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. [email protected] [email protected] 1