On the nature of ensembles from gravitational path integrals

TL;DR

The paper tackles whether gravitational path integrals with wormholes imply an ensemble of theories and what kinds of theories such an ensemble could contain. It analyzes a solvable $d=2$ topological gravity model, leveraging the CSCC (baby-universe) sector and $\alpha$-states to define ensemble components through $\zeta_\alpha$ that factorize over disconnected boundaries, while the full no-boundary path integral $\zeta$ does not. A key finding is that positivity of the CSCC Hilbert space is guaranteed in the model, but sectors with CS-boundaries can exhibit positivity violations depending on the eigenvalues $Z_d$, leading to a non-factorizing structure; a surgical approach is proposed to keep positivity while benefiting from ensemble-averaged observables for other properties. The work clarifies how bulk ensembles might be compatible with holography by isolating positivity-preserving sectors and using ensemble averages for uncorrelated observables, offering a concrete toy model to guide future UV-complete constructions and extensions to include EOW branes and higher dimensions.

Abstract

Spacetime wormholes in gravitational path integrals have long been interpreted in terms of ensembles of theories. Here we probe what sort of theories such ensembles might contain. Careful consideration of a simple $d=2$ topological model indicates that the Hilbert space structure of a general ensemble element fails to factorize over disconnected Cauchy-surface boundaries, and in particular that its Hilbert space ${\cal H}_{N_{CS\partial}}$ for $N_{CS\partial}$ Cauchy-surface boundaries fails to be positive definite when the number $N_{CS\partial}$ of disconnected such boundaries is large. This suggests a generalization of the AdS/CFT correspondence in which a bulk theory is dual to an ensemble of theories that deviate from standard CFTs by violating both locality and positivity (at least under certain circumstances). Since violations of positivity are undesirable, we propose that positivity-violating elements of the ensemble be removed when studying physics in asymptotically AdS spacetimes (or in other contexts in which Cauchy surfaces have asymptotic boundaries), perhaps reducing the ensemble to a single standard CFT. Nevertheless, properties of any remaining CFTs that are uncorrelated with positivity of ${\cal H}_{N_{CS\partial}}$ at large $N_{CS\partial}$will agree with those of typical elements of the full ensemble and may be computed using the ensemble average. On the other hand, elements that violate positivity at large $N_{CS\partial}$ can still have a positive-definite cosmological sector with $N_{CS\partial}=0$. Such elements then define a basis for a Hilbert space describing such cosmologies. In contrast to the cases in which Cauchy-surfaces are allowed to have boundaries, we argue that the resulting Hilbert space need not decohere into single-state theories. As a result, familiar physics might be more easily recovered from this new scenario.

Figures (3)

• Figure 1: Observables that are not diagaonal in $\alpha$ will not commute with observables at Cauchy-surface boundaries (CS-boundaries). A general such observable will thus fail to commute with the Hamiltonian at any CS-boundary and will thus depend on choices of times at all CS-boundaries, even in disconnected components of the Cauchy surface. In particular, an observable $O$ that might be hoped to describe physics in a region (yellow) of a CSC-cosmology (blue), will nevertheless depend on choices of times $t_1,t_2$ at boundaries of disconnected components of spacetime (green). Operations performed at such boundaries will thus tend to decohere physics described by $O$.
• Figure 2: Simple cartoons of possible half-sources $J^1_\Sigma,J^2_\Sigma$ are shown at left as unoriented line segments with red/blue regions indicating distinct local source fields. Here $\Sigma$ is just a pair of points (without orientations). Note that $J^1_\Sigma$ is related to $J^2_\Sigma$ by a diffeomorphism that acts as a right/left reflection. Nevertheless, as shown at right, the complete source $\left(J^1_\Sigma\right)^* \cup_\Sigma J^1_\Sigma$ is not diffeomorphic to the complete source $\left(J^1_\Sigma\right)^* \cup_\Sigma J^2_\Sigma$.
• Figure 3: A simple example of a half boundary condition in $\mathcal{J}_{3,2}$, meaning that $\Sigma$ consists of 3 positively-oriented points (red) and 2 negatively-oriented points (blue). In this case, the first positive point is connected to the first negative point, while the 2nd and 3rd positive points source EOW branes of flavors $I_1$ and $I_2$ and the 2nd negative point provides a source for an EOW anti-brane of flavor $K_1$. The EOW branes and anti-branes are distinguished in the figure by the orientations of the associated line segments.