Green's functions (many-body theory)

Green's functions (many-body theory)
Name	Green's functions (many-body theory)
Field	Theoretical physics
Introduced	1950s
Notable users	Richard Feynman, Lev Landau, Ludwig Boltzmann, Julian Schwinger, John Bardeen, Lev Davidovich Landau, P. W. Anderson

Green's functions (many-body theory) Green's functions are propagators used to describe excitations and correlations in interacting quantum systems. They connect early developments by George Green with operator techniques of Julian Schwinger and diagrammatics of Richard Feynman, and underpin methods employed by Lev Landau, John Bardeen, P. W. Anderson, and institutions like Bell Labs and CERN.

Introduction

Green's functions trace lineage to work by George Green and later formalization by Julian Schwinger, Sin-Itiro Tomonaga, Freeman Dyson, and Richard Feynman. They serve as central tools in theoretical frameworks advanced at Princeton University, Harvard University, University of Cambridge, and Landau Institute for Theoretical Physics. Historical milestones include contributions from Lev Landau, Lev Davidovich Landau, P. W. Anderson, John Bardeen, Nikolay Bogoliubov, and developments associated with conferences at CERN and institutes such as Institute for Advanced Study.

Formulation in Many-Body Quantum Mechanics

In second quantization the time-ordered Green's function is defined using creation and annihilation operators introduced in curricula at Massachusetts Institute of Technology, University of Chicago, Columbia University, and Stanford University. The Matsubara formalism for finite temperature was developed following work by Tomonaga and used in research at Kyoto University and Tokyo Institute of Technology. Retarded and advanced propagators connect to causality principles applied in analyses at Los Alamos National Laboratory and Argonne National Laboratory. The Lehmann representation relates spectral weights to matrix elements often discussed in seminars at Princeton University and Yale University.

Diagrammatic Expansion and Feynman Diagrams

Diagrammatic perturbation theory uses rules laid out by Richard Feynman and formalized by Freeman Dyson and Julian Schwinger, linking to vertex functions studied by Miguel A. Virasoro and groups at Max Planck Institute for Physics. Wick's theorem, associated historically with courses at University of Cambridge and University of Oxford, contracts operators to build diagrams used in calculations at Brookhaven National Laboratory and Rutherford Appleton Laboratory. Skeleton diagrams, parquet equations, and linked-cluster theorems were refined by researchers at Bell Laboratories and Princeton University and applied in collaborations with teams from Lawrence Berkeley National Laboratory and ETH Zurich.

Self-Energy, Dyson Equation, and Approximations

The Dyson equation, introduced by Freeman Dyson, expresses the full propagator in terms of the self-energy; its use is standard in curricula at Imperial College London and University of California, Berkeley. Approximations such as Hartree–Fock, GW, and T-matrix were developed across projects at Bell Labs, IBM Research, and Los Alamos National Laboratory and influenced work by John Bardeen and P. W. Anderson. Diagram resummation techniques, renormalization group methods from Kenneth Wilson at Cornell University, and conserving approximations formulated by Gordon Baym and Leo Kadanoff are widely employed in analyses at MIT and Stanford University.

Spectral Functions, Analytic Properties, and Sum Rules

Spectral functions, with formulations used in spectroscopies at Argonne National Laboratory and SLAC National Accelerator Laboratory, are constrained by sum rules derived in contexts studied at Princeton University and Harvard University. Analytic continuation from Matsubara frequencies to real frequencies is a nontrivial step tackled in collaborations involving Max Planck Institutes and École Normale Supérieure, with methods inspired by work at University of Tokyo. Kramers–Kronig relations appear in treatments connected to labs at Bell Labs, Brookhaven National Laboratory, and Lawrence Livermore National Laboratory.

Numerical Methods and Computational Techniques

Quantum Monte Carlo techniques, developed by groups at Los Alamos National Laboratory and Sandia National Laboratories, compute Green's functions for models investigated at Harvard University and Princeton University. Dynamical mean-field theory (DMFT), introduced by researchers associated with Rutgers University and applied at ETH Zurich, combines impurity solvers like numerical renormalization group (NRG) from University of Stuttgart and continuous-time Monte Carlo from University of Vienna. Exact diagonalization and tensor network approaches trace development to groups at California Institute of Technology, Max Planck Institutes, and Perimeter Institute for Theoretical Physics.

Applications in Condensed Matter and Nuclear Physics

In condensed matter, Green's functions underpin theories of superconductivity by John Bardeen and collaborators at University of Illinois at Urbana–Champaign and explain quasiparticle dynamics studied at Argonne National Laboratory, Brookhaven National Laboratory, and Bell Labs. In nuclear physics, propagators are central to self-consistent Green's function methods developed at Oak Ridge National Laboratory and TRIUMF, and to ab initio approaches advanced at Lawrence Livermore National Laboratory and Fermi National Accelerator Laboratory. Applications extend to correlated electron systems studied at Max Planck Institutes, low-dimensional materials characterized by Columbia University and EPFL, and ultracold atomic gases investigated at JILA and MIT.

Category:Quantum many-body theory