We present a general theory for the calculation of the single-particle time-correlation function $\ensuremath{\alpha}(t)=〈[a,{a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)]〉$ which is the canonical average of the commutator between the particle annihilation
a and creation ${a}^{\ifmmode\dagger\else\textdagger\fi{}}(t)$ at time
t. The theory is based on the projection-operator method. The complex spectral function $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ is expressed in terms of the natural frequency of oscillation $\ensuremath{\Omega}$ and the width function $\stackrel{^}{\ensuremath{\gamma}}(\ensuremath{\omega})$. From the analytical property of $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ in the complex $\ensuremath{\omega}$ plane for the weak-coupling limit, the long-time behavior of the correlation function $\ensuremath{\alpha}(t)$ and the relaxation functions is obtained. For a harmonic oscillator immersed in a heat bath, the perturbation calculations for $\ensuremath{\Omega}$ and $\stackrel{^}{\ensuremath{\gamma}}(\ensuremath{\omega})$ are given in the power of the coupling constant. By means of this series, the spectral function $\stackrel{^}{\ensuremath{\alpha}}(\ensuremath{\omega})$ for a single normal mode of an anharmonic system is explicitly calculated as a function of the frequency $\ensuremath{\omega}$ and the temperature
T. As a possible application of the results the electrical conductivity due to a localized mode is discussed.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.