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

Abstract

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