On the Fock Transformation in Nonlinear Relativity

TLDR: In this article, a new deformed Poisson bracket was proposed, which leads to the Fock coordinate transformation by using an analogous procedure as in Deformed Special Relativity, and a canonical transformation with which the new coordinates and momentum satisfy the usual Poisson brackets and therefore transform like the usual Lorentz vectors.

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 $p_{\mu} x^{\mu} $, 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.