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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 G₂ (U) which appears in these equations by various functionals of G (U). If the approximation for G₂ (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 ({) }ₔ=₀, 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, G₂ (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) '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, Vₒ (U). This potential is obtained by summing bubbles composed of two G (U) 's. The resulting L equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to |{Vₒ|}^2.
Baym et al. (Sun,) studied this question.