Review Reports - Windowed Quantum Field Theory: Domain-Restricted Actions, Standard Model Recovery, and the Vanishing of Delocalized Stress-Energy

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

attached

Comments for author File: Comments.pdf

Comments on the Quality of English Language

attached

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,

The manuscript proposes a windowed action prescription in which a smooth scalar $\Diamond(x)\in[0,1]$ multiplies the matter Lagrangian, producing modified Ward identities $\nabla_{\mu}(\Diamond J^{\mu})=0$ and a Bianchi-forced compensator $T^{\mathrm{comp}}_{\mu\nu}$ obeying

\begin{equation}
\nabla^{\mu}T^{\mathrm{comp}}_{\mu\nu}=-(\nabla^{\mu}\Diamond)\,T^{\mathrm{SM}}_{\mu\nu},
\end{equation}

together with the decomposition $T^{\mathrm{nl}}_{\mu\nu}=T^{\mathrm{comp}}_{\mu\nu}+T^{\mathrm{Rem}}_{\mu\nu}$. The construction is technically clean, and the parallels drawn with the Israel junction conditions and the Ashtekar--Krishnan flux laws are well chosen. The structural unification of windowed Ward identities with the gravitational compensator in Section 5 is the most original contribution, and the $\tanh$-regularisation of Eq.~(9) is well motivated. Several physical and technical points nevertheless require clarification before the paper is suitable for publication.

The first concern is the variational status of $\Diamond(x)$. The author asserts non-dynamical behaviour, yet Eq.~(14) carries a boundary contribution $\int_{\mathcal{M}}(\nabla_{\mu}\Diamond)\Pi^{\mu}\delta\Phi\sqrt{-g}\,d^{4}x$ that is not eliminated by any imposed boundary condition. How is this term reconciled with the variational principle without specifying $\delta\Phi|_{\mathcal{B}_{\epsilon}}$, and does the prescription require an analogue of the Gibbons--Hawking--York counterterm at the internal interface? Second, the rescaling argument $\chi\equiv\Diamond^{1/2}\psi$ leading to Eq.~(23) is ill-defined where $\Diamond\to 0$; the inverse $\Diamond^{-1/2}$ diverges across the boundary layer, so the cancellation that produces $(i\gamma^{\mu}\nabla_{\mu}-m)\chi=0$ is only formal. Could the author quantify the renormalisation of the spinor norm, the unitarity of the field redefinition, and its compatibility with the inner product on the Cauchy surface?

Third, the Vanishing Lemma uses a scaling estimate $\langle\nabla\Diamond\,T\rangle_{L}\sim L^{-1}T^{\Sigma}_{\mu\nu}$ but does not address phase-coherent contributions, oscillatory cancellations, or the role of stress-tensor fluctuations $\langle T_{\mu\nu}T_{\rho\sigma}\rangle$, which can dominate $\langle T_{\mu\nu}\rangle$ in vacuum states. The Lemma should be sharpened to include the variance, not merely the mean. Fourth, Eq.~(78), $\tfrac{1}{2}(\omega_{ab})_{\mu}\Sigma^{ab}\sim\partial_{\mu}$, is dimensionally heterogeneous: $\Sigma^{ab}$ is dimensionless while the spin connection has dimension of inverse length. A criterion phrased through curvature invariants $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$ versus $\partial\phi\cdot\partial\phi/m^{2}$ would be more transparent. A fifth concern is the operational interpretation of $\epsilon\sim c\tau_{\mathrm{dec}}$: how does this scale interact with the renormalisation flow when $\tau_{\mathrm{dec}}$ is itself temperature- and density-dependent, as in early-universe or near-horizon contexts?

A further question concerns the BRST sector. The author claims the ghost sector windows identically and no anomaly is introduced, yet the modified Ward--Takahashi identities will generate contact terms in $\Diamond$-dependent loop integrals. Does the windowed effective action remain BRST-invariant beyond tree level, and how is the Slavnov--Taylor identity preserved when $\nabla\Diamond\neq 0$? Finally, the high-density regime $\lambda\gtrsim\lambda_{*}(\ell)=c/\ell^{4}$ should be tied quantitatively to early-universe cosmology, near-horizon physics, and Hawking-radiation backreaction. Without explicit numerical estimates of $\rho^{\mathrm{Rem}}$ for, say, a Schwarzschild horizon of mass $M$, the falsifiability of the framework remains limited.

The English is generally acceptable, but the discussion of operational time would benefit from being tightened, and several recurring expressions should be replaced by sharper technical phrasing. I recommend major revision. The following uncited references, attached about me and literature, should be incorporated to enrich the paper:

Phys. Rev. D 104, 016022 (2021).
J. High Energy Phys. 2023, 119 (2023).
Phys. Rev. D 60, 084008 (1999).
Phys. Rev. D 105, 065007 (2022).
Int. J. Mod. Phys. A 37, 2241001 (2022).
Phys. Rev. D 97, 105008 (2018).
Proc. R. Soc. A 466, 2097 (2010).
Lett. Math. Phys. 103, 233 (2013).
Eur. Phys. J. C 75, 277 (2015).
Phys. Lett. B 780, 41 (2018).
Ann. Phys. (Berlin) 535, 2200474 (2023).

Comments on the Quality of English Language

Dear Editor,

\begin{equation}
\nabla^{\mu}T^{\mathrm{comp}}_{\mu\nu}=-(\nabla^{\mu}\Diamond)\,T^{\mathrm{SM}}_{\mu\nu},
\end{equation}

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

attached

Comments for author File: Comments.pdf

Comments on the Quality of English Language

attached

Author Response

Please see the attachment

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor,

I have read the revised manuscrript (Version May 1, 2026) and the author's point-by-point response letter, and I am satisfied that all technical concerns I raised in the first round have been answered honestly and at the right level of detail. The author has not paved over the difficulties; he has added the missing material and, where the answer is partial, said so plainly.

On the variational status of $\Diamond(x)$, the new Section 2.5 fixes the gap. The window is now declared a fixed external profile with $\delta\Diamond\equiv 0$, the two admissible variation classes are stated, and the boundary-layer integral that I flagged in the original Eq.~(14) is no longer dropped: it now produces the matching condition $(\nabla_{\mu}\Diamond)\Pi^{\mu}|_{\mathcal{B}_{\epsilon}}=0$ as Eq.~(16). The reasoning that no Gibbons–Hawking–York counterterm is needed (the metric is held fixed, only matter is varied) is correct.

The spinor rescaling argument is now stated cleanly. The domain restriction to $U_{\Diamond}\equiv\{x: \Diamond(x)>0\}$, the explicit non-unitarity statement, and the windowed inner product in Eqs.~(26)–(28) close the formal-cancellation gap I pointed to. The resulting cancellation $(i\gamma^{\mu}\nabla_{\mu}-m)\chi=0$ is an interior statement on $U_{\Diamond}$, not a global field redefinition, and the author says so.

The Vanishing Lemma has been rewritten with three labelled assumptions (A1)–(A3) and a normalised variance condition, Eq.~(70). The vacuum-state case where $\langle T_{\mu\nu}\rangle = 0$ but $\langle T_{\mu\nu}T_{\rho\sigma}\rangle\neq 0$ is now flagged explicitly, with Martín–Verdaguer and Fewster–Kontou cited at the relevant point. This is the right way to phrase the result.

My dimensional objection to the original Eq.~(78) has been resolved by the introduction of the two length scales $L_{\omega}^{-1}=(\omega_{\mu ab}\omega^{\mu ab})^{1/2}$ and $L_{\phi}^{-1}=(\partial_{\mu}\phi\,\partial^{\mu}\phi)^{1/2}/m$ in Eqs.~(90)–(91), with an equivalent curvature-invariant form $K\ell^{2}$ supplied. Both quantities now have the same dimension, and the regime classification in Eqs.~(92)–(94) is unambiguous.

The interaction between $\epsilon\sim c\tau_{\mathrm{dec}}$ and the renormalisation flow is now treated in three steps in Section 2.5.4. The schematic identity $\Gamma_{\Diamond}[\mu]=\int_{\mathcal{M}}\Diamond(x)\,\mathcal{L}_{\mathrm{eff}}[\mu,g]\sqrt{-g}\,d^{4}x+\mathcal{B}_{\epsilon}[\mu]$ in Eq.~(17) makes the point correctly: $\Diamond$ does not run, the bulk $\beta$-functions are standard, and varying environments mean varying profiles rather than rerunning the RG. The condition $\mu\epsilon\gg 1$ for the scale separation is also stated.

The BRST beyond-tree-level concern is handled in Section 4.3.1 by the modified Slavnov–Taylor identity, Eq.~(47), $\mathcal{S}(\Gamma_{\Diamond})=\int d^{4}x\sqrt{-g}\,(\nabla_{\mu}\Diamond)\,\mathcal{C}_{\Diamond}^{\mu}$, with support confined to $\mathcal{B}_{\epsilon}$. The author labels this "schematically" rather than claiming an all-orders derivation, which I take to be accurate. The interior identity is recovered exactly where $\nabla\Diamond=0$, and the boundary contact terms are $\mathcal{O}(\epsilon/L)$.

The new Section 7.7 supplies the quantitative content I asked for. The Schwarzschild horizon scaling, Eqs.~(81)–(85), gives a definite bound $\rho^{\mathrm{Rem}}\lesssim \rho_{\mathrm{Pl}}^{(m)}(\epsilon/r_{s})^{4}$, and the suggestive coincidence that $\kappa$ and $T_{H}$ are both proportional to $1/M$ throughout evaporation, Eqs.~(86)–(87), is offered as an open question rather than a finished claim. The early-universe analysis correctly distinguishes a high event-rate density from a causally irreversible boundary, and explains why the Lemma still applies in radiation domination.

A few small items remain. The label "schematically" attached to Eq.~(47) should be retained but accompanied by a short sentence stating which order in perturbation theory the identity has actually been checked at. The reference to Mignemi 2015 in support of the EUP-corrected Eq.~(2) is fine, but the curvature length $L_{R}$ should be tied to a definite background curvature rather than left generic. These can be handled in a minor revision and do not affect my overall recommendation.

The manuscript has improved in clarity and in technical sharpness. The author's response letter is detailed and frank about which results are exact and which are structural characterisations. I recommend acceptance after a short minor revision pass on the two items above.