REVIEW 2 major objections 4 minor 21 references
Still learning about space dimensionality: from the description of hydrogen atom by a generalized wave equation for dimensions D$\geq$3
T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash
Pith's one-line read The hydrogen atom's measured ground-state energy, -0.5 hartree, is claimed to be a consequence of space being three-dimensional.
desk verdict The paper's central claim that hydrogen's measured -0.5 Ha selects D=3 and n=1 is not established; it rests on an unjustified m=n assumption and an approximation that gives -0.11 Ha for the key case. 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
The machinery is the parameterized family of wave equations $(-1)^n\Delta^n\psi - (\alpha/r^\beta)\psi=E\psi$ together with the generalized Poisson equation $(\Delta)^m G(r)=-4\pi\delta(r)$. The Green function fixes the potential as $\alpha(D,m)/r^{D-2m}$, so the Coulomb potential is one member of a larger class. The load-bearing step is identifying $m=n$, 'for consistency': it makes $\beta=D-2n$ and, with the requirement that the effective potential have a minimum, yields $2n<D<4n$. The $1/N$ expansion supplies the closed-form approximate energy for this family. The combination of the Green-function result, the $m=n$ identification, and the $1/N$ energy formula reduces the question 'why three dimensions?' to a search over integer pairs $(D,n)$.
What would settle it
Using the paper's own Eqs. (4) and (6), evaluate the predicted ground-state energy for every integer pair $2n<D<4n$ and compare with the measured $-0.5$ hartree. The central claim is falsified if any pair other than $(3,1)$ yields the measured value, or if an exact numerical solution of $(-1)^n\Delta^n\psi - \alpha(D,n)r^{-(D-2n)}\psi=E\psi$ for small admissible pairs places another pair within experimental precision of $-0.5$ hartree.
Extended reading notes
Core claim
The paper's central claim is that the threefold nature of space and the value of the hydrogen ground-state energy are two sides of the same parameter choice. For the generalized wave equation $(-1)^n\Delta^n\psi - (\alpha/r^\beta)\psi = E\psi$, the potential is fixed by the Green function of the iterated Poisson equation, giving $\alpha(D,m)/r^{D-2m}$. Setting $m=n$ for consistency, the ground-state energy from the $1/N$ expansion becomes a function of $(D,n)$, and the conditions $\beta>0$ and $E<0$ become $2n<D<4n$. The only integer pair with $D=3$ is $(3,1)$, the ordinary Schrödinger hydrogen atom; every other admissible pair predicts a bound state at least $10^3$ times less bound. The paper presents this as the first framework in which the measured value $-0.5$ hartree is understood as a consequence of $D=3$ and $n=1$, noting that the $1/N$ estimate at $N=3$ is only $-0.11$ hartree because of the method's limited accuracy at small $N$.
Load-bearing premise
The load-bearing premise is that the integer $m$ appearing in the Poisson equation for the potential is equal to the integer $n$ appearing in the wave equation; the paper assumes $m=n$ 'for consistency' rather than proving it, and all of its allowed-dimension conclusions depend on that equality.
Editorial extensions
If this is right
- In three-dimensional space, the only admitted hydrogen bound state has $n=1$; any other Laplacian power fails to give a negative-energy solution.
- Allowed pairs with $n>1$ predict ground-state binding energies at least $10^3$ times smaller than $0.5$ hartree, so such states would look like Rydberg atoms rather than ordinary hydrogen.
- Dimensions $D=4,5,6$ admit no bound hydrogen state at any $n$, because they fall outside the window $2n<D<4n$.
- The standard Schrödinger pair $(3,1)$ is also the most deeply bound of all allowed pairs, which the authors read as nature favoring both three-dimensional space and the ordinary Laplacian kinetic term.
Reading between the lines
- The uniqueness of $(3,1)$ is conditional on the assumed equality $m=n$; the paper's own alternative with $m=1$ admits $D=3,5,6,7$ for $n=3$, so the conclusion changes if that consistency choice is relaxed.
- The steep falloff in predicted energies with increasing $D$ implies that any surviving higher-dimensional hydrogen states would be extremely weakly bound; high-precision searches for near-threshold Rydberg-like lines in atomic spectra would be a direct way to test their absence.
- A sharper test would replace the $1/N$ approximation with exact numerical solutions for small $(D,n)$: the exact $(3,1)$ energy is the textbook $-0.5$ hartree, and the key check is that every other admissible pair stays orders of magnitude less bound.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes to study the hydrogen atom in D-dimensional space using a generalized wave equation with an iterated Laplacian Δ^n and a Coulomb-like potential r^{-β}, Eq. (1). Starting from a 1/N expansion result taken from Ref. [8], the authors fix the potential through the generalized Poisson equation (3) and its Green function (4), and then impose m = n 'for consistency' to obtain the ground-state energy E0(D, n), Eq. (6). From the constraint 2n < D < 4n they conclude that the only bound state in D = 3 has n = 1, and that the measured value -0.5 Ha is a consequence of D = 3 and n = 1. Table 1 lists E0 for several (D, n) combinations.
Significance. If the central claim were established, the paper would provide a striking connection between the dimensionality of space and the numerical value of a physical constant, going beyond Ehrenfest's stability argument. The manuscript has the merit of making the parameter dependence explicit, deriving the generalized potential from the Green function, and including the alternative m = 1 case as a comparison. However, the conclusion rests on an unjustified identification m = n, and the numerical result for the physically relevant case does not match the measured value. The paper's own equations show that the special status of (D = 3, n = 1) disappears when m = 1, so the significance claimed in the abstract is not realized in the present form.
major comments (2)
- [Equations (3), (4), (6), (9) and summary item 1] The identification m = n is an assumption, not a consequence of the equations. The power n of the iterated Laplacian in the wave equation (1) and the power m in the generalized Poisson equation (3) are logically independent; Eq. (4) fixes the potential for each m but does not constrain the kinetic term. The manuscript asserts m = n 'for consistency' without proof. This assumption is load-bearing because it determines which (D, n) pairs are allowed. Indeed, with m = 1, Eq. (9) gives E0 < 0 for D < 2(n + 1), so for n = 2 the case D = 3 is admitted (2 < D < 6), directly contradicting summary item 1, which states that the only possibility of a bound state in D = 3 is n = 1. Thus the uniqueness of (D = 3, n = 1) is an artifact of the choice m = n rather than a property of the generalized wave equation.
- [Table 1 footnote and Eq. (6)] For the physically relevant case (D = 3, n = 1), Eq. (6) yields E0 = -0.11 Ha, not the measured -0.5 Ha. The footnote attributes the discrepancy to the modest accuracy of the 1/N expansion when N = 3, but the final paragraph nevertheless claims that the measured value -0.5 Ha is 'a consequence' of D = 3 and n = 1. No derivation of the measured value is supplied, and D = 3 is precisely the low-N case where the 1/N approximation is least reliable. Consequently, the numerical part of the central claim is unsupported as stated.
minor comments (4)
- [Abstract and summary] There are several typographical errors: 'generalizatio n' in the abstract, 'in summmary' before the numbered list, 'seams' should be 'seems', and 'proporcional' should be 'proportional'.
- [Eq. (4)] The domain of validity of Eq. (4) should be stated explicitly: for the gamma function Γ(D/2 - m) to be well defined one needs D/2 - m not to be a non-positive integer, which is essentially the condition β = D - 2m ≥ 0. This is presumably intended but not stated.
- [Eq. (6)] The derivation of Eq. (6) from Eqs. (2) and (4) is not shown; since this formula is central to the numerical results, a few lines of algebra would help the reader verify the substitution β = D - 2n.
- [References] Reference [6] is incomplete: the paper by Tangherlini is missing the year, and the word 'in' in the title is misspelled as 'inn'.
Circularity Check
The uniqueness of (D=3,n=1) is an artifact of imposing m=n, and the claimed -0.5 Ha value is imported, not derived.
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other
[After Eq. (5), before Eq. (6): the assumption m=n]
"Inspired in what happens in 3 dimensions, we will assume throughout the Letter the validity of Gauss’ law and, for consistency, that m = n. So, the ground state energy , associated with equation (1) for a potential given by equation (4), is obtained by substitutingβ = D − 2n in equation (2), yielding [Eq. (6)]"
The paper's entire selection of (D=3,n=1) follows from the unproved identification m=n. Equations (1) and (3) have logically independent powers; Gauss' law and the Green function (4) fix the potential for each m but do not constrain the kinetic term's n. The paper itself shows the alternative m=1 leads to Eq. (9) with E0<0 for D<2(n+1), so for n=2, D=3 is allowed, contradicting summary item 1. Thus the 'only n=1 in D=3' result is not a consequence of the generalized wave equation but of the ad hoc constraint m=n; the central prediction reduces to this assumption by construction.
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renaming known result
[Table 1 footnote and final paragraph]
"(∗) This value is the same found in Ref. [9] and should be compared to the well known result −0.5 Ha. The discrepancy found here is due to the quite modest accuracy obtained by the 1/N expansion when in the sequel one puts N = 3. However, this fact does not invalidate the analysis made in the text since the accuracy of the method is much better for high N values. ... the measured value for the energy of the hydrogen atom ground state, −0.5 Ha, is a consequence of the very particular combination D = 3 and n = 1."
Eq. (6) evaluated at (D=3,n=1) gives −0.11 Ha, not −0.5 Ha. The paper dismisses this discrepancy as 1/N inaccuracy, then claims the measured −0.5 Ha is 'a consequence' of D=3,n=1. No calculation in the paper produces −0.5 Ha; the value is imported from the standard Schrödinger hydrogen atom and re-labeled as a framework output. The value part of the central claim is therefore the known result, not a derived prediction from the generalized equation.
full rationale
The paper is not dominated by self-citation: the energy formula Eq. (2) comes from external Ref. [8] and the Green function Eq. (4) from Ref. [16]. The circularity is instead in the construction of the central conclusion. First, the authors impose m=n 'for consistency' with no derivation, and the allowed-region inequality 2n<D<4n then makes (D=3,n=1) unique. The paper itself shows that with the equally standard choice m=1 the constraints become 2<D<2(n+1), which admits D=3 for n=2; hence the uniqueness of n=1 in D=3 is an artifact of the assumed identification, not a robust prediction of the generalized wave equation. Second, the claim that the measured value −0.5 Ha is explained by D=3,n=1 is not supported by Eq. (6), which gives −0.11 Ha at that point; the disagreement is waived by a footnote, and the exact value is effectively taken from the known Schrödinger solution. These two steps make the central claim substantially circular, though the treatment is transparent about the m=n assumption and does not rely on self-citation chains.
Assumptions & free parameters
free parameters (1)
- m = n
assumptions (3)
- domain assumption The 1/N expansion ground-state energy formula (Eq. 2) from Ref. [8] is valid for all dimensions D, including D=3 where the method's accuracy is poor.
- domain assumption Gauss' law is valid in D dimensions, and the generalized Poisson equation (3) with an iterated Laplacian is the correct way to generalize the Coulomb potential.
- ad hoc to paper The equality m=n is a valid restriction of the parameter space.
Cite this review
Pith. "Pith review of Still learning about space dimensionality: from the description of hydrogen atom by a generalized wave equation for dimensions D$\geq$3." pith.science (2026). https://pith.science/paper/HRMLMO6Z
@misc{pith2026200913473,
author = {Pith},
title = {Pith review of: Still learning about space dimensionality: from the description of hydrogen atom by a generalized wave equation for dimensions D$\geq$3},
year = {2026},
howpublished = {\url{https://pith.science/paper/HRMLMO6Z}},
note = {Machine review of arXiv:2009.13473}
}
abstract
The hydrogen atom is supposed to be described by a generalization of Schr\"{o}dinger equation, in which the Hamiltonian depends on an iterated Laplacian and a Coulomb-like potential $r^{-\beta}$. Starting from previously obtained solutions for this equation using the $1/N$ expansion method, it is shown that new light can be shed on the problem of understanding the dimensionality of the world as proposed by Paul Ehrenfest. A surprisingly new result is obtained. Indeed, for the first time, we can understand that not only the sign of energy but also the value of the ground state energy of hydrogen atom is related to the threefold nature of space.
Reference graph
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Reviewed August 27, 2026 · model on record in the stance chip above.
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