Lorentzian Contour Prescription

• Lorentzian contour prescription is a rule for selecting complex integration paths that render amplitudes, path integrals, or correlators well defined by encoding the iε choice.
• It employs complexification and contour deformation techniques—such as Picard–Lefschetz theory and generalized Pochhammer contours—to manage oscillatory and divergent behavior.
• Its applications span quantum cosmology, string theory, and discrete gravity, with differing contour choices that directly impact physical interpretations and predictions.

A Lorentzian contour prescription is a rule for choosing an integration contour in a complexified variable so that a Lorentzian amplitude, path integral, modular integral, or correlator is well defined in the physical domain. In the literature, the relevant variable may be the lapse $N$, the long-tube parameter $y=\mathrm{Im}\,\tau$, the loop energy $k^0$, spatial loop momenta, or moduli-space coordinates. The common purpose is to implement the Lorentzian $i\epsilon$ choice, control oscillatory or noncompact integrals, select physically relevant saddles, and encode the correct branch structure at thresholds or degenerations (Feldbrugge et al., 2017, Dorronsoro et al., 2017, Wang, 16 Jun 2026, Eberhardt et al., 2024, Anselmi et al., 3 Mar 2025).

1. Definition and general structure

In the settings considered here, the prescription begins from a Lorentzian object rather than from a Euclidean one. In minisuperspace quantum cosmology, this means an integral of the form

$$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$

with $\mathcal C$ a contour in the complexified configuration space, including the lapse (Dorronsoro et al., 2017). In genus-one string theory, it means a modular integral whose non-separating degeneration must be treated by a Lorentzian continuation of the long proper time variable $y$ (Wang, 16 Jun 2026). In real-time holography, it means that a contour in the complex time plane is “filled in” by Lorentzian bulk segments on real branches and Euclidean bulk segments on imaginary branches, with matching conditions at the corners (0805.0150).

The prescription is therefore not a single formula. It is a family of constructions that share a common architecture: complexify the relevant variable, identify singularities or degeneration loci, deform the original Lorentzian contour without crossing forbidden singularities, and define the amplitude on the deformed contour. Depending on the problem, the deformed contour may be a Lefschetz thimble, a vertical tail in moduli space, a generalized Pochhammer contour, or a contour in the complex energy plane (Feldbrugge et al., 2017, Eberhardt et al., 2024, Eichmann et al., 2019).

A recurring distinction is between the original Lorentzian contour and its convergent representative. In Picard–Lefschetz theory, the original contour is decomposed into steepest-descent thimbles attached to critical points. In string modular integrals, the original semi-infinite cusp strip is replaced by a compact-domain integral plus a universal analytic tail. In amplitude theory with complex poles, a Minkowskian contour may require simultaneous deformation of both the energy contour and the spatial momentum domain (Anselmi et al., 3 Mar 2025).

2. Field-theoretic and geometric interpretations

One influential interpretation rewrites Feynman’s $i\epsilon$ prescription as a slight complex deformation of the Lorentzian metric. In flat space,

$\Delta_F(E,p)=\frac{i}{E^2-p^2-m^2+i\epsilon}$

can be written using

$$(\eta_\epsilon)^{ab}=\eta^{ab}-i\epsilon\,V^aV^b+O(\epsilon^2),$$

so that the causal prescription is encoded in an “almost real” complex metric rather than only in a contour for the energy variable (Visser, 2021). Visser extends this to curved spacetime and then to fluctuating geometries, where the relevant contour becomes a contour in the space of acceptable complex metrics satisfying the $y=\mathrm{Im}\,\tau$0-form, $y=\mathrm{Im}\,\tau$1-form, and $y=\mathrm{Im}\,\tau$2-form acceptability conditions. In that framework, real Lorentzian metrics lie on the boundary of the acceptable region, and the Lorentzian contour prescription is a deformation from that boundary into the interior of the acceptable region (Visser, 2021).

In scattering theory, the same issue appears in a more concrete analytic form. For Lorentz-invariant integral equations, branch cuts in complex invariant variables obstruct a naive Euclidean integration path. The scalar model of Bethe–Salpeter and scattering equations in Lorentz-invariant variables develops cuts in $y=\mathrm{Im}\,\tau$3, and the physical amplitude is obtained only after explicit contour deformations in those variables. The second Riemann sheet is then accessed directly by the two-body unitarity relation rather than by Euclidean data alone (Eichmann et al., 2019).

The problem becomes sharper in theories with complex poles. Four inequivalent amplitude prescriptions are identified: textbook Wick rotation by analytic continuation of the external momenta from Euclidean to Lorentzian signature, the Lee–Wick–Nakanishi prescription, the fakeon prescription, and direct Minkowski spacetime integration. The paper concludes that only the fakeon prescription is physically viable, because mixed Euclidean-Lorentzian prescriptions for internal and external momenta in loop integrals break Lorentz invariance, while other prescriptions forfeit either the optical theorem, analyticity, or power-counting renormalizability (Anselmi et al., 3 Mar 2025). In that setting, the Lorentzian contour prescription is not merely technical; it defines which quantum theory one obtains.

3. Minisuperspace quantum cosmology and lapse contours

In quantum cosmology, the key variable is the lapse $y=\mathrm{Im}\,\tau$4. The Lorentzian no-boundary wave function can be written as an integral over complexified lapse, and Picard–Lefschetz theory is used to deform the contour into a sum of Lefschetz thimbles (Dorronsoro et al., 2017). A central contour is

$y=\mathrm{Im}\,\tau$5

meaning the real axis from $y=\mathrm{Im}\,\tau$6 to $y=\mathrm{Im}\,\tau$7 with a detour below the pole at $y=\mathrm{Im}\,\tau$8. In the classical domain, the contributing saddles are the two saddles in the lower half-plane, and their sum yields a real WKB wave function that satisfies the Wheeler–DeWitt equation exactly (Dorronsoro et al., 2017). The same contour, with scalar matter included, leads to a real wave function describing an ensemble of asymptotically classical, inflationary universes with nearly Gaussian fluctuations (Dorronsoro et al., 2017).

A different Lorentzian starting point appears in the minisuperspace treatment that advocates the positive-lapse contour $y=\mathrm{Im}\,\tau$9 and then uses Picard–Lefschetz theory to determine the contributing thimbles. In that construction, the dominant no-boundary saddle gives a semiclassical factor that is precisely the inverse of the Hartle–Hawking result, so the Lorentzian contour selects a suppression $k^0$0 rather than the Euclidean enhancement $k^0$1 (Feldbrugge et al., 2017). The disagreement is not about a local sign alone; it comes from different original contours and different saddle selection rules.

A third proposal regularizes the lapse integral before any contour deformation. The vanishing initial size $k^0$2 is imposed through a Gaussian representation of the delta function,

$k^0$3

and the limit $k^0$4 is taken only after all functional integrations (Yamada, 12 Nov 2025). For fixed $k^0$5, the lapse integral is absolutely convergent along or slightly below the real axis, because

$k^0$6

This allows a “purely Lorentzian” contour for the tunneling wave function and a “nearly Lorentzian” contour from $k^0$7 to $k^0$8 passing below the singularity for the Hartle–Hawking wave function, while avoiding excursions into $k^0$9 and keeping perturbations Gaussian and suppressed (Yamada, 12 Nov 2025).

The same contour logic extends to higher-derivative minisuperspace models. In Lorentzian quantum cosmology with an $i\epsilon$0 correction, the path integral reduces, after gauge fixing and integrating fluctuations, to an ordinary integral over the constant lapse $i\epsilon$1, and Picard–Lefschetz theory again selects the relevant thimbles (Narain et al., 2019). In the perturbative regime analyzed there, the $i\epsilon$2 term shifts the saddle positions and modifies the saddle weights, but the qualitative contour structure inherited from Einstein gravity remains the same (Narain et al., 2019).

These constructions are also the locus of major controversy. One disputed issue is whether the physically correct contour must enter $i\epsilon$3 or remain on or below the real axis. Another is whether the no-boundary path integral should define a solution of the Wheeler–DeWitt equation or merely a Green’s function. The papers do not converge on a single answer, but they do agree that the choice of contour is part of the definition of the Lorentzian theory, not an afterthought (Dorronsoro et al., 2017, Yamada, 12 Nov 2025, Feldbrugge et al., 2017).

4. String theory: modular integrals and moduli-space contours

In genus-one string theory, the Lorentzian contour prescription enters through the cusp $i\epsilon$4 of the torus moduli space. In the Type IIB torus vacuum, the truncated modular domain is decomposed as

$i\epsilon$5

with $i\epsilon$6 the compact “keyhole” region and $i\epsilon$7 the semi-infinite strip (Wang, 16 Jun 2026). For a Fourier mode $i\epsilon$8, integration over the strip projects to diagonal modes by

$i\epsilon$9

so only $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$0 survives in the cusp. The Lorentzian $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$1 prescription is then implemented by deforming the long-tube variable $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$2 from the real interval into the contour

$$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$3

which is Euclidean up to height $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$4 and then vertical in the imaginary direction (Wang, 16 Jun 2026).

The crucial result is that, for $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$5,

$$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$6

so the Lorentzian tail is exactly the generalized exponential integral $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$7. The regularized modular block becomes

$$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$8

which is precisely the Manschot–Wang $$\Psi[h_{ij},\chi]\equiv \int_{\mathcal C}\mathcal D g\,\mathcal D\phi\; e^{iS[g,\phi]/\hbar},$$9-regularized block (Wang, 16 Jun 2026). In this formulation, the Lorentzian contour prescription and the $\mathcal C$0-regularized modular integral are two descriptions of the same cusp prescription. The paper applies this sector by sector to the unprojected Type IIB torus integrand, defines

$\mathcal C$1

and shows that the GSO-projected combination vanishes term by term because $\mathcal C$2 (Wang, 16 Jun 2026).

At genus zero, the contour problem appears on the moduli space of punctured spheres. Tree-level string amplitudes are usually defined on Euclidean moduli space, but physical scattering requires Lorentzian worldsheets and Minkowskian kinematics. The proposed solution is a compact Lorentzian contour on the complexified moduli space, built from generalized Pochhammer contours based on the combinatorics of associahedra (Eberhardt et al., 2024). For the four-point open-string amplitude, the Euclidean interval is replaced by a contour of the form

$\mathcal C$3

so the poles are moved from divergences of the integral to explicit factors $\mathcal C$4 (Eberhardt et al., 2024). For general $\mathcal C$5, the contour $\mathcal C$6 is assembled over the faces of the associahedron, and for closed strings the left- and right-moving contours are combined by a KLT kernel (Eberhardt et al., 2024). This makes the poles and factorization structure manifest directly in physical kinematics.

The same real-time logic appears in holography. In the real-time gauge/gravity prescription, a boundary contour in complex time is filled by Lorentzian AdS solutions on real segments and Euclidean AdS caps on imaginary segments, with continuity of fields and matching of conjugate momenta at the corners. In the scalar example, these matching conditions fix the contour in the complex $\mathcal C$7-plane to be exactly the Feynman contour and reproduce the correct $\mathcal C$8 insertions in the Lorentzian CFT two-point function (0805.0150).

5. Discrete Lorentzian gravity, entropy, and cases without contour deformation

Not every Lorentzian path integral requires a nontrivial contour prescription of the continuum type. A striking counterexample is the Lorentzian spinfoam path integral for entanglement entropy in covariant loop quantum gravity. There the amplitude is built from Lorentzian $\mathcal C$9 EPRL/KKL data and sums over a family of $y$0-complexes obtained by stacking faces on a root $y$1-complex (Han, 30 Oct 2025). The resulting entropy obeys a geometric area law,

$y$2

with $y$3 independent of the underlying $y$4-complexes, and reproduces the Bekenstein–Hawking formula for $y$5 after relating the coupling constant of the sum over $y$6-complexes to the Barbero–Immirzi parameter (Han, 30 Oct 2025). The paper explicitly states that this provides “a Lorentzian path integral approach to gravitational entropy without the need for contour prescriptions” (Han, 30 Oct 2025).

A different discrete Lorentzian example reaches the opposite conclusion. In a $y$7-dimensional simplicial minisuperspace model of de Sitter entropy, the path integral begins as a genuinely Lorentzian integral over real lapse, including both positive and negative lapse, but the action becomes multivalued when light-cone irregular configurations and contractible closed timelike curves appear (Dittrich et al., 2024). The contour must then be deformed in the complexified lapse-squared variable $y$8, and the fluctuation convergence criterion is used to select the allowed side of the branch cuts. In that model, the Lorentzian contour can be deformed onto a Euclidean saddle, and the dominant contribution comes from configurations with contractible closed timelike curves that encircle the boundary of the disc (Dittrich et al., 2024).

Lorentzian Regge calculus exhibits the same structural issue. Bulk edges can become arbitrarily large while boundary edges remain small, producing spikes, spines, and irregular light-cone structures. In the light-cone regular sector, the oscillatory Lorentzian contour is safe against the spike and spine divergences that plague Euclidean Regge calculus; in light-cone irregular sectors, however, the Regge action develops branch cuts along the Lorentzian contour, and one must choose the side of the branch cut that suppresses asymptotically large irregular configurations if one wants a meaningful Lorentzian path integral (Borissova et al., 2024).

These examples show that the necessity of a Lorentzian contour prescription is formulation-dependent. In continuum and simplicial settings with complexified proper times, degenerations, or irregular light-cone structures, contour data remain essential. In the spinfoam state-sum studied in loop quantum gravity, the integration cycle is non-singular and no contour ambiguity arises (Han, 30 Oct 2025).

6. Common themes, controversies, and open directions

Across these literatures, several structural features recur. First, the Lorentzian contour prescription is usually local to a specific degeneration or singular variable: the lapse $y$9, the long-tube modulus $i\epsilon$0, a loop-energy variable, or a boundary divisor in moduli space. Second, the prescription is constrained by physical principles that differ by context: the optical theorem in field theory with complex poles, the Wheeler–DeWitt equation in quantum cosmology, modular invariance and causal propagation in string theory, or factorization and KLT structure at tree level (Anselmi et al., 3 Mar 2025, Dorronsoro et al., 2017, Wang, 16 Jun 2026, Eberhardt et al., 2024).

A common misconception is to identify a Lorentzian contour prescription with a mere Wick rotation. The papers repeatedly reject that simplification. In gravity, Wick rotation is not a symmetry operation and does not by itself determine which saddles contribute (Feldbrugge et al., 2017). In field theory with complex poles, analytic continuation from Euclidean space can violate the optical theorem, while a purely Minkowskian prescription can violate power-counting renormalizability (Anselmi et al., 3 Mar 2025). In string theory, the Lorentzian contour at tree level is compact and combinatorial rather than a naive continuation of the real Euclidean integration region (Eberhardt et al., 2024).

A second controversy concerns analyticity. In ordinary local QFT, analyticity is tightly bound to contour deformation and causality. In the fakeon framework, by contrast, the physically viable prescription replaces ordinary analytic continuation by average continuation, preserving Lorentz invariance and the optical theorem at the cost of analyticity (Anselmi et al., 3 Mar 2025). This suggests that analyticity is not always the principle that fixes the Lorentzian contour.

The principal open directions are explicit in the cited work. Extending Picard–Lefschetz contour prescriptions from minisuperspace to full superspace remains open (Dorronsoro et al., 2017). Higher-genus string amplitudes require more complicated multi-cusp and multi-parameter contour constructions (Wang, 16 Jun 2026, Eberhardt et al., 2024). Lorentzian Regge and spinfoam formulations still leave open how broadly discrete Lorentzian path integrals can replace continuum contour prescriptions (Borissova et al., 2024, Han, 30 Oct 2025). A plausible implication is that “Lorentzian contour prescription” is best understood not as a single universal algorithm, but as a class of causality-preserving, convergence-preserving, and factorization-compatible rules adapted to the analytic geometry of each problem.