$κ$-entropic statistical paradigm for relativistic corrections to the Heisenberg principle

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 $κ$-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.

• The paper formulates a novel relativistic extension of the Heisenberg Uncertainty Principle using κ-deformed Kaniadakis entropy.
• The derived commutator integrates Lorentz invariance and yields a minimal position uncertainty tied to the Compton wavelength.
• The framework imposes strict phenomenological bounds on κ and opens new avenues for high-precision relativistic quantum measurements.

$\kappa$-Entropic Statistical Paradigm for Relativistic Corrections to the Heisenberg Principle

Introduction and Motivation

The paper addresses the structural incompatibility of the standard Heisenberg Uncertainty Principle (HUP) with Special Relativity (SR), particularly in the regime where particle velocities are substantial yet not ultra-relativistic. Conventional quantum mechanics, anchored in the HUP, fails to incorporate relativistic statistical effects. Previous partial treatments—including the ultra-relativistic regime and Generalized Uncertainty Principle (GUP) scenarios motivated by quantum gravity—do not adequately capture corrections relevant to the domain where SR influences are experimentally accessible. The authors propose a new formalism utilizing the $\kappa$-deformed Kaniadakis entropy, a Lorentz-motivated generalization of Boltzmann-Gibbs-Shannon entropy, to construct a relativistic extension of the Heisenberg algebra.

Kaniadakis Statistical Framework

Kaniadakis statistics emerges from fundamental requirements of Lorentz invariance in entropy measures and compositions, producing power-law distributions observed in cosmic rays and other relativistic systems. The entropy functional, parameterized by $\kappa$, is

$S_\kappa = -\sum_i n_i \ln_\kappa n_i \,,$

where $\ln_\kappa(y) = (y^\kappa - y^{-\kappa})/(2\kappa)$. Maximizing $S_\kappa$ subject to normalization and mean energy yields the stationary distribution: $n_i \propto \exp_\kappa(-\beta E_i).$ Here, $$\exp_\kappa(y) = (\sqrt{1+\kappa^2 y^2} + \kappa y)^{1/\kappa}$$ introduces power-law tails. The parameter $\kappa \in (-1,1)$ controls deviations, and Boltzmann-Gibbs statistics are recovered as $\kappa \to 0$. This formalism is related to observed distributions for cosmic rays and other relativistic phenomena, strengthening its phenomenological justification.

Algebraic Derivation of Relativistic Uncertainty Principle (RUP)

Recognizing theoretical precedents—including GUP-Tsallis correspondences—where coherent states and probability amplitudes correspond to deformed statistical distributions, the authors generalize the commutator at algebraic level to

$\kappa$0

with $\kappa$1 to be determined by imposing that the minimum-uncertainty wavefunctions are $\kappa$2-Gaussians: $\kappa$3 Through variational analysis and operator ordering considerations, the physically admissible commutation relation emerges: $\kappa$4 This is an exact operator identity encoding relativistic effects via the statistical deformation parameter $\kappa$5 and an intrinsic momentum scale $\kappa$6.

In the weakly relativistic regime, the commutator admits a perturbative expansion: $\kappa$7 resulting in

$\kappa$8

Formally analogous to quadratic GUP relations, this is conceptually distinct: the correction is rooted in Lorentzian statistics, not quantum gravity.

Physical Implications

Minimal Position Uncertainty

The quadratic $\kappa$9 dependence yields a nonzero minimum position uncertainty: $\kappa$0 Unlike the GUP, where this scale is universal (Planck length), here it depends on particle mass and $\kappa$1, typically corresponding to the Compton wavelength. This explicitly links the minimal localization to the kinematics of relativistic quantum particles, aligning with the Landau–Peierls argument that localization below Compton scale is forbidden due to pair creation thresholds.

Phenomenological Constraints

Precision experiments probing the fine-structure constant impose stringent bounds on $\kappa$2. By reformulating the RUP as $\kappa$3, with $\kappa$4 modified appropriately, the predicted deviation in $\kappa$5 must not exceed the experimental uncertainty. Setting the characteristic momentum scale to the electron’s in hydrogen yields: $\kappa$6 for $\kappa$7, restricting $\kappa$8 to an intermediate regime where relativistic corrections remain perturbative.

Comparison with Alternative Approaches

• Landau–Peierls: The RUP recovers the Compton wavelength as a minimal length in a purely algebraic way, matching the original operational argument about localization and pair creation.
• Amelino-Camelia & Vestuti: Their operational approach yields a similar quadratic correction in $\kappa$9, but does not modify the operator structure; the present work promotes such corrections to the fundamental commutator.
• Putra–Alrizal: Their velocity-dependent uncertainty relation, with corrections scaling as $S_\kappa = -\sum_i n_i \ln_\kappa n_i \,,$0, is conceptually distinct and not directly mappable to the momentum-dependent structure in the RUP, though both account for relativistic amplification of uncertainty bounds.

Implications and Future Directions

Practically, the paradigm offers a model for relativistic corrections to quantum uncertainty relations relevant for high-precision quantum measurements operating outside the Galilean regime (e.g., relativistic atomic systems, high energy spectroscopy). Theoretical implications concern the structural modification of the quantum algebra by statistical deformations, potentially creating routes for manifest Lorentz-invariant uncertainty relations and connections to $S_\kappa = -\sum_i n_i \ln_\kappa n_i \,,$1-deformed algebras.

The authors suggest further developments could include Lorentz-covariant realization (treating space and time symmetrically), embedding within more general entropic frameworks (interpolating between Kaniadakis, Tsallis, and Boltzmann-Gibbs), and potential experimental tests in regimes where relativistic statistical effects are accessible. The construction is poised as a bridge between phenomenological, algebraic, and statistical treatments of quantum uncertainty under SR.

Conclusion

This paper constructs a mathematically consistent and phenomenologically testable relativistic extension of the Heisenberg algebra using $S_\kappa = -\sum_i n_i \ln_\kappa n_i \,,$2-deformed Kaniadakis statistics. The resulting Relativistic Uncertainty Principle not only structurally modifies the canonical commutator, but yields a minimal position uncertainty governed by relativistic kinematics. Precision measurements, such as atomic spectroscopy, severely constrain the deformation parameter $S_\kappa = -\sum_i n_i \ln_\kappa n_i \,,$3. The approach innovatively unifies statistical and quantum mechanical paradigms, providing a foundation for further theoretical explorations and experimental probes into the interrelation of quantum uncertainty and SR.