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[phi, Sigma] of the field phi on a closed 3D surface Sigma that bounds a finite region R of Minkowski spacetime. The dependence of W[phi, Sigma]on Sigma is governed by a local covariant generalization of the Schrodinger 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[phi, Sigma]. The dependence of W[phi, Sigma] on the geometry of Sigma expresses the dependence of the transition amplitudes on the relative location of the particle detectors. In a gravitational theory. background independence implies that W[phi, Sigma] is independent of Sigma. However, the detectors' relative location is still coded in the argument of W[phi], because the geometry of the boundary surface is determined by the boundary value phi of the gravitational field. This observation clarifies the physical meaning of the functional W[phi] 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(Sigma)>, which codes the dynamics, front the Minkowski vacuum \0(M)>, 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.