Conformal geometry as a gauge theory of gravity: covariant equations of motion & conservation laws

A. Micu; C. Condeescu; D. M. Ghilencea

REVIEW 2 major objections 4 minor 2 cited by

Conformal geometry as a gauge theory of gravity: covariant equations of motion & conservation laws

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Pith's one-line read The most general quadratic Weyl-geometry action has fully covariant equations of motion and dual conservation laws.

desk verdict Genuinely new covariant equations of motion and conservation laws for the most general Weyl quadratic gravity, with the conservation laws anchored in the Noether symmetries; the main weakness is that the explicit algebra proving the key identities is not shown. read the letter →

arxiv 2412.16548 v3 pith:HPHNEMZN submitted 2024-12-21 hep-th gr-qchep-ph

C. Condeescu , D. M. Ghilencea , A. Micu This is my paper · ORCID

classification hep-thgr-qchep-ph

keywords Weylconformalgeometryquadraticgravitygaugetheoryofdilatationsymmetryconservationlawsenergy-momentumtensorcurrentdimensionalregularisation

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 tries to establish that Weyl conformal geometry, read as a gauge theory of the Weyl group (Poincaré transformations together with local dilatations), supports a complete quadratic gravity theory whose equations of motion and conservation laws are manifestly Weyl gauge covariant. For the most general action $S=\int d^4x\,\sqrt{g}\,(\alpha_1 R^2+\alpha_2 C^2+\alpha_3 F^2+\alpha_4 G)$, it derives the covariant field equations $W_{\mu\nu}=0$ and $B_\mu=0$ in explicit tensor form, and proves that the energy-momentum tensor $W_{\mu\nu}$ and the Weyl gauge current $j_\mu$ are conserved. The conservation laws hold both with the Weyl gauge covariant derivative $\hat{\nabla}$ and with the ordinary Riemannian derivative $\mathring{\nabla}$, and they are extended to arbitrary dimension by a gauge-invariant regularisation. If the argument is right, this gives a usable covariant formalism for a realistic gauge theory of gravity that reduces to general relativity with a massive vector and a cosmological constant after spontaneous symmetry breaking.

What carries the argument

The engine is the Weyl gauge covariant derivative $\hat{\nabla}_\mu$, defined by $\hat{\nabla}_\mu X = \nabla_\mu X + q_X\,\omega_\mu X$ or equivalently through the space-time charge, which is non-affine (no connection $\hat{\Gamma}$ exists for fields of arbitrary charge) yet automatically metric, $\hat{\nabla}_\mu g_{\alpha\beta}=0$. Its commutator on a vector is $[\hat{\nabla}_\mu,\hat{\nabla}_\nu]V^\rho = R^\rho{}_{\sigma\mu\nu}V^\sigma + q_V F_{\mu\nu}V^\rho$, with no torsion term, and it obeys integration-by-parts rules and Bianchi identities that make it the exact analogue of $\mathring{\nabla}$ in Riemannian geometry. This derivative carries all variations of the curvature tensors, reducing the equations of motion to covariant combinations of $R_{\mu\nu}$, $R$, $F_{\mu\nu}$ and their $\hat{\nabla}$ derivatives. A secondary mechanism is the geometric regulator $R^{(d-4)/2}$ in the $d$-dimensional action, which keeps the action Weyl gauge invariant in arbitrary dimension.

What would settle it

Check the claimed off-shell identities (65) and (68) on a generic smooth metric and Weyl field, for each curvature invariant $i$; because the paper derives these as algebraic identities, one explicit configuration where $\hat{\nabla}_\mu W^{(i)\mu\nu} - \frac{1}{2} F^{\mu\nu} B^{(i)}_\mu \neq 0$ or $\hat{\nabla}_\mu B^{(i)\mu} - 2\,\mathrm{tr}\,W^{(i)} \neq 0$ falsifies the conservation laws.

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Extended reading notes

Core claim

The central claim is that the general quadratic action of Weyl conformal geometry has well-defined manifestly covariant equations of motion, not only after translating to Riemannian variables, and that its symmetries force two conservation laws at once. Introducing the non-affine Weyl gauge covariant derivative $\hat{\nabla}_\mu$ and using it to vary the action, the paper obtains $W_{\mu\nu}=0$ and $B_\mu=0$, with the explicit tensors given in (52)-(59). It then derives the identities $\hat{\nabla}^\mu W_{\mu\nu}^{(i)}=\frac{1}{2} F_{\mu\nu}B^{(i)\mu}$ and $\hat{\nabla}^\mu B_\mu^{(i)}=2\,\mathrm{tr}\,W^{(i)}$, which on shell become $\hat{\nabla}^\mu W_{\mu\nu}=0$, $\hat{\nabla}^\mu j_\mu=0$, and the trace identity $\mathrm{tr}\,W_{\mu\nu}=0$. Because $\hat{\nabla}$ and $\mathring{\nabla}$ differ by a term proportional to $\omega_\mu$ contracted with the trace, the same trace identity makes the Riemannian conservation laws $\mathring{\nabla}^\mu W_{\mu\nu}=0$ and $\mathring{\nabla}^\mu j_\mu=0$ follow as well. The same structure is shown to hold for the regularised action in arbitrary $d$ dimensions, where the Euler term contributes for $d\neq 4$.

Load-bearing premise

The load-bearing premise is that the charge assignments and torsion-free commutator (21) of the Weyl gauge covariant derivative $\hat{\nabla}$ are correct, since every equation of motion and conservation law is written through that derivative.

Editorial extensions

If this is right

The explicit covariant equations (60)-(61) can be used directly in conformal geometry, without first rewriting the action in Riemannian variables.
In the spontaneously broken phase, the conservation laws persist in Riemannian form, so the massive vector $\omega_\mu$ and the dilaton $\varphi$ obey standard energy-momentum conservation.
In arbitrary dimension, the regularised action remains Weyl gauge invariant, making the formalism suited to dimensional regularisation and keeping the theory free of the Weyl anomaly.
In the conformal-gravity limit ($\alpha_1=\alpha_3=0$), the Weyl current vanishes identically and the conserved tensor $W^{(c)}_{\mu\nu}$ is traceless, reproducing conformal gravity from the covariant formulation.
The same conservation structure holds in any dimension because it follows from diffeomorphism and dilatation invariance rather than from $d=4$ identities.

Reading between the lines

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

A natural extension not pursued here is to couple matter: demanding simultaneous conservation of $W_{\mu\nu}+T_{\mu\nu}$ with both $\hat{\nabla}$ and $\mathring{\nabla}$ would impose strong consistency conditions on any matter sector, including the Standard-Model embedding.
The $\epsilon$-dependent terms in the $d$-dimensional equations suggest that quantum corrections generate Weyl-invariant higher-dimensional operators suppressed by powers of $R$; the authors note this but do not classify the full operator basis.
The geometric interpretation of $\omega_\mu$ as part of the connection, combined with its Riemannian conservation as a matter-like current, offers a route to reinterpret the dark-matter candidate of this theory as a geometric effect rather than a new particle species.
One could test the formalism's usefulness by deriving the conserved current for the non-perturbative Weyl-DBI action, whose leading order matches the quadratic action studied here.

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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

What you should know: this paper computes the manifestly Weyl-gauge-covariant equations of motion and conservation laws for the most general quadratic action in Weyl conformal geometry, in four and then arbitrary dimensions, using the authors' gauge-covariant (non-affine) formalism. That is a real computation, and the working is mostly careful and reproducible.

What is new: the covariant tensors W^(i)_μν and B^(i)_μ in (52)-(59) and the conservation identities (65) and (68), plus the d-dimensional generalizations. The paper earns credit for the clean Noether argument in Section 4.2: it shows that (65) follows from invariance under gauged covariant diffeomorphisms and (68) from dilatation invariance, which is why the conservation laws hold with respect to both the Weyl-covariant and Riemannian derivatives. The spot-checks in Section 5 pass: the Gauss-Bonnet combination vanishes and conformal gravity reproduces known equations.

Now the soft spots, in proportion. The stress-test note worries that (65)-(68) are asserted rather than derived. That is not quite fair: Section 4.2 gives a symmetry derivation of the total conservation law, and since each term in the action is separately invariant, the split per term follows. What is missing is the explicit algebraic demonstration that the long tensor expressions actually satisfy (65)-(68) when you expand the commutators and Bianchi identities. The text says "after some algebra" and leaves it at that. For a paper whose headline is precisely these identities, that gap is worth closing. A referee should ask for an appendix (or a supplementary file) with the full derivation, especially for the d-dimensional case where (154) is also asserted. There is also a minor unstated assumption: the regulator R^{(d-4)/2} in the d-dimensional action requires R ≠ 0, which is not mentioned.

Neither issue looks load-bearing: the symmetry argument is sound, the explicit checks pass, and the sign conventions are consistent. But the paper would be much more useful if the algebra were shown.

Who it's for: anyone working on Weyl quadratic gravity, conformal geometry as a gauge theory, or scale-invariant gravity. It deserves a serious referee; I would send it out with a request for the detailed derivation of (65)-(68) and a note on the R ≠ 0 assumption. Given the stakes, I'd want the referee to verify one of the long identities independently.

Referee Report

2 major / 4 minor

Summary. This paper studies Weyl conformal geometry as a gauge theory of the Weyl group in a manifestly Weyl gauge covariant formalism. The authors compute the equations of motion for the most general quadratic action S = ∫ d^4x √g [α1 R^2 + α2 C^2 + α3 F^2 + α4 G] in four dimensions, obtaining covariant tensors W_μν and B_μ, and then generalize the computation to arbitrary d dimensions using the geometric regulator R^{(d-4)/2}. They further derive conservation laws for the energy-momentum tensor and for the Weyl gauge current, in both Weyl-covariant and Riemannian forms, and illustrate the results with Gauss-Bonnet, R^2+F^2, and conformal gravity examples.

Significance. If correct, the paper provides a useful systematic toolkit: explicit covariant equations of motion (52)-(59) and (145)-(152), a clean Noether derivation of the conservation laws from gauged diffeomorphisms in Section 4.2, and dimension-independent statements. The checks in Section 5 (Gauss-Bonnet combination vanishes; conformal gravity reproduces known equations) support the main calculations. The central off-shell identity (65), however, is misprinted with an index/sign error, and the d-dimensional analogue (154) is asserted rather than proved; these issues must be fixed before the paper can be used as a reference.

major comments (2)

[§4.1, Eq. (65)] The identity is stated as ∇̂^μ W^{(i)}_{μν} = (1/2) F^{μν} B^{(i)}_μ. This is inconsistent with the derivation in Section 4.2. From δg_{μν} = -∇̂_μξ_ν - ∇̂_νξ_μ and δω_μ = F_{μν}ξ^ν, the Noether argument gives δS = ∫√g[-2∇̂^μW_{μν} + F_{μν}B^μ]ξ^ν, hence 2∇̂^μW_{μν} = F_{μν}B^μ. Since F^{μν}B_μ = -F_{μν}B^μ for the antisymmetric F, Eq. (65) as written has the wrong sign and index placement. A direct check for i=3, using W^{(3)}_{μν}=2F_{μρ}F_ν^ρ - (1/2)g_{μν}F^2 and B^{(3)} of Eq. (58), confirms ∇̂^μW^{(3)}_{μν} = (1/2)F_{μν}B^{(3)μ}, not (1/2)F^{μν}B^{(3)}_μ. The on-shell conservation laws (67) and (71) are unaffected because the right-hand side vanishes when B_μ=0, but the off-shell identity must be corrected, and the same correction carries to (154).
[§6.2, Eq. (154)] The d-dimensional identities are introduced with the statement 'calculations are rather long but one can show' and are not backed by the symmetry argument that is already present in Section 4.2. Since these identities are the d-dimensional extension of the paper's main result, the authors should present the derivation (which the symmetry argument in Section 4.2 makes straightforward and dimension-independent) and verify the corrected index structure. As written, Eq. (154) inherits the sign/index error of Eq. (65).

minor comments (4)

[§4.1, before Eq. (65)] The sentence 'After some algebra with commutators (21) and Bianchi identities (27)' is superseded by the symmetry derivation in Section 4.2; please add a forward reference so the reader can find the proof.
[§4.1, Eq. (72)] Because F^{νμ} is antisymmetric, the index placement in 4α3∇̂_νF^{νμ} = j^μ is essential; please add one sentence explaining how this form follows from B_μ=0 after lowering indices, to prevent the type of sign ambiguity that appears in Eq. (65).
[Affiliation line] The affiliation line contains a typo ('Physic s'); please correct it.
[§5.1, Eq. (121)] The claim B^{(G)}_μ ≡ 0 is stated as 'easy to check'; since it is used as a consistency check, please show the one-line cancellation using Eqs. (56)-(59).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: energy-momentum and Weyl-current conservation are Noether-type consequences of the invariant action within the authors' Weyl-covariant formalism; the load-bearing identities (65), (68), (154) are asserted, not proven, but this is an omitted-proof concern, not a reduction to inputs.

full rationale

Score 2 reflects reliance on the authors' own Weyl-covariant formalism ([10,19,22]) as a framework, not a circular derivation. The paper's central targets — W_mu nu = 0, B_mu = 0 in eqs. (52)-(59) and the conservation laws (67), (71), (74), (78) — are obtained by varying the action (29) and by Noether-type arguments from diffeomorphism and dilatation invariance; no parameter is fitted and no target equation is inserted into the input. The identities (65) and (68), which connect the divergences of W^(i) and B^(i), are asserted as "after some algebra" (Section 4.1) rather than proved; their d-dimensional versions are introduced as "calculations are rather long but one can show" (Section 6.2). This is a genuine omitted-proof/correctness flag: if a commutator sign or contraction in (D-12)-(D-14) is wrong, the conservation laws fail. But omission of an algebraic proof is not the same as assuming the conclusion; the identities are not definitions, fits, or renamed copies of the action. Self-citations to [10,19,22] supply the charge table, commutator (21), and integration-by-parts rule (26), but those objects are restated explicitly in this paper, so the subsequent computation is independently checkable. The Gauss-Bonnet check (Section 5) is an internal consistency test, not a circular validation. Hence no circular step is established.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper contributes no fitted parameters and introduces no new entities. It relies on the Weyl gauge covariant formalism (developed in the authors' prior work), on the claim that the action (29)/(139) is the most general quadratic Weyl invariant action, and on the R^{(d-4)/2} regulator for arbitrary dimension. The first two are assumptions of the research program; the third is a regularization choice that also requires R ≠ 0. These are stated in the paper, though the R ≠ 0 condition is not.

assumptions (4)

domain assumption The Weyl gauge covariant derivative ∇̂ (eq 17) with the charge assignments in Section 2 defines the physical derivative; its commutator has no torsion term (eq 21).
This is the foundational postulate of the manifestly covariant formalism, developed in the authors' prior work [10,19,22]; the paper's equations of motion and conservation laws all use ∇̂.
domain assumption The action (29)/(139) is the most general Weyl gauge invariant quadratic action; C_μνρσ = ˚C_μνρσ and G transforms covariantly (eqs D-6, D-8, 23).
Maximality and the covariance properties of the two non-trivial terms come from the conformal geometry framework; they enter through the coefficient map (33)/(143).
ad hoc to paper The geometric regulator R^{(d-4)/2} preserves Weyl gauge invariance in arbitrary d and defines the quantum-corrected action.
This regulator is a choice specific to this program (from [22]); other regularizations would give different ε-dependent terms in the d-dimensional equations of motion. It also requires R ≠ 0.
standard math Standard Riemannian geometry identities (Bianchi, commutators) hold for the objects when translated to the ˚∇ picture.
Used in Sections 4.1 and 4.3 to convert between Weyl and Riemannian conservation laws.

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Pith. "Pith review of Conformal geometry as a gauge theory of gravity: covariant equations of motion & conservation laws." pith.science (2026). https://pith.science/paper/HPHNEMZN

@misc{pith2026241216548,
  author       = {Pith},
  title        = {Pith review of: Conformal geometry as a gauge theory of gravity: covariant equations of motion & conservation laws},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HPHNEMZN}},
  note         = {Machine review of arXiv:2412.16548}
}

abstract

We study Weyl conformal geometry as a general gauge theory of the Weyl group (of Poincar\'e and dilatations symmetries) in a manifestly Weyl gauge covariant formalism in which this geometry is automatically metric and physically relevant. This gives a realistic (quadratic) gauge theory of gravity, with Einstein-Hilbert gravity recovered in its spontaneously broken phase, motivating our interest in this geometry. For the most general action we compute the manifestly Weyl gauge covariant equations of motion and present the conservation laws for the energy-momentum tensor and Weyl gauge current. These laws are valid both in Weyl conformal geometry (with respect to the Weyl gauge covariant derivative) but also in the Riemannian geometry equivalent picture (with respect to its associated covariant derivative). This interesting result is a consequence of gauged diffeomorphism invariance of the former versus usual diffeomorphism invariance of the latter. These results are first derived in $d=4$ dimensions. We then successfully derive the conservation laws and equations of motion in Weyl conformal geometry in arbitrary $d$ dimensions, while maintaining manifest Weyl gauge invariance/covariance. The results are useful in physical applications with this symmetry.

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Works this paper leans on

50 extracted references · 16 canonical work pages · cited by 2 Pith papers

[1]

(An English version by D

Hermann Weyl, Gravitation und elektrizit¨ at, Sitzungs berichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin (1918) , pp.465. (An English version by D. H. Delphenich is currently availab le at: http://www.neo-classical-physics.info/spacetime-structure.html). This work includes Einstein’s critical report, stating tha t the spacing bet...

work page 1918
[2]

Eine neue Erweiterung der Relativit¨ atst heorie

Hermann Weyl “Eine neue Erweiterung der Relativit¨ atst heorie” (“A new exten- sion of the theory of relativity”), Ann. Phys. (Leipzig) (4) 59 (1919), 101-

work page 1919
[3]

Raum, Zeit, Materie

Hermann Weyl “Raum, Zeit, Materie”, vierte erweiterte A uﬂage. Julius Springer, Berlin 1921 “Space-time-matter”, translated from German b y Henry L. Brose, 1922, Methuen & Co Ltd, London, www.gutenberg.org/ebooks/43006 (Project Gutenberg License)

work page 1921
[4]

Quantum gravity from Weyl conformal ge ometry,

D. M. Ghilencea, “Quantum gravity from Weyl conformal ge ometry,” [arXiv:2408.07160 [hep-th]]

arXiv
[5]

Invariant Variation Problems,

E. Noether, “Invariant Variation Problems,” Gott. Nach r. 1918 (1918) 235 [Transp. Theory Statist. Phys. 1 (1971) 186] doi:10.1080/004114571 08231446 [physics/0503066]. Gott. Nachr. 1918 (1918) 235

arXiv 1918
[6]

Invariant theoretical interpretation of i nteraction,

R. Utiyama, “Invariant theoretical interpretation of i nteraction,” Phys. Rev. 101 (1956), 1597-1607 doi:10.1103/PhysRev.101.1597

doi:10.1103/physrev.101.1597 1956
[7]

Lorentz invariance and the gravitation al ﬁeld,

T. W. B. Kibble, “Lorentz invariance and the gravitation al ﬁeld,” J. Math. Phys. 2 (1961), 212-221 doi:10.1063/1.1703702

doi:10.1063/1.1703702 1961
[8]

Weyl transformations and poincare gauge in variance,

A. Bregman, “Weyl transformations and poincare gauge in variance,” Prog. Theor. Phys. 49 (1973), 667-692 doi:10.1143/PTP.49.667

work page doi:10.1143/ptp.49.667 1973

Show all 50 references

[9]

A gauge theory of the Weyl group,

J. M. Charap and W. Tait, “A gauge theory of the Weyl group, ” Proc. Roy. Soc. Lond. A 340 (1974), 249-262 doi:10.1098/rspa.1974.0151

1974
[10]

The Gauge Theory of Weyl Group and its Interpretation as Weyl Quadratic Gravity,

C. Condeescu and A. Micu, “The Gauge Theory of Weyl Group and its Interpretation as Weyl Quadratic Gravity,” [arXiv:2408.07159 [hep-th]]

arXiv
[11]

Supergravity,

D. Z. Freedman and A. Van Proeyen, “Supergravity,” Camb ridge Univ. Press, 2012, ISBN 978-1-139-36806-3, 978-0-521-19401-3 doi:10.1017/ CBO9781139026833. 39

2012
[12]

Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves,

P. D. Mannheim and D. Kazanas, “Exact Vacuum Solution to Conformal Weyl Gravity and Galactic Rotation Curves,” Astrophys. J. 342 (1989), 635-638 doi:10.1086/167623; D. Kazanas and P. D. Mannheim, “General Structure of the Grav itational Equa- tions of Motion in Conformal Weyl ...

1989 doi
[13]

Gauge Theory of the Confor- mal and Superconformal Group,

M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Gauge Theory of the Confor- mal and Superconformal Group,” Phys. Lett. B 69 (1977), 304-308 doi:10.1016/0370- 2693(77)90552-4

1977 doi
[14]

The unexpected resurgence of Weyl geometry in the late 20-th cen- tury physics,

E. Scholz, “The unexpected resurgence of Weyl geometry in the late 20-th cen- tury physics,” Einstein Stud. 14 (2018), 261-360 doi:10.1007/978-1-4939-7708-6 11 [arXiv:1703.03187 [math.HO]]

2018 arXiv
[15]

Long range forces and broken symmetries ,

P. A. M. Dirac, “Long range forces and broken symmetries ,” Proc. Roy. Soc. Lond. A 333 (1973), 403-418 doi:10.1098/rspa.1973.0070

1973
[16]

Towards a Theory of Space-Time Structure at Very Short Distances,

L. Smolin, “Towards a Theory of Space-Time Structure at Very Short Distances,” Nucl. Phys. B 160 (1979), 253-268 doi:10.1016/0550-3213(79)90059-2

1979 doi
[17]

Spontaneous breaking of Weyl quadrat ic gravity to Einstein action and Higgs potential,

D. M. Ghilencea, “Spontaneous breaking of Weyl quadrat ic gravity to Einstein action and Higgs potential,” JHEP 1903 (2019) 049 [arXiv:1812.08613 [hep-th]]. D. M. Ghi- lencea, “Stueckelberg breaking of Weyl conformal geometry and applications to grav- ity,” Phys. Rev. D 101 (2...

2019 arXiv
[18]

Interaction forces in electrod ynamics and in the ﬁeld theory of nuclear forces,

E. C. G. Stueckelberg, “Interaction forces in electrod ynamics and in the ﬁeld theory of nuclear forces,” Helv. Phys. Acta 11 (1938) 299

1938
[19]

Weyl quadrat ic gravity as a gauge theory and non-metricity vs torsion duality,

C. Condeescu, D. M. Ghilencea and A. Micu, “Weyl quadrat ic gravity as a gauge theory and non-metricity vs torsion duality,” Eur. Phys. J. C 84 (2024) no.3, 292 doi:10.1140/epjc/s10052-024-12644-6 [arXiv:2312.1338 4 [hep-th]]

2024
[20]

Non-metric geometry as the origin of m ass in gauge theories of scale invariance,

D. M. Ghilencea, “Non-metric geometry as the origin of m ass in gauge theories of scale invariance,” Eur. Phys. J. C 83 (2023) no.2, 176 doi:10.1140/epjc/s10052-023-11237-z [arXiv:2203.05381 [hep-th]]

2023 arXiv
[21]

Weyl gauge theories of gr avity do not predict a second clock eﬀect,

M. P. Hobson and A. N. Lasenby, “Weyl gauge theories of gr avity do not predict a second clock eﬀect,” Phys. Rev. D 102 (2020) no.8, 084040 doi:10.1103/PhysRevD.102.084040 [arXiv:2009.06407 [gr -qc]]

2020 arXiv
[22]

Weyl conformal geometry vs Weyl anoma ly,

D. M. Ghilencea, “Weyl conformal geometry vs Weyl anoma ly,” JHEP 10 (2023), 113 doi:10.1007/JHEP10(2023)113 [arXiv:2309.11372 [hep-th ]]

2023 arXiv
[23]

Twenty years of the Weyl anomaly,

M. J. Duﬀ, “Twenty years of the Weyl anomaly,” Class. Quan t. Grav. 11 (1994), 1387- 1404 doi:10.1088/0264-9381/11/6/004 [arXiv:hep-th/930 8075 [hep-th]]. M. J. Duﬀ, “Observations on Conformal Anomalies,” Nucl. Phys. B 125 (1977), 334-348 doi:10.1016/0550-3213(77)90410-2

1994 doi
[24]

Photon correctio ns to the graviton prop- agator,

D. M. Capper, M. J. Duﬀ and L. Halpern, “Photon correctio ns to the graviton prop- agator,” Phys. Rev. D 10 (1974), 461-467 doi:10.1103/PhysRevD.10.461. 40

1974 doi
[25]

Trace anomalies in dimensiona l regularization,

D. M. Capper and M. J. Duﬀ, “Trace anomalies in dimensiona l regularization,” Nuovo Cim. A 23 (1974), 173-183 doi:10.1007/BF02748300

1974 doi
[26]

Nonlocal Conformal A nomalies,

S. Deser, M. J. Duﬀ and C. J. Isham, “Nonlocal Conformal A nomalies,” Nucl. Phys. B 111 (1976), 45-55 doi:10.1016/0550-3213(76)90480-6

1976 doi
[27]

Geometric classiﬁcation of conformal anomalies in arbi- trary dimensions,

S. Deser and A. Schwimmer, “Geometric classiﬁcation of conformal anomalies in arbi- trary dimensions,” Phys. Lett. B 309 (1993), 279-284 doi:10.1016/0370-2693(93)90934- A [arXiv:hep-th/9302047 [hep-th]]

1993 arXiv
[28]

Conformal Invari ance in Quantum Gravity,

F. Englert, C. Truﬃn and R. Gastmans, “Conformal Invari ance in Quantum Gravity,” Nucl. Phys. B 117 (1976), 407-432 doi:10.1016/0550-3213(76)90406-5

1976 doi
[29]

Standard Model in Weyl conformal geom etry,

D. M. Ghilencea, “Standard Model in Weyl conformal geom etry,” Eur. Phys. J. C 82 (2022) no.1, 23 doi:10.1140/epjc/s10052-021-09887-y [ar Xiv:2104.15118 [hep-ph]]

2022 arXiv
[30]

Scal e-independent R2 in- ﬂation,

P. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, “Scal e-independent R2 in- ﬂation,” Phys. Rev. D 100 (2019) no.12, 123516 doi:10.1103/PhysRevD.100.123516 [arXiv:1906.03415 [gr-qc]]

2019 arXiv
[31]

Weyl R 2 inﬂation with an emergent Planck scale,

D. M. Ghilencea, “Weyl R 2 inﬂation with an emergent Planck scale,” JHEP 10 (2019), 209 doi:10.1007/JHEP10(2019)209 [arXiv:1906.11572 [gr- qc]]

2019 arXiv
[32]

Gauging scale symmetry and inﬂation: Weyl versus Palatini gravity,

D. M. Ghilencea, “Gauging scale symmetry and inﬂation: Weyl versus Palatini gravity,” Eur. Phys. J. C 81 (2021) no.6, 510 doi:10.1140/epjc/s10052-021-09226-1 [arXiv:2007.14733 [hep-th]]

2021 arXiv
[33]

A New Type of Isotropic Cosmologica l Models Without Singular- ity,

A. A. Starobinsky, “A New Type of Isotropic Cosmologica l Models Without Singular- ity,” Phys. Lett. B 91 (1980), 99-102 doi:10.1016/0370-2693(80)90670-X

1980 doi
[34]

Testing Weyl geometric gravi ty with the SPARC galactic rotation curves database,

M. Cr˘ aciun and T. Harko, “Testing Weyl geometric gravi ty with the SPARC galactic rotation curves database,” [arXiv:2311.16893 [gr-qc]]. P . Burikham, T. Harko, K. Pim- samarn and S. Shahidi, “Dark matter as a Weyl geometric eﬀect, ” Phys. Rev. D 107 (2023) no.6, 064008 doi:...

2023 arXiv
[35]

Black hole solution s in the quadratic Weyl conformal geometric theory of gravity,

J. Z. Yang, S. Shahidi and T. Harko, “Black hole solution s in the quadratic Weyl conformal geometric theory of gravity,” Eur. Phys. J. C 82 (2022) no.12, 1171 doi:10.1140/epjc/s10052-022-11131-0 [arXiv:2212.0554 2 [gr-qc]]

2022
[36]

Foundations of the new ﬁeld theor y,

M. Born and L. Infeld, “Foundations of the new ﬁeld theor y,” Proc. Roy. Soc. Lond. A 144 (1934) no.852, 425-451 doi:10.1098/rspa.1934.0059

1934
[37]

An Extensible model of the electron,

P. A. M. Dirac, “An Extensible model of the electron,” Pr oc. Roy. Soc. Lond. A 268 (1962), 57-67 doi:10.1098/rspa.1962.0124

1962
[38]

Introductory Notes on N on-linear Elec- trodynamics and its Applications,

For a review, see D. P. Sorokin, “Introductory Notes on N on-linear Elec- trodynamics and its Applications,” Fortsch. Phys. 70 (2022) no.7-8, 2200092 doi:10.1002/prop.202200092 [arXiv:2112.12118 [hep-th] ]

2022 arXiv
[39]

Aspects of Born- Infeld theory and string / M theory,

For a review in relation to string theory: G. W. Gibbons, “Aspects of Born- Infeld theory and string / M theory,” AIP Conf. Proc. 589 (2001) no.1, 324-350 doi:10.1063/1.1419338 [arXiv:hep-th/0106059 [hep-th]] . 41

2001 arXiv
[40]

Weyl gauge invariant DBI action in con formal geometry,

D. M. Ghilencea, “Weyl gauge invariant DBI action in con formal geometry,” [arXiv:2407.18173 [hep-th]]

arXiv
[41]

Standard Model in confor mal geome- try: Local vs gauged scale invariance,

D. M. Ghilencea and C. T. Hill, “Standard Model in confor mal geome- try: Local vs gauged scale invariance,” Annals Phys. 460 (2024), 169562 doi:10.1016/j.aop.2023.169562 [arXiv:2303.02515 [hep- th]]

2024
[42]

I. L. Buchbinder, I. L. Shapiro, ”Introduction to QFT wi th applications to Quantum Gravity”, Oxford University Press, Oxford 2021

2021
[43]

Scale-Indepen dent Inﬂation and Hierarchy Generation,

P. G. Ferreira, C. T. Hill and G. G. Ross, “Scale-Indepen dent Inﬂation and Hierarchy Generation,” Phys. Lett. B 763 (2016), 174-178 doi:10.1016/j.physletb.2016.10.036 [arXiv:1603.05983 [hep-th]]

2016 arXiv
[44]

No ﬁfth force in a scale invariant universe,

P. G. Ferreira, C. T. Hill and G. G. Ross, “No ﬁfth force in a scale invariant universe,” Phys. Rev. D 95 (2017) no.6, 064038 doi:10.1103/PhysRevD.95.064038 [arXiv:1612.03157 [gr-qc]]

2017 arXiv
[45]

Inertial Spont aneous Symmetry Breaking and Quantum Scale Invariance,

P. G. Ferreira, C. T. Hill and G. G. Ross, “Inertial Spont aneous Symmetry Breaking and Quantum Scale Invariance,” Phys. Rev. D 98 (2018) no.11, 116012 doi:10.1103/PhysRevD.98.116012 [arXiv:1801.07676 [hep -th]]

2018 arXiv
[46]

Weyl Current, S cale-Invariant In- ﬂation and Planck Scale Generation,

P. G. Ferreira, C. T. Hill and G. G. Ross, “Weyl Current, S cale-Invariant In- ﬂation and Planck Scale Generation,” Phys. Rev. D 95 (2017) no.4, 043507 doi:10.1103/PhysRevD.95.043507 [arXiv:1610.09243 [hep -th]]

2017 arXiv
[47]

Higgs-Dilaton Cos- mology: From the Early to the Late Universe,

J. Garcia-Bellido, J. Rubio, M. Shaposhnikov and D. Zen hausern, “Higgs-Dilaton Cos- mology: From the Early to the Late Universe,” Phys. Rev. D 84 (2011), 123504 doi:10.1103/PhysRevD.84.123504 [arXiv:1107.2163 [hep- ph]]

2011 arXiv
[48]

Dimensionally dependent identities,

D. Lovelock, “Dimensionally dependent identities,” P roc. Cambridge Philos. Soc., 68, 345–350
[49]

Manifestly scale-invariant regular ization and quantum eﬀective operators,

D. M. Ghilencea, “Manifestly scale-invariant regular ization and quantum eﬀective operators,” Phys. Rev. D 93 (2016) no.10, 105006 doi:10.1103/PhysRevD.93.105006 [arXiv:1508.00595 [hep-ph]]; D. M. Ghilencea, Z. Lalak and P. Olszewski, “Two-loop scale-invariant scalar potential...

2016 arXiv
[133]

Delphenich is currently ava ilable at this link: http://www.neo-classical-physics.info/spacetime-structure.html)

(an English version by D.H. Delphenich is currently ava ilable at this link: http://www.neo-classical-physics.info/spacetime-structure.html)

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

(An English version by D

Hermann Weyl, Gravitation und elektrizit¨ at, Sitzungs berichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin (1918) , pp.465. (An English version by D. H. Delphenich is currently availab le at: http://www.neo-classical-physics.info/spacetime-structure.html). This work includes Einstein’s critical report, stating tha t the spacing bet...

work page 1918

[2] [2]

Eine neue Erweiterung der Relativit¨ atst heorie

Hermann Weyl “Eine neue Erweiterung der Relativit¨ atst heorie” (“A new exten- sion of the theory of relativity”), Ann. Phys. (Leipzig) (4) 59 (1919), 101-

work page 1919

[3] [3]

Raum, Zeit, Materie

Hermann Weyl “Raum, Zeit, Materie”, vierte erweiterte A uﬂage. Julius Springer, Berlin 1921 “Space-time-matter”, translated from German b y Henry L. Brose, 1922, Methuen & Co Ltd, London, www.gutenberg.org/ebooks/43006 (Project Gutenberg License)

work page 1921

[4] [4]

Quantum gravity from Weyl conformal ge ometry,

D. M. Ghilencea, “Quantum gravity from Weyl conformal ge ometry,” [arXiv:2408.07160 [hep-th]]

arXiv

[5] [5]

Invariant Variation Problems,

E. Noether, “Invariant Variation Problems,” Gott. Nach r. 1918 (1918) 235 [Transp. Theory Statist. Phys. 1 (1971) 186] doi:10.1080/004114571 08231446 [physics/0503066]. Gott. Nachr. 1918 (1918) 235

arXiv 1918

[6] [6]

Invariant theoretical interpretation of i nteraction,

R. Utiyama, “Invariant theoretical interpretation of i nteraction,” Phys. Rev. 101 (1956), 1597-1607 doi:10.1103/PhysRev.101.1597

doi:10.1103/physrev.101.1597 1956

[7] [7]

Lorentz invariance and the gravitation al ﬁeld,

T. W. B. Kibble, “Lorentz invariance and the gravitation al ﬁeld,” J. Math. Phys. 2 (1961), 212-221 doi:10.1063/1.1703702

doi:10.1063/1.1703702 1961

[8] [8]

Weyl transformations and poincare gauge in variance,

A. Bregman, “Weyl transformations and poincare gauge in variance,” Prog. Theor. Phys. 49 (1973), 667-692 doi:10.1143/PTP.49.667

work page doi:10.1143/ptp.49.667 1973