Green's function (many-body theory)

Green's function (many-body theory)
Name	Green's function
Field	Quantum field theory, Condensed matter physics
Discovered	Julian Schwinger, Lev Landau, Alexei Abrikosov
Related concepts	Feynman diagram, Self-energy, Kubo formula

Green's function (many-body theory). In quantum field theory and condensed matter physics, Green's functions are fundamental tools for describing the behavior of interacting many-particle systems. They encode information about particle propagation, excitation spectra, and response functions, providing a bridge between microscopic Hamiltonians and observable quantities. The formalism was extensively developed by figures like Julian Schwinger and Lev Landau, and it underpins modern treatments of phenomena such as superconductivity and the fractional quantum Hall effect.

Definition and basic properties

The single-particle Green's function is defined as the expectation value of time-ordered products of creation and annihilation operators in the Heisenberg picture. For fermions, a common definition is \(G(1,2) = -i \langle T \psi(1) \psi^\dagger(2) \rangle\), where \(T\) is the time-ordering operator, and the angle brackets denote an average over the exact many-body ground state or a thermodynamic ensemble. This function describes the probability amplitude for a particle added at spacetime point 2 to propagate to point 1. Key analytical properties include causality, which is enforced by the time-ordering, and relationships to physical observables like the density of states. The advanced and retarded versions of the Green's function are directly related to response theory and are crucial for calculating transport coefficients.

Types of Green's functions

Several distinct Green's functions are used depending on the context and the time contour chosen for the calculation. The **causal** or **time-ordered** Green's function is standard in zero-temperature ground state theory. The **retarded** and **advanced** Green's functions are vital for real-time response and spectroscopy problems, as their Fourier transforms yield directly measurable spectral functions. For systems at finite temperature, the **Matsubara Green's function**, defined on the imaginary-time axis, is a cornerstone of the finite-temperature quantum field theory formalism. The **Keldysh contour** Green's function, or **nonequilibrium Green's function**, is essential for describing systems driven out of equilibrium, such as those under an applied bias in mesoscopic physics.

Equation of motion and Dyson equation

The Green's function satisfies an equation of motion derived from the Heisenberg equation of motion for the field operators. This equation links the Green's function to higher-order correlation functions, leading to an infinite hierarchy of coupled equations. The **Dyson equation** provides a closed, self-consistent framework by introducing the **self-energy** \(\Sigma\), a complex function that encapsulates all interaction effects. In operator form, it is written as \(G = G_0 + G_0 \Sigma G\), where \(G_0\) is the non-interacting Green's function. Solving the Dyson equation is equivalent to summing an infinite series of certain **Feynman diagrams**, typically those that are one-particle irreducible. This equation is central to perturbative approaches like **GW approximation** and non-perturbative methods such as **dynamical mean-field theory**.

Spectral representation and Lehmann representation

A powerful representation of the Green's function is its spectral decomposition, which expresses it in terms of the exact energy eigenvalues and eigenstates of the many-body Hamiltonian. For the retarded function, this is \(G^R(\omega) = \int d\omega' \frac{A(\omega')}{\omega - \omega' + i\eta}\), where \(A(\omega)\) is the **spectral function**. The **Lehmann representation**, developed for the finite-temperature case, provides an explicit formula showing that the poles of the Green's function correspond to the exact excitation energies of the system, such as quasiparticle peaks and **incoherent backgrounds**. The spectral function satisfies sum rules and is directly probed by techniques like **angle-resolved photoemission spectroscopy**.

Applications in many-body physics

Green's functions are indispensable in calculating properties of correlated electron systems. They are used to compute the **electrical conductivity** via the **Kubo formula**, describe **superconducting pairing** through the anomalous Gor'kov Green's function, and analyze **magnetic ordering** in the **Hubbard model**. In **nuclear matter** and **quantum chromodynamics**, they describe the propagation of nucleons and quarks. The formalism is also critical for understanding **quantum impurity problems** via the **Anderson model** and for developing **ab initio** electronic structure methods that go beyond **density functional theory**, such as the **Bethe-Salpeter equation** for optical absorption.

Connection to other formalisms

The Green's function formalism is deeply intertwined with other major techniques in many-body theory. It provides the generating functional for **Feynman diagram** expansions in perturbative quantum field theory. The **partition function** and all thermodynamic potentials can be expressed as functionals of the Green's function, a relationship formalized in the **Luttinger-Ward functional**. Through the **fluctuation-dissipation theorem**, the Green's function is linked to dynamical correlation functions measured in **neutron scattering** or **Raman spectroscopy**. Furthermore, the **density matrix** and the **one-particle reduced density matrix** can be obtained from the equal-time limit of the Green's function, creating a bridge to **quantum chemistry** methods.

Category:Quantum field theory Category:Condensed matter physics Category:Mathematical physics