Ward Identities for Interacting Electronic Systems
  A Ward-Takahashi identity, as a consequence of gauge invariance and in a form that relates self-energy to the two-particle Bethe-Salpeter scattering kernel, was first derived by Vollhardt and W\"{o}lfle for a system of independent particles moving in a random medium. This is generalized to a class of interacting electronic systems in materials with or without random impurities, following a procedure previously used for classical waves transport in disordered media. This class of systems also possesses other symmetry properties such as invariance under time translations and local spin rotations, which imply local conservation laws for energy and spin current. They imply additional Vollhardt-W\"{o}lfle type identities. We present non-perturbative derivations of these identities, and consider the constraints they impose on the relationship between the self-energy and the two-particle scattering kernel. 
H. T. Nieh, Ping Sheng, Xiao-Bing Wang