Keldysh Functional Integral Overview

• Keldysh functional integral is a real-time method that doubles fields on forward and backward branches to generate expectation values and correlation functions.
• It unifies coherent Hamiltonian dynamics with dissipative processes, enabling analysis of both equilibrium and non-equilibrium systems within a single framework.
• The formalism facilitates deriving effective actions and kinetic equations, providing insights into operator growth, phase transitions, and non-equilibrium phenomena.

The Keldysh functional integral, also called the Schwinger–Keldysh or closed-time-path functional integral, is a real-time representation of quantum dynamics in which expectation values and correlation functions are generated by doubling fields on a forward and a backward time branch. Because it evolves the density matrix rather than a pure-state amplitude, it treats response, fluctuations, coherent Hamiltonian dynamics, dissipation, and external drive within a single action, with a normalized stationary generating functional $Z=1$ playing the role of a non-equilibrium analogue of a partition function (Sieberer et al., 2015).

1. Closed-time-path construction

The defining object is the doubled time evolution of the density matrix. In the Hamiltonian case,

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$

and in the Markovian open-system case one starts from the Lindblad equation

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$

This is why the formalism naturally carries two branches, usually denoted $+$ and $-$: the forward branch acts on $\rho$ from the left, and the backward branch acts from the right (Sieberer et al., 2015).

In coherent-state language, the contour action takes the general form

$$S = \int_{t_0}^{t_f} dt \left( \psi_+^* i\partial_t \psi_+ - \psi_-^* i\partial_t \psi_- - i\,\mathcal{L}(\psi_+^*,\psi_+,\psi_-^*,\psi_-) \right),$$

so coherent evolution and dissipative processes enter a single real-time functional. For stationary problems the contour is extended from $t_0\to-\infty$ to $t_f\to+\infty$, and trace preservation implies

$$Z=\int \mathscr{D}[\psi_+,\psi_+^*,\psi_-,\psi_-^*]\,e^{iS}=1,$$

which encodes normalization and probability conservation (Sieberer et al., 2015).

The same doubling appears already in the elementary harmonic-oscillator construction. There the generating functional is

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$0

showing directly that the Keldysh functional integral generates expectation values rather than transition amplitudes (BenTov, 2021). In resonant inelastic light scattering, the same closed contour arises because the scattering probability contains both a process and its complex conjugate, and therefore requires time ordering and anti-time ordering simultaneously (Lee, 2013).

2. Classical/quantum fields and Green-function structure

A practical formulation is obtained by the Keldysh rotation to “classical” and “quantum” variables,

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$1

The terminology is structural rather than mnemonic: $\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$2 can acquire a finite expectation value, while $\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$3 encodes response and noise (Sieberer et al., 2015).

In this basis the two-point function has the canonical matrix form

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$4

The exact zero in the lower-right corner is a hallmark of causality and probability conservation (Sieberer et al., 2015). The corresponding operator identities separate spectral information from occupations: $\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$5 The spectral function is

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$6

while equal-time Keldysh correlators encode occupancies, for example

$\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$7

for a damped cavity mode (Sieberer et al., 2015).

In source language, differentiation with respect to the two contour sources produces the four basic real-time correlators. The $\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$8 derivative yields the Feynman correlator, the $\rho(t)=U(t,t_0)\,\rho(t_0)\,U^\dagger(t,t_0),$9 derivative the Dyson or anti-time-ordered correlator, and the mixed derivatives yield the two Wightman functions (BenTov, 2021). This branch-resolved generating structure is why the formalism can handle response functions, spectral functions, and intensity correlations within a single functional integral.

3. Equilibrium as symmetry and non-equilibrium as symmetry breaking

A central development in the modern formulation is the identification of thermal equilibrium with a discrete symmetry of the Schwinger–Keldysh action. For bosonic fields,

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$0

with $$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$1. This transformation combines complex conjugation, time reversal, and an imaginary-time shift by $$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$2, and the action is invariant under $$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$3 if and only if the dynamics is in thermal equilibrium (Sieberer et al., 2015).

This symmetry is equivalent to the KMS condition and implies the Gibbs stationary state. Its Ward–Takahashi identities generate the full hierarchy of fluctuation-dissipation relations, beginning with

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$4

In the classical limit this reduces to the equilibrium symmetry of Langevin and response functionals, with the Rayleigh–Jeans form

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$5

appearing at high temperature (Sieberer et al., 2015).

The same framework clarifies non-equilibrium. Generic driven or Markovian open systems violate the thermal symmetry because drive, loss, or coupling to reservoirs with different thermodynamic parameters breaks detailed balance (Sieberer et al., 2015). A contour-reparametrized formulation makes this explicit: for a time-dependent protocol $$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$6, the broken-symmetry term $$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$7 measures irreversibility on a single quantum trajectory and obeys fluctuation theorems such as

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$8

together with the Crooks-type relation for its probability distribution (Aron et al., 2017). In this sense, non-equilibrium dynamics is represented as symmetry breaking of the Keldysh generating functional rather than as a mere absence of a Gibbs density matrix.

4. Effective actions, auxiliary fields, and controlled approximations

Much of the power of the Keldysh functional integral lies in integrating out microscopic degrees of freedom and rewriting the theory in terms of collective fields. The resulting effective actions are typically nonlocal and nonlinear, and often generated by a trace log or a determinant.

In weakly interacting quantum dots described by the single-impurity Anderson model, the interaction can be decoupled in the spin channel by a continuous Hubbard–Stratonovich magnetization field. After integrating out fermions, one obtains

$$\partial_t \rho = -i[H,\rho] + \sum_\alpha \gamma_\alpha \left(L_\alpha \rho L_\alpha^\dagger - \frac{1}{2}\{L_\alpha^\dagger L_\alpha,\rho\}\right).$$9

and in the nonmagnetic case the action is even,

$+$0

A second-order expansion in the magnetization field gives a quadratic Keldysh action valid for weak electron-electron interaction $+$1 at low temperature. Within that regime the theory reproduces the unitary conductance limit

$+$2

which the paper emphasizes as a key advantage over several strong-coupling slave-boson expansions (Smirnov et al., 2012).

In the finite-$+$3 Kondo problem, the local impurity Hilbert space is rewritten in a slave-particle basis with bosons for empty and doubly occupied states and fermions for singly occupied states. After imposing the exact constraint and integrating out fermions, the tunneling action has the universal form

$+$4

Expanding around the zero slave-boson configuration yields a quadratic effective Keldysh action for the empty- and double-occupancy bosons. This gives an analytically solvable theory valid near and above $+$5, more precisely for $+$6 and, in the comparison with NRG reported there, reliably for $+$7 in the strong finite-$+$8 regime $+$9 (Smirnov et al., 2011).

A different collective-field construction appears in infinite-range quantum spin glasses, where disorder averaging produces a nonlocal quartic term that is decoupled by a bilocal Hubbard–Stratonovich field $-$0. In the late-time glassy limit, and under strong separation of time scales, the replica-free Dyson–Keldysh equations reproduce the algebra of Parisi ultrametric matrices. The stationary glass thereby spontaneously breaks thermal symmetry, or the Kubo–Martin–Schwinger relation of a state in global thermal equilibrium (Lang et al., 2024).

5. Driven-open systems, semiclassical limits, and stationary distributions

For driven open many-body systems, the Keldysh functional integral merges quantum optics, many-body physics, and statistical mechanics. Starting from the Lindblad generator, the real-time action treats coherent drive and dissipation on equal footing and makes it possible to diagnose non-equilibrium through symmetry rather than through a few selected observables (Sieberer et al., 2015).

At long wavelengths, Markovian noise often generates a finite low-energy Keldysh component, and the theory reduces to a semiclassical MSR/Langevin form. For driven polaritons, the infrared action becomes equivalent to a stochastic equation with white noise,

$-$1

with

$-$2

Within this framework the ordered phase of driven-dissipative Bose condensation carries a dissipative Goldstone mode,

$-$3

three-dimensional systems can flow to an effective equilibrium fixed point, and lower-dimensional condensates can instead display KPZ scaling (Sieberer et al., 2015).

For quadratic many-body Lindbladians, the Keldysh formulation also separates dynamical spectra from stationary statistics. The non-Hermitian single-particle matrix $-$4 determines the poles of the retarded and advanced Green’s functions, while the Hermitian distribution matrix $-$5 is fixed by a Lyapunov equation,

$-$6

for fermions, with the corresponding bosonic formula carrying $-$7 factors. This means that relaxation modes and the stationary state are distinct objects. A further consequence emphasized there is that exceptional points are dynamical singularities of $-$8, whereas $-$9 remains analytic across them (Thompson et al., 2023).

The open-system symmetry perspective extends to quantum Brownian motion on a deformed complex-time contour. After integrating out the Caldeira–Leggett bath, the influence functional is invariant in equilibrium under

$\rho$0

and the fluctuation-dissipation relation follows as a consequence of that symmetry. Under external driving, the symmetry-breaking terms lead to a Jarzynski-like identity

$\rho$1

with $\rho$2 a trajectory-dependent quantum work functional (Yeo, 2019).

6. Applications, extensions, and scope of the formalism

The range of applications is unusually broad. In resonant inelastic light scattering, the cross section can be rewritten as a four-current correlation function on the Keldysh–Schwinger contour, which eliminates explicit sums over unknown intermediate states and permits ordinary real-time Feynman-diagram perturbation theory with self-energy and vertex corrections. Applied to one $\rho$3-phonon Raman scattering in graphene, the resulting expression agrees with the conventional Fermi-golden-rule result obtained by Basko, and the same contour construction extends directly to pumped non-equilibrium systems (Lee, 2013).

In microcavity polaritons, the Keldysh formalism treats coherent exciton–photon hybridization together with independent excitonic and photonic baths. After integrating out the baths, one obtains effective damping rates $\rho$4, where $\rho$5 is a cross-damping term generated by direct upper-polariton decay. The saddle-point dynamics then takes a Josephson-like form, but with an additional dissipative coupling not present in treatments with only simple bare decay channels (Rahmani et al., 2016).

The Kadanoff–Baym–Keldysh or NEGF framework also appears in first-principles nanomaterials modeling. In the 2025 nanocrystal study, nonadiabatic couplings extracted from DFT-based non-adiabatic molecular dynamics are reinterpreted as a time-dependent external potential, allowing the Keldysh self-energies $\rho$6 and $\rho$7 to be converted into Boltzmann collision integrals for exciton-phonon relaxation. Combined with Bethe–Salpeter exciton wavefunctions and radiative processes, this yields a kinetic theory of phonon-mediated relaxation and photoluminescence for several $\rho$8-nm semiconductor chalcogenide nanocrystals (Griffin et al., 22 Mar 2025).

More recent extensions push the formalism in two different directions. One direction is conceptual: the 2026 Schwinger–Keldysh formulation of Krylov dynamics treats Krylov complexity as an in-in observable, rewrites Lanczos coefficients as an effective Hamiltonian on Krylov phase space, and uses the Keldysh sector to study fluctuations and large deviations of operator growth (Murugan et al., 2 Feb 2026). The other direction is mathematical: the 2025 rigorous construction for fermions proves convergence of the discrete-time Grassmann Gaussian Keldysh representation in the time-continuum limit, analyticity of the effective action, and explicit connected-correlation bounds, with the dissipative case displaying time behavior not necessarily restricted to short times (Aretz et al., 3 Aug 2025).

A useful boundary of the concept is that not every real-time functional integral is a Keldysh functional integral. The supercoherent-state Hubbard-model construction with deformed measure, Jackson integration, and a $\rho$9-adic limit explicitly does not introduce a contour-ordered Keldysh path integral, forward/backward branches, classical/quantum fields, or explicit nonequilibrium Green’s functions; its emphasis is the deformation of the measure rather than the closed-time-path structure (Zharkov, 2012).

Taken together, these developments define the Keldysh functional integral less as a single approximation scheme than as a real-time organizing principle. Its characteristic features are contour doubling, causal Green-function structure, the separation of response and correlations, and the ability to pass from microscopic Hamiltonians or Lindbladians to effective actions, kinetic equations, Langevin limits, symmetry diagnostics, and, in some settings, rigorous constructive control.