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Diffeomorphism invariance of the effective gravitational action

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper shows that the Fradkin–Vilkovisky path-integral measure, with its $g^{00}$ factors, is the diffeomorphism-invariant one, while the Fujikawa measure is not.

desk verdict A careful but scheme-dependent calculation showing FV measure invariant and Fujikawa not; worth refereeing, but the conclusion hinges on a time-slicing convention that a covariant regulator could evade. read the letter →

arxiv 2506.05100 v1 pith:FUCD7CMD submitted 2025-06-05 hep-th gr-qchep-ph

Carlo Branchina , Vincenzo Branchina , Filippo Contino , Riccardo Gandolfo , Arcangelo Pernace This is my paper
classification hep-thgr-qchep-ph MSC 81T2083C45 PACS 04.60.-m04.62.+v
keywords pathintegralmeasurediffeomorphisminvarianceeffectiveactionquantumgravityFradkin-VilkoviskyFujikawatime-orderingparameterg^{00}factors
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 asks which path-integral measure makes the quantum effective gravitational action invariant under coordinate changes, or diffeomorphisms. It argues that the Fradkin–Vilkovisky measure, which contains factors of the inverse-metric component $g^{00}(x)$, is the diffeomorphism-invariant one, while the Fujikawa measure, built only from $(-g)^{1/4}$, is not. The reason is that a coordinate change also changes the lattice and the time-ordering parameter used to define the path integral, producing nontrivial factors $A$ and $B$; invariance holds when $A B C E = 1$, which the Fradkin–Vilkovisky measure realizes and the Fujikawa measure does not. This matters because it determines whether the $g^{00}$ factors are a defect or a necessary ingredient, and it undercuts recent claims that the Fujikawa measure is the invariant choice.

What carries the argument

The central object is the product over spacetime points $\prod_x M(g(x))\,d\phi(x)$ entering the path integral, together with the four factors $A$, $B$, $C$, $E$ that describe its transformation under a diffeomorphism. The nontrivial content is in $A$ and $B$: they encode the fact that writing the same path integral in a new coordinate frame changes the time-ordering parameter, from $\hat{x}^0$ to $x^0$, and the lattice of points used to define the product, and this is not a mere relabeling. $A$ is computed from the change in the phase-space measure after integrating out momenta, $B$ from the Jacobian relating $\prod_{\hat{x}} M(\hat{g}(\hat{x}))$ to $\prod_x M(\hat{g}(\hat{x}(x)))$, and $C$ from expressing $M(\hat{g}(\hat{x}(x)))$ in terms of $g(x)$. The identity $A B C E = 1$ is the criterion for invariance; for the Fradkin–Vilkovisky measure the $g^{00}$ factors in $B_{FV}$ and $C_{FV}$ cancel the nontrivial $A$, while for the Fujikawa measure no such cancellation occurs.

What would settle it

Repeat the transformation analysis with a manifestly covariant lattice discretization of the path-integral measure that treats all spacetime points symmetrically and does not select $x^0$ as the time-ordering parameter; if the factors $A$ and $B$ then both equal $1$ and the identity $A B C E = 1$ fails for the Fradkin–Vilkovisky measure, the paper's central claim would be falsified.

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Extended reading notes

Core claim

For a scalar field in a gravitational background, the paper tracks the effective action $\Gamma[g]$ through an infinitesimal diffeomorphism $x \to \hat{x} = x + \varepsilon(x)$. It isolates four transformation factors: $A$ and $B$ arise from switching the product over lattice sites and the time-ordering parameter from the new frame back to the old frame, $C$ comes from rewriting $M(\hat{g})$ in terms of $g$, and $E$ is the field Jacobian, which equals $1$ for a scalar. The paper computes these factors and obtains $A B_{FV} C_{FV} E = 1$ for the Fradkin–Vilkovisky measure $M_{FV}=(-g^{00})^{1/2}(-g)^{1/4}$, while for the Fujikawa measure $M_{Fuji}=\mu(-g)^{1/4}$ the product equals $\exp[\delta^{(4)}(0)\int d^4x\,(\partial_\mu\varepsilon^\mu \log[\mu/(-g^{00})^{1/2}] - (g^{0\mu}/g^{00})\partial_\mu\varepsilon^0)] \neq 1$. Hence the effective action is diffeomorphism invariant with the Fradkin–Vilkovisky measure and not with the Fujikawa measure. The paper further argues that the opposite conclusion in a recent work comes from treating the lattice and time-ordering switch as a trivial relabeling and from mishandling $\log(-\Box + m^2)$.

Load-bearing premise

The argument assumes that defining the path integral requires choosing a time-ordering parameter and a spacetime lattice, and that switching between lattices under a coordinate change is not a trivial relabeling; if a manifestly covariant discretization made $A$ and $B$ trivial, the Fradkin–Vilkovisky measure would no longer be singled out as the invariant one.

Editorial extensions

If this is right

  • If the paper is right, the $g^{00}$ factors in the Fradkin–Vilkovisky measure are not a flaw but the mechanism that cancels non-invariant terms coming from time ordering and from the correct evaluation of $\log(-\Box+m^2)$.
  • The effective action computed with the Fujikawa measure acquires an extra non-covariant term proportional to $\delta^{(4)}(0)$, so it cannot be diffeomorphism invariant.
  • The recent claim that the Fujikawa measure is invariant is attributed to two omissions: missing the $A$ and $B$ lattice-switch factors and mishandling the distributional evaluation of $\log(-\Box+m^2)$.
  • The phase-space measure that yields the Fujikawa configuration-space measure, $\prod_x (g^{00})^{-1/2}\,d\pi\,d\phi$, is not the natural Liouville measure and should be rejected.
  • Renormalization-group equations for gravity built on the Fujikawa measure, including terms such as $\Lambda^4\sqrt{g}$ and $\Lambda^2\sqrt{g}R$, are not reliable consequences of the measure choice.

Reading between the lines

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

  • If one adopted a manifestly covariant discretization of the path integral with no preferred time-ordering parameter, the factors $A$ and $B$ would likely both become $1$; then the Fradkin–Vilkovisky measure would no longer be singled out as invariant, so the conclusion is tied to the canonical lattice and time-ordering scheme.
  • The same $A B C E$ criterion should apply to graviton fluctuations, and computing the spin-2 analogues of $A$, $B$, $C$ would test whether the pure-gravity conclusion inherits the scalar result or needs its own cancellation.
  • Both the non-invariant exponent and the cancellation involve $\delta^{(4)}(0)$, suggesting the result may be sensitive to how the divergent product over spacetime points is regulated; checking regulator independence would sharpen the claim.
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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

2 major / 4 minor

Summary. The paper investigates whether the effective gravitational action is invariant under diffeomorphisms when the path integral measure is the Fradkin-Vilkovisky (FV) measure or the Fujikawa measure. The authors define the path integral on a spacetime lattice with a chosen time-ordering parameter, and then track the factors A, B, C, and E that appear when the integration variables are changed from one coordinate frame to another. Their central calculation shows that for the FV measure the product ABCE equals 1 (Eq. 4.4), so the effective action is diffeomorphism invariant, while for the Fujikawa measure ABCE is not 1 (Eq. 4.7), so the effective action is not diffeomorphism invariant. The paper also criticises a recent claim by Bonanno, Falls, and Ferrero that the Fujikawa measure is the invariant one, arguing that the latter work misses the non-trivial lattice/time-ordering factors.

Significance. If the result holds, it would resolve a long-standing controversy in favour of the FV measure and would directly invalidate the diffeomorphism-invariance claims for the Fujikawa measure made in Ref. [17]. The paper's strength is its explicit, step-by-step calculation of the transformation factors A, B, C, and E, including the exact cancellation for the FV measure and the explicit non-cancellation for the Fujikawa measure. The identification of the lattice and time-ordering subtlety is a useful clarification that is absent from many formal treatments. However, the universality of the claim is limited by the specific non-covariant lattice prescription used, and the paper does not fully address whether a manifestly covariant regularization would alter the conclusions.

major comments (2)
  1. [Section 2, Eq. (2.9); Section 3, Eqs. (3.12), (3.18), (3.20); Section 4, Eqs. (4.4), (4.7)] The factors A and B are computed under the assumption that the path integral must be defined on the non-covariant lattices E1/E2 with time-ordering parameters x0 and \hat x0, and that the transition E2→E1 is not a trivial relabeling; this assumption is imported from Refs. [13,19] and is not derived in the paper. Because a manifestly covariant discretization could make A=B=1 and alter the cancellation in Eq. (4.4) and the non-cancellation in Eq. (4.7), the central claim that the FV measure is diffeomorphism invariant while the Fujikawa measure is not is scheme-dependent as stated. Please either justify the uniqueness of this lattice prescription or explicitly qualify the claim as holding within the canonical time-slicing framework.
  2. [Section 5, Eqs. (5.3)-(5.4)] The assertion that Tr log(-□+m²) contains a non-covariant term δ^(4)(0)∫d⁴x log(g⁰⁰), attributed to Refs. [13,20], is not derived in this paper. Since standard spectral determinants of the covariant operator -□+m² are diffeomorphism invariant, this is a regulator-dependent claim that underlies the paper's critique of [17] and the statement that the FV measure is necessary to compensate non-invariant terms. The authors should either derive this term within the framework of Sections 2–4 or present it as a consequence of the same time-slicing scheme, rather than as an established fact.
minor comments (4)
  1. [Eq. (3.22)] The metric transformation in Eq. (3.22) has the Jacobian inverted and the wrong sign: for a coordinate change \hat x^μ = x^μ + ε^μ, the covariant metric at first order transforms as \hat g_{μν} = g_{μν} - g_{ρν}∂_μ ε^ρ - g_{μσ}∂_ν ε^σ, not as written. Although the final expression for C_FV in Eq. (4.2) is consistent with the correct transformation of g⁰⁰ and the density weight of (-g)^{1/4}, the intermediate formula should be corrected.
  2. [Section 3, Eqs. (3.14)-(3.17)] The factors δ^(4)(0) are treated as finite constants without specifying the regularization; the paper would benefit from a sentence stating that in a lattice regularization δ^(4)(0) equals the inverse cell volume and is finite.
  3. [Section 1] The introduction refers to the 'euclidean effective action', but the metric signature used throughout is (-,+,+,+); please clarify the intended signature or usage.
  4. [Sections 2-4] A summary table of the definitions of A, B, C, E and their values for the FV and Fujikawa measures would improve readability and help the reader follow the cancellation in Eq. (4.4) and the non-cancellation in Eq. (4.7).

Circularity Check

0 steps flagged · score 2.0 of 10

Central invariance computation is explicit and non-circular; only peripheral self-citations to companion papers.

full rationale

The central claim is obtained by an explicit factor-by-factor computation, not by assuming the target result. The paper defines the lattice/time-ordering framework, derives the transformation factors A (Eq. 3.18), B (Eq. 3.20), C (Eq. 3.24) and E=1, and then evaluates the products A B C E separately for the FV measure (Eq. 4.4) and the Fujikawa measure (Eq. 4.7). No parameter is fitted and no prediction is renamed from an input; the invariance condition A B C E = 1 is a computed outcome. The external inputs (the canonical lattice framework from [13,19] and the δ(0) log g00 term from [20]) are prior independent works, not self-citations. The authors' own companion papers [2,3,25] are used only in peripheral remarks about the one-loop formula and the absence of the asymptotic-safety fixed point; they do not carry the measure-invariance proof. A possible reliance of the A and B factors on the chosen non-covariant lattice is a scheme-dependence/correctness concern rather than circularity. The score of 2 reflects the presence of minor non-load-bearing self-citations, not a circular derivation.

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

No fitted numerical parameters. The arbitrary mass scale mu in the Fujikawa measure is an input constant, not fitted. The central calculation rests on four domain assumptions about the canonical path integral and its regularization; no new entities are introduced.

assumptions (4)
  • domain assumption The path integral requires a time-ordering parameter and a lattice; the measure Q_x is defined only after this choice.
    Section 2, discussion of lattices E_1 and E_2; the basis for the non-trivial factors A and B.
  • domain assumption The phase-space measure is the Liouville measure Q_x d pi(x) d phi(x); the configuration-space measure follows by integrating over momenta.
    Section 3, Eq. (3.1); this selects the FV measure over the Fujikawa one.
  • domain assumption The product over points is regularized via exp[delta^(4)(0) integral d^4x log(...)] and total-derivative terms are dropped.
    Eqs. (3.14), (3.17), (3.19), (3.23); the formal regularization underlying the result.
  • domain assumption The Green's function of -Box + m^2 has a distributional singularity that produces delta^(4)(0) integral d^4x log(g^00) in the Tr log (taken from [20]).
    Section 5, Eq. (5.4); the cancellation of this term is load-bearing for the FV invariance claim at one loop.

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Cite this review

Pith. "Pith review of Diffeomorphism invariance of the effective gravitational action." pith.science (2026). https://pith.science/paper/FUCD7CMD

@misc{pith2026250605100,
  author       = {Pith},
  title        = {Pith review of: Diffeomorphism invariance of the effective gravitational action},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUCD7CMD}},
  note         = {Machine review of arXiv:2506.05100}
}
abstract

We investigate on the diffeomorphism invariance of the effective gravitational action, focusing in particular on the path integral measure. In the literature, two different measures are mainly considered, the Fradkin-Vilkovisky and the Fujikawa one. With the help of detailed calculations, we show that, despite claims to the contrary, the Fradkin-Vilkovisky measure is diffeomorphism invariant, while the Fujikawa measure is not. In particular, we see that, contrary to naive expectations, the presence of $g^{00}$ factors in the Fradkin-Vilkovisky measure is necessary to ensure the invariance of the effective gravitational action. We also comment on results recently appeared in the literature, and show that formal calculations can easily miss delicate points.

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Forward citations

Cited by 2 Pith papers

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    Spectral hard and smooth cutoffs on the covariant Laplacian yield Einstein-Hilbert RG flows with a non-Gaussian UV-attractive fixed point, supporting asymptotic safety.

  2. Gravity and the Higgs boson mass

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Reference graph

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