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

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.

arXiv comment41 pages, 7 figures; corrected typos; v2 resolves a puzzle raised in v1, about a bound on the JT closed universe Hilbert space with an observer

References in corpus (8)

• Replica Wormholes and the Entropy of Hawking Radiation
• The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
• Transcending the ensemble: baby universes, spacetime wormholes, and the order and disorder of black hole information
• Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture
• Observations of Hawking radiation: the Page curve and baby universes
• Gravity factorized
• Closed universes in two dimensional gravity
• Algebras and Hilbert spaces from gravitational path integrals: Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT

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