Pith. sign in

REVIEW 3 major objections 6 minor 39 references

Relational dynamics in Brans–Dicke theory are set by the full canonical clock pair, not the clock variable alone, restoring Jordan–Einstein equivalence after reduction.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · grok-4.5

2026-07-30 10:26 UTC pith:IZR4J65C

load-bearing objection Solid canonical clarification: naive relational reduction fails to commute with the Jordan–Einstein map because of the p_ϕ shift by p_h/ϕ; adapting the full clock pair restores equivalence. Incremental but clean and useful. the 3 major comments →

arxiv 2607.23852 v1 pith:IZR4J65C submitted 2026-07-26 gr-qc

Canonical Clock Sectors and Relational Frame Equivalence in Brans--Dicke Theory

Yaser Tavakoli , Jerzy Lewandowski This is my paper
classification gr-qc
keywords Brans–Dicke theoryJordan frameEinstein framerelational deparametrizationcanonical clock sectorreduced Hamiltonianscalar-tensor gravityproblem of time
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Brans–Dicke gravity can be written in two conformal frames that look different but describe the same classical physics. At the level of the action and the ordinary ADM phase space they are equivalent, yet once you pick the scalar field as an internal clock and reduce the Hamiltonian constraint, the resulting physical Hamiltonians appear to disagree. The paper shows that this disagreement is not a physical inequivalence. It comes from treating only the clock configuration variable as the same object while ignoring that its conjugate momentum mixes with the gravitational momentum under the conformal map. By first rewriting the Jordan-frame variables so that the entire clock pair matches the Einstein-frame one, then reducing, the two physical Hamiltonians coincide. The claim is that relational evolution is governed by the canonical clock sector (T, P_T), and that aligning that sector before deparametrization restores frame equivalence for reduced dynamics and for reduced phase-space quantization.

Core claim

The apparent mismatch of reduced Hamiltonians between Jordan and Einstein frames after using the Brans–Dicke scalar as clock is an artifact of mismatched canonical clock sectors. Once the clock is represented by the adapted pair (Φ, P_Φ) = (ϕ, p_ϕ − p_h/ϕ) before reduction, the reduced Hamiltonians become identical, so relational frame equivalence is restored.

What carries the argument

The frame-adapted canonical chart inside the Jordan phase space: Q_ab = ϕ h_ab, P^{ab} = p^{ab}/ϕ, Φ = ϕ, P_Φ = p_ϕ − p_h/ϕ. This chart aligns the full clock sector with its Einstein-frame counterpart before deparametrization, so that H*_phys = H̃_phys (Theorem 2).

Load-bearing premise

The Brans–Dicke scalar must serve as a good global clock: it must evolve monotonically so the Hamiltonian constraint can be solved for its momentum on a definite branch of phase space.

What would settle it

Construct an explicit homogeneous or inhomogeneous Brans–Dicke solution in which the scalar is non-monotonic or the Poisson bracket {ϕ, H_N} vanishes on an open set, then check whether the adapted and Einstein-frame reduced Hamiltonians still coincide after any local deparametrization.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Canonical equivalence of unreduced constrained systems does not automatically survive relational deparametrization; the clock sector must be transformed as a pair.
  • Reduced phase-space quantization of Brans–Dicke theory yields the same physical evolution in both frames only when the adapted clock momentum is used.
  • Frame dependence in scalar–tensor gravity after reduction is diagnosed by whether (T, P_T) is aligned, not by whether T itself is invariant.
  • The same clock-sector criterion supplies a template for checking relational equivalence under other canonical transformations in generally covariant theories.

Where Pith is reading between the lines

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

  • The same non-commutativity R ∘ C ≠ C ∘ R should appear in any scalar–tensor or f(R) model whose conformal map mixes scalar and metric momenta.
  • Operational definitions of quantum clocks that track only the configuration variable T, without its conjugate momentum, would systematically misidentify frame inequivalence.
  • Extending the construction to matter clocks or to inhomogeneous modes would test whether the adapted-sector principle remains sufficient outside pure Brans–Dicke gravity.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript studies Jordan–Einstein frame equivalence in Brans–Dicke theory at four levels: covariant action, standard ADM phase space, extended Dirac phase space, and relationally deparametrized phase space. The authors show (i) the conformal map is canonical on the standard ADM phase space (Θ_E = Θ_J, Eq. (4.3)) but not on the extended phase space (Eq. (4.9)), reconciling the Garay/Deruelle and Gionti et al. results; (ii) naive deparametrization with the scalar clock ϕ yields reduced Hamiltonians differing by p_h/ϕ (Theorem 1, Eq. (5.10)), so reduction and conformal transformation do not commute; (iii) transforming the full clock pair to (Φ, P_Φ) = (ϕ, p_ϕ − p_h/ϕ) before reduction restores equality of the reduced Hamiltonians (Theorem 2, Eq. (6.15)). The conceptual claim is that relational dynamics are governed by the canonical clock sector (T, P_T), not by T alone.

Significance. If the analysis holds, the paper makes a useful and clean contribution to a genuinely confused corner of the literature: it localizes the disagreement between prior Hamiltonian frame-equivalence results to the choice of phase space, and it identifies the precise mechanism (momentum mixing p̃_ϕ = p_ϕ − p_h/ϕ) by which naive relational reduction appears to break frame equivalence. The derivation is parameter-free: everything follows from the standard Brans–Dicke ADM constraints (3.8)–(3.9) and the known conformal map, with no fitting, no new entities beyond a change of canonical chart, and no ad hoc assumptions. The two theorems are stated with explicit hypotheses and proved step by step, and the load-bearing algebra (symplectic cancellation (4.3), the constraint solution (5.4)–(5.5), and the identity Ω̃ = Ω/√ϕ in (5.8)) is elementary and independently checkable — I verified it, including the full-gradient version of (5.8), where the coefficient identity 3/2 − ω̄ = −ω does the work. The clock-sector concept is a transferable lesson for reduced phase-space quantization in scalar–tensor gravity generally. The significance is moderated by the fact that Theorem 2 is close to definitional

major comments (3)
  1. [§VI, Theorem 2 (Eqs. (6.8)–(6.15))] Theorem 2 is correct but very close to definitional: the adapted chart (6.8) is literally the Einstein-frame chart (Q_ab = h̃_ab, P^ab = p̃^ab, P_Φ = p̃_ϕ), so H*_phys = H̃_phys follows by construction, as the authors themselves note ('By construction, P_Φ = p̃_ϕ'). The substantive results of the paper are Theorem 1 and the clock-sector diagnosis; Theorem 2 repackages them. This matters because the abstract and §VII frame the result as 'restoring the equivalence of the reduced relational dynamics,' which a reader could misread as a nontrivial recovery of an equivalence that had been lost. Classically, frame equivalence was never in doubt; what the paper actually shows is that the naive reduced Hamiltonians describe evolution with respect to two different canonical embeddings of the clock, and it identifies the invariant clock sector. I ask the authors to state this plainly in the abstrac
  2. [§V, Eqs. (5.1)–(5.6) and (5.9)] The scope of the reduction needs sharper statement. The gauge χ(x) = ϕ(x) − t ≈ 0 (5.1) forces D_aϕ = 0 on the gauge surface, and the reduced Ω expressions actually used ((5.6), (5.9)) are the homogeneous truncations. The general formulas (5.5) and (5.8) retain gradient terms, so a reader is left unsure whether the theorems are proved for the full field theory or only for the homogeneous-scalar sector. Since (5.8) in fact holds with the full gradient terms (the coefficients match via 3/2 − ω̄ = −ω), the identity itself is general; but the physical Hamiltonian as a functional on the reduced phase space is only obtained in the homogeneous-clock truncation. Please state explicitly which statements are sectorial and which are general, and comment on what the reduction describes when ϕ is inhomogeneous (a field-valued clock gives a local, 'multi-fingered' time; a citation to the bubble-time/m
  3. [§V, Eqs. (5.1)–(5.3)] The admissibility conditions for the scalar clock — non-degeneracy {ϕ, H_N} ≠ 0 and monotonicity on a definite-sign branch of the clock momentum — are stated, but their bearing on the main theorems should be made explicit. Theorem 1's conclusion H̃_phys = H_phys + p_h/ϕ and Theorem 2's restoration both hold only on phase-space sectors where the constraint can be solved globally for the clock momentum and the sign σ in (5.4)/(5.7) is fixed. Where the clock fails (e.g., recollapsing scalar, or p_ϕ − p_h/ϕ crossing zero), the reduced Hamiltonians and the restored equivalence are only local. A short remark in §VII acknowledging that the equivalence result is sectorial, and ideally a characterization of the sectors (e.g., in terms of the sign of P_Φ), would make the claims precise without weakening them.
minor comments (6)
  1. [§III.B, Eqs. (3.15)–(3.16)] There is an index-position inconsistency: (3.15) states Ñ^a = ϕ N^a while (3.16) states Ñ^a = N^a. Presumably one of these is the lower-index shift covector (Ñ_a = ϕ N_a). Please fix the index placement so the two equations are consistent.
  2. [§VI.A, after Eq. (6.6)] 'we obtain a uniquely unique result' — redundant phrasing; delete one 'unique'.
  3. [Throughout] Spelling alternates between 'deparametrization' and 'deparameterization' (e.g., abstract vs. §V heading and §VII); please standardize. Also 'Section. VII' and 'Section. IIIB' have a stray period after 'Section'.
  4. [§V, Eq. (5.4)] The sign convention p_ϕ = −H_phys with the σ = ±1 branch should be commented on: the orientation of relational time (choice of σ) is fixed frame-independently only because P_Φ = p̃_ϕ; a one-sentence remark would help readers see that the branch choice does not reintroduce frame dependence.
  5. [§VII] The five-line summary table of equivalence levels is helpful; consider adding the hypothesis of each line (e.g., 'standard ADM phase space, ω̄ ≠ 0' for the canonical-equivalence line, 'monotonic scalar sector' for the last line) so the table is self-contained.
  6. [References] The multiple-choice problem of time is cited via Kuchař, Isham, and Anderson; given the role of clock momentum here, a reference to work on relational clocks in quantum cosmology where clock momentum branching is explicit (e.g., Hoehn–Smith–Lock or Gielen–Menéndez-Pidal on clock dependence in quantum cosmology) would strengthen the outlook paragraph, though this is at the authors' discretion.

Circularity Check

1 steps flagged

Mild definitional character in Theorem 2: the adapted chart is the Einstein-frame chart by construction, so restored equivalence is tautological once defined; Theorem 1 and the diagnostic content are independent and non-circular.

specific steps
  1. self definitional [Sec. VI.A eqs. (6.2)/(6.8); Theorem 2 proof, eqs. (6.13)–(6.15)]
    "The clock variables are therefore defined as Φ := ϕ, P_Φ := p_ϕ − p_h/ϕ. ... Under the canonical embedding between the Jordan and Einstein descriptions, one has P_Φ ≡ p̃_ϕ. Consequently, solving the Hamiltonian constraint for the adapted clock momentum gives P_Φ = −H*_phys, while the Einstein-frame reduction gives p̃_ϕ = −H̃_phys. Since the clock momenta are identical canonical variables, P_Φ = p̃_ϕ, the two reduced Hamiltonians necessarily coincide, H*_phys = H̃_phys."

    P_Φ is defined to be exactly the Einstein-frame scalar momentum p̃_ϕ (cf. eq. 3.25). Therefore H*_phys = H̃_phys is true by the definition of the adapted chart, not by an independent dynamical derivation. The paper presents this coincidence as 'restoring' relational frame equivalence (Theorem 2 / abstract), but once the chart is chosen to match the Einstein-frame clock sector, equality of the reduced Hamiltonians is automatic. The step is transparent and does not contaminate Theorem 1.

full rationale

The paper's load-bearing diagnostic result (Theorem 1) is a parameter-free algebraic computation from the standard Brans–Dicke ADM constraints and the known conformal map: naive deparametrization with the same configuration clock ϕ yields H̃_phys = H_phys + p_h/ϕ because p̃_ϕ = p_ϕ − p_h/ϕ. That identity is independently checkable and is not imposed by fit, ansatz, or self-citation. Theorem 2 then 'restores' equivalence by introducing the adapted pair (Φ, P_Φ) := (ϕ, p_ϕ − p_h/ϕ), which is defined to equal the Einstein-frame clock momentum, so H*_phys = H̃_phys holds by construction of the chart. The paper is transparent about this ('By construction, P_Φ ≡ p̃_ϕ') and does not hide a fitted prediction or a self-citation chain. Citations (Garay, Deruelle, Gionti, Rovelli/Isham on the problem of time) are standard background, not uniqueness theorems that force the reduced-Hamiltonian identity. No data fitting, no smuggled ansatz, no renaming of an empirical pattern. The only circularity is the mild, openly definitional character of the restoration step; the central conceptual claim—that relational dynamics track the full canonical clock sector (T, P_T)—is a framing of that definitional alignment, not a circular derivation of new dynamics. Score 2 reflects one minor self-definitional presentation step that is not load-bearing for the diagnostic content.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 1 invented entities

Load-bearing structure is standard constrained Hamiltonian GR plus the Brans–Dicke action and the usual Weyl map. No fitted parameters. Main extra assumptions are domain choices: scalar as monotonic internal clock, work on standard (not extended) ADM phase space for the reduction theorems, homogeneous scalar for simplified Ω, and f(ϕ)>0 so the conformal map is invertible. The ‘frame-adapted canonical chart’ is a constructed change of coordinates, not a new physical entity.

axioms (6)
  • domain assumption Brans–Dicke action in Jordan frame with constant ω, f(ϕ)=ϕ, and invertible conformal map Ω²=ϕ (f>0) to Einstein frame.
    Sec. II; defines the theory and ensures one-to-one frame map.
  • domain assumption Standard ADM phase space with lapse/shift as multipliers; hypersurface-deformation algebra closes for ω̄≠0.
    Secs. III–IV; reduction theorems are stated on Γ_J not the extended Dirac space.
  • domain assumption Scalar gauge χ=ϕ−t≈0 is admissible: {ϕ,H_N}≠0 and monotonic definite-sign sectors exist so H_N can be solved for p_ϕ (or P_Φ).
    Sec. V eqs. 5.1–5.3; without this, reduced H_phys is not defined as claimed.
  • domain assumption Homogeneous clock (D_a ϕ=0) when simplifying Ω and Ω̃ for the explicit reduced Hamiltonians.
    Sec. V after (5.5); used to drop spatial-gradient terms in the physical Hamiltonians.
  • standard math Canonical transformations preserve physical equivalence of unreduced constrained systems; relational equivalence requires matching full clock pairs (T,P_T).
    Standard symplectic geometry / deparametrization; used throughout Secs. IV–VI.
  • domain assumption Boundary GHY terms and matter do not affect the bulk canonical clock-sector analysis under compact slices or standard fall-off.
    Sec. II.A; declared so bulk Poisson structure suffices.
invented entities (1)
  • Frame-adapted canonical chart Γ* = {Q_ab, P^{ab}; Φ, P_Φ} with P_Φ := p_ϕ − p_h/ϕ and Q_ab := ϕ h_ab no independent evidence
    purpose: Align Jordan-frame clock momentum with Einstein-frame p̃_ϕ before deparametrization so reduced Hamiltonians coincide.
    Sec. VI; a coordinate chart on the existing Jordan ADM phase space, not a new degree of freedom. independent_evidence false only in the sense it is a definition; it is falsifiable by checking whether H*_phys equals H̃_phys.

pith-pipeline@v1.2.0-grok45-kimik3 · 20673 in / 3245 out tokens · 57496 ms · 2026-07-30T10:26:22.230614+00:00 · methodology

0 comments
read the original abstract

We investigate the equivalence of the Jordan and Einstein frames of the Brans--Dicke theory before and after relational deparametrization of the constrained Hamiltonian system. Although the two conformal formulations are equivalent at the covariant level and related by a canonical transformation on the standard ADM phase space, their equivalence after reduction with respect to an internal clock is nontrivial. Using the Brans--Dicke scalar as a relational clock, we show that the apparent discrepancy between the reduced Hamiltonians does not indicate a physical inequivalence of the two frames. Instead, it arises from a mismatch in the canonical embedding of the clock sector prior to reduction. While the scalar configuration variable is preserved under the conformal transformation, its conjugate momentum is shifted by a contribution involving the gravitational momentum trace. Consequently, relational dynamics are determined not by the clock variable $T$ alone, but by the complete canonical clock pair $(T,P_T)$. We construct a frame-adapted canonical chart in which the clock sector is consistently transformed prior to deparameterization. The resulting reduced Hamiltonian coincides with that of the Einstein frame, restoring the equivalence of the reduced relational dynamics. Our results identify the canonical clock sector as the fundamental structure governing relational evolution in Brans--Dicke theory and provide a general framework for understanding frame dependence in scalar--tensor gravity and reduced phase-space quantization.

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

39 extracted references · 26 linked inside Pith

  1. [1]

    Brans and R

    C. Brans and R. H. Dicke, Mach’s principle and a relativistic theory of gravitation, Phys. Rev. 124, 925 (1961)

    1961

  2. [2]

    R. H. Dicke, Mach’s principle and invariance under transformation of units, Phys. Rev.125, 2163 (1962)

    1962

  3. [3]

    Fujii and K

    Y. Fujii and K. Maeda,The scalar-tensor theory of gravitation, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2007)

    2007

  4. [4]

    Capozziello, P

    S. Capozziello, P. Martin-Moruno, and C. Rubano, Physical non-equivalence of the Jordan and Einstein frames, Phys. Lett. B689, 117 (2010), arXiv:1003.5394 [gr-qc]

    Pith/arXiv arXiv 2010

  5. [5]

    R. V. Wagoner, Scalar tensor theory and gravitational waves, Phys. Rev. D1, 3209 (1970)

    1970

  6. [6]

    Faraoni and S

    V. Faraoni and S. Nadeau, The (pseudo)issue of the conformal frame revisited, Phys. Rev. D 75, 023501 (2007), arXiv:gr-qc/0612075

    Pith/arXiv arXiv 2007

  7. [7]

    E. E. Flanagan, The Conformal frame freedom in theories of gravitation, Class. Quant. Grav. 21, 3817 (2004), arXiv:gr-qc/0403063

    Pith/arXiv arXiv 2004

  8. [8]

    Faraoni and E

    V. Faraoni and E. Gunzig, Einstein frame or Jordan frame?, Int. J. Theor. Phys.38, 217 (1999), arXiv:astro-ph/9910176

    Pith/arXiv arXiv 1999

  9. [9]

    Deruelle and M

    N. Deruelle and M. Sasaki, Conformal equivalence in classical gravity: the example of ’veiled’ General Relativity, Springer Proc. Phys.137, 247 (2011), 1007.3563

    Pith/arXiv arXiv 2011

  10. [10]

    Doménech and M

    G. Doménech and M. Sasaki, Conformal frame dependence of inflation, JCAP04, 022, 1501.07699

    Pith/arXiv arXiv

  11. [11]

    A. Y. Kamenshchik and C. F. Steinwachs, Question of quantum equivalence between Jordan frame and Einstein frame, Phys. Rev. D91, 084033 (2015), arXiv:1408.5769 [gr-qc]

    Pith/arXiv arXiv 2015

  12. [12]

    Falls and M

    K. Falls and M. Herrero-Valea, Frame (In)equivalence in Quantum Field Theory and Cosmol- ogy, Eur. Phys. J. C79, 595 (2019), arXiv:1812.08187 [hep-th]. 22

    Pith/arXiv arXiv 2019

  13. [13]

    Pandey and N

    S. Pandey and N. Banerjee, Equivalence of Jordan and Einstein frames at the quantum level, Eur. Phys. J. Plus132, 107 (2017), arXiv:1610.00584 [gr-qc]

    Pith/arXiv arXiv 2017

  14. [14]

    Sharma, G

    M. Sharma, G. S. Vicente, L. L. Graef, R. O. Ramos, and A. Wang, Quantum geometric formulation of Brans-Dicke theory for Bianchi I spacetime, Phys. Rev. D111, 043501 (2025), arXiv:2309.01080 [gr-qc]

    Pith/arXiv arXiv 2025

  15. [15]

    Artymowski, Y

    M. Artymowski, Y. Ma, and X. Zhang, Comparison between Jordan and Einstein frames of Brans-Dicke gravity a la loop quantum cosmology, Phys. Rev. D88, 104010 (2013), arXiv:1309.3045 [gr-qc]

    Pith/arXiv arXiv 2013

  16. [16]

    Artymowski and J

    M. Artymowski and J. Mielczarek, Quantum Hubble horizon, Eur. Phys. J. C79, 632 (2019), arXiv:1806.03924 [gr-qc]

    Pith/arXiv arXiv 2019

  17. [17]

    P. A. M. Dirac,Lectures on Quantum Mechanics(Yeshiva University, 1964)

    1964

  18. [18]

    Henneaux and C

    M. Henneaux and C. Teitelboim,Quantization of Gauge Systems(Princeton University Press, 1992)

    1992

  19. [19]

    Thiemann,Modern Canonical Quantum General Relativity(Cambridge University Press, 2007)

    T. Thiemann,Modern Canonical Quantum General Relativity(Cambridge University Press, 2007)

    2007

  20. [20]

    L. J. Garay and J. Garcia-Bellido, Jordan-Brans-Dicke quantum wormholes and Coleman’s mechanism, Nucl. Phys. B400, 416 (1993), arXiv:gr-qc/9209015

    Pith/arXiv arXiv 1993

  21. [21]

    Deruelle, Y

    N. Deruelle, Y. Sendouda, and A. Youssef, Various Hamiltonian formulations of f(R) gravity and their canonical relationships, Phys. Rev. D80, 084032 (2009), arXiv:0906.4983 [gr-qc]

    Pith/arXiv arXiv 2009

  22. [22]

    L. Järv, P. Kuusk, M. Saal, and O. Vilson, Invariant quantities in the scalar-tensor theories of gravitation, Phys. Rev. D91, 024041 (2015), arXiv:1411.1947 [gr-qc]

    Pith/arXiv arXiv 2015

  23. [23]

    S. J. Gabriele Gionti and S. J, Canonical analysis of Brans-Dicke theory addresses Hamiltonian inequivalence between the Jordan and Einstein frames, Phys. Rev. D103, 024022 (2021), arXiv:2003.04304 [gr-qc]

    Pith/arXiv arXiv 2021

  24. [24]

    Galaverni and G

    M. Galaverni and G. G. S. J., Jordan and Einstein frames from the perspective ofω=-3/2 Hamiltonian Brans-Dicke theory, Phys. Rev. D105, 084008 (2022), arXiv:2110.12222 [gr-qc]

    Pith/arXiv arXiv 2022

  25. [25]

    Rovelli, What Is Observable in Classical and Quantum Gravity?, Class

    C. Rovelli, What Is Observable in Classical and Quantum Gravity?, Class. Quant. Grav.8, 297 (1991)

    1991

  26. [26]

    Rovelli, Partial observables, Phys

    C. Rovelli, Partial observables, Phys. Rev. D65, 124013 (2002), arXiv:gr-qc/0110035

    Pith/arXiv arXiv 2002

  27. [27]

    Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen

    B. Dittrich, Partial and complete observables for Hamiltonian constrained systems, Gen. Rel. Grav.39, 1891 (2007), gr-qc/0411013

    Pith/arXiv arXiv 2007

  28. [28]

    K. V. Kuchař, Time and interpretations of quantum gravity, Int. J. Mod. Phys. D20, 3 (2011)

    2011

  29. [29]

    C. J. Isham, Canonical quantum gravity and the problem of time, NATO Sci. Ser. C409, 157 (1993), arXiv:gr-qc/9210011

    Pith/arXiv arXiv 1993

  30. [30]

    Anderson, Problem of Time in Quantum Gravity, Annalen Phys.524, 757 (2012), arXiv:1206.2403 [gr-qc]

    E. Anderson, Problem of Time in Quantum Gravity, Annalen Phys.524, 757 (2012), arXiv:1206.2403 [gr-qc]

    Pith/arXiv arXiv 2012

  31. [31]

    X.ZhangandY.Ma,NonperturbativeLoopQuantizationofScalar-TensorTheoriesofGravity, Phys. Rev. D84, 104045 (2011), arXiv:1107.5157 [gr-qc]

    Pith/arXiv arXiv 2011

  32. [32]

    Zhang and Y

    X.-D. Zhang and Y. Ma, Loop Quantum Brans-Dicke Theory, J. Phys. Conf. Ser.360, 012055 (2012), arXiv:1111.2215 [gr-qc]

    Pith/arXiv arXiv 2012

  33. [33]

    Dyer and K

    E. Dyer and K. Hinterbichler, Boundary Terms, Variational Principles and Higher Derivative Modified Gravity, Phys. Rev. D79, 024028 (2009), arXiv:0809.4033 [gr-qc]

    Pith/arXiv arXiv 2009

  34. [34]

    G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D15, 2752 (1977)

    1977

  35. [35]

    J. W. York, Jr., Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett.28, 1082 (1972). 23

    1972

  36. [36]

    York, Boundary terms in the action principles of general relativity, Found

    J. York, Boundary terms in the action principles of general relativity, Found. Phys.16, 249 (1986)

    1986

  37. [37]

    R. L. Arnowitt, S. Deser, and C. W. Misner, Canonical variables for general relativity, Phys. Rev.117, 1595 (1960)

    1960

  38. [38]

    R. L. Arnowitt, S. Deser, and C. W. Misner, The Dynamics of general relativity, Gen. Rel. Grav.40, 1997 (2008), arXiv:gr-qc/0405109

    Pith/arXiv arXiv 1997

  39. [39]

    G. J. Olmo and H. Sanchis-Alepuz, Hamiltonian Formulation of Palatini f(R) theories a la Brans-Dicke, Phys. Rev. D83, 104036 (2011), arXiv:1101.3403 [gr-qc]

    Pith/arXiv arXiv 2011