Mathematical foundations and validity boundary of the complex Langevin method for complex-valued path-integral weights

In 1983 Parisi proposed evading the sign problem of Euclidean path integrals by simulating the Langevin stochastic dynamics of a field theory after complexifying the fields, so that a complex (non-positive) action still drives a well-defined real stochastic process whose time averages formally equal the complex expectation values. The method is today the only technique producing nonperturbative results in four-dimensional QCD at high baryon chemical density -- presently limited to the deconfined phase for physical quark masses -- yet it is known to converge to wrong answers in certain regimes ('wrong convergence'), and its mathematical foundation has never been established. The open core of the complex Langevin program comprises: (i) existence and uniqueness (non-explosion) of the complexified Langevin process for all Langevin times; (ii) necessary and sufficient, checkable conditions on the action under which long-time averages provably equal the intended complex expectation values; (iii) an a priori characterization of which integration cycle a convergent process computes when the original integral diverges or vanishes; (iv) the boundary in parameter space (e.g. the complex coupling plane of $g\phi^4$ theory) separating correct convergence, instability (runaway trajectories and non-convergence), and wrong convergence (convergent but incorrect averages); and (v) reliable extension to real-time (Schwinger-Keldysh) path integrals beyond a few oscillation periods. Items (i)-(iv) are the open core of the program initiated by Parisi (1983); item (v) is the later real-time extension of the complex Langevin method, not an item of the 1983 paper. Necessary conditions, a-posteriori correctness criteria, and stabilization techniques (gauge cooling, kernels, thimble-inspired regularizations) are highly developed; the rigorous core remains open (Seiler 2018; Berger et al. 2021; Aarts and Sexty 2026).

Background

Monte Carlo evaluation of Euclidean path integrals rests on importance sampling, which requires a non-negative Boltzmann weight $e^{-S}$. In many central physical settings the action is complex or the weight is not positive: QCD at finite baryon chemical potential (the determinant of the Dirac operator becomes complex), Chern-Simons and theta-term theories (topological terms that make the Euclidean action imaginary or complex), and real-time (Minkowski) correlators, for which the Schwinger-Keldysh closed-time-path formalism is the standard tool. The cancellations between positive and negative (or oscillatory) contributions then make the signal-to-noise ratio decay exponentially with volume -- the sign problem, one of the defining obstacles of lattice field theory.

A seemingly formal escape was proposed by Parisi (1983), building on the stochastic quantization of Parisi and Wu (1981). For a real action $S$ one writes the field as a stochastic process obeying a Langevin equation, $d\phi = -\nabla S(\phi)\,dt + d\omega$, with $\omega$ a real Wiener process (white noise) of strength two, $\mathbb{E}[\omega(t)\,\omega(t')] = 2\min(t,t')$, the standard stochastic-quantization convention in which the stationary density is $e^{-S}$. The drift performs gradient descent on $S$ while the noise re-excites the field; the one-time probability density obeys a Fokker-Planck equation (the diffusion equation governing the probability density of a stochastic process) and relaxes to $e^{-S}$, so Langevin-time averages of observables equal path-integral averages. The real case is mathematically well understood: the Fokker-Planck Hamiltonian is self-adjoint and non-negative, and the process is ergodic under standard conditions. Parisi's observation was that nothing in the Langevin equation requires $S$ to be real if one complexifies the field, $\phi \to z = \phi + i\psi$ with $\phi,\psi$ real: since $S$ is holomorphic, the drift $-\nabla S(z)$ is well defined on the complexified configuration space, while the noise remains real (it only acts along the $\phi$ directions). Klauder (1983) independently applied Langevin dynamics to systems whose effective action is complex after integrating out fermionic degrees of freedom.

The resulting complex Langevin (CL) method is only formally justified. The density $p(\phi,\psi,t)$ of the process on the complexified space is non-negative and obeys a generalized Fokker-Planck equation, but the object that should reproduce the path integral is a reduced complex density $P(\phi,t)$, defined only implicitly by matching moments, and it is not a probability density: it is complex and may be singular. The formal derivation showing that CL time averages equal the complex expectation values assumes existence and uniqueness of the relevant $t\to\infty$ limits and of the stationary density. The generalized Fokker-Planck operator is neither self-adjoint nor normal, so none of the spectral theory that underwrites real stochastic quantization applies; Parisi derived a necessary spectral condition (non-negative real parts of the eigenvalues of the analytically continued Fokker-Planck Hamiltonian) and tested the method numerically on the quartic oscillator with complex coupling $g$, finding correct results for positive real $g$, 'rather confused' behavior for purely imaginary $g$, and convergent but incorrect answers ('wrong convergence') for $g$ with $\Re(g)<0$ and $\Im(g)\neq 0$.

Subsequent work confirmed and sharpened the danger. Hamber and Ren (1985) and Ambjørn and Yang (1985) showed that the naive method can converge to wrong answers, that trajectories can run away and reach $|z|=\infty$ at finite Langevin time, and that singularities (poles) of the drift in the complexified space are a mechanism of failure. A revival around 2005-2013 made the method practical: adaptive and implicit stepsize control tamed runaways; Aarts, Seiler and Stamatescu (2010) reduced the question of correctness to boundary terms at infinity and at singularities of the drift and posed the analytic questions (semigroup well-posedness, boundary terms, spectrum) in modern form; gauge cooling (Seiler, Sexty and Stamatescu 2013) stabilized QCD with heavy quarks at finite density by using the gauge symmetry to pull the complexified gauge field back toward the gauge orbit. A family of a-posteriori correctness criteria was developed and cross-compared: fulfillment of Schwinger-Dyson identities (the exact integration-by-parts identities that the expectation values produced by any correct method must satisfy), size of boundary terms, decay of the drift in the imaginary directions, histograms, and localization of the density in the imaginary directions (Aarts, Giudice and Seiler 2013; Salcedo 2016; Salcedo and Seiler 2019; Seiler, Sexty and Stamatescu 2024; Mandl 2026).

Two further lines shape the current frontier. First, the connection to Lefschetz thimbles (the manifolds of steepest descent of the weight $e^{-S}$ through its critical points in the complexified configuration space, along which the oscillatory weight becomes damped): when boundary terms vanish, CL computes a linear combination of integration-cycle (thimble) integrals (Salcedo and Seiler 2019; Hansen, Mandl, Seiler and Sexty 2025), which explains what convergent CL averages mean when the original real-contour integral diverges or its normalization vanishes; thimble-inspired weight regularizations extend the region of validity (Boguslavski, Hotzy and Müller 2025). Second, Mandl, Seiler and Sexty (2025) proved necessary and sufficient conditions for a given equilibrium density to yield correct expectation values -- a sharp characterization that can detect incorrectness but cannot certify correctness a priori. Meanwhile the real-time program (Alvestad, Larsen and Rothkopf 2021; Alvestad, Rothkopf and Sexty 2025) pushes CL onto Schwinger-Keldysh contours with kernel modifications (multiplicative deformation of drift and noise by a holomorphic matrix 'kernel' field), currently reaching only a few elementary oscillation periods (Aarts and Sexty 2026). What has never been obtained, in more than forty years, is the rigorous core: a theorem that the complexified process is well posed (non-explosive) and that, under checkable hypotheses on $S$, its long-time averages equal the intended complex expectation values -- or a rigorous characterization of when this must fail.

Problem Statement

Setup. Let $S: \mathbb{C}^N \to \mathbb{C}$ be a holomorphic action (e.g. a lattice or continuum field action with $N$ real components, complexified as $z = \phi + i\psi$), with formal weight $\rho(x) = e^{-S(x)}$ on the real submanifold $M = \mathbb{R}^N$ and formal complex expectation values

$$\langle O \rangle = \frac{\int_M O(x)\, e^{-S(x)}\, dx}{\int_M e^{-S(x)}\, dx}$$

for holomorphic observables $O$ (interpreted where the integrals converge; see part (c) otherwise). The complex Langevin equation is

$$dz_i = -\,\partial_i S(z)\, dt + d\omega_i, \qquad i = 1, \dots, N,$$

with $\omega$ a real $N$-dimensional Wiener process of strength two, $\mathbb{E}[\omega_i(t)\,\omega_j(t')] = 2\,\delta_{ij}\min(t,t')$ (the standard stochastic-quantization convention, under which the stationary density for a real action is $e^{-S}$): the drift is holomorphic, but the noise acts only along the real directions. In components, $d\phi_i = -\mathrm{Re}\,\partial_i S(z)\,dt + d\omega_i$ and $d\psi_i = -\mathrm{Im}\,\partial_i S(z)\,dt$. The one-time density $p(\phi, \psi, t) \ge 0$ on $\mathbb{R}^{2N}$ then obeys the generalized (degenerate) Fokker-Planck equation

$$\partial_t p = \sum_{i=1}^N \left[ \partial_{\phi_i}^2 p + \partial_{\phi_i}\!\left( \mathrm{Re}\,\partial_i S(\phi + i\psi)\, p \right) + \partial_{\psi_i}\!\left( \mathrm{Im}\,\partial_i S(\phi + i\psi)\, p \right) \right],$$

with no diffusion in the $\psi$ directions. The object that should reproduce the path integral is not $p$ but a reduced complex density $P(\phi, t)$ on the real submanifold, defined -- following Parisi -- only implicitly by matching its moments against the complex moments of the full density: for every holomorphic polynomial $O$,

$$\int_{\mathbb{R}^N} O(\phi)\, P(\phi, t)\, d\phi = \int_{\mathbb{R}^{2N}} O(\phi + i\psi)\, p(\phi, \psi, t)\, d\phi\, d\psi,$$

equivalently $\int P(\phi,t)\,\phi^{n} = \int p(\phi,\psi,t)\,(\phi+i\psi)^{n}$ for all multi-indices $n$ (interpreted in a weak, distributional sense where needed). $P$ is complex and need not be a probability density, and the existence of a solution of this infinite moment problem for all $t$ is itself part of what must be proved. Formal manipulations then suggest that $P$ obeys the ordinary complex Fokker-Planck equation on the real submanifold, with stationary solution proportional to $e^{-S}$, so that $\lim_{t \to \infty} P \propto e^{-S}$ and Langevin-time averages of $O(z(t))$ equal $\langle O \rangle$.

Problem (five linked parts; a complete resolution requires all five, and each part is an independently attackable sub-question):

(a) Well-posedness. Prove or disprove: under checkable hypotheses on $S$ (growth at infinity, pole/singularity structure, holomorphy domain), the complexified Langevin equation admits a unique solution for all $t \ge 0$ with probability one (no finite-time explosion), and the induced evolution of densities and observables defines well-specified semigroups on natural function spaces. Currently not even a partial theorem of this type is known.

(b) Correct convergence. Find necessary and sufficient, checkable-in-principle conditions -- analytic conditions on $S$, spectral conditions on the generalized Fokker-Planck operator, or decay conditions on the equilibrium density that are a-priori consequences of $S$ -- under which: the equilibrium density exists and is unique; all boundary terms (at infinity and at singularities of the drift) vanish; and Langevin-time averages of every holomorphic polynomial observable equal the formal complex expectation value $\langle O \rangle$, with convergence to the correct limit rather than a spurious one ('wrong convergence' excluded by hypothesis rather than detected after the fact).

(c) Interpretation. When the original real-contour integral diverges or its normalization vanishes ($\int_M e^{-S} = 0$), determine a priori which integration cycle (which linear combination of Lefschetz thimbles) a convergent complex Langevin process computes. The integration-cycle theorem answers this in principle once the process is known to be correct with vanishing boundary terms; the open part is the a priori prediction for realistic interacting theories, where unwanted cycles may contribute.

(d) Validity boundary. For the specified model families -- the complex-coupling quartic oscillator ($S$ with coupling $g \in \mathbb{C}$) and the heavy-dense QCD lattice theory at finite chemical potential (the limit of QCD in which the spatial hopping of the quarks is neglected) -- characterize, in each family's parameter space, the region in which the method is provably correct, the region in which it is unstable (runaway or non-convergent), and the region in which it converges incorrectly, separating within the last the part detectable by known a-posteriori criteria from the part undetectable by them.

(e) Real-time extension. Determine whether modified processes -- kernelled complex Langevin (drift and noise premultiplied by a holomorphic kernel), complex-noise deformations, contour deformations combined with CL -- can be made provably reliable for real-time (Schwinger-Keldysh) path integrals, including a controlled statement of the achievable extent in physical time beyond the few oscillation periods currently reachable.

All five parts share one obstruction: the process has non-restoring complex drift driven by degenerate (real-direction-only) noise, and its generator is non-self-adjoint and non-normal, so it falls outside existing stochastic-analysis theory.

References

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