Lorentzian Path-Integral Approach

• Lorentzian path-integrals are real-time functional integrations that explicitly retain the Lorentzian signature and causal structure in quantum gravity frameworks.
• The method uses complex contour deformations via Picard–Lefschetz theory to achieve convergence and a correct semiclassical limit, crucial for quantum cosmology and black hole analyses.
• Applications extend to false-vacuum decay, holography, and singularity resolution while addressing challenges like gauge ambiguities and conformal-mode instabilities.

The Lorentzian path-integral approach is a real-time functional integral formulation central to quantum gravity, quantum cosmology, semiclassical tunneling, gravitational thermodynamics, and quantum field theory in curved spacetime. Distinguished from the Euclidean approach by the explicit retention of Lorentzian signature and causal structure, the Lorentzian path integral provides a manifestly unitary and causal framework whose convergence properties and correct semiclassical limit are regulated via contour deformation in complexified configuration space, generally following Picard–Lefschetz theory. The Lorentzian method underpins recent advances in quantum cosmology, black hole stability, vacuum decay, and quantum gravitational entropy, and serves as the foundation for modern treatments of tensor networks, holography, and topology change.

1. Mathematical Formulation and Regularization

The Lorentzian gravitational path integral, written schematically as

$$Z = \int_{\mathcal C} \mathcal{D}g_{\mu\nu}\, e^{i S[g_{\mu\nu}]}$$

is a priori only conditionally convergent due to rapid phase oscillations from the $i$ in the exponent. In mini-superspace reductions (parametrized by a constant lapse $N$), this reduces to

$$G(q_1; q_0) = \int dN\, \int_{q(0)=q_0}^{q(1)=q_1} \mathcal{D}q\, e^{i S[q, N]}$$

with $S[q, N]$ the parametrized action (e.g., Einstein–Hilbert plus boundary terms in closed FLRW).

Absolute convergence is achieved by one of two strategies:

• Picard–Lefschetz theory: The original real-$N$ contour is deformed in the complex plane to a sum over "Lefschetz thimbles," each passing through a relevant saddle point with $\mathrm{Re}(iS)$ strictly decaying away from the saddle (Feldbrugge et al., 2017).
• Regulator insertion: A delta function $\delta(q_0)$ enforcing a vanishing initial scale factor is represented as a narrow Gaussian $\exp[-q_0^2/(2\sigma^2)]$; the $\sigma\to0$ limit is taken only after all path integrals are performed, ensuring quadratic suppression of the integrand for large $|N|$ without shifting the $N$-contour into the upper half-plane (Yamada, 12 Nov 2025).

For fixed $\sigma>0$, the lapse integral is absolutely convergent along (or just below) the real axis for all $N$ by exponential damping of the integrand, obviating the need for excursions into $\mathrm{Im}\,N>0$. This establishes rigorous existence of the Lorentzian path integral in mini-superspace or similar truncations.

2. Boundary Conditions, Initial States, and Saddle-Point Structure

The initial quantum state in the Lorentzian path integral is generally specified via a boundary condition or insertion. In quantum cosmology, the "creation from nothing" boundary is imposed by $\delta(q_0)$, regularized as a sharply peaked Gaussian. In Gauss–Bonnet and higher-derivative theories, Robin or mixed boundary conditions (linear combinations of scale factor and its momentum) can be implemented by appropriate boundary surface terms, allowing for an exact realization of the Hartle–Hawking or Vilenkin initial states (Ailiga et al., 23 Jul 2024, Narain, 2021, Narain, 2022).

The structure and number of relevant saddle points in the complex lapse plane are contingent on these boundary conditions and the properties of the underlying action. Saddle points can be purely Lorentzian, Euclidean, or genuinely complex, with their contributions determined by intersection numbers from Picard–Lefschetz analysis (Feldbrugge et al., 2017, Ghosh et al., 2023). For the no-boundary condition in closed FRW or related models, the relevant saddle yields either the tunneling (Vilenkin) or Hartle–Hawking wave function depending on the choice of lapse-contour.

3. Convergence and Suppression of Fluctuations

A central achievement of the Lorentzian path-integral approach is the robust suppression of inhomogeneous perturbations. For scalar and tensor modes, the quadratic path integral yields, after integrating over initial data weighted by the Bunch–Davies vacuum, a Gaussian-suppressed final wave function

$$\Psi_n(\chi_{n,1}) \propto \sqrt{\frac{\gamma_1}{2\pi}} \exp\left[ -\frac{1}{2} \gamma_1 \chi_{n,1}^2 \right] \exp\left[ -\frac{1}{2} \ln\left(\frac{f_*'(\eta_1)}{f_*'(\eta_0)}\right) \right]$$

with $\mathrm{Re}\,\gamma_1 = n > 0$, enforcing suppression (Yamada, 12 Nov 2025). The inclusion of a physically motivated initial vacuum for perturbations ensures that Lorentzian path-integral wave functions naturally avoid the unsuppressed growth of fluctuations observed when contours are deformed deep into $\mathrm{Im}\,N > 0$.

4. Semiclassical Limits, Tunneling vs. No-Boundary, and Emergence of Time

The semiclassical ($\hbar \to 0$) evaluation singles out dominant saddle points by stationary phase, yielding the tunneling or Hartle–Hawking results depending on the contour. For fixed $N > 0$, the relevant saddle lies in the lower half-plane, reproducing Vilenkin's tunneling exponent

$$\Psi_T(q_1) \propto \exp\left[ -\frac{4\pi^2}{H^2} (H^2 q_1 - 1)^{3/2} - i \cdots \right]$$

(Yamada, 12 Nov 2025). For contours traversing around the origin and picking up two complex-conjugate saddles (e.g., $N \in (-\infty, +\infty)$ below the pole), their interference yields the Hartle–Hawking wave function

$$\Psi_{HH}(q_1) \propto \exp\left[ +\frac{4\pi^2}{H^2} \right] \cos \left[ \frac{4\pi^2}{H^2} (H^2 q_1 - 1)^{3/2} \right]$$

with fluctuations still suppressed (Yamada, 12 Nov 2025, Ailiga et al., 23 Jul 2024, Narain, 2021). The Euclidean-to-Lorentzian crossover, governed by the position of the saddle(s) and the contour deformation, is entirely fixed by the geometry of the flow in the $N$-plane and is not an artifact of an ad hoc Wick rotation.

In extended and higher-derivative models such as Gauss–Bonnet, the topological term introduces nonperturbative weightings and can "lock in" the no-boundary initial condition via stationary-phase arguments on the Robin data (Ailiga et al., 23 Jul 2024). In some tunings, transition amplitudes can become independent of initial conditions up to fixed regularizing choices (Narain, 2022).

5. Broader Applications and Formulations

Beyond mini-superspace quantum cosmology, the Lorentzian path-integral approach has been adapted to a wide spectrum of problems:

• False-vacuum decay dynamics: The nucleation rate and tunneling amplitudes for critical and off-critical bubbles are computed by Lorentzian path integrals with real-time "bounce" solutions. Picard–Lefschetz prescription determines which complex lapse saddle (e.g., $N_s = -i\pi\rho_b/2$) contributes, yielding exponents in precise agreement with Euclidean instanton methods and generalizing to configurations inaccessible in purely Euclidean quantum field theory (Hayashi et al., 2021, Matsui, 2021).
• Quantum tunneling and WKB: The Lorentzian Picard–Lefschetz method directly generalizes WKB by summing over complex lapses that enforce the classical constraint, providing a consistent and semiclassically accurate wave function in both quantum mechanics and gravity (Matsui, 2021).
• Black-hole thermodynamics: The gravitational thermal partition function is constructed from the Lorentzian path integral. The stability and contribution of black-hole saddles are governed by the sign of the Hessian of the (singular) constrained surface-term action, encoding the standard thermodynamic criteria (heat capacity, moment of inertia, permittivity) (Chen, 14 Jan 2025).
• Gauge theory, entropy, and lattice/Regge gravity: Lorentzian path-integral techniques have been adapted to spinfoam models, three- and four-dimensional Regge calculus, and computations of geometric entropy without ambiguous conformal-factor pathologies (Han, 30 Oct 2025, Borissova et al., 2023, Dittrich et al., 4 Mar 2024, Han et al., 2013).
• Holography and quantum information: Lorentzian path-integral optimization and complexity are encoded via Hartle–Hawking wave functions in AdS/dS, providing a unified bridge between gravitational path integrals and the selection of tensor network geometries in dual CFTs (Boruch et al., 2021).

6. Implications for Singularities, Topology Change, and Nonlocal Theories

The Lorentzian path integral provides a dynamical mechanism for singularity resolution in quantum gravity. Singular configurations (e.g., classical black-hole metrics with divergent curvature) are excluded by destructive interference: in higher-curvature or nonlocal gravity actions, the action diverges on such metrics, leading to infinitely rapid phase oscillations that suppress their contribution (Borissova et al., 2020, Calcagni et al., 22 Feb 2024). For regular black holes, the action remains finite, and such configurations survive as quantum-corrected effective metrics.

The approach also makes rigorous sense of topology-changing amplitudes and wormholes; complexification of integration variables and the identification of contributing thimbles enables a precise accounting of contributions from Lorentzian, Euclidean, and genuinely complex saddles within a unified sum (Loges et al., 2022, Usatyuk, 2022). In Jackiw–Teitelboim gravity, the Lorentzian path integral admits degenerate metrics at isolated interaction points corresponding to topology change, with amplitudes governed by an analytic continuation of lightcone-diagram moduli and localized insertions in the dilaton action (Usatyuk, 2022).

7. Critical Assessment and Limitations

While the Lorentzian path-integral approach, when formulated with the appropriate convergence prescriptions, provides a conceptually and mathematically robust framework, several challenges and frontiers persist:

• Extension beyond mini-superspace: The Picard–Lefschetz machinery becomes infinitely dimensional in full field theory and is only partly controlled at this stage; infinite-dimensional thimble decompositions pose nontrivial functional-analytic challenges.
• Gauge and measure ambiguities: Nontrivial gauge-fixing (including Faddeev–Popov and Batalin–Vilkovisky structure) and the correct choice of path-integral measure (especially in discretized/Regge approaches) remain subtle, particularly in four dimensions (Borissova et al., 2023).
• Ambiguity in thimble summation: Physical interpretation of which combinations of saddles are to be retained ("tunneling vs. no-boundary") sometimes remains tied to boundary physics or initial data, and not uniquely fixed by internal path-integral structure.
• Conformal-mode stabilization and nonlocality: Nonlocal gravitational actions and suitable gauge choices are necessary to evade Euclidean conformal instability, a role naturally played by the oscillatory nature of the Lorentzian path integral (Calcagni et al., 22 Feb 2024).

The Lorentzian path-integral approach, by combining rigorous complex-contour methods, unitarity, causal structure, and broad physical applicability, now constitutes a central tool in the mathematical and conceptual foundations of quantum gravity, quantum cosmology, and semiclassical gravitational physics. Recent advances, notably in regularization schemes (Yamada, 12 Nov 2025), detailed contour analysis (Feldbrugge et al., 2017), and entropy derivations (Han, 30 Oct 2025), continue to extend its reach across theoretical physics.