Anti-de Sitter momentum space
Theoretical Physics, Blackett Laboratory, Imperial College, London, SW7 2BZ, United Kingdom(Dated: December 8, 2014)We investigate the anti-de Sitter (AdS) counterpart to the well studied de Sitter (dS) model forenergy-momentum space, viz \-momentum space" space (with a structure based on the propertiesof the -Poincare Hopf algebra). On the basis of previous preliminary results one might expect thetwo models to be \dual": dS exhibiting an invariant maximal spatial momentum but unboundedenergy, AdS a maximal energy but unbounded momentum. If that were the case AdS momentumspace could be used to implement a principle of maximal Planck-scale energy, just as several studiesuse dS momentum space to postulate of maximal Planck-scale spatial momentum. However severalunexpected features are uncovered in this paper, which limit the scope of the expected duality,and interestingly they take di erent forms in di erent coordinatizations of AdS momentum space.\Cosmological" AdS coordinates mimic the dS construction used for -momentum space, and pro-duce a Carrol limit in the ultraviolet. However, unlike the -momentum space, the boundary of thecovered patch breaks Lorentz invariance, thereby introducing a preferred frame. In \horospherical"coordinates we achieve full consistency with frame independence as far as boost transformationsare concerned, but nd that rotational symmetry is broken, leading to an anisotropic model for thespeed of light. Finally, in \static" coordinates we nd a way of deforming relativistic transforma-tions that successfully enforces frame invariance and isotropy, and produces a Carrol limit in theultraviolet. However, the phenomenological implications appear to be too weak for any realisticchance of detection. Our results are also relevant for a long-standing debate on whether or not coor-dinate rede nitions in momentum space lead to physically equivalent theories: our three proposalsare evidently physically inequivalent, leading to alternative models of Planck-scale e ects. As acorollary we study the UV running of the Hausdor dimension of momentum space in the rst andthird model, obtaining di erent results.I. INTRODUCTION