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

We study coherent states associated to a generalized uncertainty principle (GUP). We separately analyze the cases of positive and negative deformation parameter β, showing that the ensuing probability distribution is a Tsallis distribution whose non-extensivity parameter q is monotonically related to β. Moreover, for β<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 β parameter on cosmological time is derived for the reheating epoch. The obtained β is consistent with values predicted by both string-theory models and the naturalness principle. Further salient issues, including derivation of new β-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)].