Field Theory of Many-Body Lindbladian Dynamics

We review and further develop the Keldysh functional integral technique for the study of Lindbladian evolution of many-body driven-dissipative quantum systems. A systematic and pedagogical account of the dynamics of generic bosonic and fermionic Lindbladians is presented. Our particular emphasis is on unique properties of the stationary distribution function, determined by the Lyapunov equation. This framework is applied to study examples of Lindbladian dynamics in the context of band theory, disorder, collisionless collective modes, and mean-field theory.