Incorporating the Cosmological Constant in a Modified Uncertainty Principle

Abstract

This study explores the cosmological constant problem and modified uncertainty principle within a unified framework inspired by a void-dominated scenario. In a recent paper~\cite{Yusofi:2022hgg}, voids were modeled as spherical bubbles of similar average sizes, and the surface energy on the voids' borders was calculated across various scales in a heuristic manner. We show that this results in a significant discrepancy of approximately $\mathcal{O}(+122)$ between the cosmological constant values from the minimum to the maximum radii of bubbles. Furthermore, when considering the generalized form of the uncertainty principle with both minimum and maximum lengths, i.e. $\Delta X \Delta P \geq \frac{\hbar}{2} \frac{1}{1- \beta \Delta P^2} \frac{1}{1- \alpha \Delta X^2}$, a similar order of discrepancy is observed between $\alpha_{\rm max}$ and $\alpha_{\rm min}$, indicating that $\alpha\propto\beta^{-1}\propto\Lambda\propto{\rm length}^{-2}~(m^{-2})$. As a primary outcome of this finding, we offer a novel uncertainty principle that incorporates a non-zero cosmological constant.