Schwinger-Keldysh Approach to Disordered and Interacting Electron Systems: Derivation of Finkelstein's Renormalization Group Equations

We develop a dynamical approach based on the Schwinger-Keldysh formalism to derive a field-theoretic description of disordered and interacting electron systems. We calculate within this formalism the perturbative RG equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein using replicas. The major simplifying feature of this approach, as compared to Finkelstein's is that instead of $N \to 0$ replicas, we only need to consider N=2 species. We compare the dynamical Schwinger-Keldysh approach and the replica methods, and we present a simple and pedagogical RG procedure to obtain Finkelstein's RG equations.