  We study density matrices in quantum gravity, focusing on topology change. We
argue that the inclusion of bra-ket wormholes in the gravity path integral is
not a free choice, but is dictated by the specification of a global state in
the multi-universe Hilbert space. Specifically, the Hartle-Hawking (HH) state
does not contain bra-ket wormholes. It has recently been pointed out that
bra-ket wormholes are needed to avoid potential bags-of-gold and strong
subadditivity paradoxes, suggesting a problem with the HH state. Nevertheless,
in regimes with a single large connected universe, approximate bra-ket
wormholes can emerge by tracing over the unobserved universes. More drastic
possibilities are that the HH state is non-perturbatively gauge equivalent to a
state with bra-ket wormholes, or that the third-quantized Hilbert space is
one-dimensional. Along the way we draw some helpful lessons from the well-known
relation between worldline gravity and Klein-Gordon theory. In particular, the
commutativity of boundary-creating operators, which is necessary for
constructing the alpha states and having a dual ensemble interpretation, is
subtle. For instance, in the worldline gravity example, the Klein-Gordon field
operators do not commute at timelike separation.
