Theory of many-fermion systems. II. The case of Coulomb interactions

In a recent paper a general field-theoretical description of many-fermion systems with short-range interactions has been developed. Here we extend this theory to the case of disordered electrons interacting via a Coulomb potential. A detailed discussion is given of the Ward identity that controls the soft modes in the system, and the generalized nonlinear $\sigma$ model for the Coulombic case is derived and discussed.