Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: new route to non-extensive thermostatistics

TL;DR

This paper explores coherent states linked to a generalized uncertainty principle, revealing their connection to Tsallis distributions and non-extensive thermodynamics, with implications for cosmology, quantum gravity, and fundamental physics.

Contribution

It introduces a novel framework connecting GUP coherent states with Tsallis entropy, providing new insights into non-extensive thermodynamics and quantum gravity models.

Findings

Tsallis distribution describes GUP probability distribution for positive and negative deformation parameters.

Reformulation of GUP in terms of Tsallis entropy-power uncertainty relations.

Derived cosmological and quantum gravity implications consistent with existing theories.

Abstract

We study coherent states associated to a generalized uncertainty principle (GUP). We separately analyze the cases of positive and negative deformation parameter $\beta$, showing that the ensuing probability distribution is a Tsallis distribution whose non-extensivity parameter $q$ is monotonically related to $\beta$. Moreover, for $\beta <0$ (corresponding to $q<1$), we reformulate the GUP in terms of a one-parameter class of Tsallis entropy-power based uncertainty relations, which are again saturated by the GUP coherent states. We argue that this combination of coherent states with Tsallis entropy offers a natural conceptual framework allowing to study quasi-classical regime of GUP in terms of non-extensive thermodynamics. We substantiate our claim by discussing generalization of Verlinde's entropic force and ensuing implications in the late-inflation epoch. Corresponding dependence of the $\beta$ parameter on cosmological time is derived for the reheating epoch. The obtained $\beta$ is consistent with values predicted by both string-theory models and the naturalness principle. Further salient issues, including derivation of new $\beta$-dependent expressions for the lowest possible value of the spin and Immirzi parameter in Loop Quantum Gravity, and connection of our proposal with the Magueijo--Smolin doubly special relativity are also discussed. This article provides a more extended and comprehensive treatment of our recent letter [Phys. Rev. D 105, L121501 (2022)].