Self-consistency in Theories with a Minimal Length

TL;DR

This paper clarifies the relationships among three approaches to theories with a minimal length scale, showing their equivalences and how they can be integrated into a comprehensive framework.

Contribution

It demonstrates the equivalence between Deformed Special Relativity and Generalized Uncertainty Principle approaches and explains how to derive Modified Dispersion Relations from them.

Findings

Deformed Special Relativity and GUP are equivalent.

All three features are necessary for a consistent minimal length theory.

The paper provides translation methods between the approaches.

Abstract

The aim of this paper is to clarify the relation between three different approaches of theories with a minimal length scale: A modification of the Lorentz-group in the 'Deformed Special Relativity', theories with a 'Generalized Uncertainty Principle' and those with 'Modified Dispersion Relations'. It is shown that the first two are equivalent, how they can be translated into each other, and how the third can be obtained from them. An adequate theory with a minimal length scale requires all three features to be present.