Gravitational Path Integral

• Gravitational Path Integral is a coarse-grained statistical sum over spacetime geometries that encodes semiclassical gravito-hydrodynamics and entropy fluctuations.
• It employs hydrodynamic variables and white-noise modular Hamiltonian techniques to capture stochastic fluctuations and nonlocal entropic effects.
• This formulation connects black hole microstate counting, quantum thermodynamics, and cosmological path integrals through an effective, entropic approach to Einstein’s equations.

The gravitational path integral (GPI) is a central but subtle construct in quantum gravity, integrating over the space of spacetime geometries—typically modulo diffeomorphisms—to define transition amplitudes, partition functions, microcanonical or canonical statistical sums, and observables such as black hole entropy. Its rigorous formulation, physical interpretation, and computational efficacy vary dramatically across spacetime dimension, background geometry, and quantum regime. The contemporary viewpoint, as exemplified in recent work, reframes the GPI as a hydrodynamically coarse-grained, statistical object, encoding semiclassical gravito-hydrodynamics rather than a literal quantization of spacetime geometry (Banks, 15 Jan 2026).

1. Hydrodynamic Formulation and Statistical Character

The foundational proposal is that the Euclidean gravitational path integral should not be interpreted as an analytic continuation of a quantum-mechanical amplitude but as a coarse-grained partition functional with fluid-like degrees of freedom. Specifically, one considers nested families of causal diamonds along a timelike geodesic, using as hydrodynamic variables:

• $h_{ab}(t, x^i)$: the metric on the maximal-area codimension-2 surfaces (“screens”) at proper time $t$,
• $\pi^{ab}(t, x^i) = \sqrt{h} K^{ab}$: their conjugate momenta (extrinsic curvature components),
• $S_{vv}(t, x^i)$: a stochastic source encoding fluctuations on the stretched horizon.

The resulting functional integral takes the form:

$$Z_{\rm hydro} = \int [Dh\, D\pi\, D S_{vv}]\, \exp\left\{ - \left( S_{\rm cl}[h, \pi] + \delta S[h, S_{vv} ] \right) \right\}$$

where $S_{\rm cl}$ is a hydrodynamic action derived from the Einstein equations in Gaussian null coordinates, and $\delta S$ incorporates nonlocal, entropic “wormhole”-like effects from white-noise modular Hamiltonian fluctuations. The functional measure is the natural Liouville measure for $(h,\pi)$ and a Gaussian defined by the two-point function of $S_{vv}$ (Banks, 15 Jan 2026).

This approach is physically grounded in three coarse-grained postulates:

• Covariant Entropy Principle (CEP): The modular Hamiltonian $K$ of an “empty diamond” of area $A$ satisfies $\langle K \rangle = (\Delta K)^2 = A/4G_N$, encoding Bekenstein-Hawking entropy and its statistical variance.
• Hydrodynamic Hamiltonian Flow: The slow variables $(h_{ab}, \pi^{ab})$ evolve via a Hamiltonian flow generated by the Einstein evolution and constraint equations; in Gaussian null coordinates, the system is manifestly Hamiltonian with clear separation of kinematics and dynamics.
• Carlip–Solodukhin Fluctuations: Each stretched horizon supports a cutoff 2d CFT of central charge $c = A/4G_N$, whose null-null stress-tensor $S_{vv}$ is modeled as white noise, enforcing non-trivial quantum stochasticity in the effective hydrodynamics.

Collectively, these prescriptions yield a stochastic gravito-hydrodynamics whose statistical weight is the gravitational path integral.

2. Modular Hamiltonian, Entropy, and Fluctuations

The modular Hamiltonian $K_\diamond$ plays several structural roles:

• Equilibrium mean: Its expectation value yields the “empty” system’s entropy, $S_0 = \langle K \rangle = A/4G_N$.
• Source of fluctuations: Its variance, $(\Delta K)^2 = A/4G_N$, quantifies the intensity of hydrodynamic white noise, providing the reservoir for entropic (stochastic) excitations.
• Analytic continuation: Under $t \to t + i\beta$, one computes spectral form factors of the type $$Z(t + i\beta) = \langle E | U(t + i\beta) | E \rangle$$, with both disconnected (classical) and connected (fluctuational, “wormhole”) contributions. The connected piece encodes fluctuations of $K_\diamond$ that are nonzero off-shell or at nonzero temperature.

Physically, energy or stress localized in the diamond appears as entropy deficits, encoding Einstein’s equations in an entropic (hydrodynamic) form—a realization of Jacobson’s gravitational thermodynamics (Banks, 15 Jan 2026).

3. Derivation and Structural Elements

The derivation proceeds by:

• Starting with Jacobson’s thermodynamic derivation of Einstein’s equation from the Covariant Entropy Principle: $$\delta S = \delta\langle K\rangle = R_{mn}k^m k^n/(4G_N)$$.
• Identifying hydrodynamic variables in Gaussian null coordinates, with canonical momenta and area elements tied directly to geometric quantities.
• Writing the t-evolution Hamiltonian and recasting it in Lagrangian form, suitable for path integral quantization and imitation of Schwinger-Keldysh or imaginary-time contours.
• Treating $S_{vv}$ as a stochastic variable with Gaussian white-noise statistics, the functional integration over which yields an influence functional $\delta S_{\rm fluct}$ encoding nonlocal stochastic “wormhole” effects.
• The resulting path integral $$Z_{\rm hydro} = \int [Dh\, D\pi] \exp\{-S_{\rm cl}[h,\pi] - \delta S_{\rm fluct}[h]\}$$ recovers, in saddle-point approximation, the classical Einstein action and, at one-loop and beyond, stochastic/entropic effects corresponding to Euclidean or Lorentzian wormholes and nonlocal correlators.

The measure is fixed: Liouville form for phase space variables $(h,\pi)$, and a white-noise determined Gaussian for $S_{vv}$.

4. Topology, Saddle Structure, and Limits

Only two-dimensional Euclidean “wormhole” topologies emerge as representatives of the nonlocal entropic influence functional once modular fluctuations are properly incorporated. For $d>3$, higher-dimensional genus sums are not required or well-defined—and real models never implement a literal microscopic sum over all spacetime topologies in $d$-dimensions (Banks, 15 Jan 2026). For fixed-area black holes (e.g., eternal AdS black holes), the dominant connected contribution is the two-boundary Lorentzian wormhole metric:

$$ds^2 = -( \rho - i \epsilon )^2 dt^2 + d\rho^2 + d\Sigma_{d-2}^2$$

and its analytic continuation matches the SFF “ramp.” In JT gravity, the same geometry appears as the “double trumpet.”

If one imposes a random matrix interpretation or considers stable AdS black holes, higher-genus 2d surfaces contribute, but for de Sitter or unstable configurations these phases are unobservable. This sharply limits the sum-over-topologies in the GPI to minimal representative topologies—physical models sum over only those connected to the dominant entropic effect under the coarse-grained hydrodynamic mapping.

5. Role in Quantum Gravity and Open Questions

The gravitational path integral, in this hydrodynamic formalism, is a coarse-grained statistical sum, not a sum over fundamental quantum amplitudes of microscopic spacetime configurations. Microscopic quantum gravitational theories must match:

• The Covariant Entropy Principle (empty-diamond area law),
• The structure and variance of modular Hamiltonian fluctuations (white noise, $A/4G_N$ scaling),
• The correct induced Hamiltonian flow for gravitational data across nested maximal-area “screens.”

Open challenges include a microscopic account of the cutoff CFT structure on stretched horizons, the fate and extension of the double-cone wormhole to higher genus and dimension (for deeper random-matrix universality), and a robust theory of coarse-grained hydrodynamics and modular fluctuations in quantum de Sitter space (where no observer can persist for a recurrence or scrambling time).

6. Broader Implications and Connections

The GPI serves as a unifying statistical functional for diverse phenomena:

• Black hole microstate counting: In supersymmetric sectors, the GPI precisely counts BPS microstates by summing over BPS-saturated geometries, with subleading wormhole or orbifold corrections necessary for matching to string theory (Boruch et al., 2023, Iliesiu et al., 2022).
• Quantum-corrected thermodynamics: Including off-shell, Gaussian, and higher fluctuations enriches the black hole phase structure, introducing loop-corrected free-energy landscapes and altering phase transitions (Liu et al., 18 Jun 2025).
• Cosmological path integrals: In minisuperspace reductions, care with gauge fixing, conformal (“CKV”) zero modes, and BV quantization is required. Exact and semiclassical path integral calculations reproduce microcanonical ensemble sums, the Hartle-Hawking proposal, and include all relevant quantum corrections (Barvinsky, 2010, Ailiga et al., 2024).
• Random tensor networks and entanglement wedges: The path integral, together with fixed-geometry or random tensor decompositions and replica methods, yields a statistical derivation of generalized entanglement wedges (Bousso-Penington), directly linking gravitational and information-theoretic constructs through the geometry of replica domain walls (Kaya et al., 11 Jun 2025).

In summary, the gravitational path integral, as modernly understood, is not a literal sum over microscopic spacetime geometries but rather the "partition function" of a fundamentally stochastic, hydrodynamic effective theory that reproduces classical Einstein dynamics as its saddle point, and whose fluctuations and topological features encode genuine collective, entropic, and modular structure required for any consistent quantum gravitational completion (Banks, 15 Jan 2026).