  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.
