REVIEW 2 major objections 5 minor 31 references
Charged current induced electron-proton scattering and the axial vector form factor
T0 review · 2 major / 5 minor · reviewed 2026-07-13 · grok-4.5
Pith's one-line read Coherent frames can exist with zero lower Beurling density on certain solvable Lie groups, so the usual density lower bound needs a localization assumption.
desk verdict Clean sharpness result: zero lower Beurling density coherent frames exist for square-integrable projective representations of unimodular exponential solvable groups, so localization cannot be dropped from the density lower bound. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
Central extension G_σ of G that converts the projective representation into an ordinary representation, followed by restriction to a closed simply-connected non-unimodular subgroup H isomorphic to the affine or Grélaud group; continuous Parseval frames of translates on H are discretized and transferred back to produce a zero-density frame on G.
What would settle it
Exhibit a connected simply-connected unimodular solvable Lie group of exponential growth whose every square-integrable projective representation forces every coherent frame to have strictly positive lower Beurling density, or show that the continuous-frame discretization step fails for the central-extension groups used in the construction.
Extended reading notes
Core claim
Let (π, H_π) be an irreducible square-integrable projective representation of a connected, simply connected unimodular solvable Lie group G of exponential growth. Then there exist a vector η in H_π and a discrete set Γ subset G with lower Beurling density D−(Γ) = 0 such that the coherent system π(Γ)η is a frame for H_π. Consequently the localization condition η ∈ B_π cannot be removed from the density lower bound for frames.
Load-bearing premise
The argument needs non-unimodular subgroups (affine or Grélaud type) that admit continuous frames of translates which can be discretized, and that these frames survive the passage through the central extension.
Editorial extensions
If this is right
- The localization hypothesis η ∈ B_π is indispensable for the density lower bound D−(Γ) ≥ d_π on unimodular amenable groups.
- Stability of frames under weak limits of translates need not hold without localization.
- Frame-measure formulas that relate density to frame bounds fail once localization is dropped.
- Zero-density coherent frames can be realized inside subgroups of dimension at most three.
Reading between the lines
- The same zero-density phenomenon may appear for other non-type-R solvable groups once continuous frames of translates are known for their non-unimodular subgroups.
- Practical sampling schemes on exponential solvable groups cannot rely solely on Beurling density; some form of localization or decay on the generating vector must be verified separately.
- The construction suggests that density theorems for projective representations should be restated in terms of the associated central extension, where ordinary representation theory applies.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that every irreducible square-integrable projective unitary representation of a connected, simply connected unimodular solvable Lie group of exponential growth admits a coherent frame whose index set has vanishing lower Beurling density (Theorem 1.2). The argument reduces the projective case to a genuine representation of a unimodular central extension G_σ of exponential growth, embeds a closed simply-connected nonunimodular subgroup H isomorphic to the affine or Grélaud group (Lemma 3.4), obtains a continuous Parseval frame of translates for the regular representation of H, discretizes it, transfers the frame via unitary equivalence of restricted regular representations (Theorem 3.1), and pulls it back through the coefficient map of the associated representation of G_σ. Zero density follows because the image of H is a proper closed nonunimodular subgroup, so G cannot be covered by finitely many right translates of a compact set. A short final section constructs a concrete class of unimodular exponential groups admitting square-integrable representations modulo the centre to which the result applies.
Significance. The result is a clean and useful negative theorem: it shows that the localisation hypothesis η ∈ B_π cannot be dropped from the density lower bound for coherent frames on unimodular amenable groups (Theorem 1.1), and that several related structural results (stability under weak limits of translates, frame measure formulae) likewise require localisation. The reduction through central extensions is carefully adapted to the projective setting, where genuine square-integrable representations are known not to exist for the groups under consideration. The existence proof is non-constructive but rests on standard structural theorems used within their stated hypotheses; if correct, it settles a natural open question in the density theory of coherent systems.
major comments (2)
- In the proof of Theorem 1.2 (Section 3.3), the claim that the closed simply-connected subgroup H of G_σ isomorphic to the affine or Grélaud group necessarily satisfies H ⊆ G × {1} is asserted in a single sentence (“as H is simply connected”). Because the circle factor is central and compact, the projection of H onto T must indeed be trivial, but this deserves an explicit one-line argument (e.g., continuous homomorphisms from a simply-connected exponential group into T are trivial). Without it the subsequent identification Γ = j(Λ) and the transfer of the frame from λ_{G_σ}(Λ) to π(Γ)η are not fully justified.
- The same proof invokes [11, Thm. 0.2] for a continuous Parseval frame of translates for λ_H and [10, Thm. 1.3] (or [5, Thm. 3.4]) for discretisation. Both results are stated for genuine regular representations of nonunimodular groups. While the reduction to G_σ makes the application formally legitimate, a brief remark confirming that the hypotheses of those theorems (type-I, exponential, existence of admissible vectors) remain satisfied for the concrete affine/Grélaud subgroups of G_σ would close a potential gap for readers who do not consult the cited works.
minor comments (5)
- Notation for the universal cover in the proof of Lemma 3.4 is typographically garbled (“eS”, “eH”, “e_S”). Standard notation (tilde or widehat) would improve readability.
- In Section 2.1 the formal-degree identity is written with d_π^{-1} on the right-hand side; some authors place d_π on the left. A parenthetical consistency note with the convention of [1] would avoid confusion.
- The phrase “projective regular representations” appears in the introduction; the body works exclusively with ordinary regular representations of the central extension. Aligning the terminology would prevent a minor ambiguity.
- Reference [5] is listed as a 2026 preprint (arXiv:2603.10423). If it remains unpublished, a short statement that the discretisation can equally be taken from the published [10] alone would future-proof the citation.
- Section 4 ends abruptly after recovering the Heisenberg example of [28]. A single sentence indicating that the same construction yields infinitely many non-isomorphic groups would strengthen the “class of examples” claim.
Circularity Check
No significant circularity: pure existence proof from external structural theorems, not forced by self-definition or fitted inputs.
full rationale
The paper's central claim (Theorem 1.2) is an existence result: for irreducible square-integrable projective representations of connected simply-connected unimodular solvable Lie groups of exponential growth, there exist coherent frames whose index set has vanishing lower Beurling density. The derivation chain reduces the projective case to a genuine representation of a unimodular central extension G_σ, embeds a closed simply-connected nonunimodular subgroup H isomorphic to the affine or Grélaud group (Lemma 3.4, via Jenkins growth and absolute closedness), obtains a continuous Parseval frame of translates for λ_H from Führ [11], discretizes it via Freeman–Speegle [10], transfers by unitary equivalence of regular representations (Theorem 3.1, Herz/Dixmier), and projects back via the coefficient map of π_σ. Zero density then follows because the image of H is a proper closed nonunimodular subgroup, so G cannot be covered by finitely many right translates of a compact set. None of these steps redefine the target as an input, fit parameters to data, or load-bear on an unverified self-citation uniqueness theorem. Self-citations to the authors' related density work ([6,8,9,23]) appear only as motivation and contrast for why localization cannot be dropped from Theorem 1.1; they do not force the existence construction. The argument is self-contained mathematical existence from cited external hypotheses and is not circular.
Assumptions & free parameters
assumptions (6)
- domain assumption For unimodular amenable groups, frames π(Γ)η with η in the localization space B_π satisfy D−(Γ) ≥ d_π (Theorem 1.1 / cited [6,8,9,23]).
- standard math Connected Lie groups of exponential growth have Lie algebras not of type R and contain subalgebras isomorphic to the affine or Grélaud algebra (Jenkins [19]).
- domain assumption Affine and Grélaud groups admit continuous Parseval frames of translates for the regular representation ([11, Thm. 0.2]) that can be discretized to discrete frames ([10, Thm. 1.3]).
- standard math If λ_H has infinite multiplicities, then λ_H ≃ λ_G|H (Theorem 3.1, via Herz and Dixmier quasi-equivalence).
- domain assumption Square-integrable genuine unitary irreps do not exist for connected simply connected unimodular solvable Lie groups ([4, Cor. 4.2]); projective/square-integrable-mod-center setting is the correct one.
- standard math Any Borel cocycle is similar to an analytic cocycle ([31, Cor. 7.30]), so the analytic case covers the general projective representation.
Cite this review
Pith. "Pith review of Charged current induced electron-proton scattering and the axial vector form factor." pith.science (2026). https://pith.science/paper/R4XZYGYR
@misc{pith2026260400764,
author = {Pith},
title = {Pith review of: Charged current induced electron-proton scattering and the axial vector form factor},
year = {2026},
howpublished = {\url{https://pith.science/paper/R4XZYGYR}},
note = {Machine review of arXiv:2604.00764}
}
abstract
We investigate the total scattering cross section($\sigma$), the differential cross section$\left(\frac{d\sigma}{dQ^2}\right)$, the longitudinal($A_L(E_e,Q^2)$) and perpendicular($A_P(E_e,Q^2)$) spin asymmetries of the polarized target proton, as well as the longitudinal($P_L(E_e,Q^2)$), perpendicular($P_P(E_e,Q^2)$), and transverse($P_T(E_e,Q^2)$) polarization components of the final neutron, in the weak charged current induced electron-proton scattering relevant to the future experiments at the Thomas Jefferson National Accelerator Facility(JLab) and Mainz Microtron(MAMI). The analysis is performed assuming time-reversal(T) invariance as well as without assuming T invariance, allowing for a nonvanishing transverse polarization component of the final nucleon, perpendicular to the production plane. Numerical results are presented for the above mentioned observables, and their sensitivities to the various parameterizations of the axial vector form factor $g_1(Q^2)$ and a nonzero weak electric form factor $g_2(Q^2)$ are examined. We find that the cross section depends strongly on the parameterizations used for the axial vector form factor. Moreover, the dipole parameterization of $g_1(Q^2)$ with a higher value of the axial dipole mass $M_A$ simulates the apparent enhancement in $\sigma$ obtained using the non-dipole parameterizations like the $z$-expansion and Faddeev equation form. The cross sections are found to depend only weakly on the weak electric form factor $g_2(Q^2)$, which is associated with the violation of G-invariance. On the contrary, the spin observables both $A_{L,P}(E_e, Q^2)$ and $P_{L,P}(E_e, Q^2)$ are found to be strongly dependent on $g_2(Q^2)$. This study may be useful in the analysis of the neutrino oscillation experiments to provide an alternative constrain on the parameterization of axial vector form factor, which currently has large uncertainties.
Reference graph
Works this paper leans on
-
[1]
P. Aniello. Square integrable projective representations and square integrable representations modulo a relatively central subgroup.Int. J. Geom. Methods Mod. Phys., 3(2):233–267, 2006
2006
-
[2]
G. Arsac. Sur l’espace de Banach engendre par les coefficients d’une repr´ esentation unitaire.Publ. D´ ep. Math., Lyon, 13(2):1–101, 1976
1976
-
[3]
B. Bekka. Square integrable representations, von Neumann algebras and an application to Gabor analysis. J. Fourier Anal. Appl., 10(4):325–349, 2004
2004
-
[4]
Beltit ¸˘ a and D
I. Beltit ¸˘ a and D. Beltit ¸˘ a. Square-integrable representations and the coadjoint action of solvable Lie groups. Forum Math., 37(3):693–715, 2025
2025
-
[5]
M. Bownik and P.-T. Yu. Uniform discretization of continuous frames. Preprint, arXiv:2603.10423, 2026
arXiv 2026
-
[6]
Caspers and J
M. Caspers and J. T. van Velthoven. Overcompleteness of coherent frames for unimodular amenable groups.Ark. Mat., 61(2):277–299, 2023
2023
-
[7]
Dixmier.� � -algebras., volume 15 ofNorth-Holland Math
J. Dixmier.� � -algebras., volume 15 ofNorth-Holland Math. Libr.Elsevier (North-Holland), Amsterdam, 1977
1977
-
[8]
Enstad and S
U. Enstad and S. Raum. A dynamical approach to sampling and interpolation in unimodular groups. Trans. Am. Math. Soc., 378(3):1975–2006, 2025
1975
Show all 31 references
-
[9]
Enstad and J
U. Enstad and J. T. van Velthoven. Coherent systems over approximate lattices in amenable groups.Ann. Inst. Fourier, 75(6):2293–2315, 2025
2025
-
[10]
Freeman and D
D. Freeman and D. Speegle. The discretization problem for continuous frames.Adv. Math., 345:784–813, 2019
2019
-
[11]
H. F¨ uhr. Admissible vectors for the regular representation.Proc. Am. Math. Soc., 130(10):2959–2970, 2002
2002
-
[12]
F¨ uhr.Abstract harmonic analysis of continuous wavelet transforms, volume 1863 ofLect
H. F¨ uhr.Abstract harmonic analysis of continuous wavelet transforms, volume 1863 ofLect. Notes Math. Berlin: Springer, 2005
2005
-
[13]
F¨ uhr, K
H. F¨ uhr, K. Gr¨ ochenig, A. Haimi, A. Klotz, and J. L. Romero. Density of sampling and interpolation in reproducing kernel Hilbert spaces.J. Lond. Math. Soc., II. Ser., 96(3):663–686, 2017
2017
-
[14]
F¨ uhr and V
H. F¨ uhr and V. Oussa. Groups with frames of translates.Colloq. Math., 167(1):73–91, 2022
2022
-
[15]
Fujiwara and J
H. Fujiwara and J. Ludwig.Harmonic analysis on exponential solvable Lie groups. Springer Monogr. Math. Tokyo: Springer, 2015
2015
-
[16]
M. Goto. Absolutely closed Lie groups.Math. Ann., 204:337–341, 1973
1973
-
[17]
Gr¨ ochenig
K. Gr¨ ochenig. The homogeneous approximation property and the comparison theorem for coherent frames. Sampl. Theory Signal Image Process., 7(3):271–279, 2008
2008
-
[18]
Hilgert and K.-H
J. Hilgert and K.-H. Neeb.Structure and geometry of Lie groups. Springer Monogr. Math. Berlin: Springer, 2012
2012
-
[19]
J. W. Jenkins. Growth of connected locally compact groups.J. Funct. Anal., 12:113–127, 1973
1973
-
[20]
Kaniuth and A
E. Kaniuth and A. T.-M. Lau.Fourier and Fourier-Stieltjes algebras on locally compact groups, volume 231 ofMath. Surv. Monogr.Providence, RI: American Mathematical Society (AMS), 2018
2018
-
[21]
Mitkovski and A
M. Mitkovski and A. Ramirez. Density results for continuous frames.J. Fourier Anal. Appl., 26(4):26,
-
[22]
H. Omori. Homomorphic images of Lie groups.J. Math. Soc. Japan, 18:97–117, 1966
1966
-
[23]
Papageorgiou and J
E. Papageorgiou and J. T. van Velthoven. Counting function estimates for coherent frames and Riesz sequences.Ann. Mat. Pura Appl. (4), 204(4):1469–1491, 2025
2025
-
[24]
A. M. Perelomov. Coherent states for arbitrary Lie group.Commun. Math. Phys., 26:222–236, 1972
1972
-
[25]
Pogorzelski, C
F. Pogorzelski, C. Richard, and N. Strungaru. Leptin densities in amenable groups.J. Fourier Anal. Appl., 28(6):36, 2022. Id/No 85. COHERENT FRAMES WITH ZERO BEURLING DENSITY 9
2022
-
[26]
Ramanathan and T
J. Ramanathan and T. Steger. Incompleteness of sparse coherent states.Appl. Comput. Harmon. Anal., 2(2):148–153, 1995
1995
-
[27]
J. L. Romero and J. T. van Velthoven. The density theorem for discrete series representations restricted to lattices.Expo. Math., 40(2):265–301, 2022
2022
-
[28]
Rosenberg
J. Rosenberg. Square-integrable factor representations of locally compact groups.Trans. Am. Math. Soc., 237:1–33, 1978
1978
-
[29]
R. Tessera. Large scale Sobolev inequalities on metric measure spaces and applications.Rev. Mat. Iberoam., 24(3):825–864, 2008
2008
-
[30]
J. T. van Velthoven. Density conditions for coherent state subsystems of nilpotent Lie groups. InExtended abstracts 2021/2022. Methusalem lectures, Ghent, Belgium, pages 219–227. Cham: Birkh¨ auser, 2024
2021
-
[31]
V. S. Varadarajan. Geometry of quantum theory. 2nd ed. New York etc.: Springer-Verlag., 1985. Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, PO Box 1-764, 014700 Bucharest, Romania Email address:������������������������� ���������������������� F aculty of M...
1985
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