On the Fock Transformation in Nonlinear Relativity

Abstract

In this paper, we propose a new deformed Poisson brackets which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity. We therefore derive the corresponding momentum transformation which is revealed to be different from previous results. Contrary to the earlier version of Fock's nonlinear relativity for which plane waves cannot be described, our resulting algebra keeps invariant for any coordinate and momentum transformations the four dimensional contraction pµx µ , allowing therefore to associate plane waves for free particles. As in Deformed Special Relativity, we also derive a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors. Finally, we establish the dispersion relation for Fock's nonlinear relativity.

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

“…We have also established the expression of the Lorentz force in this context. The R-Lorentz algebra, established in [2], is restored in the presence of the electromagnetic field by adding to the angular momentum the electromagnetic field contribution which generates the Dirac magnetic monopole. Contrary to the DSR case [15], the angular magnetic momentum is affected by the parameter deformation.…”

“…In order to establish Maxwell's equations in the context of Fock's nonlinear relativity, let us first search for the analogous of relations (10) and (12) in the framework of R-Minkowski spacetime. For this purpose, let us recall that it is shown in [2] that the following canonical variables…”

“…where γ = (1−u 2 /c 2 ) −1/2 , α R = 1+[(γ − 1)ct − γxu/c] /R and R being the universe radius. This transformation was recently reproduced from an appropriate deformed Poisson brackets [2] by using an analogous procedure as in [3], where the coordinate transformation of Deformed Special Relativity [4,5,6,7] is recovered. It is shown then that in addition to the constant c which represents the speed of light at the limit R → ∞, transformation (1) keeps also invariant the universe radius.…”

From Fock’s transformation to de Sitter space

“…The corresponding coordinate transformation is not linear and it is proved later that it can be constructed by deforming the Poisson brackets [7]. By following the same way, we showed recently [8] that also the Fock coordinate transformation [9],…”

Dirac’s Equation in R-Minkowski Spacetime

“…By following the same way as in DSR [6], we recently proposed an appropriate deformed Poisson brackets [7] {x µ , x ν } = 0,…”