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A new time-dependent quantum theory based on Tsallis' distribution

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Pith's one-line read The Tsallis q-exponential, Wick-rotated to imaginary time, defines a unitary quantum evolution whose leading corrections are set by (1−q)².

desk verdict A clean but formally narrow q-deformed evolution construction whose central advertised result is undermined by an internal sign contradiction that the authors must resolve. read the letter →

arxiv 2505.05419 v1 pith:P4JD2JYH submitted 2025-05-08 quant-ph math-phmath.MP

Won Sang Chung , Georg Junker , Luis M. Nieto , Hassan Hassanabadi This is my paper · ORCID
classification quant-phmath-phmath.MP MSC 81Q0581R50 PACS 03.65.-w
keywords q-deformedquantummechanicsTsallisdistributionq-exponentialtime-dependentSchrödingerequationGaussianwavepacketharmonicoscillatorunitaryevolutiondeformedBoltzmannfactor
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 proposes a new deformed quantum dynamics by taking Tsallis' q-deformed Boltzmann factor, rotating inverse temperature into imaginary time (β→it), and normalising the resulting operator to make it unitary. The evolution operator is exp{−i arctan[εtH]/ε} with ε=1−q, and the accompanying time-dependent q-deformed Schrödinger equation has effective Hamiltonian H[1+ε²t²H²]^{-1}. The authors work out two textbook examples: a free Gaussian wave packet, whose spreading is slowed by an ε² correction, and a displaced Gaussian in a harmonic trap, whose probability density acquires oscillatory ε² corrections. The paper matters because it gives the Tsallis factor a dynamical role in quantum mechanics and produces explicit, computable deviations from standard evolution that could be searched for in simple systems.

What carries the argument

The central object is the normalised Tsallis evolution operator $U_q(t)=e_q(-it\hat H)/|e_q(-it\hat H)|$, built from the Tsallis q-exponential $e_q(z)=[1+(1-q)z]^{1/(1-q)}$. The identity $e_q(-iz)=[1+(1-q)^2z^2]^{1/[2(1-q)]}\exp\{-i\arctan[(1-q)z]/(1-q)\}$ converts it into the manifestly unitary form $\exp\{-i\arctan[\varepsilon t\hat H]/\varepsilon\}$. The companion mechanism is the q-product $\otimes_q$, which expresses how two non-normalised q-evolution operators combine; after renormalisation this yields the time-dependent effective Hamiltonian $\hat H_{\rm eff}(t)=\hat H[1+\varepsilon^2t^2\hat H^2]^{-1}$ and hence the time-dependent q-deformed Schrödinger equation. The effective-Hamiltonian identity carries the argument: it turns the deformed group law into a differential equation and identifies $(\varepsilon t)^2$ as the leading correction scale.

What would settle it

Compute the integral in Eq. (22) exactly for finite $\varepsilon$: if the exact width of the freely evolving Gaussian packet has no term of order $(\varepsilon t)^2$, the prediction in Eq. (29) is wrong. Alternatively, test the composition rule directly on a superposition state, where Eq. (9) predicts that evolving for time $t_1+t_2$ gives a different result from evolving for $t_1$ and then $t_2$ at order $\varepsilon^2$.

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Extended reading notes

Core claim

The paper's central claim is that the ordinary evolution operator $e^{{−itH}}$ can be replaced by the normalised Tsallis operator U_q(t)=e_q(−itH)/|e_q(−itH)|=exp{−i arctan[εtH]/ε}, and that this defines a valid unitary evolution with effective Hamiltonian H_eff(t)=H[1+ε²t²H²]^{-1}. Since the effective Hamiltonian depends on absolute time, the resulting dynamics is not standard quantum mechanics in disguise; all deviations start at order ε²=(1−q)². For a free Gaussian initial state, the paper derives the width σ(t)²=σ²(1+t²/τ²)[1−15ε²(t/τ)³(3−6t²/τ²+t⁴/τ⁴)/(1+t²/τ²)³+O(ε⁴)], so the deformation slows the spreading of the packet. For the same Gaussian placed in a harmonic well, the probability density acquires corrections proportional to ε²(ωt)³ with oscillations at up to six times the trap frequency. For Hamiltonian eigenstates the evolution is only a phase, so the deformation is visible only in superpositions of different energies.

Load-bearing premise

The construction assumes that analytically continuing the Tsallis factor to imaginary time and then normalising the result yields a legitimate quantum evolution, even though the resulting dynamics depends on absolute time and does not satisfy the usual composition law $U_q(t_1+t_2)=U_q(t_1)U_q(t_2)$.

Editorial extensions

If this is right

  • If $\varepsilon=1-q$ is nonzero, a free Gaussian wave packet spreads more slowly than in standard quantum mechanics, with a leading deviation proportional to $\varepsilon^2(t/\tau)^3(3-6t^2/\tau^2+t^4/\tau^4)$.
  • In a harmonic trap, the probability density of a displaced Gaussian develops extra harmonics up to $6\omega$, all proportional to $\varepsilon^2(\omega t)^3$; these harmonics are a distinctive fingerprint of the deformed phase.
  • Because eigenstates evolve by a pure phase, any observable effect of the deformation requires an initial state that superposes at least two Hamiltonian eigenstates.
  • For positive Hamiltonians the evolution freezes at large times: $\psi(x,t)\to e^{-i\pi/(2\varepsilon)}\psi(x,0)$, so the deformation eventually stops the dynamics.
  • For small $\varepsilon$ the corrections are suppressed at short times, so experimental bounds on wave-packet spreading or oscillator harmonics translate directly into upper bounds on $|1-q|$.

Reading between the lines

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

  • The paper does not discuss it, but the failure of the semigroup law means energy is not conserved under $H_{\rm eff}(t)$; a system prepared at different absolute times would evolve differently, a sharp difference from standard quantum mechanics.
  • One testable extension is to look for the predicted $6\omega$ sideband in the oscillating probability density; standard harmonic motion has no such frequency, so it would isolate the q-deformation from ordinary dynamics.
  • The same construction could be applied to other deformed Boltzmann factors; each would yield its own effective Hamiltonian and its own wave-packet signature, allowing the deformation mechanism to be distinguished from generic small corrections.
  • The large-time freezing result relies on $H>0$; for free particles with zero-energy plane-wave components, or for Hamiltonians with continuous spectrum down to zero, the $t\to\infty$ limit needs a separate treatment, and the freezing may not occur.
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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

2 major / 6 minor

Summary. The paper constructs a q-deformed quantum evolution operator by applying the inverse Wick rotation β→it to the Tsallis q-exponential and then normalizing to enforce unitarity. This yields U_q(t) = exp{-(i/ε) arctan(εtH)} with ε=1-q, and a time-dependent q-deformed Schrödinger equation i∂_t ψ = H[1+ε²t²H²]^{-1}ψ. As applications, the authors compute, to second order in ε, the time evolution of a free Gaussian wave packet and of a Gaussian wave packet in a harmonic oscillator potential, reporting O(ε²) corrections to wave-packet spreading and probability densities. The abstract and conclusion advertise these corrections as the concrete predictions of the new framework.

Significance. If correct, the paper offers a simple one-parameter family of unitary time evolutions with an explicitly time-dependent effective Hamiltonian, and the resulting O(ε²) corrections are concrete and falsifiable. The algebraic derivation of U_q(t) is elementary and internally consistent, and the perturbative integrals in Sections 3 and 4 are plausible. However, the internal sign contradiction in the free wave-packet result and the unexamined validity of the eigenfunction expansion in the harmonic-oscillator example mean that the advertised predictions are not currently reliable. The proposal is a novel toy model rather than a derivation from known principles; its significance is primarily formal, and it would need a consistent set of predictions to be useful.

major comments (2)
  1. [Section 5 (Conclusion) vs. Section 3, Eq. (29) and Fig. 1] The paper's central free-particle result is internally contradictory. Equation (29) gives ⟨x̂²⟩/σ² = (1 + t²/τ²)[1 − 15ε² (z/(1+z²))³ (3 − 6z² + z⁴) + O(ε⁴)] with z = t/τ, which for small z yields a negative O(ε²) correction (reduced width, slowed spreading), while for 0.742 ≲ z ≲ 2.334 the correction is positive (enhanced spreading). Nevertheless, the conclusion in Section 5 states that 'for the free Tsallis-deformed quantum dynamics the spreading of such a wave packet is increased by this deformation,' and Figure 1's caption claims a slowing down for all plotted times. These statements are mutually incompatible, and at least one of Eq. (29), the figure interpretation, or the conclusion must be wrong. Because the sign of the O(ε²) free-particle correction is advertised in the abstract as one of the paper's two concrete results, this is a load-bearing inconsistency that must be resolved before the paper can be considered.
  2. [Section 4, Eqs. (35)–(44)] The harmonic-oscillator example uses a small-ε expansion of exp{−(i/ε) arctan[εωt(n+1/2)]} for every n in the eigenfunction expansion (35). The Taylor expansion of arctan is only valid for |εωt(n+1/2)| < 1; for larger n the argument lies outside the radius of convergence. Although the Gaussian coefficients A_n decay superexponentially, the paper provides no bound showing that the truncated series plus the remainder is controlled for the relevant range of ε, ω, and t. The resulting expression (44) is therefore a formal asymptotic series whose quantitative reliability for the reported parameter values (e.g., Fig. 2) has not been established. This needs either a justification (e.g., a bound on the remainder or an explicit restriction on εωt and the state width) or a clear statement that the results are only formal.
minor comments (6)
  1. [Section 2, Eq. (9)] The second equality in Eq. (9) is written with a denominator 1 + ε²(t1+t2)²H², but Eq. (6) shows that the normalization factor is [1 + ε²(t1+t2)²H²]^{1/(2ε)}; the exponent is missing. This should be corrected for consistency.
  2. [Section 2, Eq. (14)] The statement that the long-time limit requires the original Hamiltonian to be bounded from below by H > 0 is not satisfied by the free Hamiltonian used in Section 3, which has spectrum [0,∞). The claim should be restricted to the positive spectral subspace, or the limiting argument should be amended to address the zero-energy component explicitly.
  3. [Section 3, text after Eq. (29)] The sentence lists 'ε2 = 0.0005 (purple) and ε = 0.001 (brown)'; the last entry should read 'ε² = 0.001' to match the other entries and the figure caption.
  4. [Section 4, Fig. 2 caption] The caption states 'ωt = π/2ω' for the right panel, which is dimensionally inconsistent; it should be 'ωt = π/2' or 't = π/(2ω)'.
  5. [Throughout] There are several typos: 'Tallis' for 'Tsallis' in Section 1; 'uncertainly relation' for 'uncertainty relation' in Section 3; and 'Spinger' for 'Springer' in reference [16].
  6. [Section 2, general remark] The paper does not explicitly discuss the fact that U_q(t) does not form a one-parameter unitary group and that the time-dependent Schrödinger equation (11) is therefore not time-translation invariant. Eq. (9) already hints at this, but the physical consequences (e.g., the absence of a standard conserved energy) are not discussed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the q-deformed evolution operator is stipulated and all subsequent results are derived consequences, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's construction begins by explicitly stipulating a q-deformed evolution operator obtained from the Tsallis q-exponential via the inverse Wick rotation and a normalization for unitarity, Eq. (3), leading to U_q(t) = exp{-i arctan[εtH]/ε}, Eq. (7). No parameter is fitted to data, and no quantity is returned as a prediction from the same data upon which it was based. The time-dependent q-deformed Schrödinger equation, Eq. (11), is obtained by direct differentiation of this stipulated evolution operator: i∂_t U_q(t) = H[1 + ε^2 t^2 H^2]^{-1} U_q(t). This is the standard relationship between an evolution operator and its Schrödinger equation, not a circular reuse of a fitted input. The free Gaussian wave packet result, Eq. (29), and the harmonic oscillator probability density, Eq. (44), are derived from this definition through explicit integrals and generating functions; they are consequences of the postulate, not inputs to it. The paper contains no load-bearing self-citations: the cited Tsallis and q-algebra references are external standard literature, and none is invoked to force the central choice. The reported internal inconsistency between the conclusion that the free wave-packet spreading is 'increased by this deformation' and the expression in Eq. (29) / Fig. 1 indicating a slowing of spreading is a genuine correctness concern, but it is not circularity: it does not involve a prediction reducing by construction to its inputs. Therefore, under the circularity criteria specified, the derivation chain is self-contained and the circularity score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The only genuinely free parameter is ε = 1-q; all other inputs such as m, σ, ω, and a are standard physical or initial parameters. The construction rests on functional calculus for unbounded operators and on the legitimacy of the Wick rotation, which are standard tools but not fully justified here. No new physical entities are introduced.

free parameters (1)
  • epsilon = 1 - q
    Deformation parameter introduced by hand in Section 2 (Eqs. (1)-(7)); it controls all deviations from standard quantum mechanics and is not fixed by any data or independent constraint.
assumptions (4)
  • domain assumption The Tsallis q-exponential and the q-product identity e_q(x) ⊗_q e_q(y) = e_q(x+y) extend from real scalars to unbounded Hermitian operators.
    Used in Section 2, Eq. (9), to discuss composition of evolution operators; no domain or convergence conditions are stated.
  • domain assumption The inverse Wick rotation β → it can be applied to e_q(-βH), and the resulting operator-valued arctan is defined by the principal branch.
    Section 2, Eqs. (3)-(7); the paper assumes a functional calculus for arctan(εtH) on L2(R) without discussing spectrum or branch issues.
  • ad hoc to paper The original Hamiltonian is bounded from below and positive, H > 0, to take the t → ∞ limit.
    Stated just before Eq. (14); the free Hamiltonian has zero-energy plane waves, so the strict positivity assumption is not satisfied by the first example.
  • domain assumption The small-ε expansion is uniformly valid for the times considered, and the O(ε²) truncation preserves unitarity to the stated order.
    Eqs. (23)-(29) and (40)-(44) truncate at O(ε²); the paper notes this is valid for small εt but does not quantify the error terms.

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Pith. "Pith review of A new time-dependent quantum theory based on Tsallis' distribution." pith.science (2026). https://pith.science/paper/P4JD2JYH

@misc{pith2026250505419,
  author       = {Pith},
  title        = {Pith review of: A new time-dependent quantum theory based on Tsallis' distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P4JD2JYH}},
  note         = {Machine review of arXiv:2505.05419}
}
abstract

In this paper, inspired by Tsallis' probability distribution based on a $q$-deformed Boltzmann factor, we stipulate a new $q$-deformed quantum dynamics by applying the inverse Wick rotation $ \beta \rightarrow i t$ to the Tsallis-deformed Boltzmann factor. We obtain a new time-dependent $q$-deformed Schr\"odinger equation. The free time-evolution of a Gaussian wave packet and that induced by an harmonic interaction are studied within this $q$-deformed quantum mechanical framework.

Figures

Figures reproduced from arXiv: 2505.05419 by the authors.

Figure 1
Figure 1. Plot of the spreading (29) of a freely evolving Gaussian wave packet with initial width σ = 1 and time scale τ = 1, for ε 2 = 0 (pink), ε 2 = 0.0005 (purple) and ε 2 = 0.001 (brown). 6 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Plot of the probability density (43) for the oscillating wave packet at t = π/4ω (left panel) and ωt = π/2ω (right panel) with parameters a = 1 and α = 1, and for ε 2 = 0 (pink), ε 2 = 0.01 (purple), and ε 2 = 0.02 (brown). and ε 2 = 0.02 (brown); in the right panel of this figure the plot of P(x, T /4) is shown. We observe a 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

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