Physical States and Transition Amplitudes in Piecewise Flat Quantum Gravity

We show how the path integral for gravity and matter on a piecewise flat spacetime can be used to define the physical quantum gravity states and the related transition amplitudes. The physical states are given by the path integrals for open manifolds from a certain topological class, while the corresponding transition amplitudes are obtained by gluing two such open manifolds into a closed one and calculating the corresponding path integral. We also solve the problem of how to associate a quantum gravity state to an effective action. This is done by using the path integral for a closed manifold from the transition-amplitude topological class so that the state which corresponds to the effective action can be choosen to be the Hartle-Hawking or the Vilenkin wavefunction for the vacuum manifold component of the transition-amplitude manifold.