Schwinger-Keldysh EFTs: Real-Time Dynamics

• Schwinger-Keldysh EFTs are a framework that uses a closed-time path contour to derive real-time correlation functions for systems out of equilibrium.
• They systematically incorporate fluctuation-dissipation theorems and KMS symmetries to capture both unitary evolution and dissipative processes in a unified formalism.
• The formalism is pivotal for applications in cosmology, condensed matter, and relativistic hydrodynamics, enabling precise modeling of nonlocal memory effects and stochastic dynamics.

Schwinger-Keldysh Effective Field Theories (EFTs) provide a rigorous formalism for describing the dynamics of quantum and statistical systems out of equilibrium, systematically incorporating both unitary evolution and dissipative effects. Building on a closed-time-path contour (the Schwinger-Keldysh or in-in contour), these EFTs enable the derivation of real-time correlation functions—critical for nonequilibrium settings such as thermalization, quantum transport, and time-dependent cosmological backgrounds. Unlike traditional, equilibrium-focused S-matrix or Euclidean frameworks, Schwinger-Keldysh EFTs naturally encode fluctuation-dissipation theorems, dynamical Kubo-Martin-Schwinger (KMS) symmetries, and the effects of stochastic and quantum noise.

1. Schwinger-Keldysh Formulation in Time-Dependent Quantum Field Theory

The in-in (Schwinger-Keldysh, SK) path integral formalism is essential whenever the system of interest is prepared in a general density matrix (not simply the ground-state or eigenstate of the interacting Hamiltonian) and evolves out of equilibrium, as is ubiquitous in cosmological, condensed matter, or open quantum system contexts. The generating functional is defined by integrating over field configurations on a closed time contour running from an initial time $t_0$ forward to $+\infty$ and then backward, with two field copies $\Phi^+$ and $\Phi^-$. The SK generating functional reads

$$Z[J^+, J^-] = \int \mathcal{D}\Phi^+ \mathcal{D}\Phi^- \; \exp \Big\{ i (S[\Phi^+] - S[\Phi^-]) + i \int (J^+\Phi^+ - J^-\Phi^-) \Big\}$$

Expanding the fields about a time-dependent classical background $\phi(t)$ and imposing the “tadpole” condition $\langle \varphi^+(t,\mathbf{x}) \rangle = 0$, one derives an equation of motion for $\phi(t)$ incorporating all quantum (loop) corrections. This structure is crucial in situations where time-translation invariance is explicitly broken—e.g., due to interactions being turned on at a finite $t_0$, or in cosmological backgrounds where the vacuum is not the interacting one but is specified at some initial time (Collins et al., 2012).

2. Nonlocality and Boundary Effects in Time-Dependent Effective Actions

In generic SK EFTs, particularly with heavy and light field sectors, loop corrections from heavy fields lead to temporally nonlocal terms in the effective dynamics of light fields. The one-loop equation of motion for the light field $\phi(t)$ includes nonlocal integrals of the form

$$\ddot{\phi}(t) + m^2 \phi(t) + \frac{\lambda}{6} \phi^3(t) - \frac{1}{2} g^2 \phi(t) \int_{t_0}^t dt' \, \phi^2(t') \int \frac{d^3k}{(2\pi)^3} \frac{\sin(2\omega_k(t-t'))}{\omega_k^2} + \ldots = 0$$

After integrating by parts, one obtains kernels

$$K(t-t') \equiv \int \frac{d^3k}{(2\pi)^3} \frac{\cos[2\omega_k(t-t')]}{\omega_k^3}$$

These kernels produce boundary (initial time) terms such as $K_{3}^{+}(t-t_0) \phi^2(t_0) \phi(t)$ and $K_{4}^{-}(t-t_0)\dot{\phi}(t_0)\phi^2(t_0)\phi(t)$, which cannot arise in local, time-translation invariant EFTs (Collins et al., 2012). The appearance of such time-nonlocal and initial-value-dependent terms is a defining feature of SK EFT in out-of-equilibrium or nonstationary backgrounds.

Asymptotically, for late times $M(t-t_0) \gg 1$, these kernels decay as $[M(t-t_0)]^{-3/2}$ with oscillatory nonanalytic behavior, a consequence of the IR properties of the heavy-mode loop integrations. In the adiabatic limit $t_0 \to -\infty$, or for a very smooth turn-on of couplings, such nonlocal boundary effects vanish, and one recovers the standard local EFT expansion in higher-derivative operators.

3. Structure of the Schwinger-Keldysh Effective Action in Hydrodynamics and Transport

SK EFTs have been extensively employed to systematically classify allowed transport phenomena, especially in relativistic hydrodynamics. The SK construction directly implements unitarity via the condition $|Z| \leq 1$, which results in a nontrivial positivity constraint (SK-positivity) on the imaginary part of the effective action and therefore on the dissipative transport coefficients (Jensen et al., 2018).

The functional integral structure includes auxiliary “ghost” and “a-type” (difference) fields. The formalism can be promoted to a superspace description where the fundamental symmetry requirements—target-space diffeomorphisms, flavor symmetries, BRST-like (Q) symmetries, and dynamical KMS symmetry—are manifest. All allowed tensor structures (e.g., for the stress tensor and conserved currents) are classified according to their transformation under these symmetries, and only those compatible with SK-positivity and KMS symmetries are retained.

Explicitly, dissipative terms are encoded in the imaginary (noise) sector of the action, while non-dissipative (including anomaly-induced) terms reside in the real part. For example, the inclusion of ’t Hooft anomaly inflow is realized via additional Chern–Simons terms, whose supersymmetrization is necessary to preserve SK/KMS-consistent structure (Jensen et al., 2018).

4. Holographic Duals and SK Effective Actions for Dissipative Dynamics

Modern developments have constructed explicit holographic duals of the Schwinger-Keldysh effective action, leveraging the structure of eternal AdS black holes with Euclidean “caps” to mimic the SK time contour in the bulk (Boer et al., 2018). By matching conditions across Lorentzian and Euclidean caps in the bulk spacetime, one reproduces at the boundary the SK doubling, thermal initial conditions, and all symmetries (including KMS).

Key steps in the holographic construction:

• The boundary SK sources correspond to values of bulk gauge fields at the two asymptotic boundaries.
• Wilson lines in the bulk encode the “Goldstone” nature of the hydrodynamic variables; their transformation properties directly implement SK-type and KMS symmetries.
• After solving bulk equations of motion and patching the solutions across the various regions, the on-shell gravitational action yields the full quadratic (and in principle higher-order) SK effective action for charge transport or diffusion, including the fluctuation-dissipation structure and the correct analytic properties (retarded, symmetric, and advanced correlators).

Importantly, the near-horizon (membrane) region admits an IR effective action that encodes universal dissipative behavior—this functionally replaces the imposition of ingoing boundary conditions with a dynamical worldvolume action for dissipation (Boer et al., 2018).

5. Non-Gaussian Fluctuations, Stochastic Dynamics, and Higher-Order Structure

Beyond Gaussian fluctuations, SK EFTs systematically incorporate non-Gaussian noise and interactions—e.g., cubic and quartic terms in the “a-type” (fluctuation) fields generate non-Gaussian stochastic corrections, which are equivalently encoded in multiple-noise stochastic Langevin equations or in higher-derivative (fourth or higher order) Fokker-Planck operators (Lin et al., 2023). These non-Gaussianities are statistically relevant for systems far from the thermodynamic limit or subject to nonlinearities in effective couplings.

There exists an equivalence between the full SKEFT, a non-Gaussian Langevin description (with multiple, possibly non-positive-definite noise terms), and higher-order Fokker–Planck equations, provided appropriate ordering/prescription is maintained. Nontrivial moments of observables (e.g., higher-point correlators) generated by the fundamental SK action are exactly reproduced in these stochastic descriptions.

A subtlety arises when both the noise discretization scale and non-Gaussianity parameters are taken to zero simultaneously: an ambiguity (potential divergence) can occur, highlighting the necessity of proper order of limits in stochastic simulations and perturbative expansions (Lin et al., 2023).

6. SK Effective Field Theories in Hydrodynamics: Causality, Stability, and UV Sectors

The construction of SK EFTs for relativistic hydrodynamics faces stringent physical requirements absent from classical phenomenological equations. For diffusion and hydrodynamic models, simple first-order (Fick’s law or Navier-Stokes) descriptions are acausal and often unstable when subjected to stochastic fluctuations or considered in a relativistic context.

Resolution requires UV-completing the theory by adding gapped (“non-hydrodynamic”) degrees of freedom, such as in the Maxwell–Cattaneo or Müller–Israel–Stewart models. The SK action for such causal theories includes both the standard hydrodynamic sector and additional fields ensuring all real-time n-point correlation functions are stable and causal, while also manifestly realizing nonlinear dynamical KMS symmetry (Jain et al., 2023, Liu et al., 25 Nov 2024). In the classification of the effective action, various possible UV completions (“standard” and “alternate” dynamical KMS prescriptions) differ in the assignment and transformation of auxiliary fields, but all are required to yield equivalent IR physics.

Obstructions arise in attempting to quantize or stochastically complete hydrodynamic theories that only appear stable in classical or deterministically reformulated variables (e.g., certain formulations of the BDNK theory), demonstrating the necessity of SK-compatible UV sectors (Jain et al., 2023).

7. Nonlocality, Memory Effects, and Observable Consequences in Time-Dependent EFTs

A central insight from the SK EFT perspective is that, whenever a system is initialized in a non-interacting state at finite time $t_0$ or undergoes a sudden switching-on of interactions, loop effects mediated by heavy fields leave imprints as transient, nonlocal (in time) operators in the EFT (Collins et al., 2012). These operators have coefficients with oscillatory, power-law decay ($\propto [M(t-t_0)]^{-3/2}$ for mass $M$ heavy fields), encoding residual memory of the initial condition. In the strict adiabatic limit (with $t_0\rightarrow -\infty$ or adiabatic interaction turn-on), these operators vanish, recovering the equivalence with time-translation invariant local EFTs.

Such transient nonlocalities have concrete implications. In inflationary cosmology, this manifests as residual, slowly-decaying operators in the primordial perturbation spectrum, potentially observable if the elapsed duration since $t_0$ is not sufficient for decay. More broadly, these results clarify under what circumstances local EFTs suffice and when nonequilibrium dynamics or memory effects from heavy field excitation must be explicitly retained.

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This comprehensive framework underscores that Schwinger-Keldysh Effective Field Theories provide a robust, symmetry-constrained, and physically consistent approach for capturing the full content of real-time quantum field dynamics—including dissipation, statistical fluctuations, stochastic noise, time-dependent nonlocal effects, and the constraints imposed by unitarity, causality, and equilibrium thermodynamics.