Non-commutative space-time of doubly special relativity theories

Doubly Special Relativity (DSR) theory is a recently proposed theory with two observer-independent scales (of velocity and mass), which is to describe a kinematic structure underlining the theory of Quantum Gravity. We observe that there are infinitely many DSR constructions of the energy-momentum sector, each of whose can be promoted to the kappa-Poincare quantum (Hopf) algebra. Then we use the co-product of this algebra and the Heisenberg double construction of kappa-deformed phase space in order to derive the non-commutative space-time structure and the description of the whole of DSR phase space. Next we show that contrary to the ambiguous structure of the energy momentum sector, the space-time of the DSR theory is unique and related to the theory with non-commutative space-time proposed long ago by Snyder. This theory provides non-commutative version of Minkowski space-time enjoying ordinary Lorentz symmetry. It turns out that when one builds a natural phase space on this space-time, its intrinsic length parameter l becomes observer-independent.