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

Understand

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 $l_0=(3/2r_s r_\Lambda^2)^{1/3}$.