Doubly Special Relativity is just a change of variables

In a five-page-long highly efficient paper, Nosratollah Jafari and Ahmad Shariati show that the rules of the so-called "doubly special relativity" (DSR) to transform the energy-momentum vectors are nothing else than the ordinary rules of special relativity translated to awkward variables that parameterize the energy and the momentum.

In fact, there have been two main "subspecies" of DSR proposed in the literature, and our Iranian friends trivialize both of them. The DSR by Magueijo and Smolin is the simpler one.

Magueijo and Smolin demystified

Jafari and Shariati start with the realization that much like

• "p^m pⁿ g_mn"
  

is preserved by the Lorentz transformations in ordinary special relativity, a similar function

• "p^m pⁿ g_mn / ( 1 - L p⁰ )²"

where L is the Planck length - is preserved in the Magueijo-Smolin DSR. This leads them to define new variables for the energy momentum vector

• "pi^m = p^m / ( 1 - L p⁰ )"

and unless they have made a trivial error, they have shown that in terms of these new variables "pi^m", the DSR Lorentz transformations of the 4-vector reduce to the standard Lorentz transformations of special relativity. Note that all components of the energy-momentum vector have the same denominator in the redefinition which is why the direction in spacetime (or velocity) is preserved - it is only the overall magnitude of the 4-vector that is redefined.

Amelino-Camelia DSR

They also show that another version of the transformation rules, one that involves a lot of "sinh" and "cosh", is equivalent to ordinary special relativity, too. In this particular case, they find a non-linear redefinition of the generators of boosts instead of the energy-momentum vector itself. So far I have not checked this portion of the paper at all but I have not found a good reason to doubt that they are correct either.

In this Amelino-Camelia case, it is somewhat unclear whether the physical system is supposed to be invariant under the modified or unmodified boosts and what it exactly means for the system to be invariant under the modified boosts. But I guess that whatever answer we choose, we either obtain something that is equivalent to the conventional theories or inconsistent.

Conclusions

Consequently, there is no new physics in doubly special relativity, and the concept of doubly special relativity can't be used to explain why some physically sick theories such as loop quantum gravity fail to be Lorentz-invariant. Of course, this is no surprise for most physicists because everyone has always known that there can't be any fundamentally new theories that are just "partially" Lorentz-invariant. But still, it is useful to have an explicit set of formulae that prove that there is no new physics behind the papers about DSR. Note that the required field redefinitions are given by non-polynomial functions of the momenta which is why physics would look non-local in the coordinates that are dual to the bizarre DSR momenta instead of the standard ones.

What I still find a bit confusing is that I thought that the Amelino-Camelia DSR had the Poincare group that was a contraction of the quantum deformed AdS or dS symmetry. Quantum deformation looks like something different from a cheap redefinition of variables (or generators). But maybe this difference goes away in the limit of a vanishing cosmological constant, i.e. because of the contraction.

These doubts aside, DSR is certainly not the only example of an overhyped idea that is sold almost as a competitor to the whole field of high-energy physics or string theory but whose emptiness can be shown on one page or two.