Combining the Generalized and Extended Uncertainty Principles

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, the Generalized Extended Uncertainty Principle (GEUP), and analyze the implications in various regions of the [Formula: see text] plane. We also consider an alternative combination, the Extended Generalized Uncertainty Principle (EGUP), which exhibits duality between [Formula: see text] and [Formula: see text], 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.