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arxiv: 2604.13697 · v1 · submitted 2026-04-15 · 🪐 quant-ph · hep-th

kappa-entropic statistical paradigm for relativistic corrections to the Heisenberg principle

Giuseppe Gaetano Luciano , Jaume Gin\' e , Daniel Chemisana This is my paper

Pith reviewed 2026-05-10 12:44 UTC · model grok-4.3

classification 🪐 quant-ph hep-th
keywords Heisenberg uncertainty principleKaniadakis entropyrelativistic quantum mechanicskappa-deformed statisticsuncertainty relationsLorentz invariance
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The pith

A relativistic extension of the Heisenberg uncertainty principle follows from maximizing Kaniadakis entropy under Lorentz transformations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a modified position-momentum uncertainty relation that incorporates relativistic corrections in the intermediate-velocity regime. It does so by applying a variational method to the maximum-entropy principle with Kaniadakis entropy, a one-parameter deformation of ordinary statistical entropy that is induced by special relativity. A sympathetic reader cares because the construction supplies concrete predictions that can be tested with precision measurements before particles reach ultra-relativistic speeds. The work also extracts bounds on the deformation parameter from the known value of the fine-structure constant and compares the result with other proposed relativistic extensions.

Core claim

The Heisenberg algebra acquires a relativistic correction when the underlying probability distribution is obtained by maximizing the Kaniadakis entropy rather than the Boltzmann-Gibbs-Shannon entropy; the resulting uncertainty relation reduces to the standard form at low velocities and deviates measurably at intermediate speeds.

What carries the argument

The κ-deformed Kaniadakis entropy, obtained from the maximum-entropy principle applied to a Lorentz-invariant generalization of ordinary entropy, which supplies the variational principle used to deform the Heisenberg commutator.

If this is right

  • The Kaniadakis parameter is bounded by existing precision data on the fine-structure constant.
  • The deformed uncertainty relation can be compared directly with other relativistic extensions appearing in the recent literature.
  • The framework yields testable corrections to quantum limits in the velocity window between non-relativistic and ultra-relativistic regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same statistical deformation could be applied to other quantum commutators to generate consistent relativistic modifications.
  • Experimental tests might be performed with trapped ions or cold-atom interferometers operating at moderate accelerations.

Load-bearing premise

Kaniadakis entropy induced by Lorentz transformations supplies the correct statistical foundation for relativistic corrections to the Heisenberg principle in the intermediate-velocity regime.

What would settle it

A high-precision measurement of the position-momentum uncertainty product for particles at intermediate velocities that lies outside both the standard Heisenberg bound and the predicted κ-corrected bound would falsify the construction.

Figures

Figures reproduced from arXiv: 2604.13697 by Daniel Chemisana, Giuseppe Gaetano Luciano, Jaume Gin\' e.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrammatic map of uncertainty relations and their connections to deformed statistics. Solid arrows denote exten [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plot of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
read the original abstract

The Heisenberg position-momentum uncertainty relation is a cornerstone of quantum mechanics. However, its standard formulation is not fully consistent with special relativity. While partial understanding has been achieved in the ultra-relativistic regime, a comprehensive description is still lacking, particularly in the intermediate velocity domain, where particle speeds remain well below the speed of light yet relativistic corrections are expected to become appreciable. This regime constitutes the most promising arena for experimentally probing relativistic modifications of quantum uncertainty. By adopting a variational approach, in this work we derive a relativistic extension of the Heisenberg algebra within the framework of $\kappa$-deformed Kaniadakis statistics. The latter emerges from the application of the Maximum Entropy Principle to Kaniadakis entropy, a one-parameter generalization of the Boltzmann-Gibbs-Shannon entropy naturally induced by Lorentz transformations. We investigate the physical implications of the resulting uncertainty relation, deriving constraints on the Kaniadakis parameter from precision measurements of the fine-structure constant and confronting our construction with other extensions discussed in the recent literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that a variational (maximum-entropy) treatment of the one-parameter Kaniadakis entropy S_κ, asserted to be induced by Lorentz transformations, produces a relativistic extension of the Heisenberg algebra. The resulting deformed uncertainty relation is then used to extract constraints on the deformation parameter κ from precision measurements of the fine-structure constant, with comparisons to other literature extensions.

Significance. If the central mapping from the κ-maxent distribution to a modified operator algebra can be rigorously established, the work would supply a statistical-mechanics route to relativistic corrections of the uncertainty principle in the intermediate-velocity regime, where direct experimental probes are feasible. The confrontation with fine-structure-constant data offers a concrete falsifiability route, though the assumption that Kaniadakis entropy is the natural Lorentz-induced foundation remains an open modeling choice rather than a derived necessity.

major comments (1)
  1. [Derivation of the relativistic extension (following the abstract)] The manuscript applies the maximum-entropy principle to S_κ and asserts that the resulting distribution yields a deformed Heisenberg algebra, yet supplies no explicit step converting the variational condition δS_κ = 0 into a modified commutator [x, p] or the concrete uncertainty product Δx Δp ≥ ħ/2 (1 + f(κ, v/c)). This correspondence rule is load-bearing for the central claim of a relativistic extension but is not derived from Lorentz invariance or the entropy functional alone.
minor comments (2)
  1. [Abstract] The abstract repeatedly uses the phrase 'κ-deformed Kaniadakis statistics'; since the Kaniadakis entropy is already parameterized by κ, a single clarifying sentence on the terminology would reduce redundancy.
  2. [Main text] Notation for the deformed variance product and the explicit functional form of the relativistic correction term should be introduced with an equation number at first appearance to improve traceability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will revise the manuscript to incorporate the requested clarification.

read point-by-point responses

  1. Referee: [Derivation of the relativistic extension (following the abstract)] The manuscript applies the maximum-entropy principle to S_κ and asserts that the resulting distribution yields a deformed Heisenberg algebra, yet supplies no explicit step converting the variational condition δS_κ = 0 into a modified commutator [x, p] or the concrete uncertainty product Δx Δp ≥ ħ/2 (1 + f(κ, v/c)). This correspondence rule is load-bearing for the central claim of a relativistic extension but is not derived from Lorentz invariance or the entropy functional alone.

    Authors: We agree that the explicit mapping from the variational condition δS_κ = 0 to the deformed commutator requires a more detailed derivation to make the central claim fully rigorous. In the revised manuscript we will add a new subsection that starts from the κ-maxent distribution obtained by extremizing S_κ, shows how the resulting deformed probability measure induces a modified phase-space structure, and then applies the standard quantization correspondence (adapted to the κ-framework) to obtain the explicit form of [x, p] and the uncertainty product Δx Δp ≥ ħ/2 (1 + f(κ, v/c)). This step will be tied directly to the Lorentz-induced origin of the Kaniadakis entropy as established in the literature, thereby supplying the missing correspondence rule without altering the overall conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation applies the maximum-entropy principle to Kaniadakis entropy (stated as induced by Lorentz transformations) via a variational approach to obtain a deformed uncertainty relation, then extracts parameter bounds by substituting the derived relation into expressions involving the fine-structure constant. No quoted equation or step reduces the central claim to a self-definition, a fitted input renamed as prediction, or a load-bearing self-citation chain; the variational step and experimental confrontation supply independent content outside the inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only information; the central claim rests on the applicability of Kaniadakis entropy to relativistic quantum mechanics and on the validity of the variational derivation, both of which are stated without supporting detail.

free parameters (1)
  • κ (Kaniadakis deformation parameter)
    One-parameter generalization whose value is constrained by fine-structure-constant data; no numerical value or fitting procedure given in abstract.
axioms (1)
  • domain assumption Kaniadakis entropy is naturally induced by Lorentz transformations via the Maximum Entropy Principle
    Explicitly stated in the abstract as the foundation for the κ-deformed statistics.

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