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. Within this formalism we calculate the perturbative renormalization-group (RG) equations for interacting electrons expanded around a diffusive Fermi liquid fixed point, as obtained originally by Finkelstein (Zh. \'Eksp. Teor. Fiz. 84, 168 (1983) [Sov. Phys. JETP 57, 97 (1983)]) using replicas. The major simplifying feature of this approach, as compared to Finkelstein's, is that instead of →N0 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.