Electron-electron interactions in disordered metals: Keldysh formalism

We develop a field theory formalism for the disordered interacting electron liquid in the dynamical Keldysh formulation. This formalism is an alternative to the previously used replica technique. In addition, it naturally allows for the treatment of nonequilibrium effects. Employing the gauge invariance of the theory and carefully choosing the saddle point in the Q-matrix manifold, we separate purely phase effects of the fluctuating potential from the ones that change quasiparticle dynamics. As a result, the cancellation of super-divergent diagrams (double logarithms in $d=2)$ is automatically built into the formalism. As a by-product we derive a nonperturbative expression for the single-particle density of states. The remaining low-energy $\sigma$ model describes the quantum fluctuations of the electron distribution function. Its saddle-point equation appears to be the quantum kinetic equation with the appropriate collision integral along with collisionless terms. The Altshuler-Aronov corrections to the conductivity are shown to arise from the one-loop quantum fluctuation effects.