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REVIEW 4 major objections 5 minor 24 references

Classical (ontological) dual states in quantum theory and the minimal group representation Hilbert space

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper claims that applying the minimal group representation of the metaplectic group Mp(2) to circle and cylinder coherent states renders the quantum system classical, with Mp(2) as the symmetry group of classical–quantum duality.

desk verdict A small new construction of normalizable circle coherent states is buried under an unsupported claim that Mp(2) 'classicalizes' quantum systems via hand-inserted Gaussians. read the letter →

arxiv 2501.11119 v1 pith:GIRZXLQN submitted 2025-01-19 quant-ph gr-qchep-phhep-thmath-phmath.MP

Diego J. Cirilo-Lombardo (Keldysh Institute of the Russian Academy of Sciences , CONICET-UBA-INFINA) , Norma G. Sanchez (CNRS , The Chalonge - Hector de Vega International School of Astrophysics) This is my paper · ORCID
classification quant-phgr-qchep-phhep-thmath-phmath.MP MSC 81R3081R0522E7081S10 PACS 03.65.-w03.65.Ca03.65.Sq
keywords classical-quantumdualitymetaplecticgroupminimalrepresentationcirclephasestatescylindercoherentcosetWignerfunctionontologicalvariables
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 establish that classical descriptions of a quantum system are not hidden variables but dual states, and that these states become manifest when the system is projected through the minimal group representation of the metaplectic group Mp(2). Projecting Mp(2) coherent states onto the non-normalizable circle phase states yields analytic functions on the unit disk whose squared norms decay rapidly with oscillator number; for cylinder coherent states the decay is much stronger, and the generalized Wigner function becomes bell-shaped. The authors take these decays as the signature of classicalization and conclude that Mp(2) is the symmetry group of the classical–quantum duality. If the argument is right, the same mechanism would explain how quantum degrees of freedom acquire classical descriptions without invoking genuinely hidden variables.

What carries the argument

The load-bearing object is the metaplectic group Mp(2) and its minimal group representation, realized on harmonic-oscillator states through the generators T₁, T₂, T₃. The central computations are the projections ⟨φ|$Ψ^{{(±)}}$(ω)⟩ and ⟨ξ|Ψ(ω)⟩: the circle phase shift z = $ωe^{{iφ}}$ preserves analyticity in |z|<1, while the cylinder coherent states insert Gaussian screening factors that produce the claimed classicalization. The new coset coherent states, built from the group coset E(2)/T² with a fiducial vector and normalization condition Im α ≠ 0, are the device that makes the overcomplete circle states normalizable and solves the identity in a weak sense.

What would settle it

Repeat the projection calculation with cylinder states |ξ⟩ = Σ $e^{{(l−iφ)j}}$|j⟩, dropping the Gaussian factor $e^{{-j²/2}}$; if the squared norms no longer decay with factors like $e^{{-2n²}}$, then the classicalization is an artifact of the chosen states and not a consequence of the Mp(2) action.

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Extended reading notes

Core claim

The central claim is that the metaplectic group Mp(2), the double cover of Sp(2), acts as the group of classical-quantum duality: its minimal representation coherent states, when projected onto the circle phase states and the cylinder coherent states, produce analytic functions in the unit disk with sharply decaying squared norms. The even oscillator states |2n> and odd states |2n+1> span the two Mp(2) irreps H_{1/4} and H_{3/4}, so the complete Hilbert space is H_{1/4} ⊕ H_{3/4}; the projection of the total state contains exponentially decaying factors such as $e^{{-2n²}}$ and $e^{{-(2n+1)²/2}}$ in the cylinder case. The paper also constructs fully normalizable coset coherent states on the circle from the coset E(2)/T², with a complex parameter α whose imaginary part makes them normalizable, thereby repairing the non-normalizability of the old circle states. The authors read the resulting bell-shaped distributions and the rapid decay of the projections as evidence that applying the minimal group representation 'immediately classicalizes the system.'

Load-bearing premise

The claimed classicalization depends on modeling choices in the states themselves—the Gaussian factor $e^{{-j²/2}}$ in the cylinder states and the condition Im α ≠ 0 for normalizability—so if those choices were changed, the exponential decay that the paper presents as classicalization could disappear.

Editorial extensions

If this is right

  • The old overcomplete, non-normalizable circle phase states are replaced by fully normalizable coset coherent states that resolve the identity only weakly, with diagonal entries e^{-n Im α}.
  • The even and odd harmonic-oscillator sectors correspond to the two irreducible representations of Mp(2), so the full Hilbert space of the harmonic oscillator carries a complete metaplectic representation.
  • Applying the Mp(2) action to the circle states breaks time-reversal invariance, introducing an arrow of time into the otherwise reversible circle dynamics.
  • For cylinder coherent states, the projected norms decay with Gaussian factors such as e^{-2n²}, which the paper interprets as a stronger classicalization than the circle phase-space case.
  • The generalized Wigner function for the projected states is bell-shaped and closer to a classical probability distribution than the squared-norm function itself.

Reading between the lines

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

  • The Gaussian factor e^{-j²/2} placed by hand in the cylinder coherent states may be doing much of the apparent classicalization; testing the same projections with unweighted cylinder states would isolate whether Mp(2) itself contributes more than a phase relabeling.
  • The normalizability of the coset coherent states is bought by requiring Im α ≠ 0 (and Im α > 0 for the identity), so the repair of the London-state normalization problem may be a regularization rather than the discovery of a new physical degree of freedom.
  • If the Mp(2) duality is general, the same construction should transfer to other phase-space topologies and to multi-mode systems; whether the same exponential screening appears there would be a direct test of the mechanism's reach.
  • The claimed time-reversal breaking suggests that the dual states might obey a one-way, semigroup evolution; the paper does not compute such a semigroup, but the claim implies one should exist and could be looked for.
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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

4 major / 5 minor

Summary. This paper studies the London phase states |φ> = (2π)^(-1/2) Σ e^{iφ n}|n> on the circle, cylinder coherent states |ξ> = Σ e^{(l-iφ)j} e^{-j^2/2}|j>, and newly introduced coset coherent states |α,φ> on the circle. The authors compute projections of the even and odd sectors of the metaplectic coherent states |Ψ(±)(ω)> onto these states and interpret the resulting exponentially decaying overlaps as 'classicalization' induced by the Minimal Group Representation. They also claim that Mp(2) is the classical-quantum duality group and that the coset states solve the non-normalizability of the London states.

Significance. If the central claim were substantiated, the paper would offer a general group-theoretic mechanism by which quantum degrees of freedom become classical, and would identify Mp(2) as the relevant symmetry. The paper contains explicit formulas and group-theoretic background, and the projection calculations are algebraically plausible in places. However, the main interpretive conclusion is not derived from the group action: the exponential decays are built into the chosen test states, and the 'weak resolution of the identity' is not an identity. The result therefore does not, in its present form, establish the advertised classicalization mechanism.

major comments (4)
  1. [Section II.A, Eq. (2)] The relation U|j> = j|j+1> (and U†|j> = j|j-1>) is inconsistent with U = e^{iφ} being unitary. From [J,U]=U and J|j>=j|j> one obtains J(U|j>) = (j+1) U|j>, so unitarity requires U to map the j eigenspace isometrically onto the j+1 eigenspace; the coefficient cannot be j. This relation is used to identify the circle basis with the harmonic-oscillator basis and to justify the ladder-operator construction, so it is a load-bearing algebraic step.
  2. [Section V, Eqs. (19)-(21)] The cylinder coherent state (19) contains the Gaussian factor e^{-j^2/2}, introduced without derivation or physical motivation. This factor, not the Mp(2) action, produces the exponential decays e^{-2n^2} and e^{-(2n+1)^2/2} in Eq. (20): they are simply the Gaussian evaluated at j=2n and j=2n+1. Replacing this weight by another function would remove the advertised suppression. The concluding remark in Section IX that previous work is complicated by 'the introduction of a Gaussian fiducial state' applies equally to the construction used here.
  3. [Section VII, Eqs. (30)-(32)] The normalizability of the coset states requires Im α > 0 for convergence of the sum in Eq. (30), not merely Im α ≠ 0; for Im α < 0 the geometric series diverges. The 'weak resolution of the identity' in Eq. (32) is the diagonal operator diag(1, e^{-Im α}, e^{-2 Im α}, ...), which is not the identity. Thus the construction does not provide a complete orthonormal set in the usual sense, and the claim that it solves the London-state normalization problem is only achieved by imposing the needed sign on α.
  4. [Abstract; Sections IV-VI and IX] The paper's central assertion that the Minimal Group Representation 'immediately classicalizes the system' and that Mp(2) is the 'classical-quantum duality group' is not derived. The computations in Sections IV-VIII are overlap integrals between states; no dynamical mechanism, classical limit, or measurement protocol is specified that would turn an exponentially small overlap into classical behaviour. The principle of minimal group representation itself is only cited from earlier work and not defined here, so the inference from projection formulas to classicality is unsupported.
minor comments (5)
  1. [Throughout] The manuscript contains numerous typographical errors ('cilinder', 'Megaplectic', 'analiticity', 'the the', 'an principle') and inconsistent notation (e.g., sums written as 'n = 0, 1, 2..'). A thorough editorial pass is needed.
  2. [Abstract vs. Eq. (20)] The abstract lists decay factors e^{-(2n+1/2)} and e^{-(2n+1/2)^2}, but Eq. (20) contains e^{-(2n+1)^2/2}; the two forms should be reconciled.
  3. [Figure 4 caption] The caption refers to the odd sector as s = 3/2, whereas the text defines the odd sector as s = 3/4.
  4. [Eq. (22)] The approximate Wigner function contains Ei(4|z|^2) plus ln(1/|z|^4), which has a logarithmic divergence at z=0; the claim that the distribution is 'bell-shaped' needs qualification or a different approximation.
  5. [Section VI] The statement that the cylinder parameter l can be set to zero is introduced without discussion; since l appears only in the exponent of Eq. (19), its physical meaning should be clarified.

Circularity Check

3 steps flagged · score 8.0 of 10

The exponential decay and normalizability presented as Mp(2)-induced classicalization are, in the paper's own equations, the hand-inserted Gaussian e^{-j^2/2} of the cylinder state and the chosen Im α of the coset state; removing those choices removes the effect.

  1. self definitional [Section V, Eqs. (19)-(20); interpretation in Section VI]
    "In order to analyze the Coherent State of a particle in the cylinder in the context of the Minimal Group Representation, we express the coherent states as: | ξ ⟩ = ∑_{j=−∞}^{∞} e^{(l−iϕ)j} e^{−j²/2} | j ⟩ (19) ... We see that the scalar product projections ... in contrast they contain weight functions: e^{−2n²} and e^{−(2n+1)²/2}, which drastically attenuate the scalar products when n → ∞"

    The weight functions that are quoted as the sign of classicalization are exactly the Gaussian factor e^{−j²/2} inserted by hand into the test state in Eq. (19), evaluated at j = 2n and j = 2n+1. The Mp(2) part of the computation contributes only the (1−|ω|²) prefactors and the coherent sums; the n-dependent exponential decay comes entirely from the stipulated Gaussian. If the Gaussian were replaced by a flat or different weight, the predicted exponential suppression would disappear, so the claimed classicalization reduces to the ansatz by construction.

  2. self definitional [Section VII.D, Eqs. (30)-(32) and the weak-resolution computation following Eq. (32)]
    "Then, the state is fully normalizable: ϕ → ϕ′ Iff the parameter α have Im α ≠ 0 ... From Eq.(30) the normalized state coherent state ... |α, ϕ ⟩ = √(1−e^{−Im α}) ... ∑ e^{−i(ϕ−α/2)n}|n⟩ (32) ... the identity is not resolved in a strict sense, but in a weak sense, always for Im α > 0 ... = diag(1, e^{−Im α}, e^{−2 Im α}, ...)"

    The claimed resolution of the London-states problem is not a consequence of the metaplectic group action; it is produced by allowing the newly introduced free parameter α to have Im α ≠ 0 (for convergence) and Im α > 0 (for the weak identity). The weak resolution of the identity is simply the diagonal exponential e^{−n Im α}, which is the same convergence factor already put into the state by hand. The normalizability of Eq. (30) and the exponential entries of the resolution are therefore the chosen ansatz, restated as a result.

1 more flagged steps
  1. self citation load bearing [Section II (first paragraph) and Section IX, concluding item (5); Refs. [6], [7]]
    "As we showed earlier Ref [6], there is an even more general principle in the fundamental structure of quantum spacetime: the principle of minimal group representation, which allows us to obtain, consistently and simultaneously, a natural description of the dynamics of spacetime and the physical states admissible within it. ... From all the cases exhaustively studied here Mp(2) emerges as the classical-quantum duality group of symmetry."

    The central premise that a 'minimal group representation' is the correct selection principle is justified by citation to Refs. [6] and [7], which are the same authors' prior papers, and is not independently derived or tested in the present work. The concluding claim that Mp(2) is the classical-quantum duality group follows from applying that self-cited principle to states whose classicalizing properties are themselves inserted by hand in Eqs. (19) and (32). The load-bearing step is thus a self-citation chain rather than an externally established result.

full rationale

The paper's headline result—that the Minimal Group Representation of Mp(2) 'immediately classicalizes' the system—is not supported by an independent derivation. The quantitative evidence for classicalization is the rapid decay of the projections, but that decay is literally the Gaussian factor e^{−j²/2} written into the cylinder coherent state at Eq. (19); the projections in Eq. (20) merely evaluate this factor at even and odd j. Likewise, the normalizable circle states introduced in Section VII are made normalizable by choosing Im α ≠ 0, and their weak resolution of the identity is the diagonal matrix e^{−n Im α}, i.e., the convergence factor chosen in the definition of the state. These are cases where a prediction reduces, by construction, to the input ansatz. The frequent self-citations to the authors' own 'principle of minimal group representation' (Refs. [6,7]) add a further load-bearing appeal, but even without them the central classicalization claim would remain circular. A score of 8 reflects that the core conclusion is forced by the chosen definitions; a 10 would require that every technical result in the paper be definitionally empty, which is not the case because the group-theoretic computations (Mp(2)/SU(1,1) relations, coset geometry) have independent mathematical content.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The central results depend on free choices (the Gaussian damping in the cylinder states, the imaginary part of alpha, and the level-identification between |j> and |n>) and on the authors' own prior principle (MGRP). The exponential suppression interpreted as classicalization is inserted through the Gaussian ansatz rather than derived from Mp(2). No new physical entity with independent evidence is introduced.

free parameters (3)
  • Gaussian width in cylinder states = coefficient 1/2 in e^{-j^2/2}
    Introduced ad hoc in Eq. (19); the exponential decay e^{-2n^2} that is presented as classicalization originates from this choice.
  • Coset parameter alpha = Im alpha > 0 (undetermined)
    Chosen in Section VII so that the states |alpha,phi> become normalizable and so that the identity is weakly resolved; the normalizability claim is a direct consequence of this choice.
  • Cylinder parameter l = set to 0 (Section VI item iii)
    Introduced in the cylinder states in Eq. (19) and later set to zero to recover the circle variable z = omega e^{-i phi}; the 'cylinder' claims therefore assume l=0.
assumptions (4)
  • ad hoc to paper The Principle of Minimal Group Representation (MGRP) is valid and applicable
    Invoked in Sections II and IX and cited to the authors' own Refs [6,7]; no independent proof or justification is given here.
  • domain assumption The angular momentum basis |j> can be identified with the harmonic oscillator basis |n>
    Stated as '|j> ~ |n>' in Section II.A and used to connect the circle spectrum to the metaplectic oscillator states; this identification is not derived.
  • ad hoc to paper The shift relation U|j> = j|j+1> is valid
    Eq. (2) asserts this relation for the unitary shift operator U = e^{i phi}; as written it is dimensionally inconsistent and would violate unitarity for j=0, yet it is used to build the ladder operators a and a+.
  • standard math Mp(2) double covers Sp(2) and can be extended to OSp(n)
    Standard metaplectic group facts quoted in Section III.A; the OSp extension is background and not load-bearing.
invented entities (2)
  • Coset coherent states |alpha,phi>
    purpose: Render the London phase states normalizable and weakly resolve the identity.
    The states are defined with a complex parameter alpha; their normalizability comes from the imposed condition Im alpha > 0. No experimental signature is proposed.
  • Dual classical states (rename of ontological states)
    purpose: Reinterpret 't Hooft's ontological variables as classical-quantum dual partners rather than hidden variables.
    This is a relabeling with no new observable; the paper provides no falsifiable handle.

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Cite this review

Pith. "Pith review of Classical (ontological) dual states in quantum theory and the minimal group representation Hilbert space." pith.science (2026). https://pith.science/paper/GIRZXLQN

@misc{pith2026250111119,
  author       = {Pith},
  title        = {Pith review of: Classical (ontological) dual states in quantum theory and the minimal group representation Hilbert space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GIRZXLQN}},
  note         = {Machine review of arXiv:2501.11119}
}
read the original abstract

We investigate the classical aspects of Quantum theory and under which description Quantum theory does appear Classical. Although such descriptions or variables are known as "ontological" or "hidden", they are not hidden at all, but are dual classical states (in the sense of the general classical-quantum duality of Nature). The application of the Minimal Group Representation immediately classicalizes the system, Mp(2) emerging as the group of the classical-quantum duality symmetry. (Abridged)

Figures

Figures reproduced from arXiv: 2501.11119 by the authors.

Figure 1
Figure 1. FIG. 1: Graphical representation of the the norm of the proje [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Three-dimensional representation of the norm of the [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Graphical representation of the generalized Wigner [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: In the Figure we see graphically represented the squa [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]

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Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages

  1. [1]

    Clauser and Anton Zeilinger, Nobel Prize in Physics 2022, ”for experiments with entangled photons,

    Alain Aspect, John F. Clauser and Anton Zeilinger, Nobel Prize in Physics 2022, ”for experiments with entangled photons, ... and pioneering qua ntum information science” https://www.nobelprize.org/uploads/2023/10/advanced-physicsprize2022-4.pdf and Refs therein

    work page 2022

  2. [2]

    J. L. Mac Loughlin, N. G. Sanchez, ”The Photography as a Ne w Conceptual Quantum Information System”, (Oct. 2023) https://www.researchgate.net/publication/375745724 30 https://www.bibsonomy.org/bibtex/7a77b52e38f27089f47229b1193dc005 D. J. Cirilo-Lombardo and N. G. Sanchez, Entanglement and Ge neralized Berry Geometrical Phases in Quantum Gravity, Symmetr...

    work page arXiv 2024

  3. [3]

    Hopfield and Geoffrey E

    John J. Hopfield and Geoffrey E. Hinton, Nobel Prize in Physi cs 2024, ”for foundational discoveries and inventions that enable machine learning wi th artificial neural networks” https://www.nobelprize.org/uploads/2024/10/advanced-physicsprize2024-2.pdf and Refs therein

    work page 2024

  4. [4]

    where although similar functions and results in the circle are consid ered, the analyticity in the unit disk | z | < 1, is introduced differently. (3) The analytic functions induced by the action of the basic states of t he metaplectic group and the London states divide the projection of the Hilbert sp ace on the disk into even and odd functions that in al...

    work page 2022

  5. [5]

    ’t Hooft, The Hidden Ontological Variable in Quantum Harmonic Oscilla tors , Lindau 2024 Lecture, arXiv 2407.18153, (August 2024)

    G. ’t Hooft, The Hidden Ontological Variable in Quantum Harmonic Oscilla tors , Lindau 2024 Lecture, arXiv 2407.18153, (August 2024)

    arXiv 2024

  6. [6]

    London, Winkelvariable und Kanonische Transformationen in der Und u- lationsmechanik: Zeitschrift fur Physik , 37, 915-925, (1926)

    F. London, Winkelvariable und Kanonische Transformationen in der Und u- lationsmechanik: Zeitschrift fur Physik , 37, 915-925, (1926)

    work page 1926

  7. [7]

    D. J. Cirilo-Lombardo, N. G. Sanchez, Universe, 10, 22 (2 024). https://doi.org/10.3390/universe10010022

    work page doi:10.3390/universe10010022

  8. [8]

    D. J. Cirilo-Lombardo, N. G. Sanchez, Phys. Rev.D108, 12 6001, (2023)

    work page 2023

Show all 24 references
  1. [9]

    N. G. Sanchez, Phys. Rev. D 104, 123517 (2021)

    2021

  2. [10]

    N. G. Sanchez, Phys. Rev. D 107, 126018 (2023)

    2023

  3. [11]

    N. G. Sanchez, Int. J. Mod Phys D28, 1950055 (2019)

    2019

  4. [12]

    N. G. Sanchez, Int. J. Mod Phys A34, 1950155 (2019)

    2019

  5. [13]

    N. G. Sanchez, Gravit. Cosmol. 25, 91–102 (2019). https://doi.org/10.1134/S0202289319020142 31

    2019 doi

  6. [14]

    D. J. Cirilo-Lombardo, J. Math.Phys. 57 (2016) 6, 06350 3

    2016

  7. [15]

    D. J. Cirilo-Lombardo; Foundations of Physics 37 (2007 ) 919-950

    2007

  8. [16]

    D. J. Cirilo-Lombardo, Found Phys 39 (2009) 373–396

    2009

  9. [17]

    J. R. Klauder and B. S. Skagerstam, Coherent States: Applications in Physics and Mathe- matical Physics (World Scientific, Singapore, 1985)

    1985

  10. [18]

    Klauder and E

    John R. Klauder and E. C. G. Sudarshan, Fundamentals of Quantum Optics , Dover Publica- tions (2006)

    2006

  11. [19]

    A. M. Perelomov, Generalized Coherent States and Their Applications , Springer Berlin, Heidelberg (1986)

    1986

  12. [20]

    Kowalski et al J

    K. Kowalski et al J. Phys. A: Math. Gen. 29 4149 (1996)

    1996

  13. [21]

    Kowalski and J

    K. Kowalski and J. Rembielinski, Phys. Rev. A 75 052102, (2007)

    2007

  14. [22]

    J. A. Gonzalez and M. A. del Olmo, J. Phys. A: Math. Gen. 31 8841 (1998)

    1998

  15. [23]

    D. J. Cirilo-Lombardo, J. Phys. A: Math. Theor. 45, 2440 26 (2012)

    2012

  16. [24]

    Coherent states for arbitrary Lie group , A. M. Perelomov, Commun. Math. Phys. 26 (1972) 222–236]. 32

    1972

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