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| Baym-Kadanoff | |
|---|---|
| Name | Baym–Kadanoff formalism |
| Caption | Diagrammatic representation of self-energy and Green's functions |
| Field | Theoretical physics |
| Known for | Conserving approximations, many-body perturbation theory, quantum statistical mechanics |
Baym-Kadanoff
The Baym–Kadanoff formalism is a variational approach in quantum many-body theory that constructs self-consistent, conserving approximations for interacting fermion and boson systems. It provides a prescription to derive approximations for Green's functions and self-energies that respect fundamental conservation laws and thermodynamic consistency, and it has influenced work in condensed matter physics, nuclear theory, and quantum field theory. Key contributors and contexts include research by Gordon Baym, Leo Kadanoff, and follow-on developments in communities around John Schrieffer, Philip W. Anderson, Giovanni Jona-Lasinio, and institutions such as Bell Labs, Brookhaven National Laboratory, and CERN.
The formalism originated in the late 1960s and early 1970s during a period when researchers at University of Illinois at Urbana–Champaign, Massachusetts Institute of Technology, and Harvard University were consolidating diagrammatic techniques from the work of Lev Landau, Richard Feynman, Julian Schwinger, and Nikolay Bogolyubov. Early influences included L. D. Landau's quasiparticle concept, John Bardeen's work on superconductivity with Leon Cooper and Robert Schrieffer, and field-theoretic many-body methods developed by Arkady Migdal and David Pines. Baym and Kadanoff synthesized variational and diagrammatic ideas to produce a functional whose stationary conditions yield Dyson equations consistent with conservation laws, building on techniques from Schwinger–Dyson equations and the Keldysh formalism used in nonequilibrium problems studied at Stanford University and University of California, Berkeley.
The central object is the Baym–Kadanoff functional, a scalar functional of the single-particle Green's function that generalizes action principles used in the work of Julian Schwinger and Schwinger's source methods. Stationarity of the functional with respect to variations of the Green's function yields self-energy expressions consistent with Dyson's equation familiar from Freeman Dyson's work and with vertex functions appearing in the Bethe–Salpeter equation developed by Hans Bethe and Eric Salpeter. The construction invokes diagrammatic generating functionals akin to those in Migdal–Eliashberg theory and respects Ward identities associated with symmetry groups studied by Noether and implemented in contexts like electroweak theory and QCD calculations at CERN. The formalism explicitly ties to thermodynamic potentials used in studies at Los Alamos National Laboratory and to renormalization approaches advanced by Kenneth G. Wilson.
Approximations derived from truncations of the Baym–Kadanoff functional—often called Φ-derivable or conserving approximations—ensure conservation of particle number, momentum, and energy, paralleling constraints emphasized in Noether's theorem-driven analyses by researchers at Princeton University and University of Cambridge. This feature made the formalism attractive for applications in nuclear physics problems at Oak Ridge National Laboratory, heavy-ion collision modeling at GSI, and electron transport studies in mesoscale systems investigated at IBM Research. It has been applied to superconductivity problems extending BCS theory and to strongly correlated electron models like the Hubbard model and the Anderson impurity model explored at Rutgers University and University of Tokyo.
The Baym–Kadanoff approach connects to the Luttinger–Ward functional and to diagrammatic resummations used in GW approximation work pioneered by L. Hedin and groups at Bell Labs. It relates to functional renormalization group methods developed by Christof Wetterich and Timothy R. Morris and to nonperturbative techniques used in Dynamical Mean-Field Theory (DMFT) originated by researchers at Université Paris-Sud and Rutgers University. Connections also exist with nonequilibrium Green's function techniques such as the Keldysh technique used in quantum transport studies at University of Geneva and with modern quantum Monte Carlo approaches applied by groups at Princeton University and ETH Zurich.
Standard examples include the application of Φ-derivable approximations to the Hubbard model to compute spectral functions and to estimate phase boundaries between metallic and insulating states as studied in collaborations involving Georges Kotliar and Antoine Georges. Calculations of electron self-energy in the GW framework can be derived as specific truncations of the Baym–Kadanoff functional, linking to work by Friedhelm Aryasetiawan and Mark van Schilfgaarde. In nuclear matter, the formalism underpins diagrammatic summations used in computations by James P. Vary and R. Machleidt for saturation properties and equation-of-state studies relevant to research at TRIUMF and Argonne National Laboratory.
Modern extensions adapt the Baym–Kadanoff framework to nonequilibrium dynamics combining the Kadanoff–Baym equations with time-dependent DMFT pursued by teams at Max Planck Institute for the Structure and Dynamics of Matter and University of Fribourg, and to multiscale approaches that couple to density functional theory implementations developed at Oak Ridge National Laboratory and Lawrence Berkeley National Laboratory. Recent work integrates ideas from machine learning groups at Google DeepMind and Facebook AI Research for accelerated solver strategies, and hybrid methods merge Bayesian inference techniques from Stanford University and MIT with conserving-approximation constraints to improve analytic continuation and spectral reconstruction in studies relevant to materials science and astrophysics.
Category:Many-body theory Category:Quantum field theory Category:Theoretical physics