Conservation Laws and Correlation Functions

Abstract

In describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations has been developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator $G$, defined in the presence of a nonlocal external scalar field $U$. Approximations for $G(U)$ are obtained by replacing the ${G}_{2}(U)$ which appears in these equations by various functionals of $G(U)$. If the approximation for ${G}_{2}(U)$ satisfies certain simple symmetry conditions, then the $G(U)$ thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${(\frac{\ensuremath{\delta}G}{\ensuremath{\delta}U})}_{U=0}\ensuremath{\equiv}\ifmmode\pm\else\textpm\fi{}L$, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal $f$-sum rule.Several examples of conserving approximations are described. The Hartree approximation, ${G}_{2}(U)=G(U)G(U)$, generates the random-phase approximation for $L$. The Hartree-Fock approximation for $G(U)$ leads to a natural generalization of the random-phase approximation in which hole-particle ladder diagrams are summed. Another conserving approximation for $G(U)$ is obtained by expanding the self-energy to first order in the many-particle scattering matrix $T(U)$. This $T$ is obtained by summing ladder diagrams in which the sides of the ladder are composed of $G(U)'\mathrm{s}$. The resulting $L$ equation, which involves coefficients proportional to ${|T|}^{2}$, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the self-energy is expanded to first order in a dynamically shielded potential, ${V}_{s}(U)$. This potential is obtained by summing bubbles composed of two $G(U)'\mathrm{s}$. The resulting $L$ equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to ${|{V}_{s}|}^{2}$.