Canonical Realizations of Doubly Special Relativity

TL;DR

This paper clarifies the equivalence of various canonical phase space realizations of Doubly Special Relativity, explores their uncertainty principles, and discusses implications for gravity's rainbow in a unified framework.

Contribution

It demonstrates the equivalence of different canonical proposals for Doubly Special Relativity and analyzes their implications for uncertainty principles and gravity's rainbow.

Findings

Canonical proposals are shown to be equivalent.

Generalized uncertainty principles are derived and analyzed.

Extension to gravity's rainbow is discussed and compared with existing models.

Abstract

Doubly Special Relativity is usually formulated in momentum space, providing the explicit nonlinear action of the Lorentz transformations that incorporates the deformation of boosts. Various proposals have appeared in the literature for the associated realization in position space. While some are based on noncommutative geometries, others respect the compatibility of the spacetime coordinates. Among the latter, there exist several proposals that invoke in different ways the completion of the Lorentz transformations into canonical ones in phase space. In this paper, the relationship between all these canonical proposals is clarified, showing that in fact they are equivalent. The generalized uncertainty principles emerging from these canonical realizations are also discussed in detail, studying the possibility of reaching regimes where the behavior of suitable position and momentum variables is classical, and explaining how one can reconstruct a canonical realization of doubly special relativity starting just from a basic set of commutators. In addition, the extension to general relativity is considered, investigating the kind of gravity's rainbow that arises from this canonical realization and comparing it with the gravity's rainbow formalism put forward by Magueijo and Smolin, which was obtained from a commutative but noncanonical realization in position space.