The Planck length as a duality of the Cosmological Constant: S-dS and S-AdS thermodynamics from a single expression

Abstract

In this paper we suggest that the Planck length $l_{pl}$ and the Cosmological Constant scale $r_\Lambda=\frac{1}{\sqrt{\Lambda}}$ could in principle be dual each other if we take seriously the so-called q-Bargmann Fock space representation as has been previously suggested by Kempf and others and if additionally we introduce $l_{pl}$ as an ultraviolet cut-off and $r_\Lambda=\frac{1}{\sqrt{\Lambda}}$ as an infrared one. As a consequence, it is possible to demonstrate that a Generalized Uncertainty Principle (GUP) given by $\Delta X \Delta P\geq \frac{\hbar}{2}+\frac{l_{pl}^2}{2\hbar}(\Delta P)^2+\frac{\hbar}{2r_\Lambda^2}(\Delta X)^2$, can reproduce appropriately the thermodynamic behavior for both, the Schwarzschild Anti de-Sitter (S-AdS) and the Schwarzschild de-Sitter (S-dS) space without making any analytic extension for the coefficient (parameter) related to the minimum uncertainty in momentum (already suggested in the literature). This is possible if the Black Hole temperature is described with respect to the "natural" Static Observer for the S-dS case located at a distance $l_0=(3/2r_s r_\Lambda^2)^{1/3}$.