Baby Universe Hilbert Space

• Baby universe Hilbert space is a rigorous framework that organizes quantum states of topology-changing spacetimes into superselection sectors via irreducible representations.
• It employs an abelian boundary operator algebra and a third-quantization approach to recast wormhole and topology effects as weighted, factorized contributions in the gravitational path integral.
• This construction clarifies holographic consistency and black hole information by replacing ensemble averages with fixed couplings across distinct quantum sectors.

A baby universe Hilbert space is a construction within quantum gravity that encodes the possible quantum states associated with baby universes—disconnected or topologically nontrivial regions of spacetime that branch off from a "parent" universe, often through topology change such as wormhole formation. This Hilbert space formalism provides a rigorous language for summing over spacetime topologies in the gravitational path integral and for analyzing the effects of topology change, random couplings, wormholes, and related phenomena on observables in quantum gravity and holography. Multiple precise constructions exist, differing in their explicit basis, algebraic formulation, and physical interpretation, but all share a common structure: they replace the naive sum over topologies by an abelian operator algebra and a corresponding decomposition of the quantum gravity state space into superselection sectors.

1. Definition of the Baby Universe Hilbert Space: Third Quantization and Sector Decomposition

The baby universe Hilbert space $\mathcal{H}_{\rm BU}$ is defined as the span of all quantum states obtainable by "creating" baby universes via insertion of appropriate boundary or topology-changing operators in the gravitational path integral. In a precise setting, such as in (Betzios et al., 2022), $\mathcal{H}_{\rm BU}$ is constructed as follows:

• Basis: The orthonormal basis $|R\rangle$ corresponds to irreducible representations $R$ of the compact gauge group $\mathcal{G}$ on the boundary. The basis is

$\langle R|R'\rangle = \delta_{R,R'}$

• The "void" or "no-universe" state, analogous to the Fock vacuum, is

$|\Omega\rangle = \sum_{R} D_R\,|R\rangle$

where $D_R = \dim R$, the dimension of the irrep. The vacuum norm is $\langle\Omega|\Omega\rangle = \sum_{R} D_R^2$.

• Superselection: The Hilbert space decomposes into superselection sectors labeled by irreps,

$$\mathcal{H}_{\rm BU} = \bigoplus_{R} \mathcal{H}_R,\qquad \mathcal{H}_R = \mathrm{Span}\{|R\rangle\}$$

More generally, in lower-dimensional models or toy topological quantum field theories (TQFTs), alpha-states $|\alpha\rangle$ simultaneously diagonalize all commuting boundary creation operators; each alpha-label corresponds to a distinct sector with fixed values of effective couplings or observables (Marolf et al., 2020, Gardiner et al., 2020, Martynova et al., 2020).

2. Path Integral Formulation and Expectation Values

The transition from the second-quantized (standard QFT) to the third-quantized (quantum gravity) viewpoint is accomplished by interpreting the gravitational path integral as the expectation value of boundary-creation operators on $\mathcal{H}_{\rm BU}$. For a boundary partition function $Z[J]$ with sources $J$:

$$Z[J] = \frac{\langle \Omega | \hat{Z}[J] | \Omega \rangle}{\langle \Omega | \Omega \rangle}$$

Expanding in the irrep basis,

$Z[J] = \sum_{R} p_R\, Z_R[J]$

where

$$p_R = \frac{D_R^2}{\sum_{R'} D_{R'}^2},\qquad Z_R[J] = \langle R| \hat{Z}[J] | R\rangle$$

This formalism ensures the factorization of multi-boundary observables and encodes the full sum over topologies as a probabilistic mixture over superselection sectors. In the TQFT and matrix model context, analogous structures appear:

• In simple topological models, the "dimension" $Z$ of the Hilbert space on a boundary is a Poisson random variable, with basis states $|Z=d\rangle$, and the inner product reflects this distribution (Marolf et al., 2020, Gardiner et al., 2020).
• When end-of-the-world branes or matter insertions are included, the Hilbert space incorporates additional structure, e.g., complex Wishart distributions for boundary states (Gardiner et al., 2020, Peng et al., 2021).

3. Operator Algebra and Type: Abelian Structure, Null States, and Factorization

A central mathematical property of the baby universe arena is the abelian nature of the boundary-insertion algebra:

• The boundary-creation and annihilation operators all commute: $[\hat{b}_1, \hat{b}_2]=0$ for any $b_1, b_2\in\mathcal{B}$, where $\mathcal{B}$ is the set of boundary data (Casali et al., 2021, Gesteau et al., 2020).
• Consequently, all such operators are simultaneously diagonalizable; the Hilbert space decomposes into a direct sum (or integral) of one-dimensional eigenspaces:

$$\mathbb{I}_{\rm BU} = \sum_\alpha |\alpha\rangle\langle\alpha| \quad\textrm{or}\quad \int d\mu(\alpha)$$

• Null states: Many naive states constructed as linear combinations of boundary insertions may have zero norm under the inner product defined from the gravitational path integral; these are projected out in the final Hilbert space. The nontrivial structure is thus determined after quotienting by this null ideal (Marolf et al., 2020, Gardiner et al., 2020).

In summary, the baby universe Hilbert space is abelian, with a unique decomposition into superselection sectors, and can be constructed via GNS representation starting from the algebra of diffeomorphism-invariant observables and a suitable gravitational path-integral state (Gesteau et al., 2020).

4. Physical Interpretation: Replacement of $\alpha$-Parameters, Superselection Sectors, and Ensemble vs. Fixed Couplings

In the Coleman-Giddings-Strominger paradigm, baby universe creation and absorption lead to the emergence of "alpha-parameters"—undetermined integration constants encoding the values of couplings in the effective action, leading to a picture of wormhole physics as an average over a statistical ensemble of QFTs (Marolf et al., 2020, Gardiner et al., 2020, Saad, 2019). In the construction of (Betzios et al., 2022), for holographic quantum gravity with fixed microscopic data:

• There are no $\alpha$-parameters; all couplings are fixed.
• The sum over bulk topologies and wormholes is recast as a direct sum over definite quantum sectors (superselection sectors) parameterized by irreducible representations $R$.
• Correlators retain a sector-dependent factorized form:

$$\langle\Omega|\hat{Z}[J_1]\hat{Z}[J_2]|\Omega\rangle = \sum_R p_R Z_R[J_1] Z_R[J_2]$$

• The weights $p_R$ are determined by representation theory, not by dynamical or random variables.
• Wormhole-induced randomness in boundary correlators is interpreted microscopically as the effect of summing over baby-universe sectors, not over an ensemble of QFTs.

This formalism restores factorization for multi-boundary correlators in the microscopic description and maintains compatibility with holography, which forbids global symmetries and free parameters in the bulk (McNamara et al., 2020).

5. Low-Dimensional and Topological Examples: Poisson, Wishart, and Matrix Models

The structure of $\mathcal{H}_{\rm BU}$ is made explicit and computable in topological gravity and matrix model toy examples:

• For pure 2D gravity, each alpha-sector corresponds to a fixed dimension $d$ of the boundary Hilbert space, with $d\sim$ Poisson($\lambda$), and operators act as multiplication by $d$ (Marolf et al., 2020, Gardiner et al., 2020).
• With matter or brane insertions, the sectors are products of Poisson and Wishart variables, and the full Hilbert space resolves into a direct sum/integral over these occupation numbers (Peng et al., 2021).
• In the double-scaled SYK and ETH matrix model context, baby universe operators $\mathcal{B}_a$ acting on the chord Hilbert space generate all relevant gravitational topologies (disks, trumpets, double trumpets), giving a concrete operator realization of wormhole and baby universe phenomena (Okuyama, 7 Aug 2024).

These computations rigorously demonstrate sector-decomposition, null-state projection, and the precise statistical distribution of couplings emerging from the topology sum.

6. Swampland and One-Dimensionality in Higher Dimensions

In UV-complete quantum gravity in $d>3$, Swampland constraints and the absence of global $(-1)$-form symmetries forbid the existence of nontrivial baby universe sectors:

• The baby universe Hilbert space must be one-dimensional: $\dim H_{\rm baby} = 1$ (McNamara et al., 2020).
• This stems from the requirement that no parameter in the quantum gravity path integral can remain unfixed—ruling out the existence of $\alpha$-parameters or similar continuous degrees of freedom.
• The restriction is robustly realized via state-operator correspondence, which forcibly collapses any local operator in a closed universe to a multiple of the Hartle-Hawking wavefunction.

Exceptions arise only in incomplete or lower-dimensional theories (e.g., JT gravity in $d=2$), which are viewed as worldvolume or boundary-induced "quantum gravities" not subject to the $d>3$ constraint.

7. Implications for Holography, Black Holes, and Quantum Information

The baby universe Hilbert space appears centrally in resolving key aspects of quantum gravity:

• Black hole information and fine-graining: Including a baby-universe Hilbert space provides the auxiliary degrees of freedom required to purify coarse-grained density matrices, such as those describing Hawking radiation, and to realize the Page curve semiclassically (Iizuka et al., 2021, Antonini et al., 14 Jul 2025).
• Holographic encoding: The mapping from bulk closed universes (with no asymptotic boundary) to the dual CFT encodes the baby universe Hilbert space as an approximately isometric image in the CFT Hilbert space when bulk entanglement with the external region is large; in the zero-entanglement limit, the boundary only accesses a fixed, one-dimensional state (Antonini et al., 14 Jul 2025).
• Algebraic consistency: Requiring the GNS representation of the baby-universe observable algebra to be irreducible (one-dimensional) is necessary for factorization of gravitational correlators and for the absence of ensemble average phenomena in the dual CFT (Gesteau et al., 2020).
• Physical interpretation: Baby universes are interpreted microscopically as vacuoles labeled by a boundary quantum number (such as group representation), providing a controlled mechanism by which bulk topology, superselection, and the algebraic structure of observables are reconciled in quantum gravity.

In sum, the baby universe Hilbert space framework provides a mathematically rigorous and physically interpretable mechanism for organizing the quantum states of topology-changing spacetimes, for reconciling wormhole-induced statistical transfers of coupling data with the expectations of UV-complete gravity, and for implementing the sectoral and algebraic constraints required for holographic consistency. The choice of basis (irreps, alpha-states, occupation numbers) and the corresponding state in the Hilbert space encode measurable features in correlation functions, entropy, and information-theoretic measures in gravity and its holographic duals.