The Beltrami Model of De Sitter Space: From Snyder's quantized space-time to de Sitter invariant relativity

Abstract

In terms of the Beltrami model of de Sitter space we show that there is an interchangeable relation between Snyder's quantized space-time model in dS-space of momenta at the Planck length ${\ell }_P=(G\hslash c^{-3}{)}^{1/2}$ and the dS-invariant special relativity in dS-spacetime of radius $R\simeq (3{\Lambda }^{-1}{)}^{1/2}$, which is another fundamental length related to the cosmological constant. Here, the cosmological constant $\Lambda$ is regarded as a fundamental constant together with the speed of light $c$, Newton constant $G$ and Planck constant $\hslash$. Furthermore, the physics at two fundamental scales of length, the \dS-radius $R$ and the Planck length ${\ell }_P$, should be dual to each other and linked via the gravity with local dS-invariance characterized by a dimensionless coupling constant $g=\sqrt{3}{\ell }_P/R\simeq (G\hslash c^{-3}\Lambda {)}^{1/2}\sim 10^{-61}$.

15 pages. Invited talk given at `International workshop on noncommutative geometry and physics', Beijing, Nov. 7-10, 2005. To appear in the proceedings