Lorentzian Path Integral in QFT & Gravity

• Lorentzian Path Integral is a real-time functional integration method defined on Lorentzian spacetimes, using oscillatory exponentials and contour deformation.
• It employs Picard–Lefschetz theory to decompose oscillatory integrals into steepest-descent thimbles, ensuring conditional convergence in complex settings.
• Applications span quantum cosmology, black hole thermodynamics, and spinfoam models, linking semiclassical results with holographic insights.

The Lorentzian path integral is a foundational object in quantum field theory and quantum gravity, providing a real-time formulation of transition amplitudes, partition functions, and non-perturbative processes in both matter and gravitational systems. Its structure, measure, saddle-point analysis, and deformation theory have become particularly central in contemporary research, especially as new techniques clarify its relation to Euclidean counterparts and its foundational role in quantum cosmology, black hole thermodynamics, and holography.

1. Mathematical Structure and Definition

The Lorentzian path integral is a functional integral over fields (scalars, gauge fields, spinors, or metrics) defined on a spacetime manifold with Lorentzian signature, typically of the form

$$Z = \int_\mathcal{C} \mathcal{D}\phi\, e^{i S[\phi]}.$$

Here, $S[\phi]$ is the Lorentzian action, and the contour $\mathcal{C}$ specifies a real or complexified domain of integration for the fields. For quantum gravity, the central object is the gravitational path integral over Lorentzian metrics $g_{\mu\nu}$, possibly supplemented by matter fields and subject to boundary conditions or topological specifications: $$Z = \int_\mathcal{C} \mathcal{D}g_{\mu\nu}\, \mathcal{D}\text{(matter)}\, e^{i S[g, \text{matter}]}.$$ Unlike its Euclidean counterpart, the Lorentzian integral is oscillatory rather than exponentially suppressed, which leads to conditional but not absolute convergence. As a result, one must specify both the integration domain and the prescription for complexification and contour deformation.

In canonical (mini)superspace reductions, e.g., for quantum cosmology, the Lorentzian path integral is often reduced to an ordinary integral over the lapse function $N$, with boundary data on the geometry (such as the scale factor and its conjugate momentum): $$G(q_1, q_0) = \int_0^\infty dN \int_{q(0)=q_0}^{q(1)=q_1} \mathcal{D}q\, e^{i S[q,N]}.$$ This formalism is adapted and extended in gauge theories, gravity models, lattice and simplicial quantum gravity, and spinfoam models (Chen, 14 Jan 2025, Han, 30 Oct 2025, Honda et al., 2024, Held et al., 14 Jan 2026, Usatyuk, 2022, Han et al., 2013).

2. Contour Deformation and Picard–Lefschetz Theory

To resolve the oscillatory nature and possible divergences, the Lorentzian path integral is often viewed as a contour integral in complexified field space, employing Picard–Lefschetz theory to decompose the original, real integral into a weighted sum over "Lefschetz thimbles"—steepest-descent cycles attached to complex saddle points: $$\int_{\mathbb{R}^n} e^{i S(x)} dx = \sum_\sigma n_\sigma \int_{\mathcal{J}_\sigma} e^{i S(z)} dz.$$ Each thimble $\mathcal{J}_\sigma$ is defined by downward flow from a critical point (saddle) of $iS$. This approach yields absolutely convergent integrals on each thimble and allows one to identify the relevant, physically contributing saddles (Feldbrugge et al., 2017, Held et al., 14 Jan 2026, Asante et al., 2021, Hayashi et al., 2021).

Key results include:

• Selection of Dominant Saddles: Only those thimbles whose upward flow intersects the original real contour contribute. In cosmology, this determines which semiclassical geometries describe, e.g., the "no-boundary" or "tunneling" wavefunctions.
• Handling of Oscillations: Along each thimble, $\text{Re}(iS)$ decreases, so the integral converges even for highly oscillatory actions.
• Contour Ambiguities: The prescription resolves ambiguities in the Lorentzian-to-Euclidean transition and in the treatment of causality-violating or conically singular configurations (Asante et al., 2021, Borissova et al., 2023, Dittrich et al., 2024).

3. Applications in Quantum Cosmology and Gravity

3.1 Quantum Tunneling and WKB

The Lorentzian path integral embodies the real-time analogue of the WKB/instanton formulation of quantum tunneling. Saddle-point analysis via Picard–Lefschetz theory recovers conventional WKB suppression factors, with agreement between the Lorentzian and Euclidean exponents. This is established at both the level of quantum mechanics (Matsui, 2021) and in quantum cosmology models (Feldbrugge et al., 2017, Honda et al., 2024, Yamada, 12 Nov 2025, Ghosh et al., 2023).

3.2 Quantum Cosmology and No-Boundary Proposal

In (mini)superspace gravity with boundary conditions motivated by the Hartle–Hawking no-boundary proposal, the Lorentzian path integral yields, via proper contour deformation, the correct semiclassical weighting for the universe's wave function. For example, in 2D Jackiw–Teitelboim (JT) gravity, the path integral reduces to a Bessel or Hankel function, corresponding to a unique thimble passing through the relevant (often "Euclidean") saddle (Honda et al., 2024).

In four-dimensional models, the analysis shows that:

• For small final size, the dominant saddle is Euclidean and leads to exponential growth (inverse Hartle–Hawking behavior).
• For large final size, Lorentzian oscillating saddles dominate, representing classical propagation (Feldbrugge et al., 2017, Ghosh et al., 2023, Narain, 2021, Ailiga et al., 2024).
• Certain boundary conditions (e.g., Robin or Neumann) select well-behaved, perturbatively stable initial conditions and uniquely determine the path-integral measure (Ailiga et al., 2024, Narain, 2021, Narain, 2022).

3.3 Gravitational Thermodynamics and Black Holes

The gravitational thermal partition function in the Lorentzian formalism is regulated and its saddle-point contributions correspond to black hole geometries, with stability criteria determined directly by thermodynamic susceptibilities (specific heat, etc.) of the configuration. Black hole saddles contribute if and only if they are stable to variations in conserved charges (Chen, 14 Jan 2025).

Codimension-2 singularities, including conical and gauge holonomy defects, are incorporated into the action and contour structure. The formalism generalizes naturally to 3D gravity (e.g., BTZ black holes) and enables systematic inclusion of angular momentum and charge (Chen, 14 Jan 2025).

3.4 Semi-Classical Wormholes and Quantum Chaos

The Lorentzian path integral, especially in JT gravity coupled to imaginary scalars, admits wormhole saddles whose dominance depends on the analytic structure of the action and boundary conditions. Recent work demonstrates that while standard Euclidean axion wormholes are subdominant in Lorentzian AdS/CFT, JT wormholes with imaginary scalar boundary conditions do dominate the connected two-boundary partition function—mirroring complex-coupling physics in the Sachdev-Ye-Kitaev (SYK) model (Held et al., 14 Jan 2026). Stokes phenomena in the complexified contour govern the onset of this dominance.

3.5 Topology Change and Lightcone Diagrams

Lorentzian path integrals in JT gravity have been extended to sum over metrics allowing topology change, modeled on lightcone diagrams and inspired by string theory techniques. The Lorentzian genus expansion is realized as an analytic continuation of the Euclidean expansion, making topology-changing processes manifest and relating directly to phenomena such as baby universes and universality classes in random matrix models (Usatyuk, 2022).

4. Measure, Rigorous Formulation, and Stochastic Approaches

Defining a rigorous measure is complicated by the need to incorporate gauge symmetries, noncompactness, and the indefinite signature of the action. Approaches include:

• Stochastic Quantization: The path integral is recast as a Gaussian measure on $L^2$-flux Hilbert spaces generated by diffusive processes near classical solutions. This method provides a probabilistic construction equivalent to both Euclidean and Lorentzian definitions via analytic continuation and Feynman–Kac theory (Obolenskiy, 19 Nov 2025).
• Functional Measures: In nonlocal and higher-derivative gravity theories, invariant measures are constructed with careful gauge fixing and Faddeev–Popov determinants (Calcagni et al., 2024). The conformal-factor instability of the Euclidean measure is cured by appropriate Lorentzian gauge choices.

In discretized (Regge or spin foam) gravity, measure invariance under triangulation changes (e.g., via Pachner moves) is achieved in 3D, while algebraic obstructions arise in 4D, reflecting the presence of local degrees of freedom (Borissova et al., 2023).

5. Discrete and Spinfoam Realizations

Spinfoam and Regge calculus approaches provide concrete definitions of the Lorentzian path integral:

• Lorentzian Regge Calculus: Edge-length variables and dihedral angles are analytically continued into the complex plane; branch-cut structure and orientation/time-orientation determine the correct critical points and asymptotic form, yielding the Regge action at semiclassical level (Asante et al., 2021, Borissova et al., 2023).
• Spinfoam Models: The Lorentzian EPRL model is formulated as a group-theoretic sum over $SL(2,\mathbb{C})$ representations. At globally oriented and time-oriented critical points, the leading asymptotics recover $e^{i S_{\text{Regge}}}$, while nonphysical configurations give subleading or anomalous phases (Han et al., 2013, Han, 30 Oct 2025). These models bypass the need for contour prescriptions or Wick rotation, and afford combinatorial "replica" constructions for entanglement entropy computations.

6. Physical Interpretation, Fluctuations, and Quantum Corrections

Fluctuations around saddle points yield one-loop determinants and Gaussian prefactors. Criteria for the validity and dominance of specific saddles include:

• Thermodynamic and Perturbative Stability: Only those saddles stable under fluctuations (i.e., with positive-definite real part of the quadratic action) contribute dominantly (Chen, 14 Jan 2025, Held et al., 14 Jan 2026).
• Suppression of Pathological Modes: In quantum cosmology, contour choices affect whether scalar and tensor perturbations are suppressed or unsuppressed at late times; properly regulated (or Robin/Neumann) boundary conditions select Gaussian suppression and physically stable solutions (Honda et al., 2024, Yamada, 12 Nov 2025).
• Topological and Non-Perturbative Effects: The presence of topological terms (e.g., Gauss–Bonnet) can introduce additional, non-perturbative saddle points whose interference encodes genuinely quantum gravitational effects (Narain, 2021, Narain, 2022).

7. Outlook and Connections to Broader Frameworks

The Lorentzian path integral formalism, clarified via saddle-point and contour deformation methods, is central to ongoing efforts in nonperturbative quantum gravity, quantum cosmology, and black hole thermodynamics. Its synthesis of first-principles functional integration, rigorous measure constructions, combinatorial models (spin foam, Regge), and holographic dualities provides a unified language connecting semiclassical gravity, gauge/gravity duality, and statistical mechanics.

The sensitivity of Lorentzian wormhole contributions to analytic continuation and boundary data demonstrates the subtlety of nonlocal and topological quantum gravity effects, with implications for the microscopic origin of gravitational entropy, quantum chaos, and the structure of the gravitational and holographic path integral (Held et al., 14 Jan 2026, Han, 30 Oct 2025, Usatyuk, 2022). Current research continues to explore connections to universality classes, replicate statistical ensembles, and matrix-model descriptions, as well as computational techniques for higher genus and topology-changing amplitudes.

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Relevant references:

• "Lorentzian Path Integrals and Jackiw-Teitelboim wormholes with imaginary scalars" (Held et al., 14 Jan 2026)
• "Thermodynamic stability from Lorentzian path integrals and codimension-two singularities" (Chen, 14 Jan 2025)
• "Jackiw-Teitelboim Gravity and Lorentzian Quantum Cosmology" (Honda et al., 2024)
• "Lorentzian path integral for quantum tunneling and WKB approximation for wave-function" (Matsui, 2021)
• "A Stochastic Approach to the Definition of the Path Integral Measure" (Obolenskiy, 19 Nov 2025)
• "De Sitter horizon entropy from a simplicial Lorentzian path integral" (Dittrich et al., 2024)
• "Complex actions and causality violations: Applications to Lorentzian quantum cosmology" (Asante et al., 2021)
• "Lorentzian Quantum Cosmology" (Feldbrugge et al., 2017)
• "Vacuum decay in the Lorentzian path integral" (Hayashi et al., 2021)
• "Lorentzian path integral in Kantowski-Sachs anisotropic cosmology" (Ghosh et al., 2023)
• "On Gauss-bonnet gravity and boundary conditions in Lorentzian path-integral quantization" (Narain, 2021)
• "Lorentzian Robin Universe of Gauss-Bonnet Gravity" (Ailiga et al., 2024)
• "Path Integral Representation of Lorentzian Spinfoam Model, Asymptotics, and Simplicial Geometries" (Han et al., 2013)
• "Lorentzian spinfoam gravity path integral and geometrical area-law entanglement entropy" (Han, 30 Oct 2025)
• "Complex Saddles and Euclidean Wormholes in the Lorentzian Path Integral" (Loges et al., 2022)
• "Comments on Lorentzian topology change in JT gravity" (Usatyuk, 2022)
• "A new regularization scheme for the wave function of the Universe in the Lorentzian path integral" (Yamada, 12 Nov 2025)
• "Lorentzian quantum gravity via Pachner moves: one-loop evaluation" (Borissova et al., 2023)
• "Path integral and conformal instability in nonlocal quantum gravity" (Calcagni et al., 2024)
• "Surprises in Lorentzian path-integral of Gauss-Bonnet gravity" (Narain, 2022)