| Keldysh formalism | |
|---|---|
| Name | Keldysh formalism |
| Introduced | 1964 |
| Inventors | Leonid Keldysh |
| Field | Quantum field theory; Nonequilibrium statistical mechanics |
| Notable applications | Quantum transport, Quantum optics, Condensed matter physics |
Keldysh formalism
The Keldysh formalism is a theoretical framework in Quantum field theory and Nonequilibrium statistical mechanics that enables calculation of quantum correlation functions and dynamics of systems driven away from thermal equilibrium. Developed to treat time-dependent and driven quantum many-body problems, it generalizes equilibrium Green's function techniques by introducing a time contour and contour-ordered propagators, permitting systematic perturbative and nonperturbative analyses of transport phenomena and dissipative dynamics.
The formalism was introduced by Soviet physicist Leonid Keldysh in 1964 to address problems inaccessible to equilibrium methods such as the Mott insulator breakdown, nonequilibrium tunneling, and time-dependent external fields. It emerged contemporaneously with developments in many-body theory—including work by Ludvig Faddeev and expansions of the Dyson equation concept—and built on earlier ideas from Schwinger–Keldysh formalism and the Schwinger closed-time path approach. The method became widely adopted in condensed matter physics and quantum transport as experimental platforms (e.g., mesoscopic physics, quantum dots, cold atoms) began probing nonequilibrium steady states and transient dynamics.
Keldysh formalism targets systems where standard equilibrium techniques like the Matsubara formalism fail because time-translation invariance or detailed balance are broken. Examples include driven open quantum systems coupled to reservoirs (e.g., leads in quantum transport), quenched quantum many-body systems, and light-driven materials studied in pump–probe spectroscopy. Motivation includes computing observable quantities—currents, spectral functions, noise correlations, and time-resolved response—that require knowledge of real-time evolution and statistical distribution simultaneously. It is particularly valuable for describing coupling to baths represented by noninteracting leads or Lindblad equation environments and for connecting microscopic Hamiltonians to semiclassical kinetic descriptions like the Boltzmann equation.
Central to the approach is the introduction of the Keldysh contour (also called the closed-time-path) in the complex time plane, composed of a forward and backward branch that allow for contour ordering of operators. This constructs contour-ordered Green's functions G(1,2) = -i⟨T_C ψ(1) ψ†(2)⟩ that encode both causal propagation and statistical information. From the contour-ordered object one obtains the physical retarded, advanced, and Keldysh components: retarded G^R, advanced G^A, and the Keldysh Green's function G^K which carries distribution information. The formalism naturally incorporates initial density matrices and can be extended to include imaginary-time segments for thermal initial states, linking to the Thermal field theory and Matsubara Green's function approaches.
Analytic and diagrammatic techniques mirror equilibrium many-body theory but are applied on the contour. The Dyson equation is formulated for contour-ordered propagators with self-energies Σ that capture interactions and coupling to baths. After projection onto real-time components one obtains coupled Dyson equations for G^R, G^A and G^K. Standard perturbation theory uses contour Feynman rules; diagrams include interaction vertices and reservoir lines and yield self-energy approximations such as the self-consistent Born approximation or GW approximation adapted to nonequilibrium. Conserving approximations can be constructed from generating functionals analogous to the Baym–Kadanoff approach, ensuring satisfaction of continuity and conservation laws in transport calculations.
A convenient linear transformation known as the Keldysh rotation (or Larkin–Ovchinnikov rotation) rearranges Green's function components into a triangular structure separating spectral and distribution information. The Keldysh component is related to single-particle distribution functions; in weak-coupling or semiclassical limits one derives kinetic equations (quantum Boltzmann equations) for occupancies. These kinetic equations connect to Wigner transformation and gradient expansions that yield quantum-corrected transport equations useful for describing phenomena such as quantum Hall effect edge transport, nonequilibrium superconductivity (e.g., Andreev reflection), and thermalization in ultracold atomic gases.
Keldysh techniques are widely applied to quantum transport in nanostructures (quantum dots, molecular junctions), shot-noise and full-counting statistics, driven-dissipative quantum optics systems (cavity QED, Jaynes–Cummings model with reservoirs), and nonequilibrium phases in strongly correlated electron systems such as nonequilibrium superconductivity and Floquet-engineered materials. It underpins numerical and analytic methods including nonequilibrium dynamical mean-field theory (DMFT), real-time renormalization group schemes, and diagrammatic Monte Carlo approaches for transient dynamics. Experimental relevances span pump–probe spectroscopy, transport measurements in two-dimensional materials like graphene, and time-resolved studies at facilities such as XFEL light sources.
The Keldysh formalism connects to and extends several frameworks: it generalizes the Schwinger–Keldysh formalism and complements equilibrium Matsubara techniques via analytic continuation. It relates to the Lindblad master equation when tracing out baths under Markovian approximations and to semiclassical stochastic descriptions via the Martin–Siggia–Rose formalism and functional-integral methods. Modern extensions include coupling to topological systems, incorporation of strong correlations via nonequilibrium DMFT and tensor-network real-time methods, and integration with quantum information perspectives on decoherence and entanglement dynamics. Notable figures and institutions contributing to development include Alexander Altland, Jan von Delft, Gabriele Falci, research groups at Max Planck Institute for the Physics of Complex Systems and Stanford University that advanced numerical and analytic nonequilibrium techniques.