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

TL;DR

The paper addresses how Generalized Extended Uncertainty Principles (GEUP), incorporating noncommutativity of both coordinates and momenta, modify the Liouville theorem and the density of states in nonrelativistic quantum mechanics. It develops a general framework for a time-invariant, weighted phase-space measure and applies it to SdS and Yang models, deriving explicit invariant weights and exploring limiting cases that connect to GUP and EUP forms. The SdS model yields a weight $F(x,p)=1+α^2 x^2+β^2 p^2+2αβ x\cdot p$, while the Yang model provides a 1D weight $1-\frac{1}{2}(α^2 x^2+β^2 p^2)$, with fuzzy-sphere showing no change in the density of states. These results indicate that GEUP can affect thermodynamic properties via modified densities of states, with implications across various physical settings and motivating future work on higher-order GEUPs and broader realizations.

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.