Coherent states for generalized uncertainty relations as Tsallis probability amplitudes: New route to nonextensive thermostatistics
Petr Jizba, Gaetano Lambiase, Giuseppe Gaetano Luciano, Luciano Petruzziello
Abstract
We study coherent states associated with a generalized uncertainty principle (GUP). We separately analyze the cases of positive and negative deformation parameter $\ensuremath{\beta}$, showing that the ensuing probability distribution is a Tsallis distribution whose nonextensivity parameter $q$ is monotonically related to $\ensuremath{\beta}$. Moreover, for $\ensuremath{\beta}<0$ (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 one to study the quasiclassical regime of GUP in terms of nonextensive thermodynamics. We substantiate our claim by discussing the generalization of Verlinde's entropic force and ensuing implications in the late-inflation epoch. The corresponding dependence of the $\ensuremath{\beta}$ parameter on cosmological time is derived for the reheating epoch. The obtained $\ensuremath{\beta}$ is consistent with values predicted by both string-theory models and the naturalness principle. Further salient issues, including the derivation of new $\ensuremath{\beta}$-dependent expressions for the lowest possible value of the spin and Immirzi parameter in loop quantum gravity and the 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 article [Phys. Rev. D 105, L121501 (2022)].