Non Linear Lorentz Transformation and Doubly Special Relativity

TL;DR

This paper develops non-linear Lorentz transformations incorporating two invariant scales, proposing a formalism for doubly special relativity models, including a Lagrangian approach, effective metrics, and a quantization ansatz.

Contribution

It introduces a new class of doubly special relativity models using non-linear Lorentz transformations with two scales, and provides a Lagrangian and quantization framework.

Findings

Models exhibit two invariant scales: velocity and energy.

Effective metric for deformed Minkowski space derived.

Smolin model identified as part of the doubly special relativity family.

Abstract

We generate non-linear representations of the Lorentz Group by unitary transformation over the Lorentz generators. To do that we use deformed scale transformations by introducing momentum-depending parameters. The momentum operator transformation is found to be equivalent to a particle momentum transformation. The configuration space transformation is found to depend on the old momentum operator and we show that this transformation generates models with two scales, one for the velocity ($c$) and another one for the energy. A Lagrangian formalism is proposed for these models and an effective metric for the deformed Minkowski space is found. We show that the Smolin model is one in a family of doubly special relativity. Finally we construct an ansatz for the quantization of such theories.