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| Keldysh formalism | |
|---|---|
| Name | Keldysh formalism |
| Field | Theoretical physics |
| Introduced | 1964 |
| Introduced by | Leonid Keldysh |
Keldysh formalism is a theoretical framework for treating non-equilibrium quantum many-body systems that generalizes equilibrium techniques to time-dependent processes; it provides contour-ordered Green functions, a matrix structure via Keldysh rotation, and diagrammatic perturbation theory suited for transport, quantum optics, and cold atoms. The formalism sits alongside Matsubara and Schwinger methods in the development of quantum field theory and many-body physics and underpins computational approaches such as Kadanoff–Baym, GW, and dynamical mean-field theory.
The introduction traces the development of time-dependent quantum techniques from early quantum electrodynamics debates involving Paul Dirac, Richard Feynman, and Sin-Itiro Tomonaga to later advances credited to Leonid Keldysh and parallel work by Julian Schwinger and Gian-Carlo Wick; it situates the formalism within the broader lineage that includes the Feynman diagram expansion, the Dyson equation, and the emergence of non-equilibrium statistical mechanics discussed by Lev Landau, Evgeny Lifshitz, and Rudolf Peierls.
Contour-ordered Green functions are defined on a complex time contour introduced in the original works by Leonid Keldysh and formalized in treatments by Julian Schwinger and Shoichi Sakata, connecting to equilibrium Matsubara techniques associated with Shinichiro Matsubara, John Bardeen, and Lev Gor'kov; the contour integrates forward and backward segments reminiscent of approaches used by Paul Dirac and employed in analyses by Richard Feynman and Freeman Dyson. In practice, one defines lesser, greater, time-ordered, and anti-time-ordered components as in computations by Gordon Baym and Leo Kadanoff, paralleling methods developed in the textbooks of Alexander Abrikosov, Lev Pitaevskii, and Evgeny Lifshitz.
The Keldysh rotation reorganizes contour components into classical and quantum fields, a transformation discussed in expositions by Alex Kamenev, Gerry D. Mahan, and Giuseppe Stefanucci, analogous to basis changes used in treatments by Kenneth Wilson and Philippe Nozières; this yields a 2×2 matrix structure for Green functions that is exploited in works by Antoine Georges and Gabriel Kotliar on impurity models. The matrix formulation connects to causality constraints studied by Nikolay Bogoliubov, Evgeny Lifshitz, and Arthur Schawlow and echoes the operator reorganizations familiar from John von Neumann and Paul Dirac.
Diagrammatic perturbation theory within the formalism uses contour Feynman rules that extend the vacuum diagrams of Richard Feynman and the self-energy concepts of Freeman Dyson and Julian Schwinger, with prescriptions adopted in reviews by Gordon Baym, Leo Kadanoff, and Alexander Altland. Vertex rules, propagator assignments, and loop expansions mirror those employed in many-body calculations by Philippe Nozières, John Bardeen, and Lev Gor'kov, and have been applied in diagram resummations influenced by the renormalization-group work of Kenneth Wilson and the parquet schemes explored by N. E. Bickers.
The formalism is applied to quantum transport problems analyzed in landmark studies by Rolf Landauer, Supriyo Datta, and Yuriy Nazarov; to quantum optics problems connected to experiments by Arthur Schawlow, Theodor Maiman, and theoretical treatments by Roy Glauber; and to ultracold atomic gas dynamics investigated in experiments by Wolfgang Ketterle, Eric Cornell, and Carl Wieman, with theoretical modeling by Immanuel Bloch, Jaksch, and Zoller. In mesoscopic systems the approach complements scattering theories used by Markus Büttiker and noise analyses by Yuli Nazarov, while in driven-dissipative quantum optics it connects to master-equation treatments from H. J. Carmichael and Howard Wiseman.
The relation to Matsubara imaginary-time methods introduced by Shinichiro Matsubara and elaborated by John Hubbard and Nevill Mott is established through analytic continuation procedures discussed by Abrikosov, Gorkov, and Dmitri Lifshitz, while the formal equivalence and historical lineage to Schwinger’s operator approach by Julian Schwinger yield the common designation Schwinger–Keldysh in several reviews by Lev Keldysh-related historians and commentators such as Alex Kamenev and Giuseppe Stefanucci. Comparisons are also drawn to real-time path-integral implementations popularized by Richard Feynman and canonical non-equilibrium Green’s function strategies used by Gordon Baym.
Computational realizations include the Kadanoff–Baym equations formulated by Leo Kadanoff and Gordon Baym, self-energy approximations such as GW developed by Lars Hedin and refined in studies by Walter Kohn and Lu J. Sham, and non-equilibrium extensions of dynamical mean-field theory (DMFT) pioneered by Antoine Georges and Gabriel Kotliar and applied in collaborations involving Marcus Jarrell and Emanuel Gull. Numerical implementations draw on quantum Monte Carlo strategies advanced by James E. Gubernatis and continuous-time Monte Carlo methods from Philipp Werner, while matrix product state and tensor-network adaptations link to the work of Steven White, Ulrich Schollwöck, and Guifre Vidal.