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REVIEW 3 major objections 3 minor 57 references

Spacetime and Planck mass generation from scale-invariant degenerate gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Fermion quantum fluctuations in a scale-invariant, degenerate gravitational theory can simultaneously generate the Planck mass and a curved spacetime background, the paper argues.

desk verdict The combined one-loop potential is a real calculation, but the paper buys its Planck mass and spacetime with an externally inserted Rbar and a runaway C direction. read the letter →

arxiv 2411.17238 v2 pith:HUDW2MC3 submitted 2024-11-26 hep-th gr-qc

Yadikaer Maitiniyazi , Shinya Matsuzaki , Kin-ya Oda , Masatoshi Yamada This is my paper · ORCID
classification hep-thgr-qc
keywords scale-invariantgravityirreversiblevierbeinpostulatedegeneratelimitpregeometryPlanckmassgenerationColeman-Weinbergmechanismheatkernelexpansionemergenceofspacetime
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 aims to show that a gravitational theory with no dimensionful parameters can produce both the Planck mass and a curved spacetime from fermion quantum dynamics. The action is scale invariant and obeys the irreversible vierbein postulate, meaning the vierbein determinant may vanish continuously and the theory is well defined at zero vierbein. A scalar field $\phi$ coupled to spinors and to the local-Lorentz field strength develops a nonzero vacuum expectation value, and the Planck mass is $M_{\rm pl}=\xi\langle\phi\rangle$. On a constant-curvature background, the one-loop fermion effective potential has a stable minimum in the $\phi$ direction, giving $\langle\phi\rangle=0.708\Lambda_G$ and $M_{\rm pl}=0.0708\Lambda_G$ for benchmark parameters. The paper reads the potential's runaway in the vierbein direction as possible early-Universe expansion or as a signal that subleading fluctuations would stabilize spacetime; if the mechanism is right, gravity's strength and the existence of a curved background are emergent rather than input parameters.

What carries the argument

The load-bearing object is the one-loop effective potential for the conformal factor $\bar C$ and the scalar $\bar\phi$, built by integrating out the Dirac fermion on a fixed constant-curvature background. The heat-kernel expansion expresses the fermion determinant through the coefficients $a_0=4$ and $a_2=4\bar R/3$, giving the logarithmic Coleman-Weinberg structure of Eq. (32); the overall factor $\bar C^4$ comes from $\sqrt{-\bar g}=|\bar e|=\bar C^4$, so the same potential controls both the scalar vacuum and the generation of the vierbein background. The nonminimal coupling $\frac{\xi}{2}\phi^2 e^a_\mu e^b_\nu F_{ab}^{\mu\nu}$ converts the scalar vacuum expectation value into $M_{\rm pl}^2$, and the irreversible vierbein postulate, the assumption that the action stays finite in the degenerate limit $\det e=0$, keeps the $\bar C\to0$ configuration continuously connected so the theory can be treated as linear around zero vierbein rather than around a fixed background.

What would settle it

Compute Eq. (32) with $\bar R=0$ and search for stationary points with $\langle\bar\phi\rangle\neq0$ and $\langle\bar C\rangle\neq0$: the paper's own criterion $m_\phi^2=\xi\bar R>0$ says this should not occur, so a nonzero solution would falsify the claimed necessity of an assumed background curvature, while finding none would show the mechanism has no symmetry-breaking trigger without $\bar R$.

Watch

Extended reading notes

Core claim

The paper's central claim is that scalegenesis and pregeometry happen in the same transition: from a scale-invariant action that permits a continuously degenerate vierbein, the vacuum settles at $\langle\phi\rangle\neq0$ and $\langle e^a_\mu\rangle\neq0$, so the Planck mass $M_{\rm pl}^2=\xi\langle\phi\rangle^2$ and a curved background both arise from dynamics rather than from inputs. The demonstration uses the regularized one-loop effective potential (Eq. (32)): $$V_{\rm eff}(\bar\phi,\bar C)=\left[-\frac{\xi\bar R}{2}\bar\$phi^{2}$+\frac{\$\lambda$}{4!}\bar\$phi^{4}$-\frac{(y\bar\phi)^2}{16\$pi^{2}$}\left((y\bar\phi)^2-\frac{2}{3}\bar R\right)\log\frac{(y\bar\phi)^2}{\$Lambda_G^{2}$}\right]\bar $C^{4}$ .$$ For the benchmark values $\xi=0.1$, $\bar R=0.5\Lambda_G^2$, $y=0.1$, $\lambda=0.6$, the $\bar\phi$ direction has a local minimum at $\langle\bar\phi\rangle=0.708\Lambda_G$, giving $M_{\rm pl}=0.0708\Lambda_G$; the $\bar C$ direction runs away to negative infinity, which the paper interprets as possible early-Universe expansion and expects to be stabilized by subleading loop effects. Because the potential at the vacuum is negative, the generated spacetime is initially anti-de Sitter unless an additional contribution to the cosmological constant is supplied.

Load-bearing premise

The argument depends on a fixed nonzero constant background curvature $\bar R$ being present before the dynamics run, because that curvature supplies the scalar's negative mass-squared term; if $\bar R=0$, the symmetry-breaking trigger disappears and no Planck mass is generated.

Editorial extensions

If this is right

  • The Planck mass becomes a derived quantity: with the benchmark couplings $M_{\rm pl}=0.0708\Lambda_G$, so the scale $\Lambda_G$ can be fixed to reproduce the observed value instead of putting $M_{\rm pl}$ in by hand.
  • In the symmetric degenerate phase only the fermion is dynamical; the vierbein, the local-Lorentz gauge field, and $\phi$ acquire dynamics through loop effects after symmetry breaking, so effective gravity is generated rather than fundamental.
  • The vacuum energy at the generated vacuum is negative, so the emerging spacetime is anti-de Sitter at this order; matching the observed Universe requires an additional positive contribution to the cosmological constant.
  • The runaway in the $\bar C$ direction implies that the vierbein vacuum is not stable at one loop, so the next step of the program is either to treat that runaway as cosmological expansion or to include subleading fluctuations of $\phi$, $e^a_\mu$, and the gauge field to see whether a stable nonzero $\bar C$ vacuum appears.

Reading between the lines

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

  • Because a fixed nonzero $\bar R$ is inserted before the effective potential is built, the paper has not, as it stands, shown that the background curvature itself is generated; a fully dynamical version would need to derive $\bar R$ from the scalar and fermion dynamics, for example by solving the semiclassical background equations rather than fixing $\bar R$.
  • A natural extension is to compute the two-field effective potential including one-loop fluctuations of $\phi$ and the conformal factor and check whether the negative runaway in the $\bar C$ direction is lifted; a stable nonzero $\langle\bar C\rangle$ there would turn the claimed simultaneous generation into a genuine two-field vacuum.
  • The model's reliance on a nonzero $\bar R$ suggests a discriminating test: compute the effective potential at $\bar R=0$ and see whether any radiative minimum with $\langle\phi\rangle\neq0$ still exists; if none does, the Planck mass is seeded by the very curvature the theory is supposed to create.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proposes a scale-invariant gravitational action built on the irreversible vierbein postulate, which requires the action to remain finite as det e goes to zero and thereby forbids scalar and gauge kinetic terms as well as the kinetic term of the spin connection. The remaining action contains a scalar phi with nonminimal coupling xi phi^2 R, a Yukawa interaction y phi psi-bar psi, and a quartic potential. Working in conformal coordinates with background vierbein C delta^a_mu and assuming a constant background curvature Rbar, the authors compute the one-loop effective potential (32) from integrating out the spinor. For the benchmark parameters in (33), they find a local minimum in the phi direction at <phi>/Lambda_G = 0.708 and report M_pl = xi <phi> = 0.0708 Lambda_G, while the C direction is described as a runaway. The paper claims the simultaneous emergence of the Planck mass and a curved spacetime background from the dynamics.

Significance. If the central claim were established, the model would be a noteworthy mechanism for deriving both the Planck scale and a nonflat spacetime from a scale-invariant, degenerate starting point. The heat-kernel computation of the one-loop effective potential is explicit, and the connection to the pregeometry program and to earlier work on dynamically emergent gravity is useful. However, the central claims are not supported by the computed potential: the background curvature Rbar is an externally imposed dimensionful input, and the conformal-mode direction has no stable vacuum. The calculation as it stands is closer to a parameter fit for a chosen Rbar than to a demonstration of emergence, so the significance of the result is substantially weaker than the abstract suggests.

major comments (3)
  1. [Sec. III.B, Eq. (28) and Eq. (32)] The generation of the Planck mass is not dynamical: immediately after Eq. (28) the paper assumes a constant nonzero scalar curvature Rbar, and the benchmark (33) fixes Rbar = 0.5 Lambda_G^2. Because m_phi^2 = xi Rbar determines <phi> = sqrt(6 xi Rbar / lambda) and hence M_pl = xi <phi>, the Planck mass is a function of the input Rbar and the chosen couplings, not a scale produced by the dynamics. The paper itself notes that for Rbar = 0 one would only have the symmetric vacuum <phi> = 0. Since Rbar is exactly the curved-spacetime quantity that the abstract claims emerges from the dynamics, the argument is circular: the assumed background curvature supplies both the scalar VEV and the nonflat spacetime.
  2. [Sec. III.B, Eq. (32) and Fig. 2] The conformal-mode direction C has no stable vacuum. From Eq. (32), V_eff(phi, C) = C^4 V(phi), and with the benchmark parameters (33) the minimum in phi gives V(phi_min) < 0, so V_eff tends to minus infinity as C tends to infinity and the only stationary point in C is C = 0, which is a local maximum. The text concedes this with the statement that the potential diverges negatively, preventing a stable vacuum. Consequently, the paper does not exhibit any solution of the equations of motion with <C> != 0, and the claim in Sec. IV that the conformal mode acquires a nonzero vacuum expectation value is not supported by the computed potential.
  3. [Sec. III.B, after Eq. (28)] The effective potential (32) is derived under the assumption of a fixed background with constant Rbar, but no equation of motion or self-consistency condition determines Rbar in terms of <phi> or <C>. The text just before Eq. (27) states that Rbar = 0 at the degenerate limit C = 0, yet the benchmark (33) assumes Rbar = 0.5 Lambda_G^2, a nonzero value that is not derived from the model. Thus the curved background is an input rather than an emergent consequence, and the central claim of simultaneous emergence is not established by the calculation. A minimal requirement would be to solve the full background equations, including the equation for Rbar, and to show a nontrivial solution with stable C.
minor comments (3)
  1. [Sec. III.B, end of section] The text says the origin of the effective potential in the C direction is tachyonic and unstable, but the potential behaves as C^4 V(phi) with no quadratic term in C; calling this tachyonic is inaccurate, and the description should be revised.
  2. [Sec. II.C and Sec. III.A] The same symbol C is used for the constant minisuperspace ansatz in Sec. II.C and for the time-dependent scale factor Cbar(eta) in Sec. III.A; this notation should be disambiguated to avoid confusing the reader.
  3. [Sec. III.A, Eq. (27)] The paper should clarify that the scalar field phi is not dynamical at tree level owing to the irreversible vierbein postulate, and should state more explicitly that the effective potential treats phi and Cbar as classical backgrounds while only the spinor runs in the loop.

Circularity Check

2 steps flagged · score 7.0 of 10

The claimed Planck-mass and spacetime emergence is seeded by an assumed constant scalar curvature Rbar and benchmarked to fit M_pl; the C-direction has no stable vacuum.

  1. self definitional [Sec. III.B, after Eq. (28); parameter benchmark Eq. (33)]
    "Here, we assume a constant scalar curvature [50]. For m2ϕ := ξR >0, the classical potential has a stable vacuum at ⟨¯ϕ⟩ = q 6m2ϕ/λ ≠ 0, which gives rise to the Planck mass Mpl = ξ⟨¯ϕ⟩."

    The target outputs—a curved spacetime and the Planck scale—are both encoded in the assumed Rbar. m_phi^2 is defined as ξ Rbar, so <φ> = sqrt(6ξRbar/λ) and Mpl = ξ<φ> are functions of the same injected curvature. For Rbar = 0, as the paper notes, the potential is pure λφ^4 and only <φ>=0 exists, so the nontrivial vacuum is not generated but presupposed.

  2. fitted input called prediction [Sec. III.B, benchmark (33) and following text]
    "As a benchmark, we choose the following parameters: ξ = 0.1, Rbar = 0.5Λ^2_G, y = 0.1, λ = 0.6 ... ⟨¯ϕ⟩/Λ_G = 0.708. This gives the Planck mass as Mpl = ξ⟨¯ϕ⟩ = 0.0708Λ_G, matching the observed value with a suitable choice of Λ_G."

    Mpl is not predicted: it is the closed-form consequence of the chosen Rbar. With <φ> = sqrt(6ξRbar/λ), the benchmark number 0.0708Λ_G is forced by Rbar = 0.5Λ_G^2, and Λ_G is then 'suitably chosen' to match the observed Planck mass. The scale-generation claim therefore reduces to a fit of the free input Rbar (plus Λ_G normalization), not to a dynamical determination.

full rationale

The derivation of Eq. (32) itself is a standard heat-kernel computation with external references, and the authors do not hide that 'we assume a constant scalar curvature [50]' immediately after defining the tree potential. This assumption is the circular core: Rbar is a curved-spacetime background, and it is also the only source of the φ mass, m_phi^2 = ξ Rbar, which in turn sets <φ> and Mpl = ξ<φ>. If one sets Rbar = 0, as the paper itself notes around Eq. (32), the φ-direction has no nontrivial minimum and no Planck mass appears; hence the 'generation' of the Planck scale is an algebraic restatement of the injected Rbar. The benchmark (33) then fixes Rbar = 0.5Λ_G^2 and selects Λ_G afterward to match the observed Mpl, so the numerical 'prediction' 0.0708Λ_G is a parameter fit rather than a first-principles determination. Separately, the C-direction of Veff = C^4 V(φ) has no stable minimum—'the potential diverges negatively, preventing a stable vacuum'—so no nondegenerate vierbein vacuum is actually selected; the paper consigns its spacetime claim to a runaway or to unspecified subleading stabilizers. The self-citations [1,2] introduce the irreversible vierbein postulate as a working assumption, but the central circularity is the Rbar input, not the postulate.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central generation claims rest on four tuned dimensionless parameters, an assumed background curvature, and a cutoff chosen to match data. The irreversible vierbein postulate is an ad hoc working assumption. No new particle or force is introduced.

free parameters (5)
  • xi (nonminimal scalar-curvature coupling) = 0.1
    Chosen by hand as a benchmark; directly determines M_pl = xi <phi>.
  • Rbar (background scalar curvature) = 0.5 Lambda_G^2
    Assumed constant background curvature; it seeds the scalar mass m_phi^2 = xi Rbar. This is the quantity whose 'emergence' is claimed.
  • y (Yukawa coupling) = 0.1
    Chosen benchmark; controls the fermion mass and the one-loop coefficient.
  • lambda (scalar quartic coupling) = 0.6
    Chosen benchmark; determines <phi> = sqrt(6 xi Rbar / lambda).
  • Lambda_G (UV cutoff scale) = chosen so M_pl = 0.0708 Lambda_G matches the observed Planck mass
    The cutoff scale introduced to define the quantum theory; its value is fixed by requiring agreement with the observed Planck mass, so it is effectively a fitted parameter.
assumptions (6)
  • ad hoc to paper Irreversible vierbein postulate: the action must remain finite in the degenerate limit det e = 0, which forbids scalar and gauge kinetic terms and the omega kinetic term.
    Defined in Sec II.B; this is a strong technical assumption without independent empirical support.
  • domain assumption Classical scale invariance at the UV scale Lambda_G, forbidding all dimensionful parameters in the classical action.
    Motivated by Refs [6-12]; scale invariance is broken by the cutoff and log terms.
  • domain assumption The background vierbein has the minisuperspace form e^a_mu = C delta^a_mu and the scalar is spatially constant (Eq 23).
    Restricts to homogeneous conformal backgrounds; no proof is given that the vacuum takes this form.
  • ad hoc to paper The background curvature scalar Rbar is constant and nonzero (Sec III.B, citing Ref [50]).
    This assumption is load-bearing: it provides m_phi^2 and triggers symmetry breaking; the paper does not derive Rbar from the model.
  • domain assumption Quantum fluctuations of phi, e, and omega are neglected at one loop; only the fermion loop is included.
    Stated in Sec III; justified as subleading, but since the tree-level potential is unbounded in C, loop corrections could be important.
  • standard math Standard heat kernel coefficients for a Dirac fermion (a0 = 4, a2 = 4R/3) are used.
    Well-established result cited to Ref [52]; not a source of novelty or concern.

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Cite this review

Pith. "Pith review of Spacetime and Planck mass generation from scale-invariant degenerate gravity." pith.science (2026). https://pith.science/paper/HUDW2MC3

@misc{pith2026241117238,
  author       = {Pith},
  title        = {Pith review of: Spacetime and Planck mass generation from scale-invariant degenerate gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUDW2MC3}},
  note         = {Machine review of arXiv:2411.17238}
}
read the original abstract

We investigate a gravitational model based on local Lorentz invariance and general coordinate invariance. The model incorporates classical scale invariance, which forbids dimensionful parameters, and the irreversible vierbein postulate, which enables continuous degenerate limits of the vierbein, both at a specific scale. Through the dynamics of the system, we demonstrate the simultaneous emergence of the Planck mass and a curved spacetime background.

Figures

Figures reproduced from arXiv: 2411.17238 by the authors.

Figure 1
Figure 1. FIG. 1: Shape of the effective potential (20) with [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Top: the effective potential (32) at one loop as a function of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

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Works this paper leans on

57 extracted references · 34 canonical work pages

  1. [1]

    Gravitational interactions are invariant under SO(1, 3)LL × GC [1, 2]

    work page

  2. [2]

    Irreversible vierbein postulate: the action allows for the degenerate limit [3–5]

    work page

  3. [3]

    irreversible vierbein postulate

    Classical scale invariance [6, 7]. Below, we elaborate on these assumptions, explaining their definitions and the motiva- tions for adopting them. A detailed theoretical description of gravitational theories with SO(1, 3)LL × GC invariance can be found in Ref. [2]. A. LL and GC symmetries We consider a gravitational theory invariant under SO(1, 3)LL × GC ...

    work page

  4. [4]

    Matsuzaki, S

    S. Matsuzaki, S. Miyawaki, K.-y. Oda and M. Yamada, Dynamically emergent gravity from hidden local Lorentz symmetry , Phys. Lett. B 813 (2021) 135975 [ 2003.07126]

    arXiv 2021

  5. [5]

    Maitiniyazi, S

    Y. Maitiniyazi, S. Matsuzaki, K.-y. Oda and M. Yamada, Irreversible vierbein postulate: Emergence of spacetime from quantum phase transition , Phys. Rev. D 109 (2024) 106018 [2309.16230]

    arXiv 2024

  6. [6]

    A. A. Tseytlin, On the First Order Formalism in Quantum Gravity , J. Phys. A15 (1982) L105

    work page 1982

  7. [7]

    G. T. Horowitz, Topology change in classical and quantum gravity , Class. Quant. Grav. 8 (1991) 587

    work page 1991

  8. [8]

    Floreanini and R

    R. Floreanini and R. Percacci, Mean field quantum gravity , Phys. Rev. D 46 (1992) 1566

    work page 1992

Show all 57 references
  1. [9]

    Wetterich, Fine Tuning Problem and the Renormalization Group , Phys

    C. Wetterich, Fine Tuning Problem and the Renormalization Group , Phys. Lett. B 140 (1984) 215

    1984

  2. [10]

    W. A. Bardeen, On naturalness in the standard model , in Ontake Summer Institute on Particle Physics , 8, 1995

    1995

  3. [11]

    Hamada, H

    Y. Hamada, H. Kawai, K. Kawana, K.-y. Oda and K. Yagyu, Minimal scenario of criticality for electroweak scale, neutrino masses, dark matter, and inflation , Eur. Phys. J. C 81 (2021) 962 [ 2102.04617]. 15

    2021 arXiv

  4. [12]

    Kawai and K

    H. Kawai and K. Kawana, The multicritical point principle as the origin of classical conformality and its generalizations , PTEP 2022 (2022) 013B11 [ 2107.10720]

    2022 arXiv

  5. [13]

    Hamada, H

    Y. Hamada, H. Kawai, K. Kawana, K.-y. Oda and K. Yagyu, Gravitational waves in models with multicritical-point principle , Eur. Phys. J. C 82 (2022) 481 [ 2202.04221]

    2022 arXiv

  6. [14]

    Kawai, K

    H. Kawai, K. Kawana, K.-y. Oda and K. Yagyu, Quantum phase transition and absence of quadratic divergence in generalized quantum field theories , Phys. Rev. D 109 (2024) 085009 [2307.11420]

    2024 arXiv

  7. [15]

    Wetterich and M

    C. Wetterich and M. Yamada, Gauge hierarchy problem in asymptotically safe gravity–the resurgence mechanism, Phys. Lett. B 770 (2017) 268 [ 1612.03069]

    2017 arXiv

  8. [16]

    J. Kubo, M. Lindner, K. Schmitz and M. Yamada, Planck mass and inflation as consequences of dynamically broken scale invariance , Phys. Rev. D 100 (2019) 015037 [1811.05950]

    2019 arXiv

  9. [17]

    G. E. Volovik, Planck Constants in the Symmetry Breaking Quantum Gravity , Symmetry 15 (2023) 991 [ 2304.04235]

    2023 arXiv

  10. [18]

    Edery and Y

    A. Edery and Y. Nakayama, Generating Einstein gravity, cosmological constant and Higgs mass from restricted Weyl invariance , Mod. Phys. Lett. A 30 (2015) 1550152 [ 1502.05932]

    2015 arXiv

  11. [19]

    Edery and Y

    A. Edery and Y. Nakayama, Palatini formulation of pure R2 gravity yields Einstein gravity with no massless scalar , Phys. Rev. D 99 (2019) 124018 [ 1902.07876]

    2019 arXiv

  12. [20]

    Edery and Y

    A. Edery and Y. Nakayama, Restricted Weyl invariance in four-dimensional curved spacetime, Phys. Rev. D 90 (2014) 043007 [ 1406.0060]

    2014 arXiv

  13. [21]

    D. M. Ghilencea, Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential, JHEP 03 (2019) 049 [ 1812.08613]

    2019 arXiv

  14. [22]

    D. M. Ghilencea and H. M. Lee, Weyl gauge symmetry and its spontaneous breaking in the standard model and inflation , Phys. Rev. D 99 (2019) 115007 [ 1809.09174]

    2019 arXiv

  15. [23]

    Shtanov, Electroweak symmetry breaking by gravity , JHEP 02 (2024) 221 [ 2305.17582]

    Y. Shtanov, Electroweak symmetry breaking by gravity , JHEP 02 (2024) 221 [ 2305.17582]

    2024 arXiv

  16. [24]

    G. K. Karananas, M. Shaposhnikov and S. Zell, Weyl-invariant Einstein-Cartan gravity: unifying the strong CP and hierarchy puzzles , 2406.11956

    arXiv

  17. [25]

    Akama, An Attempt at Pregeometry: Gravity With Composite Metric , Prog

    K. Akama, An Attempt at Pregeometry: Gravity With Composite Metric , Prog. Theor. Phys. 60 (1978) 1900

    1978

  18. [26]

    Terazawa and K

    H. Terazawa and K. Akama, Subquark Pregeometry With Spontaneously Broken Conformal Invariance, Phys. Lett. B 97 (1980) 81. 16

    1980

  19. [27]

    Akama, PREGEOMETRY INCLUDING FUNDAMENTAL GAUGE BOSONS , Phys

    K. Akama, PREGEOMETRY INCLUDING FUNDAMENTAL GAUGE BOSONS , Phys. Rev. D 24 (1981) 3073

    1981

  20. [28]

    Akama, Quantum Mechanical Equivalence of Two Lagrangian Formalisms in ’Scalar Pregeometry’, Prog

    K. Akama, Quantum Mechanical Equivalence of Two Lagrangian Formalisms in ’Scalar Pregeometry’, Prog. Theor. Phys. 61 (1979) 687

    1979

  21. [29]

    Akama and H

    K. Akama and H. Terazawa, Pregeometric Origin of the Big Bang , Gen. Rel. Grav. 15 (1983) 201

    1983

  22. [30]

    Akama and I

    K. Akama and I. Oda, Topological pregauge pregeometry, Phys. Lett. B 259 (1991) 431

    1991

  23. [31]

    Akama and I

    K. Akama and I. Oda, Chern-Simons pregeometry, Prog. Theor. Phys. 89 (1993) 215

    1993

  24. [32]

    Akama and I

    K. Akama and I. Oda, BRST quantization of pregeometry and topological pregeometry , Nucl. Phys. B 397 (1993) 727

    1993

  25. [33]

    Akama and I

    K. Akama and I. Oda, BRST quantization of pregeometry and topological pregeometry , Adv. Appl. Clifford Algebras 4 (1994) 259

    1994

  26. [34]

    Wetterich, Pregeometry and euclidean quantum gravity , Nucl

    C. Wetterich, Pregeometry and euclidean quantum gravity , Nucl. Phys. B 971 (2021) 115526 [2101.07849]

    2021 arXiv

  27. [35]

    Wetterich, Cosmology from pregeometry, Phys

    C. Wetterich, Cosmology from pregeometry, Phys. Rev. D 104 (2021) 104040 [ 2104.14013]

    2021 arXiv

  28. [36]

    Wetterich, Pregeometry and spontaneous time-space asymmetry, JHEP 06 (2022) 069 [2101.11519]

    C. Wetterich, Pregeometry and spontaneous time-space asymmetry, JHEP 06 (2022) 069 [2101.11519]

    2022 arXiv

  29. [37]

    Floreanini and R

    R. Floreanini and R. Percacci, Topological pregeometry, Mod. Phys. Lett. A 5 (1990) 2247

    1990

  30. [38]

    G. E. Volovik, Gravity from Symmetry Breaking Phase Transition , J. Low Temp. Phys. 207 (2022) 127 [ 2111.07817]

    2022 arXiv

  31. [39]

    H. C. Ohanian, Gravitons as goldstone bosons , Phys. Rev. 184 (1969) 1305

    1969

  32. [40]

    Nakanishi and I

    N. Nakanishi and I. Ojima, Proof of the Exact Masslessness of Gravitons , Phys. Rev. Lett. 43 (1979) 91

    1979

  33. [41]

    Percacci, The Higgs phenomenon in quantum gravity , Nucl

    R. Percacci, The Higgs phenomenon in quantum gravity , Nucl. Phys. B 353 (1991) 271 [0712.3545]

    1991 arXiv

  34. [42]

    Floreanini, R

    R. Floreanini, R. Percacci and E. Spallucci, Coleman-Weinberg effect in quantum gravity , Class. Quant. Grav. 8 (1991) L193

    1991

  35. [43]

    Floreanini and R

    R. Floreanini and R. Percacci, A Multiplicative background field method ,

  36. [44]

    Floreanini and R

    R. Floreanini and R. Percacci, Quantum mechanical breaking of local GL(4) invariance , Phys. Lett. B 379 (1996) 87 [ hep-th/9508157]

    1996 arXiv

  37. [45]

    Percacci, Gravity from a Particle Physicists’ perspective , PoS ISFTG (2009) 011 17 [0910.5167]

    R. Percacci, Gravity from a Particle Physicists’ perspective , PoS ISFTG (2009) 011 17 [0910.5167]

    2009 arXiv

  38. [46]

    E. T. Tomboulis, General Relativity as the effective theory of GL(4,R) spontaneous symmetry breaking, Phys. Rev. D 84 (2011) 084018 [ 1105.5848]

    2011 arXiv

  39. [47]

    J. B. Hartle and S. W. Hawking, Wave Function of the Universe , Phys. Rev. D 28 (1983) 2960

    1983

  40. [48]

    J. J. Halliwell and J. Louko, Steepest Descent Contours in the Path Integral Approach to Quantum Cosmology. 1. The De Sitter Minisuperspace Model , Phys. Rev. D 39 (1989) 2206

    1989

  41. [49]

    Vilenkin, Approaches to quantum cosmology , Phys

    A. Vilenkin, Approaches to quantum cosmology , Phys. Rev. D 50 (1994) 2581 [gr-qc/9403010]

    1994 arXiv

  42. [50]

    S. R. Coleman and E. J. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D 7 (1973) 1888

    1973

  43. [51]

    Floreanini and R

    R. Floreanini and R. Percacci, Average effective potential for the conformal factor , Nucl. Phys. B 436 (1995) 141 [ hep-th/9305172]

    1995 arXiv

  44. [52]

    Oda, The effective potential for conformal factor and GL(4) symmetry , Commun

    I. Oda, The effective potential for conformal factor and GL(4) symmetry , Commun. Theor. Phys. 77 (2025) 015202 [ 2401.04712]

    2025 arXiv

  45. [53]

    S. B. Giddings, Spontaneous breakdown of diffeomorphism invariance , Phys. Lett. B 268 (1991) 17

    1991

  46. [54]

    Wetterich and M

    C. Wetterich and M. Yamada, Variable Planck mass from the gauge invariant flow equation , Phys. Rev. D 100 (2019) 066017 [ 1906.01721]

    2019 arXiv

  47. [55]

    D. V. Vassilevich, Heat kernel expansion: User’s manual , Phys. Rept. 388 (2003) 279 [hep-th/0306138]

    2003 arXiv

  48. [56]

    Ratra and P

    B. Ratra and P. J. E. Peebles, Cosmological Consequences of a Rolling Homogeneous Scalar Field, Phys. Rev. D 37 (1988) 3406

    1988

  49. [57]

    Wetterich, Cosmology and the Fate of Dilatation Symmetry , Nucl

    C. Wetterich, Cosmology and the Fate of Dilatation Symmetry , Nucl. Phys. B 302 (1988) 668 [1711.03844]. 18

    1988 arXiv

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Reviewed August 12, 2026 · model on record in the stance chip above.