Position space versions of Magueijo-Smolin doubly special relativity proposal and the problem of total momentum

TL;DR

This paper explores two approaches to formulate position space versions of Magueijo-Smolin doubly special relativity, addressing the total momentum problem and proposing a five-dimensional framework that incorporates two observer-independent scales.

Contribution

It introduces two distinct methods for constructing position space formulations of MS doubly special relativity, including a five-dimensional approach that resolves the total momentum issue.

Findings

Two consistent position space formulations of MS relativity are proposed.

The five-dimensional approach naturally incorporates total momentum.

The scale $k$ may relate to fundamental constants, implying a quantum of mass.

Abstract

We present and discuss two different possibilities to construct position space version for Magueijo-Smolin (MS) doubly special relativity proposal. The first possibility is to start from ordinary special relativity and then to define conserved momentum in special way. It generates MS invariant as well as nonlinear MS transformations on the momentum space, leading to consistent picture for one-particle sector of the theory. The second possibility is based on the following observation. Besides the nonlinear MS transformations, the MS energy-momentum relation is invariant also under some inhomogeneous linear transformations. The latter are induced starting from linearly realized Lorentz group in five-dimensional position space. Particle dynamics and kinematics are formulated starting from the corresponding five-dimensional interval. There is no problem of total momentum in the theory. The formulation admits two observer independent scales, the speed of light, $c$, and $k$ with dimension of velocity. We speculate on different possibilities to relate $k$ with fundamental constants. In particular, expression of $k$ in terms of vacuum energy suggests emergence of (minimum) quantum of mass.