Coherent states for generalized uncertainty relations and their cosmological implications

Abstract

We study coherent states associated with a generalized uncertainty principle (GUP). Our particular focus is on the negative deformation parameter β. We show that the ensuing coherent state can be identified with Tsallis’ probability amplitude with the non-extensivity parameter q being a monotonically increasing function of β. Furthermore, for β < 0, we reformulate the GUP in terms of a one-parameter class of Tsallis entropy 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 the quasi-classical regime of GUP in terms of non-extensive thermostatistics. We bolster this claim by discussing a generalization of Verlinde’s entropic force and the ensuing implications in the late-inflation epoch. The corresponding dependence of the β parameter on the cosmological time is derived for the reheating epoch. The obtained β is consistent with both values predicted by string-theory models and the naturalness principle.