Density matrices in quantum gravity

TL;DR

This paper explores the role of bra-ket wormholes in quantum gravity, arguing their inclusion is dictated by the global state, and examines implications for the Hartle-Hawking state and the structure of the multi-universe Hilbert space.

Contribution

It provides a detailed analysis of how bra-ket wormholes relate to the global state in quantum gravity and discusses conditions under which they emerge or are equivalent to the Hartle-Hawking state.

Findings

Bra-ket wormholes are not a free choice but depend on the global state.

The Hartle-Hawking state does not contain bra-ket wormholes.

Approximate bra-ket wormholes can emerge in certain regimes.

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.