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.