Combining the Generalized and Extended Uncertainty Principles

TL;DR

This paper explores combined forms of the Generalized and Extended Uncertainty Principles, analyzing their implications for black holes and elementary particles, including novel models with potential duality features and extended parameter ranges.

Contribution

It introduces combined uncertainty principles (GEUP and EGUP), analyzes their implications across different regimes, and links these models to black hole physics and elementary particles.

Findings

GEUP predicts a new type of strong-gravity black hole.

EGUP exhibits duality between position and momentum uncertainties.

Extended parameter analysis reveals novel features of uncertainty models.

Abstract

The Generalized Uncertainty Principle (GUP) and Extended Uncertainty Principle (EUP) are modifications to the Heisenberg Uncertainly Principle (HUP), expected to apply as the energy approaches the Planck scale. Here we consider a possible combination of these modifications (GEUP) and analyse the implications in various regions of the ($\Delta x$, $\Delta p$) plane. We also consider an alternative combination (EGUP) which exhibits duality between $\Delta p$ and $\Delta x$, showing that this has some unusual features. The parameters which describe these models are usually assumed to be positive but we extend our analysis to include negative values. All these proposals entail a link between black holes and the various types of Uncertainty Principle. In particular, the GEUP predicts a new kind of strong-gravity black hole and this implies an interesting link between black holes and elementary particles.