Pith. sign in

REVIEW 3 major objections 4 minor 1 cited by

The paper claims that the minimal-length correction to quantum momentum can turn a zero-total-momentum two-particle state into an entangled state, with complex canonical momenta appearing along the way.

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 · deepseek-v4-flash

2026-08-03 12:53 UTC pith:NNUD3I6Q

load-bearing objection The paper asks an interesting question but its central derivation is invalid: the alleged entanglement is an artifact of an unjustified projection and of treating complex roots as physical momentum eigenvalues. the 3 major comments →

arxiv 2601.02413 v3 pith:NNUD3I6Q submitted 2026-01-03 quant-ph

Minimal length: A source of quantum non-locality

H. Moradpour , S. Jalalzadeh This is my paper
classification quant-ph
keywords minimal lengthgeneralized uncertainty principleGUPcanonical momentumcomplex eigenvaluesquantum entanglementquantum non-localitymomentum operator deformation
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.

The paper asks whether the existence of a minimal length—already implied by combining quantum mechanics with gravity—can itself be a source of quantum non-locality, rather than just a correction to spectra. It studies the standard deformation of the momentum operator, P̂ = p̂(1+βp̂²), and shows that for a fixed eigenvalue P of the generalized momentum, the canonical momentum p has three roots, two of which are complex. Because of that, a generalized-momentum eigenstate can be expanded in canonical-momentum eigenstates with complex labels. The authors then consider two particles with zero total generalized momentum and show that written in the canonical-momentum basis the state becomes a superposition of paired complex-momentum states—an entangled state. That is the paper's central claim: minimal length, through the generalized uncertainty principle, may generate quantum non-locality and also give complex numbers a dynamical origin in quantum mechanics.

Core claim

On the paper's terms, the discovery is that the relation between the canonical momentum p and the generalized momentum P = p(1+βp²) is three-to-one: for each P there are three canonical-momentum roots P1, P2, P3, two of which are complex for β>0. The plane-wave solutions exp(iP_k x/ℏ) are simultaneously eigenfunctions of both operators, so the generalized-momentum eigenstate decomposes as |P⟩ = Σ α_k |p=P_k⟩. Applying the same decomposition to |−P⟩ and invoking the identity −P_k = P_l + P_m, the paper writes a two-particle state of zero total generalized momentum, |P,−P⟩, in the canonical-momentum basis as Σ α_i γ_i |P_i,−P_i⟩, an entangled state. The authors interpret this as minimal length

What carries the argument

The engine is the deformed momentum operator P̂ = p̂(1+βp̂²), where β is the GUP parameter tied to a minimal length. Its eigenvalue equation, P = p(1+βp²), is a cubic in p, giving three canonical-momentum roots and making |P⟩ a superposition of |p=P_k⟩ states. The second mechanism is the algebraic identity −P_k = P_l + P_m among the roots, which lets |−P⟩ be built from canonical-momentum states with negated complex roots; combining the two expansions turns a factorized zero-total-momentum pair state into a correlated superposition.

Load-bearing premise

The derivation assumes that the complex numbers P2 and P3 are legitimate eigenvalues of the canonical momentum operator, with corresponding states |p=P_k⟩ that form a complete, normalizable basis; since ordinary canonical momentum has only real eigenvalues, that assumption is what the entanglement claim rests on.

What would settle it

Evaluate the position-space norm of a complex-momentum plane wave, ⟨x|p=z⟩ = exp(i z x/ℏ)/√(2πℏ), for the complex roots P2, P3: the integral over x diverges, so these states are not elements of the physical Hilbert space. A direct calculation of Eq. (7) with complex Pk shows ⟨p|P⟩ contains delta functions at complex arguments; delta distributions at complex points are undefined in standard distribution theory, which would break the completeness relation (10) and collapse the construction.

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

If this is right

  • If the construction holds, any particle prepared in a definite generalized-momentum state will, when probed with an ordinary momentum measurement, appear in one of three canonical-momentum branches—two of them complex.
  • A two-particle state with zero total generalized momentum carries hidden correlations in canonical momentum; measuring one particle's canonical momentum makes the other's outcome correlated in a way not visible in the generalized-momentum basis.
  • Complex numbers cease to be an unexplained feature of quantum mechanics: they can be traced to the minimal length, so the theory predicts complex-valued intermediate labels in momentum-space decompositions.
  • The effect follows from the kinematics of the deformed momentum operator rather than from a specific Hamiltonian, so any theory with the same deformation would share the entanglement-generating property.

Where Pith is reading between the lines

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

  • Inference — the complex-momentum branches are likely unphysical: the canonical momentum operator is self-adjoint on square-integrable wavefunctions and has a purely real spectrum, so |p=P2⟩ plane waves are not normalizable and the decomposition (6) may not correspond to any state in the physical Hilbert space.
  • Inference — the entanglement is basis-dependent: the same physical state written in the generalized-momentum basis is a product state, so the non-locality is tied to choosing canonical-momentum measurements; whether it violates any Bell-type inequality is not shown and is the natural next test.
  • Inference — a concrete way to test the idea: look for the three-branch decomposition in a tabletop continuous-variable system with a simulated deformed momentum operator, e.g., an optical or cold-atom implementation of P=p(1+βp²); if the correlations predicted by Eq. (15) are absent, the effect is not physical.
  • Inference — if the construction survived Hilbert-space scrutiny, it would give a new observable signature of quantum gravity at low energies: entanglement generation from kinematics alone, independent of interaction strength.

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 / 4 minor

Summary. The manuscript argues that in the presence of a minimal length, the generalized momentum P̂ = p̂(1+βp̂²) has eigenstates that are superpositions of canonical-momentum eigenstates with generally complex eigenvalues P_i (the roots of p(1+βp²)=P). It then claims that a two-particle state |P,−P⟩ with zero total generalized momentum, when expressed in the canonical momentum basis, becomes the entangled state Σ α_i γ_i |P_i,−P_i⟩, so that the minimal length can generate quantum non-locality. The paper also suggests that complex momentum eigenvalues are a justifiable consequence of the minimal length.

Significance. If valid, the result would be conceptually important: a minimal length could entangle non-interacting particles and would motivate complex quadratures in QM. However, the central construction rests on the illegitimate use of complex generalized eigenvectors of a self-adjoint operator and on an unjustified projection; the claimed generation of entanglement is not established. The paper does not provide numerical or experimental predictions; its main contribution would be the formal argument, and that argument fails.

major comments (3)
  1. [Sec. II, Eqs. (3)–(7)] The canonical momentum operator p̂ = −iℏ∂_x on L²(R) is self-adjoint (with domain H¹(R)), hence its spectrum is real. The roots P₂ and P₃ in Eq. (5) have nonzero imaginary parts, so exp(iP_i x/ℏ) is not square-integrable and is not a legitimate generalized eigenvector in the standard rigged-Hilbert-space treatment of p̂. Consequently δ(p−P_i) in Eq. (7) is not a tempered distribution, and the expansion (6) is formal only. The statement that 'complex eigenvalues may be allowed for the canonical momentum operator due to the existence of minimal length' contradicts the self-adjointness of p̂ and is not supported by the cited literature.
  2. [Sec. II, Eqs. (9)–(10)] The derivation of Σ|α_k|²=1 presupposes both a completeness relation for the complex states |p=P_i⟩ and a delta function with complex argument. These are exactly the points that need proof; the calculation is therefore circular. Moreover, Eq. (8) is valid for real P₁,P₂ (from Kempf et al.), and its extension to complex P_i is not justified. Thus the normalization that underlies the later probabilistic interpretation is not established.
  3. [Sec. III, Eq. (15)] Even if one granted the formal complex basis, the tensor-product expansion of the product state |P,−P⟩ is Σ_{i,j} α_i γ_j |P_i,−P_j⟩. Equation (15) keeps only the diagonal terms i=j, which is not a basis change but a projection onto the subspace p₁+p₂=0. This truncation is an additional physical assumption not implied by total generalized-momentum conservation (since P_i ≠ P_j in general). Without that projection the state remains a product state; with it, the resulting state is not the same physical state. Hence the claim that minimal length 'generates' entanglement is not supported.
minor comments (4)
  1. [Sec. III, notation] The notation 'P₁ = −P₂ ≡ P' is confusing because P₁, P₂, P₃ are already defined as the roots of Eq. (5); the generalized momenta of the two particles should be denoted differently (e.g., P and −P).
  2. [Abstract and Introduction] The English is awkward throughout ('the ... difference ... is polished'); the manuscript needs careful editing before consideration.
  3. [References [33–38]] The cited works concern complex versus real Hilbert-space formulations of QM, not complex eigenvalues of observables. The conflation of 'complex numbers' as coefficients with 'complex eigenvalues' should be clarified.
  4. [Eq. (2)] The relation Δx_min = √(3β)ℏ ≡ √β₀ l_p should be checked for dimensional consistency and the parameter β₀ should be defined.

Circularity Check

1 steps flagged

Eq. (15)'s entangled state is obtained by imposing p1+p2=0, an extra condition not implied by P1+P2=0 under the GUP; the claimed minimal-length-generated nonlocality is therefore constructed rather than derived.

specific steps
  1. other [Sec. III, derivation of Eq. (15) (after Eq. (14))]
    "the generalized momentum conservation ( Pj=2 j=1 Pj = 0) in the ˆP-space also inspires the canonical momentum conservation ( Pj=2 j=1 pj =0) generating the p 1 =−p 2 condition. Accordingly, if purely quantum mechanical momentum measurements are employed (or equally, the |p⟩space is focused on), then Eqs. (6) and (14) lead us to the state 3X i=1 αiγi|Pi,−P i⟩,(15)"

    From Eqs. (6) and (14), the product state |P,−P⟩ expands as Σ_{i,j} α_i γ_j |P_i,−P_j⟩, which is still a product state. Eq. (15) keeps only i=j terms; this requires imposing p1+p2=0. But P1+P2=0 does not imply p1+p2=0 because P=p(1+βp²) is a cubic, so the map is multi-valued. Hence the diagonal entangled superposition is not a consequence of the GUP/minimal-length dynamics; it is put in by hand through an extra canonical-momentum conservation/postselection condition. The claimed 'novel quantum non-locality' is therefore equivalent to that imposed condition, not a prediction derived from minimal length.

full rationale

The paper does not fit parameters to data, and its self-citations (e.g., Refs. [12,13,17,20]) are contextual rather than load-bearing for the central calculation. The GUP momentum operator P̂=p̂(1+βp̂²) and the resolution of identity are taken from external prior work. The main circular step is in Sec. III: the author's entangled state (15) is not obtained from Eqs. (6) and (14) alone. Those equations give a product-state expansion Σ α_i γ_j |P_i,−P_j⟩ over canonical momentum eigenstates. To reach Eq. (15), the paper assumes an additional 'canonical momentum conservation' p1=−p2, justified only by saying the generalized-momentum conservation 'inspires' it. Since the GUP relation is not injective, P1+P2=0 does not entail p1+p2=0; the diagonal projection is an extra input. Thus the central claim that minimal length generates quantum non-locality reduces, by construction, to this imposed conservation/postselection condition. Separately, the use of complex roots as eigenvalues of the self-adjoint canonical momentum operator is mathematically questionable, but that is a correctness issue rather than a circularity. Overall, the result is partially circular because the alleged prediction of entanglement is effectively assumed in the projection that selects Eq. (15).

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The paper's conclusions rest on the GUP deformation from the literature, the admissibility of complex generalized eigenstates, and the completeness of that set; no numerical fitting or new physical entities are introduced.

free parameters (1)
  • β (GUP deformation parameter) = unspecified/assumed positive
    Controls the minimal length Δx_min=√(3β)ℏ; the paper's effects vanish at β=0, but no value is fitted or measured here.
axioms (4)
  • domain assumption GUP momentum operator P̂ = p̂(1+βp̂²) with position representation unchanged
    Taken from refs [1,7,23]; all conclusions depend on this deformation and on β>0.
  • ad hoc to paper Canonical momentum p̂ admits complex generalized eigenstates |p=P_i⟩ and a complete expansion (6)
    Used in Sec. II to build |P⟩ and later the alleged entangled state; conflicts with self-adjointness of p̂ on L²(R).
  • ad hoc to paper Completeness/normalization relation ∑|α_k|²=1 derived via Eq. (9)-(10)
    The integral step assumes delta-normalizability and an inner product for complex plane waves; not justified.
  • domain assumption Two-particle state with zero total generalized momentum is |P,-P⟩ = |P⟩⊗|-P⟩
    Standard momentum-conservation state; the later expansion is a product expansion, so no entanglement follows.

pith-pipeline@v1.3.0-alltime-deepseek · 5296 in / 12072 out tokens · 116443 ms · 2026-08-03T12:53:47.988158+00:00 · methodology

0 comments
read the original abstract

The narrow and subtle difference between the Hilbert spaces of operators corresponding to the canonical momentum and the generalized momentum that includes minimal length effects is polished. Consequently, complex eigenvalues may be allowed for the canonical momentum operator due to the existence of minimal length. A novel quantum entanglement generation is also reported indicating the power of theories including a minimal length in enriching the current understanding of quantum non-locality.

discussion (0)

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

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Kinematical correlations via $\kappa$-Poincar\'e coproducts

    hep-th 2026-06 unverdicted novelty 5.0

    In the classical basis the non-bijective momentum map induces branch-dependent κ-deformed back-to-back correlations for two-particle states obeying vanishing total momentum.

Reference graph

Works this paper leans on

38 extracted references · 1 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Hossenfelder, Living Rev

    S. Hossenfelder, Living Rev. Rel.16, 2 (2013)

    2013

  2. [2]

    Das and E

    S. Das and E. C. Vagenas, Phys. Rev. Lett.101, 221301 (2008)

    2008

  3. [3]

    A. F. Ali, S. Das and E. C. Vagenas, Phys. Lett. B678, 497-499 (2009)

    2009

  4. [4]

    A. F. Ali, S. Das and E. C. Vagenas, Phys. Rev. D84, 044013 (2011)

    2011

  5. [5]

    C. A. Mead, Phys. Rev.135, B849-B862 (1964)

    1964

  6. [6]

    C. A. Mead, Phys. Rev.143, 990-1005 (1966)

    1966

  7. [7]

    Kempf, G

    A. Kempf, G. Mangano and R. B. Mann, Phys. Rev. D 52, 1108-1118 (1995)

    1995

  8. [8]

    Einstein, B

    A. Einstein, B. Podolsky and N. Rosen, Phys. Rev.47, 777-780 (1935)

    1935

  9. [9]

    Bose,et al.Phys

    S. Bose,et al.Phys. Rev. Lett.119, no. 24, 240401 (2017)

    2017

  10. [10]

    Marletto and V

    C. Marletto and V. Vedral, Phys. Rev. Lett.119, no.24, 240402 (2017)

    2017

  11. [11]

    Blado, F

    G. Blado, F. Herrera and J. Erwin, Phys. Essays31, no.4, 397-402 (2018)

    2018

  12. [12]

    Aghababaei, H

    S. Aghababaei, H. Moradpour and H. Shabani, Quant. Inf. Proc.21, no.2, 57 (2022)

    2022

  13. [13]

    Aghababaei and H

    S. Aghababaei and H. Moradpour, Quant. Inf. Proc.22, no.4, 173 (2023)

    2023

  14. [14]

    Park, Nucl

    D. Park, Nucl. Phys. B977, 115736 (2022)

    2022

  15. [15]

    K. C. L. Filho, B. B. Dilem, R. O. Francisco, J. C. Fab- ris and J. A. Nogueira, Int. J. Mod. Phys. D33, no.12, 2450041 (2024)

    2024

  16. [16]

    Marletto, J

    C. Marletto, J. Oppenheim, V. Vedral and E. Wilson, [arXiv:2511.07348 [quant-ph]]

    arXiv

  17. [17]

    Moradpour, S

    H. Moradpour, S. Jalalzadeh and H. Tebyanian, Mod. Phys. Lett. A 2550004, (2025)

    2025

  18. [18]

    Pedram, Adv

    P. Pedram, Adv. High Energy Phys.2016, 5101389 (2016)

    2016

  19. [19]

    A. E. Rastegin, Annals Phys.382, 170-180 (2017)

    2017

  20. [20]

    Aghababaei, H

    S. Aghababaei, H. Moradpour, S. S. Wani, F. Marino, N. A. Shah and M. Faizal, Eur. Phys. J. C84, no.4, 404 (2024)

    2024

  21. [21]

    Aziz and R

    J. Aziz and R. Howl, Nature646, no.8086, 813-817 (2025)

    2025

  22. [22]

    A. F. Ali and A. Wojnar, Class. Quant. Grav.41, no.10, 105001 (2024)

    2024

  23. [23]

    F. Brau, J. Phys. A32, 7691-7696 (1999)

    1999

  24. [24]

    Bohm and Y

    D. Bohm and Y. Aharonov, Phys. Rev.108, 1070-1076 (1957)

    1957

  25. [25]

    Heisenberg, Zeitschrift fur Physik43, 172-198 (1927)

    W. Heisenberg, Zeitschrift fur Physik43, 172-198 (1927)

    1927

  26. [26]

    J. D. Franson, Phys. Rev. Lett62, 19 (1989)

    1989

  27. [27]

    Oppenheim, S

    J. Oppenheim, S. Wehner, Science330, 1072 (2010)

    2010

  28. [28]

    Peres, P

    A. Peres, P. F. Scudo and D. R. Terno, Phys. Rev. Lett. 88, 230402 (2002)

    2002

  29. [29]

    Altamirano, P

    N. Altamirano, P. Corona-Ugalde, R. B. Mann and M. Zych, Class. Quant. Grav.35, no.14, 145005 (2018)

    2018

  30. [30]

    M. J. W. Hall and M. Reginatto, J. Phys. A51, 085303 (2018)

    2018

  31. [31]

    M. K. D¨ oner and A. Großardt, Found. Phys.52, no.5, 101 (2022)

    2022

  32. [32]

    Spaventa, L

    G. Spaventa, L. Lami and M. B. Plenio, Quantum7, 1157 (2023)

    2023

  33. [33]

    Karam, American journal of physics 88, 39 (2020)

    R. Karam, American journal of physics 88, 39 (2020)

    2020

  34. [34]

    M. O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acin and M. Navascues, Na- ture600, no.7890, 625-629 (2021)

    2021

  35. [35]

    Z. D. Li, Y. L. Mao, M. Weilenmann, A. Tavakoli, H. Chen, L. Feng, S. J. Yang, M. O. Renou, D. Trillo and T. P. Le,et al.Phys. Rev. Lett.128, no.4, 040402 (2022)

    2022

  36. [36]

    M. C. Chen, C. Wang, F. M. Liu, J. W. Wang, C. Ying, Z. X. Shang, Y. Wu, M. Gong, H. Deng and F. Liang,et al.Phys. Rev. Lett.128, no.4, 040403 (2022)

    2022

  37. [37]

    Albert, X

    M. Albert, X. Bao and L. Chen, [arXiv:2308.16399 [quant-ph]]

    Pith/arXiv arXiv

  38. [38]

    J. L. Lancaster and N. M. Palladino, Am. J. Phys.93, no.1, 110-120 (2025)

    2025