From Fock’s transformation to de Sitter space

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 pµx µ . 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. : 03.30.+p, 98.80.Jk PACS

Maxwell’s equations in the context of the Fock transformation and the magnetic monopole

“…The corresponding first Casimir is constructed and allowed the derivation of the Klein-Gordon equation which turned out to be the Klein-Gordon equation in de Sitter Space given by its conformal metric [8]. This result established a correspondence between the R-Minkowski spacetime and de Sitter's spacetime.…”

“…constituting the phase space algebra of the R-Minkowski spacetime [8]. Here η µν = (+1, −1, −1, −1) and µ, ν = 0, 1, 2, 3.…”

Dirac’s Equation in R-Minkowski Spacetime

“…For more details, the presence of these two additional terms is justified in Appendix A. Using the representation given by (21) and (22), proposed in the R-Poincaré algebra context [8], we obtain the differential form of (30)…”

“…Also, we aim to explore more the correspondence, already established in [8], between de Sitter spacetime and the R-Minkowski spacetime. This may offer more possibilities to handle the issue of constructing physical observables in the de Sitter space [13,14].…”

“…The R-algebra, constituted by (1), (2) and (3), is completed in [8] by involving pure rotation generators,…”