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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 [186Jump To The Next Citation Point]). 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 c and a new momentum scale EDSR are invariant. Usually EDSR 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

E = ----e----, (20) 1 + cDSRe ----p----- p = 1 + c p , (21) DSR
where -1 cDSR = E DSR, E and p are the physical/measured energy and momentum, and e and p are called the ”pseudo-energy” and ”pseudo-momentum”, respectively. e and p transform under the usual Lorentz transforms, which induce corresponding transformations of E and p [163]. Similarly, the e and p for particles are conserved as energy and momentum normally are for a scattering problem.3 Given this set of rules, for any measured particle momentum and energy, we can solve for e and p and calculate interaction thresholds, etc. The invariant dispersion relation for the DSR2 boosts is given by
2 2 E2 - p2 = m--(1---cDSRE--). (22) 1 - cDSRm2
This concludes our (brief) discussion of the basics of DSR. For further introductions to DSR and DSR phenomenology see [186Jump To The Next Citation Point21Jump To The Next Citation Point1495]. We discuss the threshold behavior of DSR theories in Section 6.6.1.
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