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

Abstract

We present and discuss two different possibilities to construct position space version for MagueijoSmolin (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 fivedimensional 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.

Equilibrium properties of blackbody radiation with an ultraviolet energy cut-off

“…In [10], a momentum dependent measure has been considered which is a special case of our generalized approach. There have been many interesting attempts to incorporate the additivity of energy and momentum of composite systems in DSR [47][48][49][50][51][52][53]. In this paper, we will not follow any particular prescription as the issue is still not well settled.…”

Doubly special relativity and Finsler geometry

“…A last possibility is that the formalism work better when a five-dimensional configuration space is considered as in [9,12].…”

“…Such deformation can be obtained by suitably modifying the generators of boosts, and is not unique. The implementation of DSR in configuration space is not obvious, and has been extensively debated [6][7][8][9]. It is clear, however, that coordinate transformations consistent with DSR must be momentum dependent, and hence the possibility of a momentum (or velocity)-dependent geometry emerges.…”

Undeformed (Additive) Energy Conservation Law in Doubly Special Relativity

“…In Ref. [21] a model has been studied in which both the energy and the spatial momentum compose linearly. The energy-momentum dispersion relation analyzed in Ref.…”

“…Various possibilities have been discussed in the same Ref. [21]. The cases there taken in to the account are: k ∝ λ, k ∝ mλ and k ∝ λ/m.…”