Three-dimensional loop quantum gravity: towards a self-gravitating quantum field theory

In a companion paper, we have emphasized the role of the Drinfeld double DSU( 2) in the context of three-dimensional Riemannian loop quantum gravity coupled to massive spinless point particles. We make use of this result to propose a model for a self- gravitating quantum field theory ( massive spinless non- causal scalar field) in three- dimensional Riemannian space. We start by constructing the Fock space of the free self- gravitating field: the vacuum is the unique DSU( 2) invariant state, one- particle states correspond to DSU( 2) unitary irreducible simple representations and any multi- particles states are obtained as the symmetrized tensor product between simple representations. The associated quantum field is defined by the usual requirement of covariance under DSU( 2). Then, we introduce a DSU( 2)- invariant self- interacting potential ( the obtained model is a group field theory) and explicitly compute the lowest order terms ( in the self- interaction coupling constant.) of the propagator and of the three- point function. Finally, we compute the lowest order quantum gravity corrections ( in the Newton constant G) to the propagator and to the three- point function.