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 $\mathcal{R}$ of Minkowski spacetime. The dependence of $W[\phi ,\Sigma ]$ on $\Sigma$ 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 $W[\phi ,\Sigma ].$ 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 $W\left[\phi \right],$ 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 $W\left[\phi \right]$ 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, 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.