Conformal Group With Two Observer Independent Scales

Abstract

The Poincaré sector of a recently deformed conformal algebra is proposed to describe, after the identification of the deformation parameter with the Planck length, the symmetries of a new relativistic theory with two observer-independent scales (or DSR theory). Also a new non-commutative space-time is proposed. It is found that momentum space exhibits the same features of the DSR proposals preserving Lorentz invariance in a deformed way. The space-time sector is a generalization of the well known non-commutative κ-Minkowski space-time which however does not preserve Lorentz invariance, not even in the deformed sense. It is shown that this behavior could be expected in some attempts to construct DSR theories starting from the Poincaré sector of a deformed symmetry larger than Poincaré symmetry, unless one takes a variable Planck length. It is also shown that the formalism can be useful in analyzing the role of quantum deformations in the "AdS-CFT correspondence".One of the most studied applications of quantum groups in physics is in the description of deformed spacetime symmetries that generalize classical Poincaré kinematics beyond the Lie-algebra level. Well-known examples are the κ-Poincaré 1 and the quantum null-plane (or light-cone) Poincaré 2 algebras. The deformation parameter has been interpreted as a fundamental scale which may be related with the Planck length. In fact, these results can be seen as different attempts to develop new approaches to physics at the Planck scale 3 . It has been also recently conjectured that the κ-Poincaré algebra may provide the basis for a so-called doubly special relativity (DSR) 4,5,6,7 in which the deformation parameter/Planck length is viewed as an observer-independent length scale completely analogous to the familiar observer-independent velocity scale c, in such a manner that (deformed) Lorentz invariance is preserved 8,9,10 . In this talk I want to review briefly some related results which have been presented in greater detail in [11,12]. These results provide an analysis of the Poincaré sector of a manageable deformation of so(4, 2) introduced in [13] together with its dual in order to extract some physical implications on the associated non-commutative Minkowskian spacetime.If {J i , P µ = (P 0 , P), K i , D} denote the generators of rotations, time and space translations, boosts and dilations, the non-vanishing deformed commutation rules 1