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

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.

Spin structures and baby universes

“…In [27], Marolf and Maxfield used a baby universe picture to propose that the gravitational path integral should generically have an interpretation in terms of an ensemble average. The general discussion was illustrated by considering a simple exactly solvable topological model where the bulk path integral sums over all smooth two-dimensional manifolds with n boundaries, including both connected and disconnected manifolds, with a weight depending only on the topology of the bulk manifold.…”

“…This correlator would factorize, i.e., HH|Ẑ n |HH = HH|Ẑ|HH n , if |HH were an eigenstate ofẐ, which is only possible if the baby universe Hilbert space H BU is onedimensional. Thus, [27] showed that the bulk path integral requires a dual interpretation in an ensemble of theories (as opposed to a single theory) precisely when the baby universe Hilbert space is non-trivial. 6 In general then, in the simple topological model of [27], the gravitational path integral with n boundaries computes a non-factorizing correlator of n baby universe creation oper-atorsẐ n , and this quantity has a dual description as the average of the n th power of the partition sum evaluated in a particular ensemble of theories: HH|Ẑ n |HH = Z n , where the average on the right hand side is taken over a statistical ensemble of field theories, and…”

“…In section 2 we briefly review the topological model of [27], and introduce an extension to include a sum over spin structures. (In this section we consider a sum which weights all spin structures equally.)…”

“…As in [27], to construct the Hilbert space of baby universes, we act with formal polynomials ofẐ andẐ on |HH . This generates a substantial number of null states, which we must quotient out.…”

“…The boundary conditions for the topological model then include circular boundaries and boundary segments terminated by EOW branes at both ends. In the model of [27], the boundary segments were shown to correspond to overlaps (ψ j , ψ i ). In our model with spin structures, we can insert an operator reversing the sign of the fermions on the boundary, so we have two kinds of boundary segments, which we will show correspond to (ψ j , ψ i ) and (ψ j , (−1) F ψ i ).…”