Euclidean Path Integral Approach

Updated 25 October 2025

The Euclidean path integral approach is a framework for quantizing field theories by analytically continuing to imaginary time, which renders path integrals convergent.
It effectively manages higher-derivative and ghost modes by integrating over first derivative boundary data to yield physical, normalizable probabilities.
Applications include quantum gravity, black hole thermodynamics, and numerical evaluations using advanced techniques like neural network approximations.

The Euclidean Path Integral Approach is a framework for quantizing field theories and analyzing nonperturbative phenomena by performing path integrals over field configurations on Euclideanized (Wick-rotated) spacetimes. Formally, this involves replacing real-time evolution by imaginary-time evolution, which renders the exponential of the action real and, under proper conditions, the path integrals convergent. The Euclidean formulation is central in quantum field theory, quantum gravity, statistical mechanics, and cosmology, particularly for defining ground or thermal states, handling theories with higher-derivative ghosts, defining thermal partition functions, addressing information loss in black holes, algorithmic evaluation of path integrals, and exploring the structure of quantum field theory on nontrivial backgrounds.

1. Wick Rotation and the Euclidean Prescription

The Euclidean path integral arises by analytically continuing real (Lorentzian) time $t$ to imaginary time $T$ , i.e., $t \mapsto iT$ , leading to a Euclideanized spacetime metric. Given an action $S$ , the Lorentzian path integral involves $\exp(iS)$ , which is oscillatory; after Wick rotation the path integral becomes

$Z_E = \int \mathcal{D}q\, \exp(-S_E[q])$

where $S_E$ is the Euclidean action derived from the original $S$ by $t \to iT$ . For standard second-order systems, Euclideanization renders the action positive definite and the path integral convergent. In higher-derivative or indefinite-signature models, this procedure "tames" pathological ghostly contributions: the exponent $-S_E$ damps fluctuations even when the Lorentzian action features higher derivatives or wrong-sign kinetic terms. In general, boundary conditions in the Euclidean approach are set on initial and final field values (and, for higher-derivative systems, their derivatives up to one order less than the highest derivative).

This approach is exemplified in the quantization of higher-derivative systems, where the Euclidean path integral is constructed with boundary conditions on the field and its first derivative, as opposed to the full set of higher derivatives, and the ghost degrees of freedom are integrated out to yield well-defined, physical probabilities (Fontanini et al., 2011).

2. High-Order and Ghost-Ridden Theories

In theories containing fourth-order or sixth-order derivatives—such as higher-derivative gravity theories employed in cosmological model-building—the canonical formulation reveals additional propagating degrees of freedom, some with wrong-sign kinetic terms ("ghosts"). The Euclidean path integral approach, first systematically exploited by Hawking and Hertog for fourth-order models and extended to sixth-order in (Fontanini et al., 2011), prescribes:

Wick rotation to Euclidean time to render the action bounded, even for ghost degrees.
Construction of the path integral with a field and its first derivative as variables.
Integration over all (including ghost) modes in the Euclidean domain before analytic continuation.
Extraction of physical probabilities only after integrating over ghosts (e.g., integrating over the first derivative boundary value), avoiding reintroduction of nonphysical divergences.

For a sixth-order derivative theory (e.g., with terms $\int d^4x\, \nabla_\alpha R_{\mu\nu} \nabla^\alpha R^{\mu\nu}$ ), the action for Fourier modes in Euclidean time becomes

$S_k = \int dT\, [A (y^{(6)})^2 + B (y^{(4)})^2 + C (y^{(2)})^2 + D y^2]$

with coupling constants $A,B,C,D$ . Classical solutions are linear combinations of exponentials $y_\text{cl}(T) = \sum_i C_i e^{\lambda_i T}$ , and the path integral is constructed by integrating over quantum fluctuations around these solutions.

The Euclidean construction produces, after ghost integration, normalizable, physical probabilities for the remaining degrees—even in the presence of higher-derivative "ghost" instabilities. In Minkowski backgrounds, positivity and normalizability hold straightforwardly; in de Sitter backgrounds, subtleties arise due to the background's time dependence, but a controlled prescription remains possible (Fontanini et al., 2011).

3. Application to Quantum Gravity and Black Hole Thermodynamics

The Euclidean path integral is foundational in defining quantum gravitational amplitudes and partition functions. In semiclassical quantum gravity, it underlies computations of thermal partition functions, black hole entropy, and quantum corrections:

For black holes in anti-de Sitter (AdS), the Euclidean path integral yields a partition function $Z = \int \mathcal{D}g\, \exp(-I_E[g])$ , where $I_E$ is the classical (on-shell) Euclidean action with appropriate boundary conditions or cavities. Thermodynamic quantities like entropy $S$ , free energy $F$ , and quasilocal energy $E$ are extracted from $Z$ ; e.g., the Bekenstein-Hawking entropy $S = \pi r_+^2$ for Schwarzschild–AdS (Lemos et al., 2018).
In Schwarzschild–de Sitter spacetime, out-of-equilibrium horizons preclude a smooth Euclidean instanton for the unconstrained path integral. Instead, a constrained path integral—fixing boundary data on a finite-radius surface—produces a well-defined saddle and yields the probability for black hole nucleation in de Sitter: $P \sim e^{-(S_{dS}-S_{SdS})}$ , with $S_{dS}$ and $S_{SdS}$ the respective entropies (Draper et al., 2022).
For systems comprising black holes and self-gravitating shells in a heat reservoir, the Euclidean/semiclassical approach allows a precise computation of the total entropy and energy as functions of the gravitational radius and produces rich phase diagrams with coexisting thermodynamic phases (black hole, hot flat space, shells with or without interior black holes) (Lemos et al., 2023).

Moreover, the approach is essential in the analysis of the information loss paradox, where the Euclidean path integral is used to sum over geometries representing distinct (semi-classical and non-perturbative tunneling) histories. This sum produces a “Page curve” for entanglement entropy: early on, semi-classical (information-losing) histories dominate (entropy grows); but at late times, non-perturbative (information-preserving, horizonless) histories dominate (entropy returns to zero), restoring unitarity (Chen et al., 2021, Yeom, 2021, Chen et al., 2022, Yeom, 21 Oct 2024). The formalism naturally accounts for probabilities of such tunneling events via $p_j \sim \exp(-2\Delta S_E^j)$ , where $\Delta S_E^j$ is the difference in Euclidean action between histories.

4. Definition of Thermal Partition Functions and Subtleties

The Euclidean path integral is often used to define thermal partition functions for quantum fields on static backgrounds. On compact spatial sections, the equivalence $Z_E = Z_C$ (Euclidean = canonical partition function) holds up to inessential terms. On noncompact sections and in the presence of Killing horizons, this equality can fail:

For noncompact static manifolds without horizons, extra terms in the $Z_E$ arise, involving mass derivatives and scattering phase shifts, but these cancel with proper handling, restoring $Z_E = Z_C$ (Diakonov, 2023).
In spaces with Killing horizons (e.g., Rindler), the order of coordinate/momentum integration, boundary contributions, and the degeneracy of mode spectra can generate discrepancies. In these cases, the derivative of $\log Z_E$ with respect to inverse temperature, which should yield energy, does not coincide with standard definitions of energy (Hamiltonian, stress tensor, or sum over modes), due to the presence of UV- and horizon-divergent terms and subtleties involving boundary conditions.
A new computational prescription is proposed: add an auxiliary mass $M$ coupled via the lapse factor $g^{00}$ in the action, and perform the $M\rightarrow 0$ limit after spatial integrations. This restores $Z_E = Z_C$ even in the presence of horizons (Diakonov, 2023).

This demonstrates that while the Euclidean path integral is a universal tool, its precise application and interpretation depend on spacetime topology, boundary conditions, and the order of functional integrations.

5. Numerical and Algorithmic Evaluation: Neural Networks and Variational Dynamics

The nonperturbative evaluation of Euclidean path integrals motivates development of advanced numerical methods:

Smooth Path and Field Expansions: Rather than lattice discretizations, the path $q(\tau)$ can be represented as a sum of Gaussian (or more general) functions, $q(\tau) = \sum_{i=1}^{N_\text{sum}} q_i\, \exp(-(\tau-\tau_i)^2/\xi_i^2)$ , yielding smoother approximations to continuum path integration. Such approaches, combined with the Metropolis algorithm, give accurate ground state properties (>90% accuracy for the harmonic oscillator) and are applicable in higher-dimensional gauge theories, capturing features like confinement via Wilson loops (Sekihara, 2011).
Neural Network Expansion: The interaction term in the Euclidean path integral can be approximated by a sum of radial basis (Gaussian) functions, fit using a multi-layer perceptron with exponential nonlinearities. Each basis yields an analytically solvable Gaussian path integral; a neural network is trained to reproduce the exponential of the integrated potential. This enables efficient and accurate computation of propagators and ground state wave functions for arbitrary (including complex) potentials, with errors below a few percent for nontrivial examples (Balassa, 21 Sep 2025).

The exponential activation in the hidden layers is crucial for approximating the original path integral, since the transformation of the path integral's potential part is by nature exponential in the field variables. Extensions to potentials with imaginary parts are straightforward within a multiple-output framework with appropriately complexified weights.

6. Modular Hamiltonians, Entanglement, and Operator Reconstruction

The Euclidean path integral is a powerful tool for the construction of modular Hamiltonians and the paper of entanglement in quantum field theories:

By preparing excited states via sources for local operators in the Euclidean path integral, one may construct a family of density matrices whose modular Hamiltonians $K_\lambda$ (generators of modular flow) can be expressed perturbatively. A central result is a manifestly Lorentzian formula for $K_\lambda$ to all orders in the source, applicable to half-spaces in relativistic QFT (Balakrishnan et al., 2020).
For shape-deformed entangling surfaces, modular Hamiltonians become operator-valued functionals of integrated null energy operators. In the case of pure null deformations, the full perturbation series can be resummed, precisely matching expectations from conformal and holographic arguments.
The modular flow of local operators near the entanglement cut exhibits universal “local-boost” behavior, confirming continuity with the vacuum modular Hamiltonian in the small-distance limit.

These results enable systematic computations of entanglement entropy, relative entropy, and modular flow for non-vacuum and nontrivial geometric states, with direct implications for holographic dualities.

7. Negative Modes and Instabilities in Euclidean Quantum Gravity

When evaluating gravitational path integrals, modes with negative eigenvalues in the quadratic fluctuation operators (negative mass-squared scalars or unstable moduli) give rise to “instability phases.” The correct evaluation requires:

Identifying all directions in field space with negative quadratic action after Kaluza-Klein dimensional reduction (e.g., volume moduli).
Rotating the integration contours for each unstable mode, each rotation contributing a factor $(-i)$ in the partition function. For compact spaces $S^p\times M_q$ ,

$Z(S^p \times M_q) = i^{p+2} (-i)^{\mathcal{N}_{L,q}+\mathcal{N}_{TT,q}} |Z(S^p \times M_q)|$

where $\mathcal{N}_{L,q}$ and $\mathcal{N}_{TT,q}$ count, respectively, longitudinal scalar and transverse traceless negative modes (Ivo et al., 1 Apr 2025).

Properly handling such phases is essential for gauge invariance and for the correct physical interpretation of potential de Sitter instabilities, no-boundary wavefunction normalizations, and path integral convergence.

The physical interpretation links the instability phase to the number of “growing” (tachyonic) modes experienced by a static observer in de Sitter and a correct prescription must include only non-gauge modes and exclude zero modes associated with symmetries.

In summary, the Euclidean Path Integral Approach provides a rigorous formalism for the quantization of field theories, the analysis of gravitational thermodynamics and black hole information, algorithmic treatments of quantum path integrals, and the structural understanding of operator dynamics in quantum field theory. Its power lies in the ability to address otherwise ill-defined integrals, to control ghost and instability contributions in higher-derivative and gravity theories, and to facilitate both analytical and advanced numerical explorations of quantum phenomena. The approach remains foundational in addressing persistent theoretical puzzles, such as the black hole information paradox, and in developing new computational tools for quantum science.