Toward a mathematically consistent theory of semiclassical gravity or, How to have your wormholes and factorize, too

TL;DR

This work identifies three core inconsistencies—factorization, information-theoretic Page-curve behavior, and closed-universe realizations—that challenge semiclassical gravity when viewed through holography, the gravitational path integral, and quantum information. It proposes a modified holographic dictionary in which gravity is dual to a smooth subtheory of the full CFT, implemented via a conditional expectation $\mathcal{E}$ and an extended operator algebra $\mathscr{A}_{\text{ext}}$ built from a Q-system, together with an interaction channel $\mathcal{C}$. The extended semiclassical gravity path integral $\mathcal{Z}^{\text{ext}}_{\mathcal{E},\mathcal{C}}$ incorporates new degrees of freedom that counterbalance connected wormhole contributions, yielding factorization and a Page-curve entropy through a decomposition $\mathcal{S}(\psi_R \circ \mathcal{C}) = \mathcal{S}(\psi_R) + \mathcal{S}(\mathcal{C})$. Moreover, nontrivial closed-universe sectors arise from the irreducible sectors of the Q-system, providing a coherent picture in which background independence, refined large-$N$ limits, ensemble perspectives, and observer-like degrees of freedom all coexist within a unified algebraic-path-integral framework. If realized, this program offers a mathematically consistent route to incorporate wormholes, information recovery, and closed universes in semiclassical gravity while preserving holographic control over the theory.

Abstract

We review three well known inconsistencies in the standard mathematical formulation of semiclassical gravity: the factorization problem, the information problem, and the closed universe problem. Building upon recent work, we explore how modifying the holographic dictionary may provide the necessary freedom to resolve these three problems in a unified manner while maintaining more well established aspects of the standard correspondence. Using the modified holographic dictionary as a scaffolding, we propose a program for constructing an `extended' semiclassical gravitational path integral which (i) is manifestly factorizing, (ii) computes a von Neumann entropy which satisfies the Page curve, and (iii) incorporates new operators that create closed baby universe states. Our construction may be interpreted as imposing a semiclassical version of background independence/a no global symmetry condition, defining a modified large N limit, preparing an ensemble of dual theories, or enforcing observer rules using gravitational degrees of freedom.

Figures (8)

• Figure 1: The standard path integral of quantum field theory. The notation $\int_{\phi_2}^{\phi_1} \equiv \int_{\{\Phi \in \Gamma(M,E) \; | \; \Phi\rvert_{\Sigma_1} = \phi_1, \Phi\rvert_{\Sigma_2} = \phi_2\}}$ indicates the set of all field configurations satisfying boundary conditions $\phi_1$ at the upper boundary and $\phi_2$ at the lower boundary. If either of the boundary conditions are left empty, one should integrate over the space of fields unconstrained on the given boundary besides necessary fall-off conditions.
• Figure 2: Obstructions to the consistency of semiclassical gravity.
• Figure 3: To resolve the factorization problem, the channel $\mathcal{C}$ must include 'counterterms' which cancel the contributions of connected wormholes in $\mathcal{Z}$.
• Figure 4: To resolve the information problem, the quantum conditional entropy associated with the map $\mathcal{C}$ must contain the would be entropic contribution of the replica wormholes.
• Figure 5: To resolve the closed universe problem, the extended degrees of freedom induced from the conditional expectation $\mathcal{E}$ should furnish creation and annihilation operators for semiclassical gravitational superselection sectors.