Lorentzian Replica Framework

• The Lorentzian Replica Framework is a real-time approach that computes von Neumann entropy using n-replica path integrals without relying on Euclidean methods.
• It employs cyclic gluing of Lorentzian contours and introduces codimension-2 conical defects to construct replica wormhole saddles and derive the island rule.
• Methodologies include extremizing generalized entropy functionals and solving Lorentzian saddle-point equations to capture topology change and unitarity in quantum gravity.

The Lorentzian Replica Framework is a set of theoretical constructions and methodological advances that enable computation of entanglement entropy and analysis of replica wormhole saddles entirely in Lorentzian (real-time) spacetime signature, avoiding reliance on Euclidean path integrals or time-reflection symmetry. Originally motivated by the need to describe unitarity and information recovery in evaporating black holes—expressed by the Page curve—the framework has been developed for Jackiw–Teitelboim (JT) gravity, higher-dimensional cosmologies, quantum Regge calculus, and supersymmetric models. Essential operations include gluing branched Lorentzian contours, introducing codimension-2 conical defects, extremizing generalized entropy functionals, and constructing wormhole topologies via local saddle-point equations. The approach systematically generalizes the Euclidean replica trick, yielding the quantum extremal surface (QES) and island rules in a covariant, real-time setting.

1. Lorentzian Replica Path Integrals and Entropy Computation

In the Lorentzian framework, the von Neumann entropy of a subsystem $A$,

$$S_A = -\mathrm{Tr}(\hat\rho_A\ln\hat\rho_A) = -\lim_{n\to1}\frac{\partial}{\partial n}\ln \mathrm{Tr}\,\hat\rho_A^n,$$

is calculated by constructing a real-time $n$-replica path integral. The unnormalized trace is represented by integration over an $n$-fold Lorentzian “bra–ket” geometry $\mathcal{M}_n$ with action $I_n[g,\Phi]$ (including metric and matter fields),

$$Z[n]\equiv\int_{\mathcal{M}_n}\!{\mathcal D}g\,{\mathcal D}\Phi\;\exp\bigl(i\,I_n[g,\Phi]\bigr),\qquad S_A = -\left.\frac{\partial}{\partial n}\ln Z[n]\right|_{n=1},$$

as demonstrated explicitly for time-dependent backgrounds (Irakleous, 13 Jan 2026) and in JT gravity (Goto et al., 2020). Unlike the Euclidean approach, this construction proceeds without analytic continuation or time-reflection symmetry, thereby generalizing the applicability of the replica trick to arbitrary Lorentzian geometries including cosmologies or boundaryless settings.

2. Gluing Procedures, Conical Defects, and Topology Change

The replica manifold $\mathcal{M}_n$ is formed by cyclically gluing $n$ real-time contour segments along the chosen region $A$ at their future and past ends, leaving the complement $\bar A$ unglued. This induces codimension-2 conical defects at the fixed points (e.g., the endpoints of $A$) and, in the case of replica wormholes, swap-identifications at extremal surfaces (crotches) (Blommaert et al., 2023). The local metric near a crotch assumes singular behavior, introducing delta-function curvature,

$\sqrt{g}\,R\supset-4\pi\,\delta(x-x_{\rm sing}),$

and altering the Euler characteristic of the spacetime. Extremizing the location of the crotch with respect to the area functional places it on a quantum extremal surface (Blommaert et al., 2023). The construction of wormhole topologies is thus implemented entirely in Lorentzian signature via swap cuts and instanton-like crotches, generalizing the Lewkowycz–Maldacena Euclidean procedure.

3. Lorentzian Saddle-Point Equations and Entropy Functionals

At the saddle points, the Lorentzian equations of motion are enforced throughout the bulk, except for distributional curvature sourced at the defects. Junction conditions analogous to Israel’s formula are imposed to maintain continuity across the defect except where the localized curvature appears (Irakleous, 13 Jan 2026). For semiclassical gravity (e.g., JT gravity with a CFT), this leads to boundary particle equations of the (deformed) Schwarzian type,

$$\partial_t\{x_n(t),t\} + \kappa_n\,(\{x_n,t\}+1/2) = \text{matter flux} \sim \sum_{\mathrm{defects}}\delta(t-t_{\rm def}),$$

with $x_n(t)=e^{-i\,w_n(it)}$ and $\kappa_n=(c/24\pi)(8\pi G_N/\phi_r)$ (Goto et al., 2020). In the $n\to1$ “no-mixing” limit, the equations localize, and the dilaton profile and defect location enforce the quantum-extremal-surface condition,

$$\partial_X\Bigl[\frac{\phi(X)}{4G_N} + S_{\rm QFT}(I\cup R)\Bigr] = 0,$$

yielding the island rule and enabling computation of the Page curve.

4. Generalized Entropy, Replica Wormholes, and the Island Rule

The Lorentzian replica approach establishes a direct correspondence between on-shell gravitational action contributions from codimension-2 defects or crotches and entropy terms. Explicitly,

$$S_A = \frac{\mathrm{Area}(\chi)}{4G_N} + S_{\rm matter}(\Sigma_A),$$

where $\chi$ is the extremal surface at the defect, and $\Sigma_A$ is the entanglement wedge. Allowing for wormhole topologies (i.e., gluing through an additional codimension-1 surface $I$ or “island”) modifies the entropy functional,

$$S_A = \frac{\mathrm{Area}(\chi\cup\partial I)}{4G_N} + S_{\rm matter}(\Sigma_A\cup I),$$

with minimization and extremization reproducing the island rule (Goto et al., 2020, Irakleous, 13 Jan 2026). The formalism naturally generalizes to time-dependent, non-static backgrounds and accommodates disconnected islands and replica wormhole saddles. In the case of quantum Regge calculus (Padua-Argüelles, 25 Apr 2025), the conical defect contribution is recovered discretely from the lattice action.

5. Applications: Black Hole Evaporation, Cosmology, and SYK Models

The Lorentzian replica framework has established the detailed mechanism of entropy recovery and unitarity in evaporating black holes, yielding the unitary Page curve via QES/island saddles (Goto et al., 2020). Extension to cosmological settings (e.g., de Sitter bilayers, closed FRW universes) allows for entropy calculations without boundaries or holographic duals, under fully dynamical conditions (Irakleous, 13 Jan 2026). In quantum Regge calculus, the framework provides a discrete, triangulation-based realization of replica wormholes and the Page curve including all challenges of saddle selection, measure ambiguities, and complex contour deformation (Padua-Argüelles, 25 Apr 2025).

In supersymmetric Sachdev-Ye-Kitaev (SYK) models, the “multi-ordered trick” covers $n$-replica structure via superconformal interactions, reducing higher-order wormhole couplings to super-Schwarzian boundary actions in the deep infrared (Ge et al., 13 Aug 2025). Real-time Dyson–Keldysh equations for bilocal super-Green’s functions establish black hole and wormhole phases, and numerical evaluation confirms the entanglement capacities and modular entropies expected for replica wormhole geometries.

6. Gauge Equivalence and Relation to Euclidean Approaches

Inclusion of Lorentzian crotch or defect geometries in the gravitational path integral has been argued to be gauge-equivalent to path integration over all (mostly) Euclidean smooth spacetimes (Blommaert et al., 2023). Explicit semiclassical computations of spectral form factors, late-time two-point functions, and Page curves via Lorentzian wormhole saddles reproduce matrix-model and Euclidean results. This equivalence arises from changing integration contours or gauge choices in the metric space:

• Gauge A: (mostly) Euclidean, smooth metrics.
• Gauge B: (almost) everywhere Lorentzian metrics with $\sqrt{g}=0$ on crotches.

This suggests that Lorentzian methods provide a complete alternative perspective on topology change and unitarity in quantum gravity.

7. Challenges, Extensions, and Future Directions

Technical limitations persist regarding localization (valid mainly in the $n\to1$ limit and “no-mixing” regime), generalization to interacting matter and higher dimensions, and numerical simulation for full two-interval or multi-wormhole topologies. Quantum Regge calculus faces ambiguities in measure choice, treatment of boundaries, and specification of admissible complex metrics. Prospects for extending Lorentzian replica formulations to spinfoams and loop quantum gravity suggest a path toward nonperturbative, discrete approaches to black hole information recovery (Padua-Argüelles, 25 Apr 2025). In supersymmetric models, the challenge is to systematize higher-order modular entropies and confirm holographic matching at all replica orders (Ge et al., 13 Aug 2025).

A plausible implication is that as more refined triangulations and fully Lorentzian, boundaryless frameworks are realized, the Lorentzian replica paradigm will establish the foundational mechanism behind unitary quantum gravitational evolution, encompassing entropy dynamics in black holes, cosmologies, and strongly coupled quantum systems.