Inner Product of Closed Universe from Gravitational Path-integral

Abstract

We construct the Hilbert space of a closed universe of fixed topology in anti-de Sitter (AdS) and de Sitter (dS) Jackiw-Teitelboim (JT) gravity from the gravitational path integral. Quantizing gravity involves several subtleties, including the choice of lapse contour, path-integral measure, operator ordering, global structure of the gauge group, the implementation of gravitational constraints and the gauge symmetries at initial and final timeslices. We develop a formalism that addresses these issues and determines the inner product of physical states, providing a framework to assign probabilities to cosmological wavefunctions. For a fixed topology and a lapse contour, we compute the inner product exactly and obtain the corresponding resolution of the identity in JT gravity. We demonstrate that the co-invariant and invariant Hilbert spaces in refined algebraic quantization arise from particular gauge fixing at the initial and final timeslices. In particular, the formalism provides precise meaning for a gravitational wavefunction with fixed geometric data as a co-invariant state, including its integral measure. These constructions can be formulated within the Hamiltonian BFV-BRST formalism. We also discuss the global structure of the Hilbert space, which differs from that of BF theory and depends on the prescription adopted near the big bang/crunch. Our results imply that the cylinder geometry captures the gravitational inner product for all perturbative states, including the Hartle-Hawking no-boundary state.