  We study the transition amplitudes in state-sum models of quantum gravity in
D=2,3,4 spacetime dimensions by using the field theory over a Lie group
formulation. By promoting the group theory Fourier modes into creation and
annihilation operators we construct a Fock space for the quantum field theory
whose Feynman diagrams give the transition amplitudes. By making products of
the Fourier modes we construct operators and states representing the spin
networks associated to triangulations of spatial boundaries of a triangulated
spacetime manifold. The corresponding spin network amplitudes give the
state-sum amplitudes for triangulated manifolds with boundaries. We also show
that one can introduce a discrete time evolution operator, where the time is
given by the number of D-simplices in the triangulation, or equivalently by the
number of the vertices in the Feynman diagram. The corresponding transition
amplitude is a finite sum of Feynman diagrams, and in this way one avoids the
problem of infinite amplitudes caused by summing over all possible
triangulations.
