A comment on the paper "Deformed Boost Transformations that saturate at the Planck Scale" by N.B.Bruno,G.Amelino-Camelia, and J.Kowalski-Glikman

TL;DR

This paper presents a simplified derivation of modified dispersion relations in deformed Lorentz symmetry frameworks, clarifying the role of time derivatives and their impact on physical predictions.

Contribution

It offers an alternative, streamlined derivation of dispersion relations consistent with deformed boost transformations, highlighting non-uniqueness issues in defining time derivatives.

Findings

The simplified derivation reproduces known dispersion relations.

Non-uniqueness of time derivative choice does not affect dispersion in ta-k space.

Different time derivative choices lead to different relations in p-p0 space.

Abstract

An alternative (simplified) derivation of the dispersion relation and the expressions for the momentum-energy 4-vector $p_i,p_0$ given initially in [1] is provided. It has turned out that in a rather "pedestrian" manner one can obtain in one stroke not only the above relations but also the correct dispersion relation in $\omega-k_i$ space, consistent with the value of a velocity of a massless particle. This is achieved by considering the standard Lorentz algebra for $\omega-k_i$-space. A non-uniqueness of the choice of the time-derivative in the presence of the finite length scale is discussed. It is shown that such non-uniqueness does not affect the dispersion relation in $\omega-k_i$-space. albeit results in different dispersion relations in $p-p_0$-space depending on the choice of the definition of the time derivative.