Boundary State Stability under Spinfoam Evolution for the Quantum 4-Simplex

In the spinfoam framework for quantum gravity, we investigate the conditions to have a physical quantum state for the Barrett-Crane model for the 4d quantum gravity path integral. More precisely, we look at the simplest case of a single 4-simplex boundary and show that the requirement of working with a physical boundary state fixes the width of the semi-classical Gaussian wave-packet for the boundary 3d geometry. This is directly relevant to the graviton propagator calculations done in this framework, since the Gaussian width enters the numerical factors in front of the graviton correlations in the large scale asymptotical limit. Finally, we discuss the application of our computations to the Barrett-Crane model beyond the first order (of a single 4-simplex in the bulk) and to the more recent EPRL-FK spinfoam model.