Doubly Special Relativity in Position Space Starting from the Conformal Group

Abstract

We propose version of doubly special relativity theory starting from position space. The version is based on deformation of ordinary Lorentz transformations due to the special conformal transformation. There is unique deformation which does not modify rotations. In contrast to the Fock-Lorentz realization (as well as to recent position-space proposals), maximum signal velocity is position (and observer) independent scale in our formulation by construction. The formulation admits one more invariant scale identified with radius of three-dimensional space-like hypersection of space-time. We present and discuss the Lagrangian action for geodesic motion of a particle on the DSR space. For the present formulation, one needs to distinguish the canonical (conjugated to $x^\mu$) momentum $p^\mu$ from the conserved energy-momentum. Deformed Lorentz transformations for $x^\mu$ induce complicated transformation law in space of canonical momentum. $p^\mu$ is not a conserved quantity and obeys to deformed dispersion relation. The conserved energy-momentum $P^\mu$ turns out to be different from the canonical one, in particular, $P^\mu$-space is equipped with nontrivial commutator. The nonlinear transformations for $x^\mu$ induce the standard Lorentz transformations in space of $P^\mu$. It means, in particular, that composite rule for $P^\mu$ is ordinary sum. There is no problem of total momentum in the theory. $P^\mu$ obeys the standard energy-momentum relation (while has nonstandard dependence on velocity).