Definitions

# Doubly-special relativity

Doubly-special relativity (DSR)— also called deformed special relativity or, by some, extra-special relativity — is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of light), but an observer-independent minimum length (the Planck length).

This was first proposed in a paper by Giovanni Amelino-Camelia, though it is at least implicit in a paper by Paul Merriam. An alternate approach to doubly-special relativity theory, inspired by that of Amelino-Camelia, was proposed later by João Magueijo and Lee Smolin. There exist proposals that these theories may be related to loop quantum gravity.

The motivation to these proposals is mainly theoretical, based on the following observation: The Planck length is expected to play a fundamental role in a theory of Quantum Gravity, setting the scale at which Quantum Gravity effects cannot be neglected and new phenomena are observed. If Special Relativity is to hold up exactly to this scale, different observers would observe Quantum Gravity effects at different scales, due to the Lorentz-FitzGerald contraction, in contradiction to the principle that all inertial observers should be able to describe phenomena by the same physical laws.

There is still no experimental evidence of any departure from special relativity, but rather this is highly constrained. Nevertheless some authors suggest that the observation of high-energy cosmic rays that appear to violate the Greisen-Zatsepin-Kuzmin limit: the so-called Oh-My-God particles may be an indication of the failure of special relativity at this energy scale.

In principle, it seems difficult to incorporate an invariant length magnitude in a theory which preserves Lorentz invariance due to Lorentz-FitzGerald contraction, but in the same way that Special Relativity incorporates an invariant velocity by modifying the high-velocity behavior of Galilean transformations, DSR modifies Lorentz transformations at small distances (large energies) in such a way to admit a length invariant scale without destroying the principle of relativity. The postulates on which DSR theories are constructed are:

1. The principle of relativity holds, i.e. equivalence of all inertial observers.
2. There are two observer-independent scales: the speed of light, c, and a length (energy) scale $lambda$ ($eta=1/lambda$) in such a way that when $lambda->0$ ($eta->infinity$), special relativity is recovered.

As noted by Jerzy Kowalski-Glikman, an immediate consequence of these postulates is that the symmetry group of DSR theories must be ten dimensional, corresponding to boosts, rotations and translations in 4 dimensions. Translations, however, cannot be the usual Poincaré generators as it would be in contradiction with postulate 2). As translation operators are expected to be modified, the usual dispersion relation

$E^2-p^2=m^2$

is expected to be modified and, indeed, the presence of an energy scale, namely $eta$, allows introducing $eta$-suppressed terms of higher order in the dispersion relation. These higher momenta powers in the dispersion relation can be traced back as having their origin in higher dimensional (i.e. non-renormalizable) terms in the Lagrangian.

It was soon realized that by deforming the Poincaré (i.e. translation) sector of the Poincaré algebra, consistent DSR theories can be constructed. In accordance with postulate 1), the Lorentz sector of the algebra is not modified, but just non-linearly realized in their action on momenta coordinates. More precisely, the Lorentz Algebra

$\left[M_i,M_j\right]=iepsilon_\left\{ijk\right\} M_k$

$\left[N_i,N_j\right]=-iepsilon_\left\{ijk\right\} M_k,$

$\left[M_i,N_j\right]=-iepsilon_\left\{ijk\right\} N_k$

$\left[M_i,p_j\right]=iepsilon_\left\{ijk\right\} p_k$

$\left[M_i,p_0\right]=0$

remains unmodified, while the most general modification on its action on momenta is

$\left[N_i,p_j\right]=Adelta_\left\{ij\right\}+Bp_ip_j+Cepsilon_\left\{ijk\right\}N_k$

$\left[N_i,p_0\right]=Dp_i$

where A, B, C and D are arbitrary functions of $\left\{p_i, p_0,eta\right\}$ and M,N are the rotation generators and boost generators, respectively. It can be shown that C must be zero and in order to satisfy the Jacobi identity, A, B and D must satisfy a non-linear first order differential equation. It was also shown by Kowalski-Glikman that these constraints are automatically satisfied by requiring that the boost and rotation generators N and M, act as usual on some coordinates $eta_A$ (A=0,...,4) that satisfy

$-eta^2=eta_0^2-eta_1^2-eta_2^2-eta_3^2-eta_4^2$

i.e. that belong to de Sitter space. The physical momenta $p_mu$ are identified as coordinates in this space, i.e.

$p_mu=p_mu\left(eta_A,eta\right)$

and the dispersion relation that these momenta satisfy is given by the invariant

$m^2=eta_0^2\left(p_mu,eta\right)-eta_i^2\left(p_mu,eta\right)$.

This way, different choices for the "physical momenta coordinates" in this space give rise to different modified dispersion relations, a corresponding modified Poincare algebra in the Poincaré sector and a preserved underlying Lorentz invariance.

One of the most common examples is the so-called Magueijo-Smolin basis (Also known as the DSR2 model), in which:

$eta_mu=frac\left\{p_mu\right\}\left\{1-P_0/eta\right\}$

which implies, for instance,

$\left[N_i,P_0\right]=iP_i\left(1-frac\left\{P_0\right\}\left\{eta\right\}\right)$,

showing explicitly the existence of the invariant energy scale $P_0=eta$ as $\left[N_i,P_0=eta\right]=0$.

The theory is highly speculative as of first publishing in 2002, as it relies on no experimental evidence so far. It would be fair to say that DSR is not considered a promising approach by a majority of members of the high-energy physics community, as it lacks experimental evidence and there's so far no guiding principle in the choice for the particular DSR model (i.e. basis in momenta de Sitter space) that should be realized in Nature, if any.

DSR is based upon a generalization of symmetry to quantum groups. The Poincaré symmetry of ordinary special relativity is deformed into some noncommutative symmetry and Minkowski space is deformed into some noncommutative space. As explained before, this theory is not a violation of Poincaré symmetry as much as a deformation of it and there is an exact de Sitter symmetry. This deformation is scale dependent in the sense that the deformation is huge at the Planck scale but negligible at much larger length scales. It's been argued that models which are significantly Lorentz violating at the Planck scale are also significantly Lorentz violating in the infrared limit because of radiative corrections, unless a highly unnatural fine-tuning mechanism is implemented. Without any exact Lorentz symmetry to protect them, such Lorentz violating terms will be generated with abandon by quantum corrections. However, DSR models do not succumb to this difficulty since the deformed symmetry is exact and will protect the theory from unwanted radiative corrections — assuming the absence of quantum anomalies. Furthermore, models where a privileged rest frame exists can escape this difficulty due to other mechanisms.

Jafari and Shariati have constructed canonical transformations that relate both the doubly-special relativity theories of Amelino-Camelia and of Magueijo and Smolin to ordinary special relativity. They claim that doubly-special relativity is therefore only a complicated set of coordinates for an old and simple theory. However, all theories are related to free theories by canonical transformations. Therefore supporters of doubly-special relativity may claim that while it is equivalent to ordinary relativity, the momentum and energy coordinates of doubly-special relativity are those that appear in the usual form of the standard model interactions. It was initially speculated that ordinary special relativity and doubly-special relativity would make distinct physical predictions in high energy processes, and in particular the derivation of the Greisen-Zatsepin-Kuzmin limit would not be valid. However, it is now established that standard doubly special relativity does not predict any suppression of the GZK cutoff, contrary to the pattern explored since 1997 by Luis Gonzalez-Mestres ("weak" doubly special relativity) where an absolute local rest frame (the "vacuum rest frame") exists.