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

TL;DR

This paper explores the duality between Snyder's quantized space-time and de Sitter invariant relativity, linking fundamental length scales and constants within a unified framework based on the Beltrami model of de Sitter space.

Contribution

It establishes a relation between Snyder's quantized space-time and de Sitter invariant relativity using the Beltrami model, connecting fundamental constants and length scales.

Findings

Duality between Snyder's model and de Sitter relativity

Relation linking Planck length and de Sitter radius via gravity

Dimensionless coupling constant quantifies their connection

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 $\ell_P=(G\hbar c^{-3})^{1/2}$ 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 $\hbar$. 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 $g= \sqrt{3} \ell_P/R\simeq(G\hbar c^{-3}\Lambda)^{1/2}\sim 10^{-61}$.