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[φ,Σ] of the field φ on a closed 3D surface Σ that bounds a finite region R of Minkowski spacetime. The dependence of W[φ,Σ] on Σ is governed by a local covariant generalization of the Schrödinger 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[φ,Σ]. The dependence of W[φ,Σ] on the geometry of Σ expresses the dependence of the transition amplitudes on the relative location of the particle detectors. In a gravitational theory, background independence implies that W[φ,Σ] is independent of Σ. However, the detectors’ relative location is still coded in the argument of W[φ], because the geometry of the boundary surface is determined by the boundary value φ of the gravitational field. This observation clarifies the physical meaning of the functional W[φ] 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_Σ〉, 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.