an explicitly stated asymptotic-completeness assumption.

major comments (3)

[Section 5, Eqs. (61)-(64)] The central universal ETCR claim (64) is not actually proved. From (52) and (60) the derivation of [\ddot g, g']=0 in (61) is immediate via identity (43), but the next step to (63) is asserted: the text says 'by using the symmetry ... and the similar argument to Eqs. (53) and (58), we can also show', without writing the ansatz or the coefficient equations. The difficulty is that \ddot g is not built only from g and spatial derivatives; through the field equations (22)-(23) and expressions (92), \ddot g receives contributions from \dot K and \dot A, and the ETCRs of those fields with the variables entering \dot g are nontrivial (see Eq. (79) and Appendix A). No input such as [π_K, \ddot g]=0 or [β, \ddot g]=0 is established, and the same gap is compounded when passing to all m,n by 'proceeding along almost the same line of argument repeatedly'. The text itself concedes at the end of Section 5 that 'we have not given all the ETCRs explicitly in this article', which is difficult to reconcile with the universal statement (64). Because this vanishing-ETCR result is the paper's headline structural claim, the induction must either be completed to the next order or explicitly reduced to a stated assumption.
[Section 4, Eq. (38)] The CCRs in (38) are imposed as complete quantization rules without a Dirac-bracket analysis of the constrained system. The paper asserts that the Stückelberg-like vector field A_μ is introduced 'in order to avoid the second-class constraint' (Section 2) and cites Refs. [14-16], but the constraint algebra is not displayed. Since every ETCR in Section 5, and consequently the linearized mode analysis of Sections 6-7, is derived from these CCRs, the possibility of surviving second-class constraints is a load-bearing premise. The authors should either provide the constraint analysis or state clearly that the proof of absence of second-class constraints is taken wholesale from the cited literature and indicate where in those references it appears.
[Section 5, Eq. (53)] The symmetry-based proof of (52) and the analogous proofs of (60) and (63) rely on the assumption that the ETCR is proportional to δ^3 with coefficients that are c-numbers (or at most numerical multiples of metric operators), and the text states without proof that 'our proof can be generalized to the case where it is proportional to ∂^k δ^3'. This is an operator-ordering ansatz rather than a consequence of the CCRs. For (52) a direct derivation from (48), (51) is also given, but the footnote attached to (60) shows that even that route requires additional ETCRs such as [β_μ,β'_ν]=0 and [β_μ,π'_K^{ρσ}]=0, which are only sketched. The paper should present an explicit demonstration that derivative-of-δ^3 and operator-valued-coefficient terms cannot appear, or restrict the claimed vanishing result to the cases proved.

minor comments (5)

[Throughout] There are several typographical errors that should be corrected: 'particlular' in the abstract, 'ECTR' after Eq. (60), 'spacial' near Eq. (40), 'posseses' in Section 2, and 'caninically' in Section 6.
[Throughout] The gauge condition is called 'de Donder' in most places but 'Donder' in Eq. (59) and nearby text; please make the nomenclature uniform.
[Section 5, footnote 6] The phrase 'chicken-or-the-egg controversy' is informal and should be replaced with a precise statement about the background-dependence of the light-cone structure.
[Section 7, Eq. (131)] The statement that 'the following results do not depend on this constant c' is immediately followed by a caveat about the 4D CR [h_{μν}, \tilde A_ρ]; the sentence should be amended so that the exception is part of the main statement rather than a footnote-like qualification.
[Sections 6-7 and Abstract] The physical-content and non-unitarity conclusions are derived under the explicit assumption at the beginning of Section 6 that all elementary fields have asymptotic fields and no bound states exist. The abstract's unqualified statement that the theory 'is not unitary' should be qualified accordingly, and the final clause about recovering unitarity via ghost confinement should make clear that this is only a comment on a proposed mechanism with acknowledged unsolved problems.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's central results are derived from its CCRs, field equations, and BRST algebra rather than from fitted inputs or self-cited conclusions.

full rationale

I examined the chain leading to the headline claims. The vanishing ETCRs (52) and (60) are proved from the CCRs (38), the D-equation (41) from the de Donder gauge, and a coefficient-symmetry argument (53)-(58); the general family (64) is then asserted by an induction that is not exhibited, which is an omitted proof, not a circular reduction. The masses m^2 and M^2 in Eqs. (117) and (127) are combinations of the action parameters beta1, beta2, gamma, and kappa; nothing is fitted to a target result. The negative norm of the massive ghost is a computed coefficient in the 4D CR (166)/(189), obtained from the ETCRs and CCRs, so the unitarity conclusion is not an input. Some auxiliary ETCRs are imported from the author's earlier papers [19] and [20] (Eqs. (87), (89), (90)), but these are technical commutation relations for related de Donder-gauge formalisms and do not encode the present conclusions; under the stated rule, such independent prior results do not raise the circularity score. The text itself flags limitations: 'we have not given all the ETCRs explicitly', and footnote 6 notes that the timelike-separation statement (71) 'does not make sense' before the metric is known, a chicken-or-egg caveat. These are gaps in completeness, not evidence that any result is equivalent to its own input. I found no self-definitional, fitted-input, uniqueness-import, or ansatz-smuggling circularity.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central derivation introduces no numerically fitted parameters; masses m^2 and M^2 are derived from the action couplings. The arbitrary constants in field redefinitions are set to zero and do not affect the results. The analysis relies on the standard auxiliary-field formulation and stated physical assumptions such as asymptotic completeness. No new physical entities are introduced; K_{μν} and A_μ are standard auxiliary/Stueckelberg fields, and φ, ψ, h are field redefinitions.

free parameters (5)

a_1 = 0
Arbitrary constant in the general expression for h_{μν} (Eq. 128); set to zero for simplicity; claimed not to affect the 4D CRs.
a_2 = 0
Same as a_1.
a_3 = 0
Same as a_1.
a_4 = 0
Same as a_1.
c = 0
Arbitrary constant in the definition of tilde{A}_μ (Eq. 131); chosen zero for simplicity; paper states results are independent of c except for a footnote on one 4D CR.

assumptions (5)

standard math Identity C^2_{μνρσ} = I + 2R_{μν}R^{μν} - (2/3)R^2 with I a total derivative in four dimensions (Eq. 9)
Used in Section 2 to show that L_HD with auxiliary fields reproduces L_{R^2} + L_{C^2} up to surface terms.
domain assumption The first-order formulation with auxiliary fields K_{μν} and A_μ is equivalent to quadratic gravity and has no remaining second-class constraints
Basis for the canonical CCRs in Eq. (38); introduced in Section 2 and used throughout Sections 4-7. Not proved in this paper; follows Refs. [14-16].
domain assumption The de Donder gauge condition ∂_μ tilde{g}^{μν} = 0 (Eq. 14) is an admissible and complete gauge fixing preserving GL(4)
Adopted in Section 3 as the gauge condition for BRST quantization; all subsequent derivations depend on it.
domain assumption Asymptotic completeness: all fields have asymptotic fields and there are no bound states
Explicitly assumed at the start of Section 6 to justify extracting asymptotic fields and physical modes from the linearized Lagrangian.
standard math The physical Hilbert space is defined by the Kugo-Ojima conditions Q_B^(1)|phys> = Q_B^(2)|phys> = 0 (Eq. 159)
Standard BRST physical-state condition, used in Section 7 to identify physical modes and show non-unitarity.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Manifestly Covariant Canonical Formalism of Quadratic Gravity." pith.science (2026). https://pith.science/paper/G5HPAYCA

@misc{pith2026250509149,
  author       = {Pith},
  title        = {Pith review of: Manifestly Covariant Canonical Formalism of Quadratic Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G5HPAYCA}},
  note         = {Machine review of arXiv:2505.09149}
}

read the original abstract

We present the manifestly covariant quantization of quadratic gravity or higher-derivative gravity in the de Donder gauge condition (or harmonic gauge condition) for general coordinate invariance on the basis of the BRST transformation. We explicitly calculate various equal-time commutation relations (ETCRs), in particlular, the ETCRs between the metric tensor and its time derivatives in detail, and show that they are identically vanishing. We also clarify global symmetries, the physical content of quadratic gravity, and clearly show that this theory is not unitary and has a massive scalar, massive ghost and massless graviton as physical modes. Finally, we comment on confinement of the massive ghost, thereby recovering the unitarity of the physical S-matrix in quadratic gravity.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Vanishing Metric Commutation Relation and Higher-derivative De Donder Gauge in Quadratic Gravity
hep-th 2025-07 conditional novelty 6.0 of 10

Quadratic gravity has vanishing equal-time metric commutators in both the standard and higher-derivative de Donder gauges, making the metric look classical.

Reference graph

Works this paper leans on

29 extracted references · 19 canonical work pages · cited by 1 Pith paper

[1]

Strict Renormalizability as a Paradigm for Funda mental Physics

L. Buoninfante, “Strict Renormalizability as a Paradigm for Funda mental Physics”, arXiv:2504.05900 [hep-th] and references therein

arXiv
[2]

Renormalization of Higher Derivative Quantum Gravit y

K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravit y”, Phys. Rev. D 16 (1977) 953. 39

work page 1977
[3]

A New Type of Isotropic Cosmological Models without Singularity

A. A. Starobinsky, “A New Type of Isotropic Cosmological Models without Singularity”, Phys. Lett. B 91 (1980) 99

work page 1980
[4]

On the Quantum Field Theory of the Gravitational Int eractions

D. Anselmi, “On the Quantum Field Theory of the Gravitational Int eractions”, JHEP 06 (2017) 086

work page 2017
[5]

Quadratic Gravity

A. Salvio, “Quadratic Gravity”, Front. in Phys. 6 (2018) 77

2018
[6]

Interpretation of Quantum Mechanics with Indeﬁn ite Norm

A. Strumia, “Interpretation of Quantum Mechanics with Indeﬁn ite Norm”, MDPI Physics 1 (2019) 17

work page 2019
[7]

On Quadratic Gravity

J. E. Donoghue and G. Menezes, “On Quadratic Gravity”, Nuovo . Cim. C 45 (2022) 26

work page 2022
[8]

Kubo and T

J. Kubo and T. Kugo, ”Unitarity Violation in Field Theories of Lee-Wic k’s Complex Ghost”, PTEP 2023 (2023) 123B02

work page 2023

Show all 29 references

[9]

Making Sense of Ghosts

B. Holdom, “Making Sense of Ghosts”, Nucl. Phys. B 1008 (2024) 116696

2024
[10]

Remarks on Ghost Resonances

L. Buoninfante, “Remarks on Ghost Resonances”, JHEP 02 (2025) 186

2025
[11]

Kugo and I

T. Kugo and I. Ojima, ”Local Covariant Operator Formalism of N onabelian Gauge The- ories and Quark Conﬁnement Problem”, Prog. Theor. Phys. Suppl. 66 (1979) 1

1979
[12]

Nakanishi, ”Indeﬁnite Metric Quantum Field Theory of Genera l Gravity”, Prog

N. Nakanishi, ”Indeﬁnite Metric Quantum Field Theory of Genera l Gravity”, Prog. Theor. Phys. 59 (1978) 972

1978
[13]

Nakanishi and I

N. Nakanishi and I. Ojima, ”Covariant Operator Formalism of Ga uge Theories and Quan- tum Gravity”, World Scientiﬁc Publishing, 1990 and references ther ein

1990
[14]

Kawasaki, T

S. Kawasaki, T. Kimura and K. Kitago, ”Canonical Quantum Theo ry of Gravitational Field with Higher Derivatives”, Prog. Theor. Phys. 66 (1981) 2085

1981
[15]

Kawasaki and T

S. Kawasaki and T. Kimura, ”Canonical Quantum Theory of Gra vitational Field with Higher Derivatives. II”, Prog. Theor. Phys. 68 (1982) 1749

1982
[16]

Kawasaki and T

S. Kawasaki and T. Kimura, ”Canonical Quantum Theory of Gra vitational Field with Higher Derivatives. III”, Prog. Theor. Phys. 69 (1983) 1015

1983
[17]

Quantum Conformal Gravity

I. Oda and M. Ohta, “Quantum Conformal Gravity”, JHEP 02 (2024) 213

2024
[18]

Conformal Symmetry in Quantum Gravity

I. Oda, “Conformal Symmetry in Quantum Gravity”, Eur. Phys . J. C 84 (2024) 887

2024
[19]

BRST Formalism of f (R) Gravity

I. Oda, “BRST Formalism of f (R) Gravity”, arXiv:2410.20270 [hep-th]

arXiv
[20]

Quantum Scale Invariant Gravity in de Donder Gauge

I. Oda, “Quantum Scale Invariant Gravity in de Donder Gauge”, Phys. Rev. D 105 (2022) 066001. 40

2022
[21]

Quantum Theory of Weyl Invariant Scalar-tensor Gr avity

I. Oda, “Quantum Theory of Weyl Invariant Scalar-tensor Gr avity”, Phys. Rev. D 105 (2022) 120618

2022
[22]

Oda and P

I. Oda and P. Saake, ”BRST Formalism of Weyl Conformal Gravit y”, Phys. Rev. D 106 (2022) 106007

2022
[23]

BRST formalism of Weyl Invariant Gravity and Conﬁnem ent of Massive Tensor Ghost

I. Oda, “BRST formalism of Weyl Invariant Gravity and Conﬁnem ent of Massive Tensor Ghost”, PoS CORFU2023 (2024) 158

2024
[24]

Gravitation

C. W. Misner, K. S. Thorne and J. A. Wheeler, “Gravitation”, W H Freeman and Co (Sd), 1973

1973
[25]

Symmetries and Strings in Field Theory and Gravity

T. Banks and N. Seiberg, “Symmetries and Strings in Field Theory and Gravity”, Phys. Rev. D 83 (2011) 084019

2011
[26]

Introduction to String and Superstring Theory II, in From the Planck Scale to the Weak Scale

M. E. Peskin, “Introduction to String and Superstring Theory II, in From the Planck Scale to the Weak Scale”, TASI 1986, ed. by H. E. Haber, World Scien tiﬁc (1987) 277

1987
[27]

General Relativity

R. M. Wald, “General Relativity”, The University of Chicago Pres s, 1984

1984
[28]

Introduction to Quantum Field Theory with Appli- cations to Quantum Gravity

I. L. Buchbinder and I. L. Shapiro, “Introduction to Quantum Field Theory with Appli- cations to Quantum Gravity”, Oxford University Press, 2021

2021
[29]

Kawasaki and K

S. Kawasaki and K. Kimura, ”A Possible Mechanism of Ghost Conﬁ nement in a Renor- malizable Quantum Gravity”, Prog. Theor. Phys. 65 (1981) 1767. 41

1981

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Strict Renormalizability as a Paradigm for Funda mental Physics

L. Buoninfante, “Strict Renormalizability as a Paradigm for Funda mental Physics”, arXiv:2504.05900 [hep-th] and references therein

arXiv

[2] [2]

Renormalization of Higher Derivative Quantum Gravit y

K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravit y”, Phys. Rev. D 16 (1977) 953. 39

work page 1977

[3] [3]

A New Type of Isotropic Cosmological Models without Singularity

A. A. Starobinsky, “A New Type of Isotropic Cosmological Models without Singularity”, Phys. Lett. B 91 (1980) 99

work page 1980

[4] [4]

On the Quantum Field Theory of the Gravitational Int eractions

D. Anselmi, “On the Quantum Field Theory of the Gravitational Int eractions”, JHEP 06 (2017) 086

work page 2017

[5] [5]

Quadratic Gravity

A. Salvio, “Quadratic Gravity”, Front. in Phys. 6 (2018) 77

2018

[6] [6]

Interpretation of Quantum Mechanics with Indeﬁn ite Norm

A. Strumia, “Interpretation of Quantum Mechanics with Indeﬁn ite Norm”, MDPI Physics 1 (2019) 17

work page 2019

[7] [7]

On Quadratic Gravity

J. E. Donoghue and G. Menezes, “On Quadratic Gravity”, Nuovo . Cim. C 45 (2022) 26

work page 2022

[8] [8]

Kubo and T

J. Kubo and T. Kugo, ”Unitarity Violation in Field Theories of Lee-Wic k’s Complex Ghost”, PTEP 2023 (2023) 123B02

work page 2023