On the Hilbert Space of Quantum Gravity from the Gravitational Path Integral

Does the gravitational path integral define a valid Hilbert space for open and closed universes? In this talk, I will review the positivity criteria that the gravitational path integral has to satisfy in order for all open and closed universe states to have positive norm. Although the positivity criteria appear to be highly constraining, I will describe several cases in which they are miraculously satisfied. This leads to a well defined Hilbert space where we can meaningfully discuss transition probabilities among different closed and open universe states. Using the no-boundary state as a case study, I will explain how to compute transition probabilities from this state when properly accounting for non-perturbative corrections in the gravitational path integral. Surprisingly, the resulting probabilities drastically differ from the traditional approach of Hartle and Hawking.