The Hilbert space of de Sitter quantum gravity

Abstract

We obtain solutions of the Wheeler-DeWitt equation with positive cosmological constant for a closed universe in the large-volume limit. We argue that this space of solutions provides a complete basis for the Hilbert space of quantum gravity in an asymptotically de Sitter spacetime. Our solutions take the form of a universal phase factor multiplied by distinct diffeomorphism invariant functionals, with simple Weyl transformation properties, that obey the same Ward identities as a CFT partition function. The Euclidean vacuum corresponds to a specific choice of such a functional but other choices are equally valid. Each functional can be thought of as specifying a "theory" and, in this sense, the space of solutions is like "theory space". We describe another basis for the Hilbert space where all states are represented as excitations of the vacuum that have a specific constrained structure. This gives the finite $G_N$ generalization of the basis proposed by Higuchi in terms of group averaging, which we recover in the nongravitational limit.