arXiv2003
We consider a local formalism in quantum field theory, in which no reference is made to infinitely extended spacial surfaces, infinite past or infinite future. This can be obtained in terms of a functional W[f,S] of the field f on a closed 3d surface S that bounds a finite region R of Minkowski spacetime. The dependence of W on S is governed by a local covariant generalization of the Schroedinger equation. Particles' scattering amplitudes that describe experiments conducted in the finite region R --the lab during a finite time-- can be expressed in terms of W. The dependence of W on the geometry of S 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 from S. 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 f of the gravitational field. This observation clarifies the physical meaning of the functional W defined by non perturbative formulations of quantum gravity, such as the spinfoam formalism. In particular, it suggests a way to derive particles' 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_S>, which codes the dynamics, from 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.