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

Author:

Petr Jizba, Gaetano Lambiase, Giuseppe Gaetano Luciano, Luciano Petruzziello

Keyword:

General Relativity and Quantum Cosmology, General Relativity and Quantum Cosmology (gr-qc), High Energy Physics - Theory (hep-th), Quantum Physics (quant-ph)

journal:

Phys. Rev. D 108, 064024 (2023)

date:

2023-08-22 16:00:00

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)].

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