"It from Bit": The Hartle-Hawking state and quantum mechanics for de Sitter observers
Ying Zhao
TL;DR
This work resolves the tension between the one-state property of closed universes and bulk quantum mechanics in de Sitter space by distinguishing the baby-universe Hilbert space (a classical probability space) from the bulk Hilbert space (the finite-dimensional quantum system seen by a de Sitter observer). It shows that quantum mechanics inside de Sitter space emerges from the classical statistics encoded in patch operators, thereby realizing Wheeler’s It from Bit idea. A concrete one-dimensional toy model with partition function $Z_L$ demonstrates that de Sitter entropy corresponds to the coarse-grained entropy of the underlying state, and that conditioning on backgrounds reduces this entropy in a controlled way. The analysis uses the gravitational path integral, patch-operator linearity, and both complex and real quantum-code descriptions to provide a coherent, $ ext{CRT}$-invariant framework for relating BU statistics to bulk QM, with broader implications for holography and cosmology.
Abstract
The one-state statement for closed universes has sparked considerable discussion. In this paper we examine its physical meaning in the context of the Hartle-Hawking state and de Sitter space. We argue that the one-state property of closed universes is fully compatible with the finite-dimensional quantum mechanics experienced by observers inside de Sitter space, and that this compatibility requires neither mixing of alpha sectors nor any modification of the rules of the gravitational path integral. The apparent tension is resolved by sharply distinguishing the baby-universe Hilbert space, namely the space of closed universes viewed from the outside, from the bulk Hilbert space that governs quantum mechanics for an observer inside a single de Sitter universe. The baby-universe Hilbert space, together with its commutative operator algebra, is not a quantum-mechanical Hilbert space: it is merely a mathematical repackaging of classical probability theory and does not carry any quantum-mechanical structure at all, a direct consequence of the one-state property of closed universes. Accordingly, attempting to formulate quantum mechanics directly on the baby-universe Hilbert space conflates classical ensemble data with the quantum mechanics experienced by bulk observers and leads to physically incorrect conclusions. By contrast, the quantum mechanics experienced by an observer inside de Sitter space emerges from the classical statistics encoded in the baby-universe Hilbert space, providing a concrete realization of Wheeler's idea of "It from Bit". We demonstrate these features by completely solving a topological toy model of one-dimensional de Sitter spacetime. Along the way we clarify the physical meaning of de Sitter entropy, showing that it corresponds to the coarse-grained entropy of the underlying state.