Keldysh action for disordered superconductors

The Keldysh representation of the functional integral for the interacting e lectron system with disorder is used to derive microscopically an effective action for dirty superconductors. In the most general case this action is a functional of the 8x8 matrix Q(t,t') which depends on two time variables, and on the fluctuating order parameter field and electric potential. We sh ow that this approach reproduces, without the use of the replica trick, the well-known result for the Coulomb-induced renormalization of the electron- electron coupling constant in the Cooper channel. Turning to the results, w e calculate the effects of the Coulomb interaction upon (i) the subgap Andr eev conductance between the superconductor and the two-dimensional dirty no rmal metal, and (ii) the Josephson proximity coupling between superconducti ve islands via such a metal. These quantities are shown to be strongly supp ressed by the Coulomb interaction at sufficiently low temperatures due to b oth zero-bias anomaly in the density of states and disorder-enhanced repuls ion in the Cooper channel.