Observers seeing gravitational Hilbert spaces: abstract sources for an abstract path integral

TL;DR

This work reframes the gravitational path integral as a map $\zeta: \mathcal{J} \to \mathbb{C}$ from a commutative $*$-algebra of abstract sources to complex numbers, and introduces partial sources and cuts to realize nontrivial $\alpha$-sectors in Hilbert spaces associated with closed universes, especially in the presence of observers. By integrating partial sources across cuts, the authors construct Hilbert spaces $\mathcal{H}_C$ that can host noncommuting operators and derive upper bounds on the traces of these operators within each $\alpha$-sector, tying the effective size of the sector to gravitational path-integral data. The framework unifies and generalizes prior observer prescriptions (spatial boundaries, prescribed observer worldlines, and external observer clones) and yields bounds that are broadly consistent with results in JT gravity and ensemble approaches, while highlighting the conditions under which $\alpha$-sectors remain finite and potentially nontrivial. These results illuminate how observer-related modifications to the gravitational path integral influence the structure of the Hilbert space of quantum gravity, and they point to concrete directions for connecting abstract source calculus to von Neumann algebras and nonperturbative state counting in gravity.

Abstract

The gravitational path integral suggests a striking result: the Hilbert space of closed universes in each superselection sector, a so-called $α$-sector, is one-dimensional. We develop an abstract formalism encapsulating recent proposals that modify the gravitational path integral in the presence of observers and allow larger Hilbert spaces to be associated with closed universes. Our formalism regards the gravitational path integral as a map from abstract objects called sources to complex numbers, and introduces additional objects called partial sources, which form sources when glued together. We apply this formalism to treat, on equal footing, universes with spatial boundaries, closed universes with prescribed observer worldlines, and closed universes containing observers entangled with external systems. In these contexts, the relevant gravitational Hilbert spaces contain states prepared by partial sources and can consequently have nontrivial $α$-sectors supporting noncommuting operators. Within our general framework, the positivity of the gravitational inner product implies a bound on the Hilbert space trace of certain positive operators over each $α$-sector. The trace of such operators, in turn, quantifies the effective size of this Hilbert space.

Figures (7)

• Figure 1: Possible examples of sources $J\in\mathcal{J}$ (black) for the gravitational path integral. In each panel, a possible bulk configuration that might appear in the path integral $\zeta(J)$ is illustrated in blue.
• Figure 2: Slicing (orange) the gravitational path integral while avoiding cutting sources (black) gives a state in the baby universe Hilbert space.
• Figure 3: Possible examples of cuts $C\in\mathcal{C}$ (red) and partial sources $J_C\in\mathcal{J}_C$ (solid black). In each panel, the complete source $(J_C'|J_C)\in\mathcal{J}$ obtained from gluing with another partial source $(J_C')^*$ (dashed black) is also shown. A possible bulk configuration that might appear in the path integral $\zeta(J_C'|J_C)$ is illustrated in blue.
• Figure 4: Slicing (orange and red) the gravitational path integral through cuts $C\in\mathcal{C}$ (red) of sources (black) gives states in sectors $\mathcal{H}_C$ of the gravitational Hilbert space.
• Figure 5: A partial source $J_{\Delta\otimes{}\Delta^*}^{1/2}=\xi_\Delta\cdot\hat{\mathscr{M}}\cdot(\xi_\Delta)^*$ (solid black), shown in each panel glued to another partial source (dashed black). A (part of a) possible bulk configuration that might appear in the path integral involving each of these glued partial sources is illustrated in blue.