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.Comment: 18 pages, accepted for publication in International Journal of Modern
Physics