Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information

TL;DR

This work clarifies how spacetime wormholes and baby universes induce ensemble-averaged structure in gravitational path integrals with AdS boundaries, recasting Z[J] observables as random variables drawn from an alpha-labeled ensemble. By solving simple topological models exactly, it shows that the asymptotically AdS Hilbert space dimension Z is a nonnegative integer random variable whose distribution is Poisson with mean λ, and that choosing an alpha-state restores factorization, effectively selecting a single ensemble member. A central role is played by null states, which impose rigidity constraints (e.g., rank bounds) that yield Page-curve-like entropy bounds within each alpha-sector, even when replica wormholes contribute to ensemble averages. The paper also extends the formalism to include end-of-the-world branes and spacetime D-branes, analyzes a third-quantized perturbation theory, and discusses the implications for black hole information and AdS/CFT, suggesting that gravity may encode a rich gauge structure across alpha-sectors while preserving observable factorization for fixed alpha.

Abstract

In the 1980's, work by Coleman and by Giddings and Strominger linked the physics of spacetime wormholes to `baby universes' and an ensemble of theories. We revisit such ideas, using features associated with a negative cosmological constant and asymptotically AdS boundaries to strengthen the results, introduce a change in perspective, and connect with recent replica wormhole discussions of the Page curve. A key new feature is an emphasis on the role of null states. We explore this structure in detail in simple topological models of the bulk that allow us to compute the full spectrum of associated boundary theories. The dimension of the asymptotically AdS Hilbert space turns out to become a random variable $Z$, whose value can be less than the naive number $k$ of independent states in the theory. For $k>Z$, consistency arises from an exact degeneracy in the inner product defined by the gravitational path integral, so that many a priori independent states differ only by a null state. We argue that a similar property must hold in any consistent gravitational path integral. We also comment on other aspects of extrapolations to more complicated models, and on possible implications for the black hole information problem in the individual members of the above ensemble.

Figures (4)

• Figure 1: The gravitational path integral with spacetime wormholes does not factorize. The top line gives a diagramatic representation of the path integrals $\langle Z[J]_1 \rangle$ and $\langle Z[J_2]\rangle$ that would naively define partition functions $Z[J_1]$ and $Z[J_2]$. The natural path integral $\langle Z[J_1]Z[J_2]\rangle$ associated with a pair of boundaries yields all terms generated by multiplying $\langle Z[J_1] \rangle \langle Z[J_2]\rangle$, but also contains additional connected contributions schematically shown as the second term in the bottom line.
• Figure 2: Slicing open a spacetime with a boundary and a handle (left) can give a disconnected geometry on the slice, including a closed 'baby universe' that has become detached from the parent asymptotically AdS universe. The baby universe does not intersect the asymptotically AdS boundary (red line) at the moment of time described by the indicated slice.
• Figure 3: In the presence of spacetime wormholes, different spatial slices of a spacetime may have different number of connected components. Here, on the slice $\Sigma_1$ we have two circular universes, but on $\Sigma_2$ we have only one. These may be thought of as different gauge choices for the same state.
• Figure 4: A spacetime contributing to an amplitude $\langle (\psi_j,\psi_i)(\psi_i,\psi_j)Z\rangle$. The solid red lines indicate asymptotically AdS boundaries, and the dashed green lines are EOW brane boundaries. The spacetime has two boundary components, each with the topology of a circle. One (solid red circle at bottom) is a single circular asymptotically AdS boundary (a $Z$-boundary). The other is formed by a pair of asymptotically AdS segments connected by a pair of EOW brane segments to form a topological circle.