  Investigations of the possibility that some novel ``quantum" properties of
spacetime might induce a modification dispersion relation focused at first on
scenarios with Planck-scale violations of Lorentz symmetry. More recently
several studies have considered the possibility of a ``doubly special
relativity", in which the modification of the dispersion relation emerges from
a framework with both the Planck scale and the speed-of-light scale as
characteristic scales of a deformation of the Lorentz transformations. For the
schemes with broken Lorentz symmetry at the Planck scale there is a large
literature on the derivation of experimental limits. We provide here a
corresponding analysis for the doubly-special-relativity framework. We find
that the analyses of photon stability, synchrotron radiation, and threshold
conditions for particle production in collision processes, the three contexts
which are considered as most promising for constraining the
broken-Lorentz-symmetry scenario, cannot provide significant constraints on
doubly-special-relativity parameter space. However, certain types of analyses
of gamma-ray bursts are sensitive to the symmetry deformation. A key element of
our study is an observation that removes a possible sign ambiguity for the
doubly-special-relativity framework. This result also allows us to characterize
more sharply the differences between the doubly-special-relativity framework
and the framework of k-Poincare Hopf algebras, two frameworks which are often
confused with each other in the literature.
