Partially Frozen Gravitational Path Integral

• The paper’s main contribution is defining partially frozen gravitational path integrals as constructions that fix a subset of variables while integrating over others.
• This methodology refines traditional quantum gravity by incorporating boundary conditions, observer-relative data, and mixed ensembles to yield nontrivial Hilbert space structures.
• Applications include controlled metric integration, factorization across internal cuts, and altered phase contours that affect semiclassical and analytic interpretations.

Searching arXiv for papers on partially frozen gravitational path integrals and closely related formulations. Searching arXiv for observer-relative and boundary-condition-based gravitational path integral work. “Partially frozen gravitational path integral” denotes a family of constructions in which the gravitational path integral is not treated as a sum over all compatible geometric data on equal footing, but instead is conditioned on, or glued across, a selected subset of variables. In the recent literature this conditioning appears in several technically distinct ways: fixing a boundary configuration rather than integrating over it, fixing the canonical momentum or a Robin combination of boundary data, restricting topology sums relative to an observer, fixing a timekeeper worldvolume, imposing a fixed-area constraint, or introducing internal cuts whose induced data are held fixed. Across these settings, the common structural feature is that some gravitational or state-preparing data remain dynamical while others are held fixed or only partially integrated over, so standard “frozen” conclusions are refined rather than simply abandoned (Chen, 21 May 2025, Krishnan et al., 2016, Ailiga et al., 2024, Anikeeva et al., 13 May 2026).

1. Conceptual scope and recurrent structures

The term does not refer to a single universally adopted formalism. Instead, the literature uses closely related constructions to describe path integrals in which only part of the gravitational configuration space is left unfixed. In AdS Hartle–Hawking language, this distinction is explicit: one may define a “fully gravitational wave function, in which the boundary configuration is integrated over,” or a “partially frozen one, in which it is fixed, as in AdS/CFT” (Anikeeva et al., 13 May 2026). In closed-universe quantum gravity, the same structural move appears when one enlarges the class of admissible state-preparing objects from complete closed sources to “partial sources,” or when one specifies an observer or timekeeper relative to which the rest of the universe is described (Chen, 21 May 2025, Wei, 26 Jun 2025).

A second recurring structure is ensemble dependence. In the standard Euclidean approach, Dirichlet boundary conditions freeze the induced boundary metric. Neumann and Robin variants instead freeze the canonical conjugate of the boundary metric, or a linear combination of position and momentum, while allowing the complementary datum to fluctuate. In Hamiltonian language this produces a mixed ensemble rather than a purely Dirichlet one (Krishnan et al., 2016, Ailiga et al., 2024).

A third structure is factorization across internal cuts. In higher-curvature gravity, the path integral can be written in a factorized form

$$Z = \int_{\mathcal T} D{\cal B}_i\, Z(\mathcal M_1|{\cal B}_i)\, Z(\mathcal M_2|{\cal B}_i),$$

where ${\cal B}_i$ is the set of interface data required by additivity of the action. Fixing ${\cal B}_i$ instead of integrating over it yields a constrained, and in this sense partially frozen, gravitational path integral (Draper et al., 2023).

Taken together, these constructions suggest that “partial freezing” is best understood as a controlled restriction of the path integral measure or gluing data, rather than as a wholesale rejection of the gravitational sum over histories.

2. Closed universes, observers, and the refinement of frozen Hilbert spaces

One major use of the idea appears in closed-universe quantum gravity. In an abstract formulation, the gravitational path integral is modeled as a map

$Z:\mathcal{S}\to \mathbb{C},$

from abstract sources to amplitudes. The key extension is the introduction of “partial sources,” which are not complete path-integral inputs by themselves but can be glued into full sources. If $p$ and $q$ are compatible partial sources, then $p\cup q\in\mathcal S$, and the path integral induces an inner product

$\langle p|q\rangle = Z(\bar p\cup q).$

After quotienting null states and completing, one obtains a gravitational Hilbert space generated by states prepared by partial sources (Chen, 21 May 2025).

This changes the usual conclusion that the Hilbert space of a closed universe in a fixed $\alpha$-sector is one-dimensional. The standard result

$\dim \mathcal H^\alpha_{\rm closed}=1$

arises when one only allows complete closed sources and therefore has no nontrivial cuts. Once one allows partial sources based on spatial boundaries, prescribed observer worldlines, or observers entangled with external systems, one instead obtains

${\cal B}_i$0

with ${\cal B}_i$1 possible inside a fixed ${\cal B}_i$2-sector. In that generalized setting, noncommuting operators can act within ${\cal B}_i$3, so the path integral is frozen with respect to superselection by ${\cal B}_i$4, but not necessarily frozen inside each sector (Chen, 21 May 2025).

A related but distinct proposal separates two sources of freezing in a closed universe. In a simple 1D gravitational path integral model, summing over metrics imposes the Wheeler–DeWitt constraint and produces the problem of time, while summing over topologies produces a one-dimensional fundamental Hilbert space, the “problem of dimension.” The paper formulates this by distinguishing a kinematic Hilbert space, a constrained Hilbert space obtained after metric summation, and a fundamental Hilbert space obtained after also summing over topologies. An observer is then defined as a subsystem whose specification results in a nontrivial Hilbert space, while a timekeeper is an observer with a specified history that can be used as a reference for the time of the environment (Wei, 26 Jun 2025).

The timekeeper-relative path integral is written schematically as

${\cal B}_i$5

so the observer worldvolume history is fixed while the ambient geometry remains summed over. This is a direct realization of a partially frozen gravitational path integral: topology summation, metric summation, and matter integration remain, but they are conditioned on a fixed observer-history datum (Wei, 26 Jun 2025).

3. Boundary conditions, ensembles, and mixed freezing of canonical data

A second major line of development concerns boundary-value problems. In the standard Dirichlet formulation, the Einstein–Hilbert action is supplemented by the Gibbons–Hawking–York term so that the variational principle is well posed when the induced boundary metric ${\cal B}_i$6 is fixed. The corresponding canonical conjugate is

${\cal B}_i$7

The Neumann action is obtained by subtracting the boundary Legendre transform,

${\cal B}_i$8

so that the variational principle is well posed when ${\cal B}_i$9 rather than ${\cal B}_i$0 (Krishnan et al., 2016).

At the covariant level this is a complete alternative to Dirichlet data, not merely a small modification of it. In Hamiltonian form, however, the Neumann action is mixed. Its boundary variation shows that the natural variables are combinations involving ${\cal B}_i$1, ${\cal B}_i$2, and ${\cal B}_i$3, so the associated ensemble is not a pure Legendre transform of the Brown–York microcanonical action. In this canonical sense the boundary data are only partially frozen: some canonical combinations are fixed, while metric-like data still fluctuate in mixed combinations (Krishnan et al., 2016).

In Lorentzian minisuperspace, the same idea becomes explicit in Robin boundary conditions for Einstein–Gauss–Bonnet gravity. With

${\cal B}_i$4

Dirichlet conditions freeze ${\cal B}_i$5, Neumann conditions freeze ${\cal B}_i$6, and Robin conditions freeze the linear combination

${\cal B}_i$7

The paper imposes Dirichlet at the final boundary and Neumann or Robin at the initial boundary. Under Robin conditions, neither ${\cal B}_i$8 nor ${\cal B}_i$9 is fixed separately; only the combination $Z:\mathcal{S}\to \mathbb{C},$0 is held fixed, while the initial scale factor is integrated over with a Gaussian boundary weight. The Robin kernel can therefore be written as a Gaussian transform of the Neumann kernel, which makes the partially frozen interpretation literal rather than metaphorical (Ailiga et al., 2024).

In that minisuperspace setting, the $Z:\mathcal{S}\to \mathbb{C},$1 limit tends to Dirichlet, while $Z:\mathcal{S}\to \mathbb{C},$2 tends to Neumann. The Hartle–Hawking branch is selected by the saddle

$Z:\mathcal{S}\to \mathbb{C},$3

and Picard–Lefschetz analysis identifies the convergent thimble for the Robin integral. The resulting picture is not one in which the initial geometry is fully fixed, but one in which the initial boundary phase space is only partly constrained (Ailiga et al., 2024).

4. Reduced and gauge-fixed path integrals

Partial freezing also appears in reduced or gauge-fixed formulations. In the higher-dimensional $Z:\mathcal{S}\to \mathbb{C},$4 deformation, the deformed partition function satisfies a diffusion-type flow equation whose kernel is identified with a Euclidean gravitational path integral between two boundaries with Dirichlet boundary conditions on the full induced metric,

$Z:\mathcal{S}\to \mathbb{C},$5

This is not a path integral with only part of the boundary metric fixed. The closest “partially frozen” feature is instead gauge-fixed: in the direct derivation the lapse is not integrated over at first, the shift is set to zero, and the integration is over the spatial metric history $Z:\mathcal{S}\to \mathbb{C},$6. Only under further interpretation does this become a gauge-fixed Hamiltonian gravitational path integral (Belin et al., 2020).

In cosmological minisuperspace, the microcanonical statistical sum is

$Z:\mathcal{S}\to \mathbb{C},$7

and after separating FRW collective variables from inhomogeneous fields one obtains

$Z:\mathcal{S}\to \mathbb{C},$8

This is not a complete freezing of non-FRW modes; those are integrated out into the effective action $Z:\mathcal{S}\to \mathbb{C},$9. Gauge fixing $p$0 removes local lapse fluctuations and leaves only a constant mode, while the residual conformal Killing symmetry requires a Batalin–Vilkovisky treatment for linearly dependent generators. For one-fold instantons the one-loop prefactor is

$p$1

The resulting path integral is therefore reduced, constrained, and gauge-fixed, but not trivialized (Barvinsky, 2010).

A more elementary example is the fixed-area ensemble in two-dimensional gravity coupled to $p$2. The genus-zero fixed-area path integral

$p$3

freezes one collective geometric variable—the total area—while still integrating over all sphere metrics of that area modulo gauge. The paper shows that this is precisely what restores the semiclassical round-sphere saddle at large $p$4 (Anninos et al., 2021).

These examples show that in reduced formalisms “partial freezing” may mean fixing lapse profiles, fixing a collective modulus, or integrating out inhomogeneous modes while keeping only a minisuperspace sector explicit. The common point is selective reduction rather than elimination of gravitational dynamics.

5. Factorization, internal boundaries, and entanglement wedges

A different realization of partial freezing arises when spacetime is cut along an internal hypersurface. In Euclidean gravity with higher-curvature $p$5 interactions, the factorized path integral takes the form

$p$6

The interface data $p$7 are determined by additivity of the action after rewriting boundary terms as bulk total derivatives. In the canonical ensemble,

$p$8

whereas in the microcanonical ensemble

$p$9

Because $q$0 and $q$1 are not part of $q$2, they may jump across the factorization surface. The paper applies this to Euclidean Schwarzschild–de Sitter, where the saddle with a discontinuous lapse has a discontinuous local Euclidean temperature but remains compatible with the constrained microcanonical path integral (Draper et al., 2023).

An internal-boundary construction also underlies the gravitational derivation of generalized entanglement wedges. For a fixed-geometry state $q$3, one defines a “hollowed” state $q$4 by removing a tiny slit around a bulk region $q$5, thereby creating two new codimension-1 boundaries $q$6 and $q$7 and fixing their induced metric to agree with the original geometry on $q$8. Replica partition functions for $q$9 then glue cyclically on $p\cup q\in\mathcal S$0 and trivially elsewhere. In the diagonal approximation,

$p\cup q\in\mathcal S$1

and the $p\cup q\in\mathcal S$2 limit yields

$p\cup q\in\mathcal S$3

Here the path integral is partially frozen because the geometry on the chosen slice and on the new internal slit boundaries is fixed, while the rest of the Euclidean geometry is integrated over (Kaya et al., 11 Jun 2025).

These factorization-based constructions show that “partial freezing” can be localized to an interface or subregion rather than imposed only at asymptotic boundaries.

6. Contours, phases, and interpretive consequences

Once some sectors are fixed and others are integrated, the contour and phase structure of the path integral becomes decisive. In the Lorentzian canonical treatment of gravity, imposing the Hamiltonian constraint by lapse integration requires

$p\cup q\in\mathcal S$4

but after integrating out momenta the configuration-space integrand develops an essential singularity at $p\cup q\in\mathcal S$5. The regulated prescription is

$p\cup q\in\mathcal S$6

so the contour must run from $p\cup q\in\mathcal S$7 to $p\cup q\in\mathcal S$8 and detour below the origin in the complex lapse plane. This result is directly relevant to partially frozen formulations: freezing some variables is compatible with the formalism, but freezing the lapse itself changes the meaning of the constrained gravitational path integral (Banihashemi et al., 2024).

The phase of the Euclidean path integral is equally sensitive to which sectors are actually integrated. For compact Einstein spaces with positive cosmological constant, a full gauge-fixed one-loop analysis gives, for connected $p\cup q\in\mathcal S$9,

$\langle p|q\rangle = Z(\bar p\cup q).$0

where $\langle p|q\rangle = Z(\bar p\cup q).$1 is the number of negative transverse-traceless tensor modes. For $\langle p|q\rangle = Z(\bar p\cup q).$2, there is one negative tensor mode, but the scalar sector contributes the extra minus sign so that the final answer is real and positive. The paper’s explicit conclusion is that

$\langle p|q\rangle = Z(\bar p\cup q).$3

is real and positive, not imaginary. This matters for partially frozen path integrals because integrating only over TT modes would give a different phase from the full path integral; freezing the scalar or conformal sector can therefore change the physical interpretation (Shi et al., 1 Apr 2025).

A later AdS study makes the same point in wave-function language by distinguishing “a fully gravitational wave function, in which the boundary configuration is integrated over,” from “a partially frozen one, in which it is fixed, as in AdS/CFT.” In that setting the fully gravitational hyperbolic-ball partition function develops a one-loop phase of

$\langle p|q\rangle = Z(\bar p\cup q).$4

whereas the partially frozen construction behaves differently (Anikeeva et al., 13 May 2026).

These contour and phase analyses show that “partial freezing” is not only a matter of which classical data are held fixed. It also changes the analytic structure of the path integral, including whether one integrates over lapse, conformal, or boundary modes, and therefore changes the resulting semiclassical interpretation.

In the current literature, the partially frozen gravitational path integral is best understood not as a single formal object but as a technically precise family of restricted gravitational sums. Observer-relative and timekeeper-relative constructions refine the closed-universe problem of time and the one-dimensionality of $\langle p|q\rangle = Z(\bar p\cup q).$5-sectors; Neumann and Robin path integrals replace fully fixed boundary metrics by mixed phase-space data; reduced and fixed-area ensembles constrain only collective variables; factorized microcanonical path integrals glue only selected interface data; and fully gravitational versus partially frozen Hartle–Hawking wave functions separate boundary integration from boundary fixation. The unifying theme is selective control over what is integrated, what is fixed, and what is glued, with the physical consequences appearing in Hilbert-space dimension, relational time, saddle structure, factorization, and one-loop phase.