Generalized Extended Uncertainty Principles, Liouville theorem and density of states: Snyder-de Sitter and Yang models

TL;DR

This paper explores how generalized uncertainty principles, incorporating quantum gravitational effects, modify phase space volume and density of states in Snyder-de Sitter and Yang models, impacting thermodynamical properties.

Contribution

It introduces a weighted phase space volume invariant under time evolution for models with GEUP, extending the Liouville theorem and proposing new higher order uncertainty principles.

Findings

Weighted phase space volume is invariant under time evolution.

GEUP modifies the density of states, affecting thermodynamics.

Special limits recover known models and new uncertainty relations.

Abstract

Modifications in quantum mechanical phase space lead to the changes in the Heisenberg uncertainty principle, which can result in the Generalized Uncertainty Principle (GUP) or the Extended Uncertainty Principle (EUP), introducing quantum gravitational effects at small and large distances, respectively. A combination of GUP and EUP, the Generalized Extended Uncertainty Principle (GEUP or EGUP), further generalizes these modifications by incorporating noncommutativity in both coordinates and momenta. This paper examines the impact of GEUP on the analogue of the Liouville theorem in statistical physics and density of states within the classical limit of non-relativistic quantum mechanics framework. We find a weighted phase space volume element, invariant under the infinitesimal time evolution, in the cases of Snyder-de Sitter and Yang models, presenting how GEUP alters the density of states, potentially affecting physical (thermodynamical) properties. Special cases, obtained in certain limits from the above models are discussed. New higher order types of GEUP and EUP are also proposed.