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Deformed algebraic structure of angular momenta: GUP perspective

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Pith's one-line read This paper derives Planck-scale corrections to the angular-momentum commutator and to hydrogen energy levels from a generalized uncertainty principle that enforces a minimal length.

desk verdict The reader's main objection is wrong—Eq. (4.4) is fine—but the hydrogen section has a genuine error, and the paper's novelty is thin. read the letter →

arxiv 2411.18901 v1 pith:LHWOUOT5 submitted 2024-11-28 gr-qc hep-thmath-phmath.MPquant-ph

Gaurav Bhandari , S. D. Pathak , Manabendra Sharma , Anzhong Wang This is my paper · ORCID
classification gr-qchep-thmath-phmath.MPquant-ph
keywords generalizeduncertaintyprincipleminimallengthangularmomentumalgebraGUP-deformedcommutatorhydrogenatomspectrumresolutionPlanck-scalecorrectionsladderoperators
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

This paper argues that if a generalized uncertainty principle (GUP) encodes a minimal length, then the angular-momentum algebra of quantum mechanics must be deformed: the standard commutator $[L_i, L_j] = i\hbar\epsilon_{ijk} L_k$ becomes $[L_i, L_j] = i\hbar\epsilon_{ijk} L_k(1 - \delta\gamma p + \epsilon\gamma^2 p^2)$, where $\gamma$ is the Planck-scale GUP parameter and $\delta,\epsilon$ are model parameters. The paper claims that $L^2$ and $L_z$ can still be diagonalized simultaneously, with eigenvalues rescaled by factors involving $\zeta = \delta\gamma p - \epsilon\gamma^2 p^2$. Because angular momentum enters the hydrogen atom Hamiltonian, the paper derives Planck-scale corrections to the hydrogen energy levels, $E_n = E_n^{(0)}[1 + 2\delta\gamma\langle p_0\rangle + \gamma^2(3\delta^2\langle p_0\rangle^2 - 2\epsilon\langle p_0^2\rangle)]$. If correct, this gives a low-energy atomic window onto quantum-gravity phenomenology: a deformed angular-momentum algebra with measurable spectral consequences. The text positions itself as a short review, but the modified commutator and its spectral application are the concrete results it aims to establish.

What carries the argument

The load-bearing mechanism is the deformed momentum variable $p_i = p_{0,i}[1 - \delta p_0 + (\epsilon+\delta^2)p_0^2]$ with unchanged coordinates $q_i = q_{0,i}$, together with the deformed commutator of Eq. (4.2). Inserting $L_i = \epsilon_{ijk}q_j p_k$ into the double commutator and pulling the momentum-dependent scalar $(1 - \delta\gamma p + \epsilon\gamma^2 p^2)$ out in front of the Levi-Civita sum produces the modified angular-momentum algebra. That same scalar, written as $(1-\zeta)$, is the engine of the spectral calculation: it rescales $L^2$, $L_z$, and the momentum $\vec p$ by expectation values, converts the hydrogen radial equation into a Laguerre equation with shifted quantum number $n = n_0(1-\langle\zeta\rangle)$, and finally produces the GUP-corrected Rydberg wavelengths. The factor's factorizability out of commutators is what makes the whole construction go through.

What would settle it

Evaluate $[L_i,L_j]$ by inserting $L_i=\epsilon_{imn}q_m p_n$ with $p_i=p_{0,i}[1-\delta p_0+(\epsilon+\delta^2)p_0^2]$ and using the full deformed bracket $[q_i,p_j]$ of Eq. (4.2), keeping all terms in which $(1-\delta\gamma p+\epsilon\gamma^2 p^2)$ moves past $q$ or $p$; if any leftover terms of order $\gamma$ survive, the simply closed algebra of Eq. (4.4) fails. On the experimental side, a precision measurement of the hydrogen $1S$-$2S$ transition that matches standard QED within the predicted shift would bound $\gamma$ below the claimed correction scale.

Watch

Extended reading notes

Core claim

The central discovery claimed is that the GUP-modified position-momentum relation forces the angular-momentum commutator to become $[L_i, L_j] = i\hbar\epsilon_{ijk} L_k(1 - \delta\gamma p + \epsilon\gamma^2 p^2)$ rather than the undeformed $so(3)$ bracket. The paper asserts that the algebraic scaffolding of the standard theory survives: the Jacobi identity still holds, $[L^2, L_j] = 0$, and $[L_i, p^2] = 0$, so eigenstates of $L^2$ and $L_z$ can still be chosen simultaneously. Their eigenvalues, however, are rescaled to $\hbar^2 l(l+1)(1-\zeta)^2$ and $\hbar m(1-\zeta)$, making the $m$-spacing of ladder steps equal to $\hbar(1-\zeta)$. Applied to the hydrogen atom, the rescaling effectively changes the principal quantum number to $n = n_0(1-\langle\zeta\rangle)$ and yields the modified energy formula, so all hydrogen lines acquire shifts controlled by the expectation values of the electron momentum and its square.

Load-bearing premise

The entire construction assumes that the momentum-dependent factor $(1-\delta\gamma p+\epsilon\gamma^2 p^2)$ can be pulled out of the commutator $[L_i,L_j]$ and treated as a commuting scalar, even though $p$ is an operator that does not commute with position in the very same GUP algebra; it also assumes the ansatz $L_k = L_{0,k}(1-\delta\gamma p+\epsilon\gamma^2 p^2)$.

Editorial extensions

If this is right

  • Hydrogen spectral lines shift by momentum-dependent amounts: $E_n = E_n^{(0)}[1 + 2\delta\gamma\langle p_0\rangle + \gamma^2(3\delta^2\langle p_0\rangle^2 - 2\epsilon\langle p_0^2\rangle)]$, so precise spectroscopy could in principle detect or bound the GUP scale.
  • Despite the deformation, $L^2$ and $L_z$ remain simultaneously diagonalizable, so the usual quantum numbers $l$ and $m$ survive, but the ladder spacing and eigenvalue scale are multiplied by $(1-\zeta)$.
  • The Jacobi identity and vanishing commutators $[L^2,L_j]=[L_i,p^2]=0$ are preserved, meaning the deformed algebra retains enough of the standard structure for hydrogen-like bound-state calculations to be solved by Laguerre polynomials.
  • The Rydberg formula becomes $1/\lambda = R_\infty |1/[n_{0,f}^2(1-\langle\zeta_f\rangle)^2] - 1/[n_{0,i}^2(1-\langle\zeta_i\rangle)^2]|$, so line ratios encode GUP parameters.

Reading between the lines

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

  • If the deformed algebra is taken literally, the same factor $(1-\zeta)$ should enter the Zeeman and Stark splittings, so those effects may offer cleaner two-level probes of $\gamma$ than the gross hydrogen shift.
  • The paper's derivation assumes the momentum-dependent factor factors out of the commutator; recomputing $[L_i,L_j]$ without that factorization, keeping terms where $p$ acts on $q$, would test whether the algebra actually closes and is the most direct next calculation.
  • A null measurement of hydrogen-line shifts at current optical-clock precision would place an upper bound on $\gamma_0$, but only if the model's $\delta$ and $\epsilon$ degeneracies can be broken by comparing several transitions.
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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 / 4 minor

Summary. The manuscript reviews several generalized uncertainty principle (GUP) models and derives a deformed angular momentum algebra from a linear-plus-quadratic GUP, obtaining the commutator [Li, Lj] = iℏεijkLk(1 − δγp + εγ²p²) and modified eigenvalues for L² and Lz. It then applies these results to the hydrogen atom, deriving corrections to the energy levels and Rydberg spectrum.

Significance. The paper provides a self-contained, pedagogically useful review of GUP models and of the angular-momentum deformation, and its comparison tables are helpful. We verified that the central derivation of Eq. (4.4) is sound: the off-diagonal terms in the deformed commutator vanish under contraction with the Levi-Civita tensors, and the factor (1 − δγp + εγ²p²) commutes with Lk because [Li, p] = 0. However, the hydrogen application contains genuine operator-ordering errors, and the final energy formula Eq. (5.11) is not reliable as written. The algebraic result is largely an adaptation of Bosso and Das (2017), so the paper's new contribution is the hydrogen application, which presently needs substantial correction.

major comments (4)
  1. [Sec. V, Eq. (5.11)] The expansion (1 − ⟨ζ⟩)^{-2} ≈ 1 + 2⟨ζ⟩ + 3⟨ζ⟩² is incorrect when ζ is an operator. The correct second-order expectation value is ⟨(1 − ζ)^{-2}⟩ = 1 + 2⟨ζ⟩ + 3⟨ζ²⟩ + O(γ³), so the γ² coefficient should be 3δ²⟨p0²⟩ − 2ε⟨p0²⟩, not 3δ²⟨p0⟩². Moreover, for hydrogen bound states ⟨p0⟩ = 0 by parity, so the linear term 2δγ⟨p0⟩ vanishes identically and the leading correction is of order γ². As written, Eq. (5.11) does not give a valid GUP correction to the hydrogen spectrum.
  2. [Sec. V, Eq. (5.3)] Replacing the operator factor (1 − ζ)^{-2} by the number (1 − ⟨ζ⟩)^{-2} inside the radial Schrödinger equation is an uncontrolled mean-field approximation. Since ζ = δγp − εγ²p² is a momentum-dependent operator, (1 − ζ)^{-2} does not act as a constant on hydrogen eigenstates; the difference ⟨(1 − ζ)^{-2}⟩ − (1 − ⟨ζ⟩)^{-2} is of order γ² and contributes at the same order as the terms retained in Eq. (5.11). The derivation should either treat (1 − ζ)^{-2} perturbatively or justify the mean-field replacement explicitly.
  3. [Sec. IV.A, Eqs. (4.14) and (4.18)-(4.22)] The ladder-operator analysis treats ζ as a c-number when it is moved into eigenvalues, for example LzL±|pλm⟩ = ℏ[m ± (1 − ζ)]L±|pλm⟩. This is only valid in a basis where p (and hence ζ) is diagonal, but the hydrogen states used in Sec. V are not eigenstates of p. The paper should clarify that the eigenvalues in Eq. (4.25) are p-dependent and that a separate averaging step is required before these results are applied to the hydrogen atom.
  4. [Sec. IV, Eq. (4.6)] The displayed Jacobi identity expression iℏ{εijk[Li, Li] + εijk[Lj, Lj] + εijk[Lk, Lk]}(1 − δγp + εγ²p²) = 0 is not a valid representation of the Jacobi identity: the indices are not summed, [Li, Li] = 0, and the expression does not follow from Eq. (4.4). While the conclusion that the Jacobi identity holds is correct, the equation as written is wrong and should be replaced by a correct evaluation.
minor comments (4)
  1. [Throughout] There are numerous typos and grammatical errors, including 'and and' in the abstract, 'teh' in Sec. IV, 'momnetum' in Sec. IV, and 'Plank' in place of 'Planck' in Sec. IV.
  2. [References] Reference [15] is incomplete: it lacks author names.
  3. [Table III] The row 'Quantum Numbers' states that l and m remain integer in the GUP case, but Eq. (4.23) redefines m → m(1 − ζ), which is not integer for nonzero γ; the table should be made consistent with the text.
  4. [Sec. V, Eq. (5.2)] The notation ⃗p|lm⟩ = ⃗p0(1 − ⟨ζ⟩)|lm⟩ is unclear; it should be explained that this is a mean-field replacement rather than an exact operator identity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the deformed angular momentum algebra is derived from the GUP commutator, and the hydrogen result follows from an explicit substitution; remaining issues are approximation errors, not circularity.

full rationale

The central commutator (4.4) is derived directly from the deformed position-momentum algebra (4.2) by expanding [Li,Lj] = ε_imn ε_jrs [q_m p_n, q_r p_s]. The non-isotropic terms proportional to p_m p_s and p_n p_r vanish upon contraction with Levi-Civita tensors, and the surviving factor G = 1 − δγp + εγ^2 p^2 legitimately factors because [Li,p^2]=0, and hence [Li,p]=0, as shown in Eq. (4.9). The eigenvalue results (4.25) follow from the standard ladder-operator analysis applied to the modified commutator, with ζ treated as a central element in each common eigenspace of p and L; they are not obtained by assuming the target eigenvalues. The hydrogen result (5.11) is an algebraic expansion of the eigenvalue equation obtained after the explicit substitution (5.2)-(5.3); no parameter is fitted to hydrogen data, and the final energy expression is a consequence of that substitution rather than an input used to define it. The uncontrolled replacement of the operator ζ by ⟨ζ⟩, and the presence of a linear term that vanishes for hydrogen eigenstates, are correctness concerns, not circularity. The self-citations in the paper are contextual and not load-bearing; the deformed angular momentum construction is attributed to external references [82–84]. The derivation chain is therefore self-contained and not circular.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim relies on the deformed GUP commutator from the literature and on two unproven assumptions about factoring momentum-dependent operators and rescaling angular momentum. The paper introduces free parameters δ, ε, γ from the GUP model. No new particles, forces, or entities are introduced.

free parameters (3)
  • gamma (GUP parameter) = gamma = gamma0/(M_Pl c), gamma0 ~ 1
    Introduced as the Planck-scale deformation scale in the deformed commutator (4.2). Not fitted, but a free parameter of the model.
  • delta = not specified
    Dimensionless coefficient of the linear momentum term in the GUP commutator (4.2). It is a free parameter of the model.
  • epsilon = not specified
    Dimensionless coefficient of the quadratic momentum term in the GUP commutator (4.2). It is a free parameter of the model.
assumptions (4)
  • domain assumption The deformed commutator [qi,pj] = iℏ(δij - γδ(pδij + pipj/p) + γ²[εp²δij + (2ε+δ²)pipj]) (Eq 4.2) is the correct quantum-gravity modification.
    Taken from Ali, Das, and Vagenas [68] and related literature; not derived in this paper.
  • domain assumption The angular momentum operator is defined as Li = ε_imn q_m p_n, with q_i unchanged and p_i deformed as in Eq (4.1).
    Standard definition extended to deformed variables; the paper assumes the same functional form for angular momentum.
  • ad hoc to paper The operator factor (1 - δγp + εγ²p²) commutes with q and p and can be factored out of the commutator [Li,Lj] (Eq 4.4).
    This is the key unsupported step; in the deformed algebra [q,p] ≠ 0, so the factorization is invalid.
  • ad hoc to paper The modified angular momentum can be written as Lk = L0,k(1 - δγp + εγ²p²) (Eq 4.5).
    Stated without derivation; not a consequence of the deformed commutator, but used to derive ladder-operator results.

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Cite this review

Pith. "Pith review of Deformed algebraic structure of angular momenta: GUP perspective." pith.science (2026). https://pith.science/paper/LHWOUOT5

@misc{pith2026241118901,
  author       = {Pith},
  title        = {Pith review of: Deformed algebraic structure of angular momenta: GUP perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHWOUOT5}},
  note         = {Machine review of arXiv:2411.18901}
}
read the original abstract

The prediction of a minimal length scale by various quantum gravity candidates (such as string/M theory, Doubly Special Relativity, Loop Quantum Gravity and others) have suggested modification of Heisenberg Uncertainty Principle (HUP), resulting in the Generalized Uncertainty Principle (GUP). In this short review, we investigate the origins of the GUP and examine higher-order models, focusing on the linear plus quadratic form of the GUP. We extend the concept of minimal length to minimal angular resolution, which plays a crucial role in modifying angular momentum and its associated algebra. A comparison is made between the standard angular momentum commutator algebra and that modified by the GUP. Finally, we review its application in the hydrogen atom spectra and and discuss future endeavors.

Figures

Figures reproduced from arXiv: 2411.18901 by the authors.

Figure 1
Figure 1. FIG. 1. In this plot, Figure (a) on the left shows a comparison between HUP and KMM GUP, where the minimal length is [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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