Abstract
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.
Highlights
One strong reason to include a sum over topologies in the gravitational path integral is that doing so gives us the HawkingPage transition in Anti-de Sitter space [2], which, famously, is the bulk dual of the confinementdeconfinement transition in gauge theory [3, 4]
While the topological classification of three- and higher-dimensional manifolds is quite complicated, in two-dimensional gravity theories, such as worldsheet string theory [7] or Jackiw-Teitelboim (JT) gravity [8,9,10], the sum over topologies reduces to a sum over the Euler characteristic of the manifold, a single integer
There has been a surge of interest in the sum over different topologies in the gravitational path integral
Summary
The gravitational path integral requires a sum over metrics on a manifold of a given topology, and a sum over manifolds of different topologies. One strong reason to include a sum over topologies in the gravitational path integral is that doing so gives us the HawkingPage transition in Anti-de Sitter space [2], which, famously, is the bulk dual of the confinementdeconfinement transition in gauge theory [3, 4].2. The bra-ket wormholes that are needed to resolve the bags-of-gold type paradoxes in de Sitter (as in section 6 of [12]) and strong subadditivity paradoxes [29] would need to “emerge" in the HartleHawking state, since they are not originally present in the definition of the Hartle-Hawking state. One way this can happen is if the density matrix of the Hartle-Hawking state, restricted to a single, late-time classical universe, contains effective wormholes obtained by tracing out the unobserved universes. Note added: While this article was nearing completion, the paper [30] appeared which has some overlap with our section 2.2
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