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A no-go theorem for large $N$ closed universes

Elliott Gesteau

TL;DR

Problem: whether a semiclassical baby universe can emerge in the large-N limit of AdS/CFT for closed universes. Method: a rigorous no-go theorem showing that, under conservative assumptions, large-N correlators of single-trace operators converge to those of a pure state in the two-copy vacuum AdS, ruling out a mixed causal wedge behind the horizon for $O(N^0)$ energy states. The proof relies on truncation bounds, Cantor diagonal extraction, and dominated convergence to construct a unique pure-state limit. The work clarifies the status of baby-universe scenarios in PETS constructions and discusses how averaging or observer-based modifications would require relaxing the theorem's assumptions. Implications: provides a mathematically precise barrier to semiclassical baby universes in standard AdS/CFT without relaxing the large-N limit axioms.

Abstract

Under conservative assumptions, it is established at a mathematical level of rigor that if correlation functions of single trace operators in a sequence of states of $O(N^0)$ energy of a two-sided holographic conformal field theory admit a large $N$ limit, then they must be described a pure state in the large $N$ Hilbert space of free field theory on two copies of vacuum AdS. This result clarifies recent discussions concerning the possible emergence of a semiclassical baby universe in the large $N$ limit of a low-energy partially entangled thermal state. The proof heavily relies on dominated convergence arguments from mathematical analysis. Some comments are also offered on the relationship between various recently proposed ways of restoring the emergence of the baby universe and the relaxation of different assumptions of the theorem.

A no-go theorem for large $N$ closed universes

TL;DR

Problem: whether a semiclassical baby universe can emerge in the large-N limit of AdS/CFT for closed universes. Method: a rigorous no-go theorem showing that, under conservative assumptions, large-N correlators of single-trace operators converge to those of a pure state in the two-copy vacuum AdS, ruling out a mixed causal wedge behind the horizon for energy states. The proof relies on truncation bounds, Cantor diagonal extraction, and dominated convergence to construct a unique pure-state limit. The work clarifies the status of baby-universe scenarios in PETS constructions and discusses how averaging or observer-based modifications would require relaxing the theorem's assumptions. Implications: provides a mathematically precise barrier to semiclassical baby universes in standard AdS/CFT without relaxing the large-N limit axioms.

Abstract

Under conservative assumptions, it is established at a mathematical level of rigor that if correlation functions of single trace operators in a sequence of states of energy of a two-sided holographic conformal field theory admit a large limit, then they must be described a pure state in the large Hilbert space of free field theory on two copies of vacuum AdS. This result clarifies recent discussions concerning the possible emergence of a semiclassical baby universe in the large limit of a low-energy partially entangled thermal state. The proof heavily relies on dominated convergence arguments from mathematical analysis. Some comments are also offered on the relationship between various recently proposed ways of restoring the emergence of the baby universe and the relaxation of different assumptions of the theorem.

Paper Structure

This paper contains 12 sections, 6 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theorem
  3. Background: the large $N$ limit and the algebraic approach to AdS/CFT
  4. Assumptions, useful facts, and theorem statement
  5. Proof
  6. Step 1: bound the error made by finite $N$ truncations
  7. Step 2: Cantor's diagonal extraction
  8. Step 3: construct a limit for some of the truncated states
  9. Step 4: construct a limit for some of the untruncated states
  10. Step 5: exploit the uniqueness of the lmit
  11. Comments on the proof
  12. How to evade the conclusion: observers, averaging, and all that

Key Result

Theorem 2.1

Under the two assumptions and two facts above, there exists a pure state $\ket{\Psi}$ in the large $N$ Hilbert space of the vacuum such that for all uniformly bounded sequences of observables $(S_N)$ of single trace observables described by a large $N$ bounded field operator $S$,

Figures (1)

  • Figure 1.1: Predictions of classical gravity for the dual of the state \ref{['eq:ASSState']}. On the left panel, which is the primary case of interest in this paper, the inverse temperatures $\beta_L,\beta_R$ are taken above the Hawking--Page transition (i.e. low temperature), which is the main case of interest of this paper. The solution of Einstein's equation predicts two copies of AdS entangled with a closed universe. On the right panel, the inverse temperatures $\beta_L,\beta_R$ are taken below the Hawking--Page transition (i.e. high temperature). The gravity dual is then thought to be a long wormhole geometry.

Theorems & Definitions (11)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2: Cantor's diagonal extraction
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 1 more