Combining the Generalized and Extended Uncertainty Principles

TL;DR

This work extends the Heisenberg Uncertainty Principle by formulating the Generalized Extended Uncertainty Principle (GEUP) and Extended Generalized Uncertainty Principle (EGUP) to unify GUP and EUP corrections and explores their black-hole implications. It derives Δx(Δp) relations, establishes real-solution conditions such as αβ < 1, and maps these to horizon radii R_H(M) and Hawking temperatures T(M) under several scenarios, including negative parameter values. A key result is that GEUP reintroduces a BHUP-type link between Compton and Schwarzschild scales, revealing a novel strong-gravity black-hole phase, while EGUP shows dual-regime behavior with a transition at α = β. The analysis highlights how allowing negative α, β broadens the parameter space and strengthens the connection between microscopic uncertainty and macroscopic black-hole thermodynamics, offering potential new quantum black-hole states and insights into the micro–macro gravity bridge.

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 ($Δx$, $Δp$) plane. We also consider an alternative combination (EGUP) which exhibits duality between $Δp$ and $Δx$, 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.

Figures (9)

• Figure 1: Form of the function $\Delta x (\Delta p)$ for (a) GUP and (b) EUP. The modulus signs are necessary if $\alpha$ or $\beta$ is negative. The broken curves apply for negative $\alpha$ and the dotted curves for negative $\beta$.
• Figure 2: Qualitative form of the function $\Delta x (\Delta p)$ in the GEUP case for $\alpha >0$ and $\beta >0$ (solid lines), $\alpha < 0$ and $\beta >0$ (broken lines), $\alpha >0$ and $\beta < 0$ (dotted lines), $\alpha <0$ and $\beta < 0$ (upper dotted plus broken-dotted plus right broken lines). The modulus signs are necessary in all but the first case.
• Figure 3: Form of the function $\Delta x (\Delta p)$ for GEUP with $\alpha = 1$ and different values of $\beta$ (left) and $\beta= 0.1$ and different values of $\alpha$ (right). Note that $\alpha\beta < 1$ is satisfied for all cases.
• Figure 4: GEUP curves for (a) $\alpha <0$ and $\beta>0$, (b) $\alpha >0$ and $\beta<0$ and (c) $\alpha <0$ and $\beta<0$. The constraint $|\alpha\beta|<1$ is maintained and $|\Delta x|$ is required if $\alpha$ and/or $\beta$ is negative (dashed lines).
• Figure 5: Form of the functions $\Delta x (\Delta p)$ in EGUP case for (a) $\alpha > \beta$ and $\alpha < \beta$ and (b) $\alpha = \beta$.