Physical states and transition amplitudes in piecewise °at quantum gravity

In this paper, we show how the path integral for gravity and matter on a piecewise °at 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 e®ective 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 e®ective action can be chosen to be the Hartle{Hawking or the Vilenkin wavefunction for the vacuum manifold component of the transition-amplitude manifold.