Dynamics of mesoscopic systems: Non-equilibrium Greenʹs functions approach

Recent developments in technologies and experiments enable us to observe dynamical behavior of very small quantum systems under conditions when it is sensitive to system parameters, internal interactions, the environment and the external time-dependent fields. This contribution deals with the status and perspectives of description of such truly non-equilibrium quantum many body systems using the non-equilibrium Greenʹs functions (NGF). The basic aim of this approach is to describe time development of the many-body system out of equilibrium from its initial state over its transient dynamics to its long time asymptotic evolution. The early stages of the transient evolution will be characterized for a broad class of the initial conditions generated in the course of Keldysh switch-on states. Gradual loss of initial correlations due to interactions and related renormalization processes in the system may, under favorable conditions, give rise to the evolution stage expressible in terms of non-equilibrium quasi-particles. This permits to reduce the description of the system behavior to a quantum transport equation. The consistency of such approximations for NGF may be judged by checking the non-equilibrium version of transport Ward identities. These identities follow from the gauge invariance of the 1st kind for NGF. To cover the whole time domain of the transient, we obtain the non-equilibrium Ward identities for NGF with initial conditions included.