From Fock's Transformation to de Sitter Space

TL;DR

This paper derives the Fock transformation from deformed Poisson brackets, constructs its Casimir, and demonstrates that the resulting spacetime is de Sitter space with a radius linked to the invariant length in the transformation.

Contribution

It introduces a new derivation of Fock's transformation using deformed algebra and shows its connection to de Sitter spacetime.

Findings

Fock transformation derived from deformed Poisson brackets.

Constructed the first Casimir of the deformed algebra.

Identified the spacetime as de Sitter space with a specific radius.

Abstract

As in Deformed Special Relativity, we showed recently that the Fock coordinate transformation can be derived from a new deformed Poisson brackets. This approach allowed us to establish the corresponding momentum transformation which keeps invariant the four dimensional contraction $p_{\mu} x^{\mu} $. From the resulting deformed algebra, we construct in this paper the corresponding first Casimir. After first quantization, we show by using the Klein-Gordon equation that the spacetime of the Fock transformation is the de Sitter one. As we will see, the invariant length representing the universe radius in the spacetime of Fock's transformation is exactly the radius of the embedded hypersurface representing the de Sitter spacetime.