Euclidean Path Integrals in QFT

• Euclidean path integrals in quantum field theory are a framework employing imaginary time to define transition amplitudes, partition functions, and correlation functions.
• They bridge statistical mechanics and quantum mechanics by converting oscillatory phase factors into positive-definite weights, enabling rigorous functional integration and regularization.
• Advanced numerical methods such as Monte Carlo simulations, smooth functional expansions, and neural network approximators are key to tackling singularities and complex actions in nonperturbative regimes.

Euclidean path integrals in quantum field theories provide a rigorous, unified framework for defining transition amplitudes, partition functions, and correlation functions in quantum mechanics and quantum field theory after a Wick rotation to imaginary time. These integrals serve as the nonperturbative definition of quantum field theory (QFT) and are foundational in the formulation of statistical mechanics, the quantization of gauge theories, and the computation of nonperturbative phenomena. The mathematical, computational, and conceptual aspects of Euclidean path integrals span regularization, the structure of the configuration space, operator correspondences, and the development of high-performance algorithms for practical calculations.

1. Mathematical Structure and Definition

The Euclidean path integral is formally written as a functional integral over field configurations $\phi(x)$ (or particle paths $x(\tau)$), weighted exponentially by minus the Euclidean action: $Z = \int \mathcal{D}\phi\, e^{-S_E[\phi]}$ or, in quantum mechanics,

$$K(x_f, x_i; T) = \int_{x(0)=x_i,\,x(T)=x_f} \mathcal{D}x(\tau)\, \exp\left[-\int_0^T d\tau\, \left(\tfrac{1}{2}m (\dot{x})^2 + V(x(\tau))\right)\right]$$

This construction is central both for defining the partition function and for representing propagators as statistical sums over histories, allowing a direct connection to the operator formalism via the relationship between the imaginary-time evolution operator $e^{-\beta H}$ and the statistical trace in the canonical ensemble (Rosenfelder, 2012).

The Wick rotation to imaginary (Euclidean) time turns the oscillatory phase $e^{iS}$ into a positive-definite weight $e^{-S_E}$, allowing the use of probabilistic and statistical methods for both analytical and numerical evaluation.

2. Functional Integral Spaces and Subtleties

The measure-theoretic setting of the path integral, especially the choice of the configuration/functional space over which the integration occurs, is critical. In the simplest treatments, the domain of integration is taken to be the set of continuous (or smooth) functions—e.g., $C[0,1]$ in the Wiener measure for Brownian motion or for free field theory. However, changes of variables, as in nonlinear interacting models, can induce singularities in the fields. For instance, a one-dimensional $\varphi^4$ theory under a nontrivial transformation can force the path integral to be taken over a union of function spaces labeled by the number and nature of singularities: $$X = \bigcup_{n\geq 0} X_n, \qquad X_n = \{\text{functions with } n \text{ singularities of the form } (t-t^*)^{-1}\}$$ This structure implies that naive manipulations may invalidate the original integration measure, leading to nontrivial consequences for the quantum theory and for renormalization (Belokurov et al., 2011).

Careful measure construction (e.g., via Kolmogorov's extension theorem for Gaussian measures, Henstock-Kurzweil or Lebesgue measure extensions on separable Hilbert spaces (Esposito et al., 29 Aug 2024)) ensures well-definedness in rigorous approaches. In practice, the necessity to cover function spaces beyond simple $L^2$ or $C^\infty$ becomes acute in models admitting highly singular field configurations.

3. Regularization Schemes and Counterterms

The continuum path integral is inherently ambiguous without precise regularization. Discretization procedures—time slicing, Fourier mode cutoffs, and dimensional extension (dimensional regularization)—all lead to different finite results unless supplemented with finite counterterms that restore physical, scheme-independent quantities.

For an O(N) extended SUSY quantum mechanics model on a curved background, the regularized Euclidean path integral for the transition amplitude is

$$\langle x, \bar\lambda | e^{-\beta H} | y, \eta \rangle = \int [Dx] [D\Psi D\bar\Psi] e^{-S[x,\Psi,\bar\Psi]}$$

with action

$$S = \int_0^\beta d\tau\, \left[\tfrac{1}{2}g_{\mu\nu}(x)\dot{x}^\mu\dot{x}^\nu + \tfrac{1}{2} \psi_{a i} D_\tau \psi^a_i + \alpha R_{abcd} \psi^a_i \psi^b_i \psi^c_j \psi^d_j + V(x)\right]$$

and the need for different local counterterms $V_{\text{CT}}$ in each regularization scheme:

• Time slicing (TS):

$$V_{\text{TS}}^{(N)} = -\left[\frac{1}{8} + \frac{\alpha N}{2}\right] R + \frac{1}{8} g^{\mu\nu} \Gamma_{\mu\lambda}^\sigma \Gamma_{\nu\sigma}^\lambda + \frac{N}{16}\omega_{\mu ab}\omega^{\mu ab}$$

• Mode regularization (MR):

$$V_{\text{MR}}^{(N)} = -\left[\frac{1}{8} + \frac{\alpha N}{2}\right] R - \frac{1}{24}(\Gamma^\mu_{\nu\lambda})^2 + \frac{N}{24}\omega_{\mu ab}\omega^{\mu ab}$$

• Dimensional regularization (DR):

$$V_{\text{DR}}^{(N)} = -\left[\frac{1}{8} + \frac{\alpha N}{2}\right] R$$

When these finite local counterterms are included, the path integral yields transition amplitudes, heat kernel coefficients, and one-loop effective actions that match operator-based results and exhibit manifest covariance (Bastianelli et al., 2011).

4. Analytical and Numerical Evaluation Techniques

Analytical progress is available in special cases (quadratic actions, free field theory, Gaussian measures), where the path integral is exactly solvable via saddle-point expansions or Gaussian integral evaluations. For generic cases (non-quadratic interactions, non-abelian gauge theories), nonperturbative approaches are required.

Monte Carlo Methods: After discretization (space-time lattice or finite-difference time slicing), the functional integral is approximated by a high-dimensional integral, which is efficiently evaluated via Monte Carlo algorithms such as the Metropolis update, using the statistical weight $e^{-S_E}$. Importance sampling and ergodic symplectic flows (variational dynamics) offer further flexibility in sampling the path space (Rosenfelder, 2012, McDearmon, 2023).

Smooth Path Representations: Instead of piecewise-linear discretization, paths or fields can be represented as finite sums of smooth functions (e.g., Gaussians), with coefficients updated using Metropolis or heat-bath sampling. This framework accurately reproduces ground state observables and is effective in quantum field theory settings (e.g., for SU(2) or U(1) gauge fields) (Sekihara, 2011, Sekihara, 2013).

Neural Network Function Approximators: Recently, radial basis function (RBF) neural networks have been used to approximate the nonlinear (non-Gaussian) interaction parts of the exponent in the Euclidean path integral. By expanding the interacting factor as a sum of Gaussians, the full path integral becomes a linear combination of analytically tractable Gaussian integrals. This approach yields high-precision results for bound state wave functions in quantum mechanics and field theory, and demonstrates efficiency in modeling phase transitions in the $\phi^4$ theory (Balassa, 21 Sep 2025, Balassa, 23 Sep 2025).

Technique	Description	Context
Monte Carlo (Metropolis)	Importance-sampled, random walk updates	General field theory on the lattice
Smooth Path (Gaussian Sum)	Functional expansion in Gaussians, Metropolis	Quantum mechanics, gauge theories
RBF Neural Networks	Expansion of interactions, explicit integration	Nonlinear quantum mechanics/QFT
Variational Dynamics	Symplectic ergodic flows in phase space	Sampling Euclidean path integral
Analytic (Gaussian Cases)	Explicit saddle/Gaussian integration	Free fields, quadratic actions

5. Applications and Physical Interpretation

Euclidean path integrals are central in the following areas:

• Quantum Field Theory: Definition of partition functions and generating functionals, calculation of n-point correlators, quantization on curved spaces, nonperturbative computation of effective actions, and computation of heat kernel coefficients (Bastianelli et al., 2011, Rosenfelder, 2012).
• Gauge Theories: Inclusion of gauge-fixing and Faddeev-Popov ghosts, quantization of non-abelian fields; care with measure and configuration space is necessary to guarantee invariance under gauge symmetry and to avoid anomalies (Esposito et al., 29 Aug 2024).
• Supersymmetric and Worldline Methods: Worldline path integrals with extended SUSY structure are applied to higher-spin fields and computation of anomalies.
• Nonlinear/Interacting Theories: Handling of path integrals with singularities, specification of function space (including discontinuous functions) necessary for correct definition of observables and renormalization (Belokurov et al., 2011).
• Statistical Mechanics and Lattice Field Theory: Evaluation of critical exponents, phase diagrams, and spontaneous symmetry breaking, where Euclidean path integrals are the foundation for lattice regularization and finite-temperature studies.
• Complex and Real-Time Extensions: While the Euclidean formulation enables stable computations, analytic continuation to real time correlators is subtle. For problems with complex actions (finite density, real-time QFT, quantum gravity with conformal factor issues), new techniques involving thimble integrations, smooth regulators, and complex stochastic dynamics (complex Langevin) are used (Feldbrugge et al., 2022, Kumar, 2023).

6. Extensions, Topology, and Advanced Mathematical Formalism

Euclidean path integrals are adaptable to a wide variety of advanced scenarios:

• Curved/Banach Manifolds and Flat Compact Spaces: For quantization on arbitrary manifolds, particularly flat compact spaces ($\mathbb{R}^n/\Gamma$), path integrals are interpreted through reproducing kernel Hilbert spaces and Feynman propagators constructed via holomorphic functions on the complexified manifold (Capobianco et al., 2015).
• Higher-Derivative and Nonlinear Actions: Path integrals with quadratic forms in derivatives of order $n$ are treated as multi-dimensional Gaussian integrals, with covariance matrices determined by Green's functions of the relevant differential operators; more generally, iterative techniques employing Riccati and matrix equations capture the free energy and propagators (Dean et al., 2019).
• Geometry and Field Quantization: The periodic structure inherent in Euclidean path integrals with respect to translation in arc length/time parameter is foundational for extending quantization schemes to extended manifolds and for the geometrical origin of field quantization via the path integral definition (Vatsya, 2014).
• Topological Sectors and Boundary Conditions: Hilbert space structures induced by Euclidean path integrals must be carefully related to hermitian inner products, with issues of orientation, time reversal, and reflection symmetry dictating the operator (e.g., $T$ or orientation reversal $\Theta$) that connects the natural bilinear pairing to the quantum mechanical sesquilinear product (Witten, 17 Mar 2025).

7. Challenges, Limitations, and Frontiers

The Euclidean approach is remarkably robust, but is subject to several fundamental and practical limitations:

• Conformal Factor and Nonpositive Actions: In quantum gravity, the unboundedness of the Euclidean Einstein-Hilbert action (the conformal factor problem) makes the Euclidean path integral ill-defined without proper treatment—at times requiring reformulation in terms of Lorentzian path integrals with special analytic continuation and the inclusion of conical singularity contributions (Marolf, 2022).
• Sign Problem and Complex Actions: In theories with chemical potential (finite density), real-time observables, or complex topological terms, the Euclidean action is complex, preventing standard importance-sampling; complex Langevin and neural network-based methods have proven effective in overcoming this challenge for select models (Kumar, 2023, Balassa, 21 Sep 2025).
• Precision and Computational Efficiency: For large lattices, strong couplings, or high-dimensional systems, efficient sampling (variance reduction, importance sampling, neural surrogates) and advanced representations (Gaussian sums, RBF expansions) are critical for tractable computations (Sekihara, 2011, Balassa, 23 Sep 2025).
• Choice of Functional Space: The mathematical underpinning of the space of integration (whether $C[0,1]$, spaces of distributions/Schwartz, Banach/Hilbert spaces, or unifications such as the KS2[H] construction) and the transition maps between overlapping charts are nontrivial, especially for rigorous constructions and when integrating over topologically nontrivial histories (Esposito et al., 29 Aug 2024).

Summary Table: Key Aspects of Euclidean Path Integrals

Aspect	Approach/Resolution	Reference(s)
Regularization/Counterterms	TS/MR/DR + finite local correction	(Bastianelli et al., 2011)
Measure/Functional Space	Extension to singular/discontinuous	(Belokurov et al., 2011, Esposito et al., 29 Aug 2024)
Numerical Evaluation	Lattice MC, smooth paths, neural nets	(1201.00552509.16953)
Gauge/SUSY/Curved space quantization	Ghosts, BRST, worldline SUSY, KS2[H]	(1103.39932408.16404)
Complex actions/sign problem	Complex Langevin, thimble/eigenthimble	(Feldbrugge et al., 2022Kumar, 2023)
Hilbert space structure	Bilinear/bra-ket, antilinear operator	(Witten, 17 Mar 2025)
Advanced field geometry	RKHS, Banach, geometric field quant.	(Capobianco et al., 2015, Vatsya, 2014)

The Euclidean path integral formalism provides a mathematically transparent, physically motivated, and computationally flexible approach for defining quantum field theories and computing their observables. Its synthesis of operator and functional methods, adaptability to advanced field structures, and foundational role in lattice and continuum nonperturbative computations make it central in both theoretical developments and applied studies spanning high-energy, condensed matter, and mathematical physics.