Consistent Evaluation of the No-Boundary Proposal

We revisit the Hartle-Hawking no-boundary proposal. To extract probabilities, one must use the gravitational path integral (GPI) to compute not only the no-boundary amplitude, but also the norms by which its square is divided. We find that this dramatically alters predictions: the probability for any closed universe is either nearly 1, or exactly 1. That is, in the Hilbert space of closed universes defined by the GPI, the states of interest in cosmology are all nearly parallel to the Hartle-Hawking state up to nonperturbative corrections in $G_N^{-1}$. We also consider a statistical interpretation of the GPI, as an average of arbitrary products of amplitudes. We find that all amplitudes are exactly 1 in this case, consistent with recent arguments that the statistical approach to the GPI with a closed boundary computes an average over one-dimensional Hilbert spaces. As an example, we illustrate the consistent evaluation of the no-boundary proposal in inflationary cosmology.

💡 Research Summary

The paper revisits the Hartle‑Hawking no‑boundary proposal with a focus on how probabilities should be extracted from the gravitational path integral (GPI). The authors emphasize that one must compute not only the no‑boundary amplitude ⟨J|∅⟩ but also the norms ⟨J|J⟩ and ⟨∅|∅⟩ that appear in the denominator of the Born rule. By evaluating these quantities in the semiclassical (saddle‑point) approximation, they find that the dominant contributions come from two disconnected saddle geometries: one associated with ⟨J|∅⟩ and the other with ⟨∅|J⟩. Because the same geometry dominates both numerator and denominator, the resulting probability P(∅→J)=|⟨J|∅⟩|²/(⟨J|J⟩⟨∅|∅⟩) is essentially unity, up to non‑perturbative corrections of order exp(−L²/G_N). This overturns the traditional view, which assumed that the norm ⟨J|J⟩ is J‑independent and that different boundary data correspond to orthogonal states.

The authors also discuss a newer “statistical” interpretation of the GPI, in which a single GPI with multiple boundary insertions is regarded as an ensemble average over inner products of many theories. In this picture, the same disconnected saddles give rise to an exact equality ⟨J|∅⟩⟨∅|J⟩ = ⟨J|J⟩⟨∅|∅⟩ for every member of the ensemble, leading to a probability that is exactly one. This aligns with recent arguments that the Hilbert space of closed universes in each ensemble member is one‑dimensional, so all normalized states differ only by a phase.

To illustrate the consequences, the paper applies the analysis to slow‑roll inflation. Using the traditional (incorrect) normalization one reproduces the classic “empty‑universe” prediction of the Hartle‑Hawking wavefunction. However, when the proper disconnected saddles are included, the probability for any inflationary configuration is again essentially one. In the statistical ensemble approach the probability is exactly one, meaning the no‑boundary state is parallel (or nearly parallel) to every simple geometric state of a closed universe. Consequently, the no‑boundary proposal loses its predictive power for distinguishing among different cosmological histories.

The discussion highlights several implications. First, the near‑parallelism of all closed‑universe states to the Hartle‑Hawking state invalidates the assumption that distinct boundary data correspond to orthogonal quantum states. Second, any observable distinction must arise from non‑perturbative corrections suppressed by exp(−L²/G_N), which are typically negligible for macroscopic curvatures. Third, the statistical interpretation provides a natural resolution to the factorization puzzles that have appeared in holographic contexts, by treating the GPI as an ensemble average rather than a single theory amplitude. Finally, the authors suggest that to recover a non‑trivial Hilbert space structure one may need to project out the no‑boundary state after assigning data to quantum states, or to employ alternative bases such as the α‑basis introduced by Marolf and Maxfield.

Overall, the paper demonstrates that a careful treatment of the GPI’s normalization dramatically changes the predictions of the Hartle‑Hawking proposal: probabilities are either nearly one (conventional approach) or exactly one (statistical approach). This calls into question the proposal’s ability to make meaningful cosmological predictions without incorporating non‑semiclassical effects or adopting a different conceptual framework for the gravitational path integral.