Conservation Laws and Correlation Functions

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 $G\left(U\right)\mathrm{}$ are obtained by replacing the $G_2\mathrm{}\left(U\right)\mathrm{}$ which appears in these equations by various functionals of $G\left(U\right)\mathrm{}$. If the approximation for $G_2\mathrm{}\left(U\right)\mathrm{}$ satisfies certain simple symmetry conditions, then the $G\left(U\right)\mathrm{}$ thus defined obeys all the conservation laws. Furthermore, the two-particle correlation function, generated as ${\left(\frac{\delta G}{\delta U}\right)}_{U=0\mathrm{}}\equiv \pm L$, 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.