The gravitational path integral from an observer's point of view

TL;DR

This work develops an observer-centric gravitational path integral to describe non-perturbative physics in quantum gravity, formulating a relational Hilbert space that accounts for a bulk observer. Using JT gravity with matter as a tractable model, the authors show that the observer enlarges the effective non-perturbative Hilbert space (e.g., $ ext{dim}(
H^{ ext{rel}}_{ ext{non-pert}})=d^2$ for a closed universe and $d^4$ for a two-sided black hole) and that the inner product remains positive semidefinite after quotienting null states. Observables dressed to the observer’s worldline reveal that non-perturbative corrections to EFT are typically suppressed until exponential times in the entropy, resolving puzzles about EFT breakdown in black hole interiors, while certain winding-geodesic effects near cosmological singularities remain subtle. The framework generalizes beyond JT gravity and offers a pathway to relational holography with worldline boundaries, potentially informing higher-dimensional quantum gravity and de Sitter contexts.

Abstract

One of the fundamental problems in quantum gravity is to describe the experience of a gravitating observer in generic spacetimes. In this paper, we develop a framework for describing non-perturbative physics relative to an observer using the gravitational path integral. We apply our proposal to an observer that lives in a closed universe and one that falls behind a black hole horizon. We find that the Hilbert space that describes the experience of the observer is much larger than the Hilbert space in the absence of an observer. In the case of closed universes, the Hilbert space is not one-dimensional, as calculations in the absence of the observer suggest. Rather, its dimension scales exponentially with $G_N^{-1}$. Similarly, from an observer's perspective, the dimension of the Hilbert space in a two-sided black hole is increased. We compute various observables probing the experience of a gravitating observer in this Hilbert space. We find that an observer experiences non-trivial physics in the closed universe in contrast to what it would see in a one-dimensional Hilbert space. In the two-sided black hole setting, our proposal implies that non-perturbative corrections to effective field theory for an infalling observer are suppressed until times exponential in the black hole entropy, resolving a recently raised puzzle in black hole physics. While the framework that we develop is exemplified in the toy-model of JT gravity, most of our analysis can be extended to higher dimensions and, in particular, to generic spacetimes not admitting a conventional holographic description, such as cosmological universes or black hole interiors.

Figures (22)

• Figure 1: Our proposal for the gravitational path integral in the presence of an observer. The worldline of the observer (depicted in red) must connect a bra and the corresponding ket when computing moments of an overlap $\langle\psi_i|\psi_j\rangle^n$. This rule can be seen as an additional boundary condition to impose when summing over geometries. (a) The overlap $\langle\psi_i|\psi_j\rangle$ between two closed universe states in pure JT gravity is unaffected by our new rules. (b) The square of an overlap $|\langle\psi_i|\psi_j\rangle|^2$ between closed universe states. The gravitational path integral sums over all possible geometries satisfying the boundary conditions at the asymptotic boundaries and at the worldline of the observer. We depict here the leading disconnected contribution and the first subleading term, a genus-zero connected geometry.
• Figure 2: (Left) A Lorentzian Big Bang-Big Crunch closed universe with a Cauchy slice of maximal length $b$ (depicted in blue). (Right) Its Euclidean counterpart has two AdS asymptotic boundaries connected by a Euclidean wormhole. The geodesic slice of minimal length $b$ is identical to the maximal slice in Lorentzian signature. This length is vanishingly small on-shell when the wormhole is not supported by matter and is finite in the presence of matter.
• Figure 3: The integration contour needed to calculate the Hilbert space dimension consists of the large defining contour of the resolvent $C_\infty$ and a small contour $C_0$ used to exclude the contributions from zero eigenvalues of the Gram matrix.
• Figure 4: Overlap between an asymptotic closed universe state $|\beta,\Delta_O^{(i)}\rangle$ and a state defined on the closed minimal geodesic (depicted in green) $|b,\Delta_O^{(i)},u\rangle$. Left: the overlap on the Euclidean cylinder. The state on the minimal closed geodesic is labeled by the length $b$ of the closed minimal geodesic and the length $u$ of a geodesic segment wrapping the cylinder once and intersecting the observer's worldline perpendicularly at both ends. Right: the cylinder can be cut along the observer's worldline and embedded in the Poincaré disk (depicted in black), with the two red lines identified. The contribution of this geometry to the path integral can be decomposed into a Hartle-Hawking wavefunction $\phi_\beta(\ell_i)$, with $\ell_i$ the length of a geodesic starting and ending at the insertion of the observer on the asymptotic boundary, and a quadrilateral.
• Figure 5: Non-perturbative corrections to the overlap $\langle b,\Delta_O^{(i)},u|\beta,\Delta_O^{(j)} \rangle$. The geodesic slice of length $b$ is depicted in green. Any higher-genus geometry contributing to the overlap can be decomposed into a genus-zero quadrilateral similar to Figure \ref{['fig:overlap']} and a region with a higher genus homologous to the boundary. The contribution of geometries with higher topology is then encoded in the Hartle-Hawking wavefunction.

Theorems & Definitions (8)

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