Stark Energy Shifts due to Quantum Gravity in RGUP Algebra

Gaurav Bhandari; S. D. Pathak

REVIEW 4 major objections 6 minor 1 cited by

Stark Energy Shifts due to Quantum Gravity in RGUP Algebra

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Applying a relativistic generalized uncertainty principle to hydrogen produces beta-dependent Stark energy shifts and bounds the deformation parameter.

desk verdict Routine RGUP-Stark calculation whose beta-dependent shifts and bound rest on an ad hoc electric-field rescaling that minimal coupling does not support. read the letter →

arxiv 2505.12985 v1 pith:UPWIHI3S submitted 2025-05-19 gr-qc hep-thmath-phmath.MPquant-ph

Gaurav Bhandari , S. D. Pathak This is my paper · ORCID

classification gr-qchep-thmath-phmath.MPquant-ph PACS 04.60.-m32.60.+i

keywords relativisticgeneralizeduncertaintyprincipleStarkeffecthydrogenatomminimallengthquantumgravityphenomenologyenergyshiftspolarizabilityboundRGUPparameter

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 extends the relativistic generalized uncertainty principle (RGUP), a deformation of quantum mechanics that enforces a minimal length while preserving Lorentz invariance, to the Stark effect in hydrogen. It claims that RGUP corrections introduce terms proportional to the RGUP parameter $\beta$ into the Stark energy shifts for both the ground state and the $n=2$ degenerate manifold, so the usual zero linear shift for the ground state becomes nonzero and the degenerate splitting acquires $\beta$-dependent contributions. It further uses the measured polarizability of atomic hydrogen to derive an upper bound $\beta < 10^{42}$, weaker than some non-relativistic bounds but obtained in a relativistic framework. If correct, the calculation identifies the Stark effect as a phenomenological probe of quantum gravity corrections in atomic spectra.

What carries the argument

The argument runs on a single perturbing operator: $V_{\text{RGUP}} = -2\beta(mc)^2\left(\frac{p^2}{2m} - \frac{p^4}{8mc^2}\right) - eEz\left(1 - \beta(mc)^2\right)$, built from the deformed momentum operator and the rescaled electric field. The parameter $\beta = \epsilon \gamma^2$ is the RGUP deformation strength, with $\gamma$ inversely proportional to the Planck mass. The paper then applies standard non-degenerate and degenerate perturbation theory to hydrogen states, using the Stetsko-Tkachuk approximation, which keeps only first-order terms in the deformation parameter and drops $O(\beta^2)$ contributions.

What would settle it

Measure the $n=2$ linear Stark splitting of atomic hydrogen with enough precision to resolve the predicted $\beta$-dependent correction $3e|E|a_0\sqrt{1 - 2c^2 m^2 \beta}$; if no such correction appears at the level set by the derived bound, Eq. (3.35) is falsified.

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Extended reading notes

Core claim

The central claim is that replacing the momentum operator with the RGUP-deformed momentum $p^\mu = p_0^\mu\left(1 + \beta p_0^\rho p_{0\rho}\right)$ and the electric field with $E_{\text{RGUP}} = (1 - \beta(mc)^2)E$ changes the Stark Hamiltonian of hydrogen. In first-order perturbation theory the paper obtains a ground-state linear shift $-2\beta(mc)^2\left(\frac{\hbar^2}{2m_e a_0^2} - \frac{5\hbar^4}{8m_e c^2 a_0^4}\right)$, a raised lower bound for the quadratic Stark shift, and $n=2$ degenerate shifts that mix the standard linear Stark term with $\beta$-dependent terms, including eigenvalues of the form $-2\beta(mc)^2(M+P) \pm 3 e |E| a_0 \sqrt{1 - 2 c^2 m^2 \beta}$. All $\beta$-dependent corrections vanish when $\beta \to 0$, recovering the standard Stark effect, and the non-relativistic limit $c \to \infty$ recovers earlier minimal-length GUP results.

Load-bearing premise

The load-bearing premise is that the RGUP-deformed derivative implies the physical electric field itself is rescaled as $E_{\text{RGUP}} = (1 - \beta(mc)^2)E$; if that rescaling is not the physical field, every $\beta$-dependent Stark shift loses its foundation.

Editorial extensions

If this is right

The ground-state hydrogen Stark shift is predicted to have a nonzero linear term proportional to $\beta$, vanishing only when $\beta = 0$.
The lower bound on the quadratic Stark shift is raised by a factor $(1 - 2\beta(mc)^2)$, so RGUP makes the field-induced energy shift slightly larger in magnitude.
The $n=2$ degenerate manifold splits into eigenvalues $0$, $-4\beta(mc)^2 P$, and $-2\beta(mc)^2(M+P) \pm 3e|E|a_0 \sqrt{1 - 2c^2 m^2 \beta}$, replacing the pure $\pm 3e|E|a_0$ linear Stark splitting.
The measured polarizability of atomic hydrogen sets the upper bound $\beta < 10^{42}$, a bound derived in the relativistic RGUP framework rather than the non-relativistic GUP framework.
All results reduce to the standard Stark effect as $\beta \to 0$ and to the non-relativistic GUP results as $c \to \infty$.

Reading between the lines

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

The beta dependence everywhere enters through the field rescaling $E_{\text{RGUP}} = (1 - \beta(mc)^2)E$; if that identification is relaxed, the predicted shifts change substantially, so a direct derivation of the rescaling from the RGUP algebra would settle the model's uniqueness.
High-precision Stark spectroscopy of hydrogen, particularly on the $n=2$ level, could in principle test the predicted beta coefficient in Eq. (3.35), though the derived bound $\beta < 10^{42}$ is too weak for observable deviations with laboratory fields.
The same Lagrangian-comparison method could be applied to other electromagnetic observables such as the Zeeman effect, AC Stark shifts, and transition amplitudes, yielding a family of beta-dependent predictions that can be checked for mutual consistency.
If a positive signal appeared, comparing the relativistic RGUP bound with non-relativistic GUP bounds would help distinguish minimal-length models that preserve Lorentz invariance from those that do not.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Punchline: this is a routine exercise in GUP phenomenology, and its central result—the beta-dependent Stark shifts and the beta<10^42 bound—rests on a field rescaling that is asserted rather than derived. I would not send this to a referee as is.

What is actually new: the specific application of the relativistic GUP (RGUP) to the Stark effect. The paper correctly recovers the standard Stark results and non-relativistic GUP limits, and the perturbation-theory setup is mostly clean. The degenerate n=2 matrix calculation is a useful worked example. The authors also compare their bound with existing GUP bounds, which is honest context.

The soft spots are not minor. The step that carries the whole paper is Eq. (3.6), where comparing the coefficient of E^2 in the deformed and undeformed electrostatic Lagrangians defines E_RGUP = (1 - beta (mc)^2) E. That is a renormalization of the field energy, not a derivation of the physical field coupling to the electron. Apply minimal coupling to Eq. (2.17) and the interaction is -e E z with no prefactor. Every E-dependent RGUP correction, including the polarizability bound (3.22) and the degenerate shifts (3.35), inherits this unsupported factor. The kinetic-term corrections are real, but they do not produce the claimed linear Stark modifications or the beta bound.

There are also smaller errors. Eq. (3.24) misses a factor of two in the denominator; as written the bound is numerically inconsistent (the quoted 10^42 does not follow from the stated numbers). The text after Eq. (3.23) says RGUP raises the lower limit of the shift, which is true but trivial because the prefactor is less than one; it is not a measurable effect without knowing beta. And the conclusion claims a previous RGUP Zeeman analysis that is not in the cited reference [35].

The paper shows competent algebra and no circular fitting: beta is an input and the bound is a one-sided estimate. But the load-bearing assumption makes the main result unsupported as presented. The factor-of-two issues are fixable; the rescaling is not, unless the authors can derive it from a gauge-invariant minimal-coupling prescription.

Who this is for: someone cataloguing GUP phenomenological bounds might want to read it as a cautionary example, but not to cite the bound. My recommendation: desk reject with a clear explanation that the field-rescaling step must be justified or removed. If they can fix that, the paper could be a minor contribution to the RGUP phenomenology literature, though the resulting bound would still be weaker than existing ones.

Referee Report

4 major / 6 minor

Summary. The paper computes quantum-gravity corrections to the Stark effect in hydrogen by working with a relativistic generalized uncertainty principle (RGUP) in Minkowski spacetime. Using the Stetsko-Tkachuk approximate representation, the authors modify the momentum operator, define a modified electric field, and derive an RGUP-perturbed Hamiltonian. They then compute the ground-state (n=1) energy shift and polarizability bound, extract an upper bound on the RGUP parameter beta, and calculate the degenerate n=2 Stark shifts from a 4x4 matrix. The paper claims that the results reduce to the standard Stark effect and to non-relativistic GUP results in the appropriate limits.

Significance. If the derivation were sound, the paper would offer a relatively simple phenomenological application of a relativistic GUP to a textbook quantum system and an order-of-magnitude constraint on the deformation parameter beta. The treatment is self-contained in the sense that beta is an input parameter rather than fitted from the Stark data, and the paper explicitly checks the beta->0 and c->infinity limits. However, the significance is currently limited by a load-bearing assumption in the definition of the modified electric field and by several algebraic errors in the central equations. The quoted bound beta<10^42 is also very weak compared with constraints already in the literature, as the authors themselves note.

major comments (4)

[Sec. III, Eq. (3.6)] The modified electric field E_RGUP = (1 - beta (mc)^2) E is introduced by comparing the coefficient of E^2 in the free electrostatic Lagrangian before and after replacing the derivative with D_mu = (1 - beta (mc)^2) partial_mu. This operation rescales the field energy; it does not derive the physical field that couples to the electron. Under the standard minimal-coupling prescription p_mu -> p_mu - e A_mu, with A=0 for a static electric field, the deformed momentum of Eq. (2.17) leaves the interaction -e E z unchanged at O(beta). Every beta-dependent term in Eqs. (3.8), (3.19), (3.22)-(3.24), and (3.33)-(3.35) inherits this unproven factor. The central claim is therefore unsupported unless Eq. (3.6) can be derived from the RGUP algebra or from a gauge-invariant action principle.
[Sec. III.1, Eq. (3.24)] Solving Eq. (3.22) for beta gives beta < [1/(2 (mc)^2)] (1 - 3 alpha_p / (16 a0^3)), not beta < [1/(mc)^2] (1 - 3 alpha_p / (16 a0^3)). The missing factor of 1/2 is an algebraic error. Although the numerical bound remains of order 10^42, the printed inequality is incorrect and the comparison with other bounds in Table I should be based on the corrected expression.
[Sec. III.1, Eqs. (3.14) and (3.16)] The completeness replacement in Eq. (3.14) is not correct as written: the sum over n != 1 should equal the sum over all states minus the n=1 contribution, |<1,0,0| V'_RGUP |1,0,0>|^2. The omitted term is O(beta^2) and may be negligible at the intended order, but the equality in Eq. (3.14) is false. In Eq. (3.16), the expansion of (V'_RGUP)^2 is also written incorrectly: the crossed term should involve z times the kinetic operator, not the separate terms 4 beta (mc)^2/(e|E|) <z> and 4 beta (mc)^2/(e|E|) <(nabla^2 z)/(2m) - ...>. The final result Eq. (3.19) may survive because the crossed term has odd parity and vanishes in the spherically symmetric ground state, but the derivation as printed needs to be repaired.
[Sec. III.2, Eqs. (3.33)-(3.35)] The degenerate-state eigenvalues contain factor-of-two errors. The diagonal element in Eq. (3.28) is -2 beta (mc)^2 P, so the decoupled states |2,1,-1> and |2,1,1> have eigenvalue -2 beta (mc)^2 P, not -4 beta (mc)^2 P as written in Eq. (3.34). For the 2x2 block mixing |2,0,0> and |2,1,0>, the average of the diagonal elements is -beta (mc)^2 (M+P), so the eigenvalues are -beta (mc)^2 (M+P) +/- 3 a0 e |E| sqrt(1 - 2 beta (mc)^2) to first order in beta, not -2 beta (mc)^2 (M+P) +/- ... as in Eq. (3.35). These errors propagate into the conclusion and need to be corrected before the degenerate shifts can be quoted.

minor comments (6)

[Abstract and Introduction] The abstract contains grammatical issues, including 'on beta the RGUP parameter' and 'enfold quantum gravitational effects'; these should be cleaned up.
[Sec. III.1, Eq. (3.16)] The notation alternates between eE and e|E| in the same expression; the vector nature of the field should be handled consistently.
[Sec. III.2, Eq. (3.32)] The last row of the matrix in Eq. (3.32) has a trailing comma after '-2 beta (mc)^2 P', which appears to be a typographical error.
[Sec. III.2, Eq. (3.35)] The expression in Eq. (3.35) is ambiguous because of the '/2' at the end; the intended numerator/denominator structure should be written unambiguously.
[References] References [47] and [49] are the same paper by Stetsko and Tkachuk and should be distinguished, and reference [5] contains the typo 'minimal of minimal length'.
[Sec. III.1, Eq. (3.23)] The quantity -8 a0^3 |E|^2 / 3 is a lower bound obtained from the n=2 denominator, not the exact standard quadratic Stark shift; the exact ground-state value is -9 a0^3 |E|^2 / 4, so the text's reference to a 'standard energy shift expression' is misleading.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: beta is an input RGUP parameter, the Stark shifts are computed from it, and the polarizability bound is a standard parameter constraint; the Eq. (3.6) field rescaling is a support gap, not a circular reuse.

full rationale

The derivation is self-contained rather than circular. The parameter beta = epsilon gamma^2 is introduced as the deformation parameter of the RGUP algebra (Eqs. (2.14)-(2.17)) and is not fitted to Stark-effect data; the energy shifts (3.11), (3.19), (3.33)-(3.35) are obtained from the perturbed Hamiltonian (3.8) by ordinary perturbation theory and reduce to standard results at beta = 0. The upper bound (3.24)-(3.25) is a parameter-estimation step: the measured polarizability alpha_p from ref. [53] is an independent experimental input used to constrain beta, not a quantity predicted from beta, so no fitted input is renamed a prediction. The references to the authors' own prior work ([25]-[27], [30], [35], [58]) are contextual and play no load-bearing role; in particular, no uniqueness theorem or result from those papers is invoked to force the present choice. One caveat, of the correctness type rather than the circularity type, is that Eq. (3.6) obtains E_RGUP by comparing coefficients in the free electrostatic Lagrangian and then inserts that same factor into the coupling -eEz in Eq. (3.8); the paper does not derive this field rescaling from the RGUP commutators or from minimal coupling. This is an unsupported modeling step, but it is not an output being recycled as an input: the Stark shifts are consequences of the stated Hamiltonian, and beta remains a free parameter throughout.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new particles or fields are introduced. The 'modified electric field' is a rescaled version of the standard field, not a new entity.

free parameters (1)

beta (epsilon gamma^2) = < 10^42 (upper bound)
The RGUP deformation parameter appearing in all shifts and in the modified electric field. The paper constrains it using the experimental polarizability of hydrogen, Eq. (3.24).

assumptions (5)

domain assumption RGUP operator algebra from Todorinov, Bosso, Das (2019), Eqs. (2.9)-(2.17), with alpha' = xi = alpha = 0.
The modified commutation relations and operator representations are taken from prior work without proof.
domain assumption Stetsko-Tkachuk first-order approximation, neglecting O(beta^2) and higher terms.
The paper keeps only linear order in beta throughout, which limits validity to very small deformation parameters.
domain assumption Hydrogen Hamiltonian H = p^2/(2m) - p^4/(8m c^2) + U as the unperturbed relativistic-corrected Hamiltonian.
The relativistic kinetic correction is expanded to order 1/c^2; this is standard but an assumption about the regime.
ad hoc to paper The modified electric field rescaling E_RGUP = (1 - beta (mc)^2) E.
Derived in Eq. (3.6) by comparing Lagrangians; not implied by the RGUP algebra, it is a modeling choice.
domain assumption Replacement of the intermediate-state sum in second-order perturbation theory by the ground-state expectation value of V'^2.
This is an approximation that yields a bound (using the minimum energy denominator), not an exact evaluation; the paper uses it to bound alpha_p and beta.

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Cite this review

Pith. "Pith review of Stark Energy Shifts due to Quantum Gravity in RGUP Algebra." pith.science (2026). https://pith.science/paper/UPWIHI3S

@misc{pith2026250512985,
  author       = {Pith},
  title        = {Pith review of: Stark Energy Shifts due to Quantum Gravity in RGUP Algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UPWIHI3S}},
  note         = {Machine review of arXiv:2505.12985}
}

abstract

In this paper, we investigate the Stark effect in the hydrogen atom under an external electric field, incorporating relativistic generalized uncertainty principle (RGUP) corrections within Minkowskian spacetime and calculate the upper bound on $\beta$ the RGUP parameter. Employing RGUP algebra and the Stetsko-Tkachuk approximation, we derive modifications to the energy spectrum for degenerate and non-degenerate states. The perturbed Hamiltonian, modified by RGUP, enfold quantum gravitational effects. Our results reveal quantum gravitational corrections to the Stark energy spectrum in the relativistic regime, with energy shifts for non-degenerate ($n=1$) and degenerate ($n \neq 1$) cases showing additional terms proportional to $\beta$. These findings reduce to standard Stark effect results and non-relativistic GUP frameworks in the limits $\beta\rightarrow 0$ and $c \rightarrow \infty $, establishing our model as a generalized framework for analyzing minimal length effects in relativistic quantum systems.

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Energy Shifts for the Ground State under RGUP Corrections ( n = 1) For the non-degenerate case, the energy shift is calculated using the RGUP-perturbed Hamiltonian (Eq. 3.7), where the RGUP terms are treated as perturbative corrections. The energy shift for the ground state of the hydrogen atom 5 is then expressed as Eshift =−2β(mc)2 ⟨1, 0, 0|∇2|1, 0, 0⟩ ...

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Reviewed August 15, 2026 · model on record in the stance chip above.

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Energy Shifts for the Ground State under RGUP Corrections ( n = 1) For the non-degenerate case, the energy shift is calculated using the RGUP-perturbed Hamiltonian (Eq. 3.7), where the RGUP terms are treated as perturbative corrections. The energy shift for the ground state of the hydrogen atom 5 is then expressed as Eshift =−2β(mc)2 ⟨1, 0, 0|∇2|1, 0, 0⟩ ...

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(3.19) We also know that the energy for nth orbit is given as En =− e2 2a0n2, (3.20) 6 and we can write −E1 +En = e2 2a0 1− 1 n2 ≥−E1 +E2 = 3 8 e2 a0 . (3.21) From Eqns.(3.13,3.19,3.21), we find the upper limit for modified polarizability parameter as αp < (1− 2β(mc)2)16a3 0 3 , (3.22) and the lower limit for the shift ∆shift >−(1− 2β(mc)2)8a3 0 3 |⃗E|2. ...

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