### 3.4 “Doubly special” relativity

Doubly special relativity (DSR), which has only been extensively studied over the past few years, is a
novel idea about the fate of Lorentz invariance. DSR is not a complete theory as it has no dynamics and
generates problems when applied to macroscopic objects (for a discussion see [186]). Furthermore, it is not
fully settled yet if DSR is mathematically consistent or physically meaningful. Therefore it is
somewhat premature to talk about robust constraints on DSR from particle threshold interactions
or other experiments. One might then ask, why should we talk about it at all? The reason is
twofold. First, DSR is the subject of a good amount of theoretical effort and so it is useful to see
if it can be observationally ruled out. The second reason is purely phenomenological. As we
shall see in the sections below, the constraints on Lorentz violation are astoundingly good in
the effective field theory approach. With the current constraints it is difficult to fit Lorentz
violation into an effective field theory in a manner that is theoretically natural yet observationally
viable.
DSR, if it can eventually be made mathematically consistent in its current incarnation, has one
phenomenological advantage - it does not have a preferred frame. Therefore it evades most of the threshold
constraints from astrophysics as well as any terrestrial experiment that looks for sidereal variations, while
still modifying the usual action of the Lorentz group. Since these experiments provide almost all of the tests
of Lorentz violation that we have, DSR becomes more phenomenologically attractive as a Lorentz
violating/deforming theory.

So what is DSR? At the level we need for phenomenology, DSR is a set of assumptions that the Lorentz
group acts in such a way that the usual speed of light and a new momentum scale are invariant.
Usually is taken to be the Planck energy - we also make this assumption. All we will
need for this review are the Lorentz boost expressions and the conservation laws, which we
will postulate as true in the DSR framework. For brevity we only detail the Magueijo-Smolin
version of DSR [210], otherwise known as DSR2 - the underlying conclusions for DSR1 [15]
remain the same. The DSR2 boost transformations are most easily derived from the relations

where , and are the physical/measured energy and momentum, and and are
called the ”pseudo-energy” and ”pseudo-momentum”, respectively. and transform under the usual
Lorentz transforms, which induce corresponding transformations of and [163]. Similarly, the
and for particles are conserved as energy and momentum normally are for a scattering
problem.
Given this set of rules, for any measured particle momentum and energy, we can solve for and and
calculate interaction thresholds, etc. The invariant dispersion relation for the DSR2 boosts is given by
This concludes our (brief) discussion of the basics of DSR. For further introductions to DSR and DSR
phenomenology see [186, 21, 14, 95]. We discuss the threshold behavior of DSR theories in
Section 6.6.1.