Kadanoff–Baym equations
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| Kadanoff–Baym equations | |
|---|---|
| Name | Kadanoff–Baym equations |
| Field | Theoretical physics |
| Introduced | 1961 |
| Derived by | Leo Kadanoff, Gordon Baym |
Kadanoff–Baym equations The Kadanoff–Baym equations provide a non-equilibrium many-body framework connecting quantum field theoretic Green functions to time evolution in interacting systems. They were developed by Leo Kadanoff and Gordon Baym and occupy a central role in the study of quantum transport, nonequilibrium statistical mechanics, and correlated electron systems. The equations link diagrammatic techniques used in Richard Feynman's perturbation theory, the Lev Landau Fermi liquid concept, and the contour methods of Leonid Keldysh to produce self-consistent evolution for two-time Green functions.
Introduction
The Kadanoff–Baym equations are integro-differential equations for two-time nonequilibrium Green functions derived within quantum many-body theory by Leo Kadanoff and Gordon Baym; they unify approaches from Julian Schwinger's closed-time-path formalism, Richard Feynman diagrams, and the Niels Bohr-era foundations of statistical mechanics. In applications they are invoked alongside models such as the Hubbard model, the Anderson impurity model, and the Bardeen-Cooper-Schrieffer theory to describe ultrafast dynamics in condensed matter, cold atoms in Wolfgang Ketterle's and Eric Cornell's laboratories, and nonequilibrium plasmas relevant to Edward Teller-type high-energy density research.
Formal derivation
Starting from the nonequilibrium generating functional introduced by Julian Schwinger and developed by Leonid Keldysh and Gert Aarts, the Kadanoff–Baym derivation uses functional differentiation of contour-ordered action to obtain equations of motion for the one-particle Green function. The derivation employs self-energy diagrams familiar from Richard Feynman's perturbation series and connects to the Migdal-type approximations used in electron-phonon problems studied by Lev Gor'kov and Alexei Abrikosov. Conservation laws associated with Emmy Noether symmetries appear when self-energies derive from a functional, linking to concepts developed by Gordon Baym and Leo Kadanoff for thermodynamic consistency.
Contour-ordered Green functions and Keldysh technique
The contour-ordered Green function formalism of Leonid Keldysh organizes time evolution on the Schwinger closed time contour and introduces lesser, greater, retarded, and advanced components that enter the Kadanoff–Baym equations. This structure parallels the matrix formulation used by Alexei Abrikosov and Lev Gor'kov in superconductivity and the nonequilibrium field-theory treatments applied by Giovanni Vignale and Giuseppe Stefanucci. The Keldysh rotation and Langreth rules, invoked by practitioners such as Henrik Bruus and Karsten Flensberg, permit conversion between contour expressions and real-time kinetic forms used in transport studies influenced by Rudolf Peierls and Nevill Mott.
Approximations and conserving approximations (Φ-derivable schemes)
Conserving approximations for the Kadanoff–Baym equations follow the Φ-derivable prescription formulated by Gordon Baym and connected to variational principles used by Richard Feynman and John Wheeler. Common self-energy choices include the second-Born approximation, the GW approximation introduced by Walter Kohn's colleagues and linked to Lars Hedin, and T-matrix schemes employed in studies by P. Nozières and G. Baym. Φ-derivable schemes ensure conservation of particle number, energy, and momentum as in analyses by Lev Landau and Igor Lifshitz, and are used in dynamical mean-field theory developments by Antoine Georges and Gabriel Kotliar.
Numerical methods and applications
Numerical solution of the Kadanoff–Baym equations uses time-stepping, matrix product state inspired compression, and parallel computation techniques explored in works by Martin Eckstein, Philipp Werner, and Nils Schlünzen. Applications span ultrafast spectroscopy experiments interpreted in the context of John Pendry-style photoemission, transport in nanojunctions related to Robert Landauer's ideas, and nonequilibrium superconductivity probed in experiments following John Bardeen's legacy. Simulations couple to realistic band structures from Walter Kohn-based density functional theory used by groups including Giulia Galli and Stefan Goedecker, and to cold-atom dynamics in traps studied by Immanuel Bloch and Jean Dalibard.
Relationship to kinetic equations and Boltzmann limit
Under quasiparticle approximations and gradient expansions the Kadanoff–Baym equations reduce to kinetic equations such as the quantum Boltzmann equation developed in the tradition of Ludwig Boltzmann and extended by Lev Landau and David Bohm. The reduction employs Wigner transforms akin to formulations by Eugene Wigner and semiclassical approximations related to J. M. Ziman's transport theory. Collision integrals derived from self-energies connect to scattering theories pioneered by Lev Landau's cohort and to the relaxation-time approximations used in Rudolf Peierls' and Nevill Mott's conductivity models.
Extensions and generalizations
Generalizations of the Kadanoff–Baym framework include nonequilibrium extensions of dynamical mean-field theory by Antoine Georges and Gabriel Kotliar, stochastic approaches influenced by H.A. Kramers and G. L. Eyink, and relativistic nonequilibrium field treatments related to Paul Dirac and Julian Schwinger. Modern work interfaces with quantum information perspectives from John Preskill and Peter Shor, with open quantum system frameworks associated with H.-P. Breuer and F. Petruccione, and with high-energy applications inspired by Edward Witten and Dirk H. Rischke.