No-Boundary Density Matrix in Quantum Cosmology

• No-Boundary Density Matrix is defined via a Euclidean path integral approach, generalizing the Hartle–Hawking state to yield a mixed state when tracing out unobserved degrees of freedom.
• It resolves paradoxes in quantum cosmology by addressing the failure of observable factorization and restoring strong subadditivity through effective emergent wormhole contributions.
• The formalism underpins cosmological applications including subregion density matrices, entanglement entropy via replica tricks, and handles non-time-orientable geometries such as elliptic de Sitter.

The no-boundary density matrix is a generalization of the Hartle–Hawking (HH) no-boundary wavefunction, constructed via the Euclidean path integral in quantum cosmology and quantum gravity. Unlike the original HH prescription, which yields a pure state in the third-quantized or “baby universe” Hilbert space, the no-boundary density matrix formalism produces a mixed state, typically by tracing out unobserved regions or degrees of freedom. This approach has become central in understanding quantum initial conditions, topology change, entropy, and observer-dependence in gravitational systems, especially in contexts where time-orientability or conventional wavefunction construction fails.

1. Hartle–Hawking State and Fundamental Construction

In a $D$-dimensional quantum gravitational theory, the third-quantized Hilbert space $\mathcal H_{\rm BU}$ consists of states characterized by their wavefunctionals on closed spatial manifolds $Y$. The traditional Hartle–Hawking state $|\rm HH\rangle\in\mathcal H_{\rm BU}$ is defined by a Euclidean path integral over all bulk $D$-manifolds $X$ whose sole boundary is $Y$: $$\Psi_{\rm HH}(Y) = \langle Y|\rm HH \rangle = \int_{X:\partial X = Y} {\cal D}g\,e^{-I[g]}$$ where $I[g]$ is the Euclidean gravity (and matter) action. This yields a pure state.

To construct a density matrix relevant for observable universes, one performs a partial trace over all unobserved "baby universes": $$\rho_{\rm HH} = \mathrm{Tr}_{\rm unobs}\,|\rm HH\rangle\langle\rm HH|$$ In the boundary basis, this gives

$$\rho_{\rm HH}[Y_1, Y_2] = \Psi_{\rm HH}(Y_1)^* \Psi_{\rm HH}(Y_2) = \left( \int_{X_1:\partial X_1=Y_1} {\cal D}g\,e^{-I[g]} \right) \left( \int_{X_2:\partial X_2=Y_2} {\cal D}g\,e^{-I[g]} \right)$$

Crucially, the sum does not include manifolds connecting $Y_1$ and $Y_2$: there are no fundamental "bra–ket" wormholes (Anous et al., 2020).

2. Mixed States and Paradox Resolution

The absence of bra–ket wormholes in $\rho_{\rm HH}$ leads to significant physical consequences. Joint observables do not factorize: $$\langle Z(\beta_1) Z(\beta_2) \rangle_{\rm HH} \neq \langle Z(\beta_1) \rangle_{\rm HH} \langle Z(\beta_2) \rangle_{\rm HH}$$ This failure gives rise to strong subadditivity and "bags-of-gold" paradoxes—violations of entropy inequalities that are otherwise restored when wormhole geometries are present. Including wormholes manually would contradict the core definition of the HH state.

Nevertheless, by tracing over unobserved universes, effective bra–ket wormholes can emerge in the coarse-grained regime. For an observed boundary $w$, the reduced density matrix is

$$\rho_{\text{1-univ}}(w, w') = \sum_Y \rho(w \cup Y, w' \cup Y)$$

In semiclassical limits, this sum can approximate a connected geometry between $w$ and $w'$, essentially mimicking the effect of wormhole insertions. This recovers strong subadditivity at the effective level but not fundamentally (Anous et al., 2020).

3. Euclidean Path Integrals and Elliptic de Sitter

In non-time-orientable backgrounds such as elliptic de Sitter ($\mathrm{dS}/\mathbb{Z}_2$), the Euclidean no-boundary construction does not produce a wavefunction but a density matrix. For $\mathbb{RP}^{d+1}$ (real projective Euclidean space), the path integral with boundary conditions on a slit yields

$$\rho_A[\varphi_-, \varphi_+] = \int_{\varphi(-0,\phi)=\varphi_-(\phi),\,\varphi(+0,\phi)=\varphi_+(\phi)} \mathcal D\varphi\,e^{-S_E[\varphi]}|_{\mathbb{RP}^2}$$

where $S_E$ is the Euclidean action. This construction is Hermitian by virtue of reflection symmetry but cannot be expressed as $|\Psi\rangle\langle\Psi|$; there is no global splitting of the manifold (Dulac et al., 30 Nov 2025).

Entanglement measures such as Rényi and von Neumann entropies are computed via the replica trick. In the 2D Dirac fermion CFT, twist operator correlators yield exact closed-form entropies and capture the effect of global time-reversal symmetry breaking and observer-dependence (Dulac et al., 30 Nov 2025).

4. Subregion Density Matrices and Cosmological Applications

The no-boundary density matrix for a subregion $\Sigma_{\rm in}$ of a spatial slice $\Sigma$ is defined by

$$\rho[\Phi^-_{\rm in}, \Phi^+_{\rm in}] = \int\!{\cal D} \Phi_{\rm out} \Psi^*[\Phi^-_{\rm in}, \Phi_{\rm out}]\,\Psi[\Phi^+_{\rm in}, \Phi_{\rm out}]$$

Semiclassically, the saddle-point geometry must satisfy no-boundary conditions in the far past and matching of fields and conjugate momenta on the traced-out region $\Sigma_{\rm out}$ (Ivo et al., 21 Sep 2024). Detailed prescriptions exist for massless scalars in fixed de Sitter, slow-roll inflation, and Coleman–de Luccia (CdL) bubble nucleation. For instance, CdL bounce solutions contribute local maxima to $\rho$ and can produce open-universe initial conditions.

A summary table of key classical solutions:

Geometry	Boundary/Trace Conditions	Density Matrix Contribution
HH Pure State	No-boundary, single full slice	Pure state, no bra–ket wormholes
Subregion (dS)	No-boundary past, matched fields/momenta on $\Sigma_{\rm out}$	Mixed state, exponential pressure to small volume
CdL Bubble	O(4)-symmetric bounce, analytic continuation	Local maximum: $e^{-I_E^{\rm bu}}$

Phenomenologically, standard HH saddles bias toward small universes, while CdL-type bounces are acceptable but subleading unless a selection principle is introduced (Ivo et al., 21 Sep 2024).

5. Nonequilibrium Formalism and KMS Periodicity

The Schwinger–Keldysh in-in formalism provides a framework for expectation values in arbitrary initial density matrix states, including the no-boundary microcanonical ensemble. For Gaussian density matrices derived from Euclidean path integrals, reflection symmetry of the instanton ensures Hermiticity. The Wightman Green’s functions satisfy the Kubo–Martin–Schwinger (KMS) periodicity: $G_{>}(t - i \beta, t') = G_{<}(t, t')$ even far from equilibrium. Analyticity in the complex time plane, accessible via a “dog-bone” contour, unifies Euclidean and Lorentzian evolution and underpins the tunneling interpretation of universe creation (Barvinsky et al., 2023).

6. Third Quantization, Hilbert Space Structure, and Operator Algebras

The baby-universe Hilbert space $\mathcal H_{\rm BU}$ and its dimension play a crucial role in density matrix properties. If $\dim\mathcal H_{\rm BU}=1$, as occurs in global quantization of certain backgrounds (e.g., $\mathrm{dS}/\mathbb{Z}_2$), the global Fock space collapses: all states are annihilated by both creation and annihilation operators, and only observer-dependent Fock spaces remain (Dulac et al., 30 Nov 2025).

In practical field theory analogies, such as worldline gravity and Klein–Gordon theory, the commutativity of boundary-creating operators is subtle. Only in Euclidean signature do these operators commute everywhere, permitting the construction of $\alpha$-states and a dual ensemble interpretation. In Lorentzian signature, non-commutativity precludes full diagonalization unless boundary points are spacelike separated (Anous et al., 2020).

7. Open Problems, Subtleties, and Future Directions

Several subtleties persist:

• Contour and normalization: The choice of complex contour is crucial for damping and regularizing high-frequency modes. Projectors on finite area and curvature are often required to produce a well-defined measure (Ivo et al., 21 Sep 2024).
• Quantum corrections and UV issues: One-loop divergences arise when sharply specifying fields at boundaries; coarse-graining and algebraic QFT approaches are necessary for rigorous definitions.
• Gauge equivalence and topology: Non-perturbative gauge equivalence might resolve paradoxes by relating the HH state to bra–ket wormhole states; alternatively, triviality of $\mathcal H_{\rm BU}$ renders all wormhole effects gauge artifacts (Anous et al., 2020, Dulac et al., 30 Nov 2025).
• Physical interpretation: The trace operation physically corresponds to focusing on a subregion, with direct analogs in AdS/CFT as well as in cosmological quantum gravity.

A plausible implication is that the no-boundary density matrix formalism, with its manifestations in Euclidean/Lorentzian analytic continuations, replica trick entropy calculations, and observer-dependent Hilbert space structure, is likely to be foundational in quantum cosmology, especially for scenarios with nontrivial topology or time-orientability obstructions. The ongoing challenge is to reconcile paradoxes and select physically relevant saddle points in the density matrix approach.