Thermodynamic Green Functions

Abstract

The quantum-field theory is designed to deal with an infinite number of degrees of freedom. This is exactly what the description of electronic excitations in condensed matter needs, at least, together with the quantum-statistical approach. Nevertheless, we start with a single quantum particle embedded in the electron gas. Propagators of electrons and holes are studied as expectation values of pairs of field operators. They allow the introduction of Green functions. The poles in frequency domain of their Fourier transforms contain information about the electronic excitations. Because of the dependence of the grand canonical statistical operator on the Hamiltonian and the inverse temperature a generalization for complex times is possible. Then one speaks about thermodynamic or Matsubara Green functions. On the single-particle level they contain the complete information about the spectral properties mediated solely by the spectral-weight function. The successive application of the equation of motion leads to a hierarchy of equations for $N$-particle Green functions. In the single-particle case it is closed introducing a self-energy of an electron that accounts for the entire electron-electron interaction. It allows the formulation of an integral equation, a Dyson equation, instead of the differential equation of motion.