Euclidean Gravitational Path Integral

• Euclidean gravitational path integral is a nonperturbative framework in quantum gravity that computes partition functions by integrating over Riemannian metrics.
• It uses semiclassical saddle-point approximations, incorporating instantons and Euclidean black holes to derive key thermodynamic and cosmological insights.
• The method bridges canonical, variational, and numerical approaches while addressing challenges such as the conformal factor problem and topology changes.

The Euclidean gravitational path integral is the central object in covariant approaches to quantum gravity, providing both a formal definition of the quantum gravitational wave function and the partition function for semiclassical gravitational thermodynamics. By summing over an appropriate class of d-dimensional Riemannian (Euclidean-signature) metrics, weighted by the exponential of minus the Euclidean action, the path integral yields a nonperturbative functional that encodes not only the semiclassical saddle points (such as instantons and black holes) but also nonperturbative effects—topology change, wormholes, and quantum geometric fluctuations. Applications include black hole thermodynamics, quantum cosmology, entropy calculations, the information paradox, and recent developments in higher-dimensional CFT/AdS dualities. Methodologically, the Euclidean approach serves as a computational and conceptual bridge between canonical quantization, variational (Hamiltonian) frameworks, and modern numerical methods. This article provides a technical review of its mathematical structure, variational formulations, semiclassical evaluation, measure factors, nonperturbative features, implications for black hole physics and information, and technical challenges arising in quantum gravity.

1. Formal Definition and Action Functionals

The basic object is the partition function or wave function formally defined as a functional integral over Euclidean metrics $g$ (and matter fields $\phi$) on a manifold $M$, with or without boundary data:

$$Z = \int \mathcal{D}g\,\mathcal{D}\phi\,\exp[-I_E(g,\phi)] \,,$$

where $I_E$ is the Euclidean action. For pure gravity with cosmological constant and matter,

$$I_E[g, \phi] = -\frac{1}{16\pi} \int_{M} d^4x\,\sqrt{g}\,(R - 2\Lambda) + I_{\mathrm{M}}[g, \phi] + \frac{1}{8\pi} \int_{\partial M} d^3 x\,\sqrt{h}\,(K - K_0) \,.$$

Here, $R$ is the Ricci scalar, $K$ is the trace of the extrinsic curvature, $K_0$ is a background subtraction to make the on-shell action finite, and $h$ denotes the induced metric on the boundary. The measure $\mathcal{D}g$ is specified by the DeWitt supermetric, Faddeev–Popov gauge fixing, and possible boundary conditions: Dirichlet (fixed induced metric), Neumann (fixed momentum), or mixed (Yeom, 2021, Yeom, 2024, Krishnan et al., 2016, McDearmon, 2023).

For configurational data specified by a simplicial complex (Regge calculus), the discrete edge-lengths $\{l_{ij}\}$ play the role of the metric, and conjugate variables are introduced in the variational phase space (McDearmon, 2023).

2. Variational and Hamiltonian Formulations

A powerful viewpoint regards the Euclidean gravitational path integral as the long-time, ergodic average of a Hamiltonian flow in an extended phase space. In such a formulation, the space of variables includes not only the edge-lengths $\{l_{ij}\}$ but also their conjugate momenta, a global scale $s$, and its conjugate. The extended phase space $\Gamma$ is coordinatized as:

$$\gamma = \{l_{ij}, \pi^{(l)}_{ij}; s, \pi^{(s)}\} \in \Gamma.$$

The extended action on $\Gamma$ is constructed to generate symplectic (Hamiltonian) flow:

$$S_{\text{ext}} = s \Biggl( \sum_{(i,j)}\frac{(\pi^{(l)}_{ij})^2}{2s^2 l_{ij}^2}V_l + \frac{(\pi^{(s)})^2}{2m_s} + S_R[q] + S^{\log}[q] - S^0 \Biggr)$$

where $S_R[q]$ is the Regge action (discrete Einstein–Hilbert analog), $S^{\log}[q]$ encodes the measure, and $S^0$ enforces constraints (McDearmon, 2023).

Hamilton's equations are then:

$$\begin{aligned} \dot{l}_{ij}(\lambda) &= +\frac{\partial S_{\text{ext}}}{\partial \pi^{(l)}_{ij}}, \qquad \dot{\pi}^{(l)}_{ij}(\lambda) = -\frac{\partial S_{\text{ext}}}{\partial l_{ij}}, \ \dot{s}(\lambda) &= +\frac{\partial S_{\text{ext}}}{\partial \pi^{(s)}}, \qquad \dot{\pi}^{(s)}(\lambda) = -\frac{\partial S_{\text{ext}}}{\partial s}. \end{aligned}$$

The fictitious "time" $\lambda$ plays the role of evolution parameter in this extended symplectic system.

Assuming ergodicity, the long-$\lambda$ time average of an observable $O[q(\lambda)]$ recovers its Euclidean path integral expectation value:

$$\langle O \rangle = \lim_{T \to \infty}\frac{1}{T} \int_0^T d\lambda\, O(q(\lambda)) = \frac{1}{Z_E} \int Dq\, O(q) e^{-S_R[q]/\hbar} F[q]$$

where $F[q]$ is the path-integral measure determined by $S^{\log}[q]$ (McDearmon, 2023).

3. Semiclassical Expansion and Instantons

The leading contribution to the Euclidean gravitational path integral comes from saddle-point (instanton) metrics solving $\delta I_E = 0$ with prescribed boundary data. The partition function is then given (in the saddle approximation) by a sum over instantons:

$$Z \simeq \sum_{\text{instantons}} A_{\text{cl}} e^{-I_E[g_\text{cl}]/\hbar}$$

with determinants $A_{\text{cl}}$ of quadratic fluctuations (one-loop prefactors):

$$A_{\text{cl}} \sim \left[\frac{\det'(\Delta_2)}{\det(\Delta_0)} \right]^{-1/2}$$

where $\Delta_2$, $\Delta_0$ are operators governing spin-2 and spin-0 metric fluctuations after gauge-fixing (Yeom, 2024, Yeom, 2021).

The dominant instantons include Euclidean black hole metrics, "trivial" (horizonless) geometries, and in certain settings, wormhole or topology-changing metrics. The semiclassical on-shell action for a four-dimensional Schwarzschild black hole is:

$S^{\text{BH}}_E = \beta M - \frac{A}{4}$

yielding entropy $S = A/4$ and reproducing black hole thermodynamics (Yeom, 2021).

4. Boundary Conditions, Ensembles, and Measure

Boundary conditions play a crucial role in defining the ensemble—canonical (Dirichlet) or microcanonical (Brown–York) or mixed (Neumann or Robin). In the Dirichlet ensemble, the boundary metric is fixed; the Gibbons–Hawking–York (GHY) term is needed for a well-posed variational principle. In the Neumann (momentum-fixed) ensemble, a Legendre transform of the action yields different boundary terms and reproduces the entropy-area law (Krishnan et al., 2016, Ailiga et al., 2024).

The functional measure is constructed using the DeWitt supermetric, gauge-fixing functionals $F[g]=0$, the Faddeev–Popov determinant $\Delta_\text{FP}[g]$, and possible ghost actions:

$$Dg \to \prod_x \prod_{a \leq b} dg_{ab}(x) \sqrt{\det G_{abcd}(x)} \delta[F(g)] \Delta_\text{FP}[g]$$

Explicitly, in Regge calculus and variational dynamics, the discrete path-integral measure $F[q]$ involves products over edge and simplex contributions, e.g., $\prod a_l[q]^{-1}$ and $1/\sqrt{k_l[l]}$ (McDearmon, 2023).

5. Applications: Black Hole Thermodynamics, Quantum Cosmology, and Information

The Euclidean gravitational path integral framework provides a first-principles basis for black hole thermodynamics, with entropy computed from saddle-point actions and precise derivation of the first law. The constrained path integral approach allows for consistent treatment of geometries with multiple horizons (e.g., Schwarzschild–de Sitter) via microcanonical constraints and on-shell evaluation of the action (Draper et al., 2022, Draper et al., 2023).

In quantum cosmology, the path integral prepares the Hartle–Hawking wave function for a given 3-geometry $h_{ij}$:

$$\Psi[h_{ij}] = \int_{g|_{\partial M} = h_{ij}} \mathcal{D}g\, e^{-I_E[g]}$$

and leads to the Wheeler–DeWitt equation $\hat{H}\Psi = 0$ as an analog of the time-independent Schrödinger equation (Yeom, 2024, McDearmon, 2023, Belin et al., 2020).

The information paradox for evaporating black holes is addressed, at least formally, by including all saddle points, including nonperturbative trivial or topology-changing geometries, in the total Euclidean path integral. In this framework, late-time dominance of trivial saddles recovers unitarity and implements the "Page curve" for entropy evolution, resolving the apparent loss of information without needing firewalls or radical modifications at the horizon (Yeom, 2024, Yeom, 2021).

6. Nonperturbative Effects and Factorization

Topologically nontrivial saddles—Euclidean wormholes, instantons connecting disconnected boundaries, and topology-changing processes—are generically present and provide nonperturbative contributions highly suppressed but crucial for phenomena such as information retrieval and ensemble averaging. These saddles can, however, induce paradoxes, notably the factorization crisis in AdS/CFT, where wormhole contributions violate exact factorization of boundary correlators and spoil cluster decomposition, suggesting either a need for an ensemble interpretation or new principles in quantum gravity (Loveridge et al., 15 Apr 2025, Loges et al., 2022, Cardoso et al., 2022, Anninos et al., 2021).

In the AdS$_3$ context, the path integral for pure gravity can be defined nonperturbatively using localization in Chern–Simons theory, with the partition function given by modular-invariant Rademacher sums and matching extremal CFT predictions at $c=24$ (Iizuka et al., 2015).

7. Technical and Conceptual Challenges

Rigorous definition of the Euclidean gravitational path integral remains elusive. The conformal factor problem renders the Euclidean action unbounded below, leading to divergent integrals. Various gauges, boundary conditions, or analytic continuation prescriptions (e.g., Picard–Lefschetz theory) are employed in controlled settings to circumvent this instability and select a convergent contour (Calcagni et al., 2024, Fontanini et al., 2011, Ailiga et al., 2024).

The choice of boundary terms and ensemble (Dirichlet, Neumann, Robin, teleparallel, first-order/Palatini) impacts the mathematical consistency and physical predictions of the ensemble. In teleparallel and first-order formalisms, the on-shell action can be rendered finite and ambiguity-free without background subtraction (Jiménez et al., 2024, 0810.0297).

Nonlocal terms, higher-curvature corrections, and dimensional reduction techniques all generalize the basic construction and are being actively explored (Draper et al., 2023, Jiménez et al., 2024, Draper et al., 2022).

Finally, numerous open problems—measure definition, universality of late-time unitarity restoration, possible sum over topologies, and the full resolution of the information paradox—remain central in current research.

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References:

(McDearmon, 2023, Yeom, 2024, Yeom, 2021, Draper et al., 2022, Draper et al., 2023, Loveridge et al., 15 Apr 2025, Jiménez et al., 2024, Iizuka et al., 2015, Fontanini et al., 2011, 0810.0297, Cardoso et al., 2022, Anninos et al., 2021, Krishnan et al., 2016, Ailiga et al., 2024, Belin et al., 2020, Loges et al., 2022, Calcagni et al., 2024, Diakonov, 2023).