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

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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 ℓₚ=(Gℏ c⁻³)¹/² and the dS-invariant special relativity in dS-spacetime of radius R≃(3Λ⁻¹)¹/², which is another fundamental length related to the cosmological constant. Here, the cosmological constant Λ is regarded as a fundamental constant together with the speed of light c, Newton constant G and Planck constant ℏ. Furthermore, the physics at two fundamental scales of length, the -radius R and the Planck length ℓₚ, should be dual to each other and linked via the gravity with local dS-invariance characterized by a dimensionless coupling constant g= √3 ℓₚ/R≃(Gℏ c⁻³Λ)¹/² 10⁻⁶¹.