Three-dimensional loop quantum gravity: physical scalar product and spin-foam models

In this paper, we address the problem of the dynamics in three-dimensional loop quantum gravity with zero cosmological constant. We construct a rigorous definition of Rovelli's generalized projection operator from the kinematical Hilbert space-corresponding to the quantization of the infinite-dimensional kinematical configuration space of the theory-to the physical Hilbert space. In particular, we provide the definition of the physical scalar product which can be represented in terms of a sum over (finite) spin-foam amplitudes. Therefore, we establish a clear-cut connection between the canonical quantization of three-dimensional gravity and spin-foam models. We emphasize two main properties of the result: first that no cut-off in the kinematical degrees of freedom of the theory is introduced (in contrast to standard 'lattice' methods), and second that no ill-defined sum over spins ('bubble' divergences) are present in the spin-foam representation.