Density matrices in quantum gravity

TL;DR

The paper analyzes density matrices in quantum gravity with topology change within a third-quantized framework, showing that the presence of bra-ket wormholes is dictated by the chosen global state (HH vs Page) rather than being freely selectable. Through the worldline gravity analogy, it highlights subtle commutativity properties of boundary-creating operators and connects these to the GNS construction, clarifying how a baby-universe Hilbert space is built from path-integral data. It compares entropy computations in the BU sector to holographic boundary entropy, arguing that bra-ket wormholes can be emergent in single-universe observables and that extreme possibilities—such as a one-dimensional BU space—are conceptually possible. These insights illuminate how wormholes influence entropy, factorization, and the interpretation of ensemble averages in quantum gravity and holography, and they identify regimes where effective wormholes may arise or be obstructed by the chosen global state.

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.

Figures (2)

• Figure 1: The difference between the Page density matrix $\rho_{\rm Page}$ and the density matrix $\rho_{\rm HH}$ corresponding to $\ket{\text{HH}}$. This figure depicts contributions to a particular matrix element of these density matrices in two-dimensional gravity where a general $Y$ is a union of disconnected circles. In red we indicated the bra and in blue the ket. In the Page density matrix (\ref{['pageconj']}) Page:1986vw, bra-ket wormholes are present, whereas they are absent in the density matrix for the Hartle-Hawking state (\ref{['proposalrho']}) Hartle:1983ai. In particular, the set of configurations that contribute to $\rho_\text{Page}(Y_1, Y_2)$ is a superset of the configurations that contribute to $\rho_\text{HH}(Y_1, Y_2)$.
• Figure 2: Representation of $\rho_{\text{1-univ}}(w,w')$ in (\ref{['rholate']}) with $\rho = \rho_{\text{HH}}$. Recall that $Y$ is not necessarily connected. For an appropriate choice of state and model, the sum over $Y$ can give rise to an approximate "effective" wormhole between $w$ and $w'$diagcylBlommaert:2019wfy.