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

TL;DR

This paper develops an abstract formalism for gravitational path integrals that incorporates observers, allowing for larger, nontrivial Hilbert spaces in closed universes and providing bounds on their effective size.

Contribution

It introduces a novel formalism treating sources and partial sources in gravitational path integrals, enabling the analysis of larger Hilbert spaces with noncommuting operators in various universe configurations.

Findings

Formalism accommodates observers and boundary conditions.

Hilbert spaces can have nontrivial $ppa$-sectors supporting noncommuting operators.

Positivity bounds the effective size of gravitational Hilbert spaces.

Abstract

The gravitational path integral suggests a striking result: the Hilbert space of closed universes in each superselection sector, a so-called $\alpha$-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 $\alpha$-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 $\alpha$-sector. The trace of such operators, in turn, quantifies the effective size of this Hilbert space.