arxiv: 2604.07508 · v1 · submitted 2026-04-08 · 🌀 gr-qc · hep-ph· hep-th

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The fall and the rise of Weyl gauge theory

D. M. Ghilencea

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:13 UTC · model grok-4.3

classification 🌀 gr-qc hep-phhep-th

keywords Weyl gauge theoryconformal geometrygravitygauge bosonEinstein-Hilbert actioncosmological constantWeyl anomalyBorn-Infeld action

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The pith

Weyl gauge theory has been revived as the only gauge theory of a spacetime symmetry with a physical gauge boson, exact geometric interpretation, and all scales of geometric origin.

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

The paper traces the initial downfall of Weyl's 1918 conformal gauge theory of gravity due to its physical interpretation and shows how the associated quadratic action has been revived into a modern physical framework. It establishes that this revived Weyl gauge theory is anomaly-free, possesses an exact geometric meaning, and produces the Einstein-Hilbert action together with a positive cosmological constant once the symmetry breaks spontaneously. A deeper Weyl-Dirac-Born-Infeld action in Weyl geometry underlies it and requires no ultraviolet regularization, with the standard Weyl gauge theory appearing as its leading term. A sympathetic reader would care because the construction supplies a gauge-theoretic account of gravity in which every dimensionful quantity emerges from geometry rather than being inserted by hand.

Core claim

The revived Weyl gauge theory is the sole gauge theory of a spacetime symmetry that contains a physical gauge boson. It is free of the Weyl anomaly, admits an exact geometric interpretation, and assigns geometric origin to all scales. In the spontaneously broken phase it yields the Einstein-Hilbert action and a positive cosmological constant. A more fundamental Weyl-Dirac-Born-Infeld gauge theory action exists in Weyl geometry that needs no UV regularization; the Weyl gauge theory is its leading-order truncation.

What carries the argument

The Weyl-Dirac-Born-Infeld gauge theory action in Weyl geometry, from which the quadratic Weyl gauge theory emerges as the leading term after geometric regularization.

If this is right

The theory generates the Einstein-Hilbert action of general relativity in its spontaneously broken phase.
A positive cosmological constant arises automatically in that phase.
Every scale in the theory, including the Planck scale, originates from geometry alone.
The construction remains free of the Weyl anomaly at the quantum level.
A more fundamental Born-Infeld-type action exists that requires no separate UV regularization.

Where Pith is reading between the lines

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

The geometric origin of scales may offer a new route to the hierarchy problem without additional fine-tuning mechanisms.
The same Weyl geometry could be extended to include other spacetime or internal symmetries, linking gravity to gauge theories of the standard model.
The absence of a need for UV regularization at the fundamental level suggests the framework could serve as a starting point for non-perturbative studies of quantum gravity.

Load-bearing premise

The revived theory supplies a genuinely physical gauge boson together with an exact geometric interpretation while remaining free of the inconsistencies that ended the 1918 version and while matching quantum and observational requirements.

What would settle it

An explicit one-loop calculation that produces a non-vanishing Weyl anomaly coefficient, or a demonstration that the dilatation gauge boson is not physical, would falsify the central claims.

read the original abstract

In 1918 Weyl introduced Weyl conformal geometry and its associated quadratic action which was the first gauge theory, of a spacetime symmetry, the Weyl gauge theory (of dilatations and Poincar\'e symmetry). The initial physical interpretation of his theory was however short-lived and led to the downfall of Weyl geometry as a physical theory. We review how this action was re-born into a physical Weyl gauge theory of gravity. This is the only gauge theory of a spacetime symmetry with a physical gauge boson, is Weyl anomaly-free, has {\it exact} geometric interpretation, with all scales of geometric origin, and generates Einstein-Hilbert action and a positive cosmological constant in its spontaneously broken phase. A more fundamental Weyl-Dirac-Born-Infeld gauge theory action exists in Weyl geometry, that does not need a UV regularisation, of which the (geometrically regularised) Weyl gauge theory is the leading order.

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.

Desk Editor's Note private letter to a colleague

This is a review recapping the history of Weyl gauge theory and arguing for its revival as a geometric framework that generates Einstein gravity plus a positive CC.

read the letter

The paper is a review of how Weyl's 1918 conformal geometry started as the first gauge theory of a spacetime symmetry, fell out of favor, and has been revived in modern work. It walks through the original quadratic action, its physical problems, and the current version that treats the Weyl gauge boson as physical, derives all scales geometrically, and reduces to Einstein-Hilbert plus positive cosmological constant after spontaneous breaking. It also sketches a more fundamental Weyl-Dirac-Born-Infeld action as a possible UV completion that avoids regularization issues. That narrative is the main contribution here; it pulls the threads together in one place rather than introducing new derivations or calculations. The writing is straightforward and the historical framing is useful for anyone who wants context without chasing every reference. The soft spots are the strong assertions that the revived theory is Weyl anomaly-free and has an exact geometric interpretation. These rest on the cited literature, and the abstract does not show fresh anomaly computations or explicit checks against the original consistency failures. If the full text only summarizes prior claims without adding verification steps or addressing the stress-test concern about one-loop invariance under dilatations, readers will still need to go back to the source papers to confirm. This is aimed at theorists working on gauge formulations of gravity or modified gravity models. Someone already following Weyl geometry will find it a convenient overview; a reader new to the area might still need the primary references for the technical details. It deserves peer review as a review article that consolidates the story and flags the conceptual advantages and open questions.

Referee Report

2 major / 2 minor

Summary. The manuscript is a review tracing the 1918 introduction by Weyl of conformal geometry and its quadratic action as the first gauge theory of a spacetime symmetry (dilatations plus Poincaré), the reasons for its initial physical rejection, and the subsequent revival of a physical Weyl gauge theory of gravity. The central claims are that this revived theory is the only gauge theory of a spacetime symmetry possessing a physical gauge boson, is Weyl anomaly-free, admits an exact geometric interpretation with all scales of geometric origin, and yields the Einstein-Hilbert action together with a positive cosmological constant upon spontaneous breaking of the Weyl symmetry; a more fundamental Weyl-Dirac-Born-Infeld action in Weyl geometry is also indicated, of which the reviewed theory is the leading-order term.

Significance. If the revival claims are substantiated, the review would usefully synthesize a gauge-theoretic approach to gravity in which scales arise geometrically and the Einstein-Hilbert term plus positive cosmological constant emerge dynamically, potentially offering a distinct route to anomaly-free quantum gravity. As a review rather than a derivation paper, its value lies in clarifying the historical and conceptual continuity between the original Weyl construction and modern formulations.

major comments (2)

[Abstract] Abstract: The statement that the revived theory 'is Weyl anomaly-free' is presented as a defining property without an explicit computation or citation to a specific section deriving the vanishing of the Weyl anomaly coefficient (typically involving the trace anomaly from the field content and representations). Classical Weyl geometry alone does not guarantee quantum anomaly cancellation for local dilatations; this claim is load-bearing for the assertion of consistency with quantum requirements and therefore requires a concrete reference or derivation to support the central revival narrative.
[Abstract] Abstract: The claim that the theory 'generates Einstein-Hilbert action and a positive cosmological constant in its spontaneously broken phase' is asserted without reference to the explicit breaking mechanism, the form of the effective potential, or the section deriving the sign and magnitude of the induced cosmological constant. Because this is presented as a key physical outcome distinguishing the revived theory, the absence of the relevant derivation or citation undermines the strength of the result.

minor comments (2)

[Abstract] The abstract refers to 'the (geometrically regularised) Weyl gauge theory' as the leading order of a more fundamental action; the precise regularization procedure and its relation to the Born-Infeld term should be clarified with a brief equation or reference in the introduction for readers unfamiliar with the construction.
The manuscript should include a short table or bullet list contrasting the original 1918 Weyl theory with the revived version on the points of gauge boson physicality, anomaly status, and scale origin to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below and will make the necessary revisions to strengthen the presentation of our claims.

read point-by-point responses

Referee: [Abstract] Abstract: The statement that the revived theory 'is Weyl anomaly-free' is presented as a defining property without an explicit computation or citation to a specific section deriving the vanishing of the Weyl anomaly coefficient (typically involving the trace anomaly from the field content and representations). Classical Weyl geometry alone does not guarantee quantum anomaly cancellation for local dilatations; this claim is load-bearing for the assertion of consistency with quantum requirements and therefore requires a concrete reference or derivation to support the central revival narrative.

Authors: We agree that the abstract would benefit from a direct reference to the supporting discussion in the manuscript. The review synthesizes results from the literature demonstrating that the specific field content in the Weyl gauge theory leads to cancellation of the Weyl anomaly, as the contributions from the gauge boson and other fields balance the trace anomaly. This is elaborated in the section on quantum aspects of the theory. We will revise the abstract to include a citation to this section, thereby providing the concrete reference requested. revision: yes
Referee: [Abstract] Abstract: The claim that the theory 'generates Einstein-Hilbert action and a positive cosmological constant in its spontaneously broken phase' is asserted without reference to the explicit breaking mechanism, the form of the effective potential, or the section deriving the sign and magnitude of the induced cosmological constant. Because this is presented as a key physical outcome distinguishing the revived theory, the absence of the relevant derivation or citation undermines the strength of the result.

Authors: We concur that referencing the relevant parts of the manuscript would clarify this key result. The spontaneous breaking of the Weyl symmetry, the form of the effective potential, and the emergence of the positive cosmological constant are derived in the sections discussing the symmetry breaking and the low-energy effective action. We will update the abstract to point to these sections, ensuring readers can easily locate the detailed derivation and the explanation for the positive sign of the cosmological constant. revision: yes

Circularity Check

0 steps flagged

Review paper with minor self-citations to prior Weyl geometry work; no load-bearing reduction of claims to inputs by construction

full rationale

This is a review summarizing the historical fall and revival of Weyl gauge theory. The abstract asserts properties such as being 'Weyl anomaly-free' and having 'exact geometric interpretation' as established features of the reborn theory, but these are framed as outcomes of the reviewed prior literature rather than new derivations performed here. No equations, fits, or self-definitional loops appear in the provided text that would reduce a 'prediction' to an input parameter or ansatz by construction. Self-citations to earlier works on the topic are present by nature of a review but are not load-bearing in a way that makes the central narrative circular; the paper remains self-contained against external benchmarks from the cited derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The provided abstract contains no new free parameters, axioms, or invented entities; the discussion concerns historical revival of an established geometric framework.

pith-pipeline@v0.9.0 · 5450 in / 1223 out tokens · 39937 ms · 2026-05-10T17:13:05.133756+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

43 extracted references · 22 canonical work pages

[1]

Hermann Weyl, Gravitation und elektrizit¨ at, Sitzungsberichte der K¨ oniglich Preussis- chen Akademie der Wissenschaften zu Berlin (1918), pp.465-480, with Einstein’s critical comment appended; (English version currently available at: http://www.neo- classical-physics.info/spacetime- structure.html)

1918
[2]

Raum, Zeit, Materie

“Raum, Zeit, Materie”, vierte erweiterte Auflage. Julius Springer, Berlin 1921 www.gutenberg.org/ebooks/43006 (Project Gutenberg License). Originally published in 1918 Berlin

1921
[3]

Eine neue Erweiterung der Relativit¨ atstheorie

“Eine neue Erweiterung der Relativit¨ atstheorie” (“A new extension of the theory of relativity”) Ann. Phys. (Leipzig) (4) 59 (1919), 101-133. (English version currently available at: http://www.neo-classical-physics.info/spacetime-structure.html)

1919
[4]

Scholz,The unexpected resurgence of Weyl geometry in late 20-th century physics,Einstein Stud.14(2018) 261 [1703.03187]

For a review: E. Scholz, “The unexpected resurgence of Weyl geometry in late 20-th century physics,” Einstein Stud.14(2018) 261 [arXiv:1703.03187 [math.HO]]

work page arXiv 2018
[5]

Long range forces and broken symmetries,

P. A. M. Dirac, “Long range forces and broken symmetries,” Proc. Roy. Soc. Lond. A333(1973) 403

1973
[6]

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. B160(1979), 253-268

1979
[7]

Invariant Variation Problems,

E. Noether, “Invariant Variation Problems,” Gott. Nachr. 1918 (1918) 235 [Transp. Theory Statist. Phys. 1 (1971) 186], Gott. Nachr. 1918 (1918) 235 8

1918
[8]

Invariant theoretical interpretation of interaction,

R. Utiyama, “Invariant theoretical interpretation of interaction,” Phys. Rev.101 (1956), 1597-1607

1956
[9]

Lorentz invariance and the gravitational field,

T. W. B. Kibble, “Lorentz invariance and the gravitational field,” J. Math. Phys.2 (1961), 212-221

1961
[10]

A gauge theory of the Weyl group,

J. M. Charap and W. Tait, “A gauge theory of the Weyl group,” Proc. Roy. Soc. Lond. A340(1974), 249-262

1974
[11]

Weyl transformations and poincare gauge invariance,

A. Bregman, “Weyl transformations and poincare gauge invariance,” Prog. Theor. Phys.49(1973), 667-692

1973
[12]

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

C. Condeescu, D. M. Ghilencea and A. Micu, “Conformal geometry as a gauge theory of gravity: Covariant equations of motion & conservation laws,” Annals Phys.480 (2025), 170125 [arXiv:2412.16548 [hep-th]]

work page arXiv 2025
[13]

The gauge theory of Weyl group and its interpre- tation as Weyl quadratic gravity,

C. Condeescu and A. Micu, “The gauge theory of Weyl group and its interpre- tation as Weyl quadratic gravity,” Class. Quant. Grav.42(2025) no.6, 065011 [arXiv:2408.07159 [hep-th]]

work page arXiv 2025
[14]

Supergravity,

D. Z. Freedman and A. Van Proeyen, “Supergravity,” Cambridge Univ. Press, 2012, ISBN 978-1-139-36806-3, 978-0-521-19401-3

2012
[15]

Exact Vacuum Solution to Conformal Weyl Grav- ity and Galactic Rotation Curves,

P. D. Mannheim and D. Kazanas, “Exact Vacuum Solution to Conformal Weyl Grav- ity and Galactic Rotation Curves,” Astrophys. J.342(1989), 635-638; D. Kazanas and P. D. Mannheim, “General Structure of the Gravitational Equations of Motion in Conformal Weyl Gravity,” Astrophys. J. Suppl.76(1991), 431-453

1989
[16]

Gauge Theory of the Con- formal and Superconformal Group,

M. Kaku, P. K. Townsend and P. van Nieuwenhuizen, “Gauge Theory of the Con- formal and Superconformal Group,” Phys. Lett. B69(1977), 304-308

1977
[17]

Ghilencea,Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential,JHEP03(2019) 049 [1812.08613]

D. M. Ghilencea, “Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential,” JHEP1903(2019) 049 [arXiv:1812.08613 [hep-th]]. D. M. Ghi- lencea, “Stueckelberg breaking of Weyl conformal geometry and applications to grav- ity,” Phys. Rev. D101(2020) 4, 045010 [arXiv:1904.06596 [hep-th]]

work page arXiv 2019
[18]

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

M. P. Hobson and A. N. Lasenby, “Weyl gauge theories of gravity do not predict a second clock effect,” Phys. Rev. D102(2020) no.8, 084040 [arXiv:2009.06407]

work page arXiv 2020
[19]

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

D. M. Ghilencea, “Non-metric geometry as the origin of mass in gauge theories of scale invariance,” Eur. Phys. J. C83(2023) no.2, 176 [arXiv:2203.05381 [hep-th]]

work page arXiv 2023
[20]

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

C. Condeescu, D. M. Ghilencea and A. Micu, “Weyl quadratic gravity as a gauge theory and non-metricity vs torsion duality,” Eur. Phys. J. C84(2024) no.3, 292 [arXiv:2312.13384 [hep-th]]

work page arXiv 2024
[21]

Weyl conformal geometry vs Weyl anomaly,

D. M. Ghilencea, “Weyl conformal geometry vs Weyl anomaly,” JHEP10(2023), 113 [arXiv:2309.11372 [hep-th]]; 9

work page arXiv 2023
[22]

Quantum gravity from Weyl conformal geometry,

D. M. Ghilencea, “Quantum gravity from Weyl conformal geometry,” Eur. Phys. J. C85(2025) 7, 815 [arXiv:2408.07160 [hep-th]]

work page arXiv 2025
[23]

Ghilencea,Standard Model in Weyl conformal geometry,Eur

D. M. Ghilencea, “Standard Model in Weyl conformal geometry,” Eur. Phys. J. C 82(2022) no.1, 23 [arXiv:2104.15118 [hep-ph]]

work page arXiv 2022
[24]

Ghilencea and C.T

D. M. Ghilencea and C. T. Hill, “Standard Model in conformal geometry: Local vs gauged scale invariance,” Annals Phys.460(2024), 169562 [arXiv:2303.02515]

work page arXiv 2024
[25]

Interaction forces in electrodynamics and in the field theory of nuclear forces,

E. C. G. Stueckelberg, “Interaction forces in electrodynamics and in the field theory of nuclear forces,” Helv. Phys. Acta11(1938) 299

1938
[26]

Conformal Invariance in Quantum Grav- ity,

F. Englert, C. Truffin and R. Gastmans, “Conformal Invariance in Quantum Grav- ity,” Nucl. Phys. B117(1976), 407-432

1976
[27]

Aspects of Quadratic Gravity,

L. Alvarez-Gaume, A. Kehagias, C. Kounnas, D. L¨ ust and A. Riotto, “Aspects of Quadratic Gravity,” Fortsch. Phys.64(2016) no.2-3, 176-189 [arXiv:1505.07657]

work page arXiv 2016
[28]

Living with ghosts,

S. W. Hawking and T. Hertog, “Living with ghosts,” Phys. Rev. D65(2002), 103515 [arXiv:hep-th/0107088 [hep-th]]

work page arXiv 2002
[29]

Einstein Gravity from Conformal Gravity,

J. Maldacena, “Einstein Gravity from Conformal Gravity,” [arXiv:1105.5632 [hep- th]]

work page arXiv
[30]

Renormalization of Higher Derivative Quantum Gravity,

K. S. Stelle, “Renormalization of Higher Derivative Quantum Gravity,” Phys. Rev. D16(1977) 953

1977
[31]

Duff,Twenty years of the Weyl anomaly,Class

M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav.11(1994), 1387-1404 [arXiv:hep-th/9308075 [hep-th]]. M. J. Duff, “Observations on Conformal Anomalies,” Nucl. Phys. B125(1977), 334-348

work page arXiv 1994
[32]

Photon corrections to the graviton propagator,

D. M. Capper, M. J. Duff and L. Halpern, “Photon corrections to the graviton propagator,” Phys. Rev. D10(1974), 461-467

1974
[33]

Trace anomalies in dimensional regularization,

D. M. Capper and M. J. Duff, “Trace anomalies in dimensional regularization,” Nuovo Cim. A23(1974), 173-183

1974
[34]

Nonlocal Conformal Anomalies,

S. Deser, M. J. Duff and C. J. Isham, “Nonlocal Conformal Anomalies,” Nucl. Phys. B111(1976), 45-55

1976
[35]

Deser and A

S. Deser and A. Schwimmer, “Geometric classification of conformal anomalies in arbitrary dimensions,” Phys. Lett. B309(1993), 279-284 [arXiv:hep-th/9302047]

work page arXiv 1993
[36]

Quantum scale invariance, cosmological con- stant and hierarchy problem,

M. Shaposhnikov and D. Zenhausern, “Quantum scale invariance, cosmological con- stant and hierarchy problem,” Phys. Lett. B671(2009), 162-166 [arXiv:0809.3406]

work page arXiv 2009
[37]

Foundations of the new field theory,

M. Born and L. Infeld, “Foundations of the new field theory,” Proc. Roy. Soc. Lond. A144(1934) no.852, 425-451 10

1934
[38]

An Extensible model of the electron,

P. A. M. Dirac, “An Extensible model of the electron,” Proc. Roy. Soc. Lond. A268 (1962), 57-67

1962
[39]

Weyl gauge invariant DBI action in conformal geometry,

D. M. Ghilencea, “Weyl gauge invariant DBI action in conformal geometry,” Phys. Rev. D111(2025) no.8, 085019 [arXiv:2407.18173 [hep-th]]

work page arXiv 2025
[40]

Unification of gravity and standard model: Weyl-Dirac-Born- Infeld action,

D. M. Ghilencea, “Unification of gravity and standard model: Weyl-Dirac-Born- Infeld action,” Phys. Rev. D112(2025) no.12, 125015 [arXiv:2508.10959 [hep-ph]]

work page arXiv 2025
[41]

Ferreira, C.T

P. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, “Scale-independentR 2 inflation,” Phys. Rev. D100(2019) no.12, 123516 [arXiv:1906.03415 [gr-qc]]

work page arXiv 2019
[42]

Weyl R 2 inflation with an emergent Planck scale,

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

work page arXiv 2019
[43]

Testing Weyl geometric gravity with the SPARC galactic rotation curves database,

M. Cr˘ aciun and T. Harko, “Testing Weyl geometric gravity with the SPARC galactic rotation curves database,” Phys. Dark Univ.43(2024), 101423 [arXiv:2311.16893 [gr-qc]]. P. Burikham, T. Harko, K. Pimsamarn and S. Shahidi, “Dark matter as a Weyl geometric effect,” Phys. Rev. D107(2023) no.6, 064008 [arXiv:2302.08289] 11

work page arXiv 2024