Minkowski vacuum in background independent quantum gravity

We consider a local formalism in quantum field theory, in which no reference is made to infinitely extended spatial surfaces, infinite past or infinite future. This can be obtained in terms of a functional W, of the field on a closed 3D surface that bounds a finite region R of Minkowski spacetime. The dependence of W, on is governed by a local covariant generalization of the Schr\"odinger equation. The particle scattering amplitudes that describe experiments conducted in the finite region R---the laboratory during a finite time---can be expressed in terms of W,. The dependence of W, on the geometry of expresses the dependence of the transition amplitudes on the relative location of the particle detectors. In a gravitational theory, background independence implies that W, is independent of. However, the detectors' relative location is still coded in the argument of W, because the geometry of the boundary surface is determined by the boundary value of the gravitational field. This observation clarifies the physical meaning of the functional W defined by nonperturbative formulations of quantum gravity, such as spinfoam formalism. In particular, it suggests a way to derive the particle scattering amplitudes from a spinfoam model. In particular, we discuss the notion of vacuum in a generally covariant context. We distinguish the nonperturbative vacuum |0_〉, which codes the dynamics, from the Minkowski vacuum |0₌〉, which is the state with no particles and is recovered by taking appropriate large values of the boundary metric. We derive a relation between the two vacuum states. We propose an explicit expression for computing the Minkowski vacuum from a spinfoam model.