Effective particle kinematics from quantum gravity

Particles propagating in de Sitter spacetime can be described by the topological BF $\mathsf{S}\mathsf{O}(4,1)$ 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 $\mathsf{S}\mathsf{O}(3,1)$, and which can be treated as a perturbation. In this paper we focus solely on topological interactions which correspond to zeroth order in this perturbative expansion. We show that in this approximation the system is effectively described by the $\mathsf{S}\mathsf{O}(4,1)$ Chern-Simons theory coupled to particles and living on the three-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.

Figure 1

Alekseev-Malkin construction, step 1.

Figure 2

Alekseev-Malkin construction, step 2.