This textbook contains a pedagogical introduction to the theory of Green's functions in and out of equilibrium, and is accessible to students with a standard background in basic quantum mechanics and complex analysis. Two main motivations prompted us to write a monograph for beginners on this topic.

The first motivation is research oriented. With the advent of nanoscale physics and ultrafast lasers it became possible to probe the correlation between particles in excited quantum states. New fields of research like, e.g., molecular transport, nanoelectronics, Josephson nanojunctions, attosecond physics, nonequilibrium phase transitions, ultracold atomic gases in optical traps, optimal control theory, kinetics of Bose condensates, quantum computation, etc. added to the already existing fields in mesoscopic physics and nuclear physics. The Green's function method is probably one of the most powerful and versatile formalisms in physics, and its nonequilibrium version has already proven to be extremely useful in several of the aforementioned contexts. Extending the method to deal with the new emerging nonequilibrium phenomena holds promise to facilitate and quicken our comprehension of the excited state properties of matter. At present, unfortunately, to learn the nonequilibrium Green's function formalism requires more effort than learning the equilibrium (zero-temperature or Matsubara) formalism, despite the fact that nonequilibrium Green's functions are not more difficult. This brings us to the second motivation.