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REVIEW 3 major objections 6 minor 30 references

Algebraic structures and deformed Schr\"{o}dinger equations from groups entropies

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Pith's one-line read Every group entropy $G$ produces a canonically deformed quantum mechanics with $[\hat{x}_G,\hat{p}_G]=i\hbar$, and the $q$- and $\kappa$-deformed equations emerge as special cases.

desk verdict A genuinely useful unification of the q- and kappa-deformed algebras under the group-entropy umbrella, with a clean G-calculus and a defensible but not uniquely fixed deformed Schrödinger equation; refereeable after fixing a few concrete errors. read the letter →

arxiv 1908.02785 v1 pith:Y3QHGGNE submitted 2019-08-07 math-ph math.GRmath.MPquant-ph

Ignacio S. Gomez , Ernesto P. Borges This is my paper · ORCID
classification math-phmath.GRmath.MPquant-ph MSC 81Q0581R5082B30
keywords groupentropyG-algebraG-derivativeG-deformedSchrödingerequationposition-dependentmassq-algebrakappa-algebraformallaw
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 tries to establish a single algebraic origin for the deformed structures used in nonextensive statistics: every group entropy $G$ defines a $G$-algebra, with $G$-sum $x\oplus_G y = G(G^{-1}(x)+G^{-1}(y))$, a $G$-product, and an associated $G$-differential calculus. From this calculus the position operator becomes $\hat{x}_G=G^{-1}(\hat{x})$ and the momentum operator $\hat{p}_G=\frac{1}{2}\{G'(G^{-1}(\hat{x})),\hat{p}\}$, and the paper proves that these satisfy the canonical commutation relation $[\hat{x}_G,\hat{p}_G]=i\hbar$. The resulting $G$-deformed Schrödinger equation is equivalent to a position-dependent-mass equation with $m(x)=m_0/[G'(G^{-1}(x))]^2$, so the $q$- and $\kappa$-deformed equations emerge as special cases and ordinary quantum mechanics as the Boltzmann-Gibbs case. A sympathetic reader would care because this turns a family of separately built deformed calculi into one $G$-parametrized structure whose physical content is a choice of effective mass.

What carries the argument

The load-bearing object is the Lazard formal group law expressed through a group entropy $G$: $\Phi(x,y)=G(G^{-1}(x)+G^{-1}(y))$, which supplies deformed addition, subtraction, multiplication, division, and their neutral and inverse elements. On this the paper builds the $G$-algebra and a $G$-calculus whose $G$-derivative is $D_G=G'(G^{-1}(x))\,d/dx$. The identity that carries the quantum argument is $[\hat{x}_G,\hat{p}_G]=i\hbar$ with $\hat{x}_G=G^{-1}(\hat{x})$ and $\hat{p}_G=\tfrac12\{G'(G^{-1}(\hat{x})),\hat{p}\}$; because $A=G'(G^{-1}(x))$ controls both the deformed derivative and the mass $m_0/A^2$, the entire deformed Schrödinger dynamics is fixed once $G$ is chosen.

What would settle it

Compute the commutator for an alternative Hermitian ordering, e.g. $\hat{p}_G^{(\delta)}=\tfrac12\{[G'(G^{-1}(\hat{x}))]^{1+\delta},\hat{p}\}$ with $\delta\neq 0$. The result is not $i\hbar$ but $i\hbar [G'(G^{-1}(\hat{x}))]^{\delta}$, so the canonical-transformation claim and the paper's $G$-deformed Schrödinger equation fail for that equally natural ordering. A measurement of the predicted asymmetric probability accumulation in the infinite well, toward the high-mass region, would select which ordering is physical.

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Extended reading notes

Core claim

The central claim is that the formal group law of group entropy theory, $\Phi(x,y)=G(G^{-1}(x)+G^{-1}(y))$, carries enough structure to deform both algebra and quantum mechanics coherently. Replacing $x$ by $x_G=G^{-1}(x)$ and $\hat{p}$ by the Hermitian $\hat{p}_G=\frac{1}{2}\{G'(G^{-1}(\hat{x})),\hat{p}\}$ preserves the canonical commutator, and the Hamiltonian built from $\hat{p}_G^2$ gives the $G$-deformed Schrödinger equation, whose only memory of the deformation is the factor $A(x)=G'(G^{-1}(x))$. In the deformed coordinate $x_G$, the same equation is just the constant-mass Schrödinger equation with $G$-derivative $D_G=A(x)\,d/dx$, so the physical content is a position-dependent mass $m(x)=m_0/A(x)^2$. The paper demonstrates the scheme on the infinite potential well: eigenfunctions are sines of $x_G$, energies scale with $1/[G^{-1}(L)]^2$, and the non-uniform spacing of zeros is generated by the $G$-sum, with explicit formulas for the Tsallis and Kaniadakis classes.

Load-bearing premise

The paper assumes, without derivation, that the Hermitian ordering of the deformed momentum is $\hat{p}_G=\tfrac12\{G'(G^{-1}(\hat{x})),\hat{p}\}$; a different Hermitian ordering would change the commutator and the entire deformed Schrödinger equation, and no physical principle fixes the choice beyond convenience.

Editorial extensions

If this is right

  • For every group entropy $G$, the operators $\hat{x}_G=G^{-1}(\hat{x})$ and $\hat{p}_G=\tfrac12\{G'(G^{-1}(\hat{x})),\hat{p}\}$ form a canonical pair, so the $G$-deformed quantum mechanics inherits the algebraic guarantees of standard quantum mechanics.
  • Choosing the Tsallis or Kaniadakis group class recovers, respectively, the $q$-algebra/$\kappa$-algebra and the $q$-deformed/$\kappa$-deformed Schrödinger equations, making the $G$-algebra the common parent structure.
  • The $G$-deformed Schrödinger equation is exactly a position-dependent-mass equation with $m(x)=m_0/[G'(G^{-1}(x))]^2$; equivalently, in the $G$-deformed coordinate $x_G$ it is a constant-mass Schrödinger equation.
  • For the infinite potential well, the eigenfunctions' zeros are spaced by the $G$-sum of the group law, and deformation breaks the density's symmetry, concentrating probability where the effective mass is largest.
  • In the classical limit the deformed probability densities tend to a delta distribution near $x=0$ for the Tsallis and Kaniadakis classes, while the standard case tends to uniform.

Reading between the lines

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

  • One could use the relation between an arbitrary group entropy $G$ and the effective-mass profile $m(x)$ to fit deformed quantum models to experimental position-dependent-mass systems; a measured $m(x)$ would then identify the group class.
  • The equivalence between a constant-mass problem in $x_G$ coordinates and a deformed problem in $x$ coordinates suggests a generator of new exactly solvable position-dependent-mass potentials: choose any solvable potential in $x_G$ and any increasing group entropy $G$.
  • The zeros-spacing formula could serve as a spectral fingerprint: counting and locating nodes of measured or calculated wavefunctions could discriminate between Tsallis and Kaniadakis deformations even when the probability densities look similar.
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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

3 major / 6 minor

Summary. The manuscript introduces a 'G-algebra' and associated 'G-calculus' built from the formal group law (group entropy) G, with deformed sum, product, derivative, and integral operations. It then proposes deformed quantum operators x̂_G = G^{-1}(x̂) and a Hermitian p̂_G = (1/2){G'(G^{-1}(x̂)), p̂}, verifies the commutator [x̂_G, p̂_G] = iℏ, and derives a G-deformed Schrödinger equation (Eq. 54). The equation is rewritten as a constant-mass equation in G-coordinates (Eq. 63), and the infinite potential well is solved, yielding a position-dependent effective mass m_G(x) = m_0/[G'(G^{-1}(x))]^2 and non-uniform zero-spacing formulas for the Tsallis and Kaniadakis classes (Eqs. 79-80). The paper claims this is a unified framework that recovers the known q- and κ-algebras and q- and κ-deformed Schrödinger equations as special cases.

Significance. If the construction were unique and the calculations correct, the paper would provide a useful unified algebraic umbrella for deformed quantum mechanics and position-dependent mass models. The algebra/calculus part is mostly clean, the special-case reductions to q- and κ-structures are worked out in detail, and the transformation leading to Eq. (63) is algebraically consistent. However, the central quantum-mechanical claim is weakened by an operator-ordering ambiguity: the deformed momentum is not uniquely fixed by the commutator, so the G-deformed Schrödinger equation, its spectrum, and the zero-spacing predictions are not determined by the group entropy G alone. In addition, the Kaniadakis zero-spacing expression (Eq. 80) contains a sign error. These issues are load-bearing for the paper's physical conclusions and require revision.

major comments (3)
  1. [Section 3.3, Eqs. (46)-(54)] The deformed momentum operator is not fixed by the canonical commutation relation. For any real smooth function h, the operator p̂'_G = (1/2){G'(G^{-1}(x̂)), p̂} + h(x̂) is Hermitian and still satisfies [G^{-1}(x̂), p̂'_G] = iℏ, because h(x̂) commutes with G^{-1}(x̂). Inserting p̂'_G into the Hamiltonian changes Eq. (54) by additional terms involving h and its derivatives, and therefore changes the eigenenergies and the zero-spacing formulas (76)-(80). The paper supplies no physical ordering principle, such as minimal coupling, equivalence-principle constraints, or a derivation from a Lagrangian, that selects h = 0. Consequently the G-deformed Schrödinger equation and its physical predictions are not determined by the group entropy G alone; they are conditional on the symmetrization choice made in Eq. (46). This is load-bearing because the concluding claims about the effective mass being 'univocally determined' and about deformed spectra and probability accumulation refer to this specific h = 0 equation. The effective-mass coefficient A(x)^2 is robust, but the full equation is not.
  2. [Section 3.4, Eq. (80)] The Kaniadakis zero-spacing formula is incorrect. With S = κL + sqrt((κL)^2 + 1), the m-th zero is x_m = (S^{m/n} - S^{-m/n})/(2κ), so the spacing is Δ(m)_κ = x_m - x_{m-1} = (1/(2κ))[S^{m/n} - S^{(m-1)/n} - S^{-m/n} + S^{-(m-1)/n}]. The printed formula has +S^{(m-1)/n} in the first bracket and, after distributing the minus sign of the second bracket, effectively +S^{-(m-1)/n}; this does not equal the difference of consecutive zeros. The Tsallis formula (Eq. 79) is correct, but the Kaniadakis illustration and any comparison based on Eq. (80) need correction.
  3. [Section 3.4, Eqs. (70)-(71)] The infinite-well solution assumes that x_G = G^{-1}(x) is an increasing diffeomorphism on [0, L] and that the potential boundaries transform as stated. The paper mentions this assumption and notes it holds for the Tsallis and Kaniadakis classes, but it does not prove monotonicity of G^{-1} for a general group entropy from the defining formal power series (Eq. 2), nor does it state the required conditions on the coefficients a_k. Since the abstract and introduction claim a general G-parametrized construction, the well example and the zero-spacing formulas are not justified for arbitrary G without either a proof of monotonicity or an explicit restriction of the scope to classes for which G^{-1} is monotone on the relevant interval.
minor comments (6)
  1. [Section 3, item (h)] The inverse of the G-product is written as ⊘_G x = exp_G(−log_G(y)), but the argument should be x; the variable y is undefined and the intended expression is exp_G(−log_G(x)).
  2. [Section 3.2.2, item (fK)] The inverse of the κ-sum is given by the q-dependent formula −x/(1 + (1−q)x), which is a copy of the q-algebra result; for the Kaniadakis class the inverse element is ⊖_κ x = −x, since x ⊕_κ (−x) = 0.
  3. [Footnote 1] The footnote says the standard case reduces to uniform zero spacing Δ(m) = L/m, but for G(t) = t the spacing is L/n, independent of m.
  4. [Eq. (74)] The index n = 0 should be excluded from the eigenfunction/eigenenergy labeling; the standard infinite-well solutions begin at n = 1, and n = 0 gives a vanishing wavefunction.
  5. [Figure 1 caption] The caption refers to m_G(x) as Eq. (75), but Eq. (75) defines the wavefunctions; the effective mass is defined in Eq. (55).
  6. [Abstract and introduction] The phrase 'the q-deformed (standard) Schrödinger equation results an special case' is ungrammatical, and the terminology alternates between 'Kappa' and 'Kaniadakis' classes; please unify the naming and fix the grammar.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the G-deformed Schrödinger equation, effective mass, and zero-spacing formulas are explicit mathematical consequences of proposed operators, not fitted inputs or self-citation imports.

full rationale

Walking the derivation chain: the paper takes a formal group law Φ(x,y)=G(G^{-1}(x)+G^{-1}(y)) as input and defines the G-algebra operations, the G-derivative (Eq. 28), and the G-integral (Eq. 31). The deformed momentum operator in Eq. (46) is explicitly introduced as a Hermitian choice ("we redefine p̂G as... that is Hermitian"), and Eq. (48) is a direct verification of the canonical commutator for that choice. The G-deformed Schrödinger equation (54) is then obtained by substituting Eq. (46) into the Hamiltonian (49) and evaluating matrix elements; it is a consequence of the postulated operator, not a quantity independently measured or fitted. The effective mass (55) is a rewriting of the coefficient A(x)^2 = [G'(G^{-1}(x))]^2, and the zero positions in Eq. (76) follow from solving D_G^2 φ + α² φ = 0 in the x_G coordinate, so the represented zero-spacing is a mathematical implication of the model, not an output secretly equal to an input. The cited group-entropy uniqueness theorem [29] is by Tempesta/Enciso, not the current authors, and it is not used to force the choice of p̂G. Refs. [22-24] are self-citations but only contextualize the q-deformed case already derived in Eq. (56). The unproven Hermitian ordering and the monotonicity assumption in Section 3.4 are modeling assumptions and would be correctness risks, but they do not make the derivation circular. No fitted parameter is renamed as a prediction, and no load-bearing step reduces to a self-citation chain.

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

The central construction rests on choosing a group entropy G; the entire content follows from the formal group law axioms. No external data are used, so the framework's value is structural unification rather than empirical prediction.

free parameters (3)
  • q (Tsallis class) = free parameter; e.g., κ = 1 - q = 100 in Fig. 2
    Appears in G_q(t) = (e^{(1-q)t}-1)/(1-q); the G-algebra, effective mass and zero spacing all depend on q.
  • κ (Kaniadakis class) = free parameter
    Appears in G_κ(t) = (e^{κt} - e^{-κt})/(2κ); controls the κ-algebra and the mass profile.
  • formal group exponential G(t) (function-valued) = power series with coefficients {a_k}, Eq. (2); unspecified except for the BG, Tsallis, Kaniadakis examples
    The entire algebraic structure and Schrödinger equation are parameterized by G; no principle selects G beyond the user's choice of entropy class.
assumptions (4)
  • standard math The operations of the G-algebra are defined as x ⊕G y = G(G^{-1}(x)+G^{-1}(y)) and x ⊗G y = expG(logG x + logG y), with G(t) the formal group exponential from Lazard theory.
    Takes the Lazard formal group representation (Eq. 1) and the group logarithm and exponential (Eqs. 4-5) as given.
  • domain assumption xG = G^{-1}(x) is assumed to be an increasing function of x (Section 3.4) so that the infinite well maps to a well of width LG = G^{-1}(L).
    Needed for the boundary conditions and probability interpretation; true for Tsallis and Kaniadakis, not generally proven.
  • ad hoc to paper The deformed momentum is required Hermitian, leading to the symmetrized ordering p̂G = (1/2){G'(G^{-1}(x̂)), p̂} (Eq. 46).
    A modeling choice; the resulting equation (54) depends on it.
  • standard math The formal group law satisfies commutativity, associativity and null-composability (C2-C4), and the group logarithm satisfies logG(x ⊗G y)=logG x + logG y (Eq. 6).
    Background from group entropy theory.
invented entities (2)
  • G-algebra, G-calculus (G-sum, G-product, G-derivative, G-integral)
    purpose: Unified deformed algebra and calculus for any group entropy
    Formal mathematical definitions; no empirical handle beyond reducing to known q and κ cases.
  • G-deformed coordinate space RG and G-dual real numbers ~RG
    purpose: Map position-dependent mass problems to constant mass problems
    Reparameterizations of the real line via G and G^{-1}; introduced for convenience.

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Cite this review

Pith. "Pith review of Algebraic structures and deformed Schr\"{o}dinger equations from groups entropies." pith.science (2026). https://pith.science/paper/Y3QHGGNE

@misc{pith2026190802785,
  author       = {Pith},
  title        = {Pith review of: Algebraic structures and deformed Schr\"odinger equations from groups entropies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3QHGGNE}},
  note         = {Machine review of arXiv:1908.02785}
}
abstract

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schr\"{o}dinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schr\"{o}dinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.

Figures

Figures reproduced from arXiv: 1908.02785 by the authors.

Figure 2
Figure 2. Top panel: the BG, Tsallis and Kappa classes for [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 1
Figure 1. Some plots of [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

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Reference graph

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