Holographic baby universes: an observable story

TL;DR

The paper reframes baby universes within an operator-algebraic framework, arguing that the appropriate description of bulk dynamics, especially with topology-changing processes, rests on the algebra of observables rather than Hilbert spaces. By introducing an extra Abelian $C^*$-algebra for baby universes and employing the GNS construction with a gravitational path integral state, the authors derive when the baby universe Hilbert space collapses to a single state (the BU hypothesis) and explain the observed cancellations via the Gelfand theory. They formulate a precise, three-part set of physical conditions that imply the BU hypothesis and discuss when these conditions fail, notably in scenarios where bulk processes influence baby-universe creation. Through two illustrative examples—a topological model and a group-theoretic analogue to $ heta$-vacua—they connect abstract algebraic structures to concrete gravitational contexts. The work also critically examines the assumptions behind Marolf–Wall/Marolf–Maxfield programs and discusses implications for AdS/CFT, suggesting that a complete quantum-gravity framework may require modifications only if the BU hypothesis fails in realistic settings.

Abstract

We formulate the baby universe construction rigorously by giving a primordial role to the algebra of observables of quantum gravity rather than the Hilbert space. Utilizing diffeomorphism invariance, we study baby universe creation and annihilation via change in topology. We then construct the algebra of boundary observables for holographic theories and show that it enhances to contain an 'extra' Abelian tensor factor to describe the bulk in the quantum regime; via the gravitational path integral we realize this extra tensor factor, at the level of the Hilbert space, in the context of the GNS representation. We reformulate the necessary assumptions for the "baby universe hypothesis" using the GNS representation. When the baby universe hypothesis is satisfied, we demonstrate that the "miraculous cancellations" in the corresponding gravitational path integral have a natural explanation in terms of the character theory of Abelian $C^\ast$-algebras. We find the necessary and sufficient mathematical condition for the baby universe hypothesis to hold, and transcribe it into sufficient physical conditions. We find that they are incompatible with a baby universe formation that is influenced by any bulk process from the AdS/CFT correspondence. We illustrate our construction by applying it to two settings, which leads to a re-interpretion of some topological models of gravity, and to draw an analogy with the topological vacua of gauge theory.

Figures (4)

• Figure 3.1: The connected sum operation. The two blue disks are removed from the hyperbolic handlebodies with red conformal boundaries. The handlebodies are then glued together along the holes.
• Figure 3.2: The gluing operation. The two red conformal boundaries are sown together.
• Figure 4.1: A two-sided semiclassical wormhole corresponding to a thermofield double state. On the right, Alice jumps into the black hole, and one can engineer an operator that acts on the left CFT and creates Bob in such a way that he will meet Alice in the black hole.
• Figure 4.2: The thermofield double state in the case where the boundaries describe two separate bulks. Assuming approximate entanglement wedge reconstruction close to the horizon, there exists an operator on the right CFT that asks whether Alice meets a "Bob-like object". This operator commutes with the operator that creates Bob.

Theorems & Definitions (9)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• Theorem 3.2
• Theorem 6.3
• Theorem 6.5
• Theorem 7.1: Proposition 4.4.1, Kadison
• Theorem 7.2: Takesaki
• Theorem 7.3: Takesaki