Quantum State of a Gravitating Region

We propose that any compact d-manifold with elliptic data, mathcal{J}, prepares a quantum state |mathcal{J}rangle on its (d-1)-boundary σ. Elliptic data consists of metric and field values, or their conjugates, but not both. No asymptotic structure is required. Inner products and traces are evaluated by the gravitational path integral with closed boundary conditions obtained by gluing elliptic data manifolds. In particular, we give a prescription for the Rényi entropies S_n of a subregion of σ. In a class of examples, we find that S_n is nonnegative and nonincreasing with n, as required for consistency. We obtain the von Neumann entropy by analytic continuation and find agreement with the minimal surface prescription of Bousso and Penington.