We present a simple algebraic argument for the conclusion that the low energy limit of a quantum theory of gravity must be a theory invariant, not under the Poincaré group, but under a deformation of it parameterized by a dimensional parameter proportional to the Planck mass. Such deformations, called κ-Poincaré algebras, imply modified energy-momentum relations of a type that may be observable in near future experiments. Our argument applies in both 2 + 1 and 3 + 1 dimensions and assumes only 1) that the low energy limit of a quantum theory of gravity must involve also a limit in which the cosmological constant is taken very small with respect to the Planck scale and 2) that in 3 + 1 dimensions the physical energy and momenta of physical elementary particles is related to symmetries of the full quantum gravity theory by appropriate renormalization depending on Λl 2 P lanck . The argument makes use of the fact that the cosmological constant results in the symmetry algebra of quantum gravity being quantum deformed, as a consequence when the limit Λl 2 P lanck → 0 is taken one finds a deformed Poincaré invariance. We are also able to isolate what information must be provided by the quantum theory in order to determine which presentation of the κ-Poincaré algebra is relevant for the physical symmetry generators and, hence, the exact form of the modified energy-momentum relations. These arguments imply that Lorentz invariance is modified as in proposals for doubly special relativity, rather than broken, in theories of quantum gravity, so long as those theories behave smoothly in the limit the cosmological constant is taken to be small.

Since the work of Mac-Dowell-Mansouri it is well known that gravity can be written as a gauge theory for the de Sitter group. In this paper we consider the coupling of this theory to the simplest gauge invariant observables that is, Wilson lines. The dynamics of these Wilson lines is shown to reproduce exactly the dynamics of relativistic particles coupled to gravity, the gauge charges carried by Wilson lines being the mass and spin of the particles. Insertion of Wilson lines breaks in a controlled manner the diffeomorphism symmetry of the theory and the gauge degree of freedom are transmuted to particles degree of freedom.

Particles propagating in de Sitter spacetime can be described by the topological BF SO(4, 1) theory coupled to point charges. Gravitational interaction between them can be introduced by adding to the action a symmetry breaking term, which reduces the local gauge symmetry down to SO(3, 1), and which can be treated as a perturbation. In this paper we focus solely on topological interactions which corresponds to zeroth order in this perturbative expansion. We show that in this approximation the system is effectively described by the SO(4, 1) Chern-Simons theory coupled to particles and living on the 3 dimensional boundary of spacetime. Then, using Alekseev-Malkin construction we find the effective theory of particles kinematics. We show that the particles action contains standard kinetic terms and the deformation shows up in the presence of interaction terms. The strength of the interactions is proportional to deformation parameter, identified with Planck mass scale.

The issue of non-perturbative background independent quantization of matrix models is addressed. The analysis is carried out by considering a simple matrix model which is a matrix extension of ordinary mechanics reduced to 0 dimension. It is shown that this model has an ordinary mechanical system evolving in time as a classical solution. But in this treatment the action principle admits a natural modification which results in algebraic relations describing quantum theory. The origin of quantization is similar to that in Adler's generalized quantum dynamics. The problem with extension of this formalism to many degrees of freedom is solved by packing all the degrees of freedom into a single matrix. The possibility to apply this scheme to field theory and to various matrix models is discussed.Comment: 21 pages, no figure