This book provides a comprehensive introduction to equilibrium and non-equilibrium Green's function methods in many-body physics. It begins with a derivation of second quantisation for relativistic systems based on the many-body relativistic Dirac equation and its non-relativistic limit. The properties of equilibrium Green's functions are then described, with discussion of the two-time and Matsubara Green's function methods. The coverage of non-equilibrium Green's function methods includes the diagrammatic techniques applicable to electrons and phonons using both the perturbation and variational approaches. Specific applications to steady-state and time-dependent quantum transport are presented in the final chapters. The book's accessible explanations, detailed derivations, and systematic treatment of the underlying theory make it a valuable resource for graduate students and early-career researchers. More than 200 problems have been included to support learning, with selected solutions available at the end of each chapter. Instructors benefit from access to the full solutions manual.

‘This book offers a clear and comprehensive introduction to equilibrium and non-equilibrium Green’s function methods, making it an essential resource for graduate students and early-career researchers. With detailed derivations, practical applications to quantum transport, and over 200 problems to reinforce learning, it provides both depth and accessibility. The inclusion of a solutions manual for instructors makes it an excellent resource as a teaching and learning tool for mastering many-body physics and Green’s function techniques.’

Rafael Gutierrez - Dresden University of Technology

‘Kantorovich provides a thorough and highly accessible introduction to equilibrium and non-equilibrium many-body quantum theory, with a particular emphasis on quantum transport. Combining detailed derivations with numerous exercises and solutions, it guides readers from fundamental concepts to advanced applications in a clear and systematic manner. This will be a valuable resource for students and researchers seeking a comprehensive treatment of Green’s function methods and quantum transport phenomena.’

Riku Tuovinen - University of Jyväskylä

‘Kantorovich writes with enthusiasm and shows great insight into the field of many-body physics. Taking on the formalism with great care, he nevertheless manages to make it accessible and comprehensive for both students and experienced theorists alike. Rather than shying away from complicated calculations, the author makes an effort to go through all the arguments and necessary steps required for producing certain results. The sometimes-lengthy derivations are genuinely helpful and instrumental for readers desiring a deeper understanding of quantum field theory and its relevance to condensed matter physics. Spanning the vast field of condensed matter, including non-equilibrium and strongly correlated electrons, the book has the potential to be a standard reference text on any physicist’s bookshelf.’

Jonas Fransson - Uppsala University