Many-Body Green's Functions for Time-Dependent Problems

Quantum many-body systems are a central feature of condensed matter physics, relevant to important, modern research areas such as ultrafast light-matter interactions and quantum information. This book offers detailed coverage of the contour Green's function formalism – an approach that can be successfully applied to solve the quantum many-body and time-dependent problems present within such systems. Divided into three parts, the text provides a structured overview of the relevant theoretical and practical tools, with specific focus on the Schwinger-Keldysh formalism. Part I introduces the mathematical frameworks that make use of Green's functions in normal phase states. Part II covers fermionic superfluid phases with discussion of topics such as the BCS-BEC crossover and superconducting systems. Part III deals with the application of the Schwinger-Keldysh formalism to various topics of experimental interest. Graduate students and researchers will benefit from the book's comprehensive treatment of the subject matter and its novel arrangement of topics.

Table of Contents:
Part I. Normal Phase; 1. Introduction; 2. The Schrödinger and Heisenberg Representations; 3. Splitting the Hamiltonian: Heisenburg and Interaction Pictures; 4. Time-dependent Quantum and Ensemble Averages: Initial Preparation of the System; 5. Quantum Averages over the Ground State and Gell-Mann-Low Theorem; 6. The Contour Idea for Time-dependent Averages: Forward and Backward Branches; 7. Closed Time Path Green's Functions; 8. Dynamics for a Correlated Initial State and Various Kinds of Contours in the Complex Time Plane; 9. Perturbation Theory: Wick's Theorem for Strings of Operators Ordered Along a Contour; 10. Non-equilibrium Diagrammatics: Feynman Rules; 11. Non-equilibrium Dyson Equations; 12. Kubo-Martin-Schwinger Boundary Conditions; 13. Converting Contour-time to Real-time Arguments; 14. Langreth Rules: Convolutions and Products; 15. The Kadanoff-Baym Equations; 16. The T-matrix Approximation in the Normal Phase; 17. Contour Diagrammatic Structure in Terms of Functional Derivatives; 18. Beyond Linear-response Theory; 19. Time-dependent Hartree-Fock Approximation and Mean-field Decoupling; 20. Miscellany and Addenda to Part I; 21. Time-dependent Version of the BCS Hamiltonian: Gor'kov Equations for the Normal and Anomalous Single-particle Green's Functions; 22. The Hamiltonian in the Nambu Representation and Role of the Hartree-Fock-BCS Self-energy; 23. Contour-ordered Green's Functions in the Nambu Representation; 24. The T-matrix Approximation in the Superfluid Phase; 25. Derivation of the Time-dependent Bogoliubov-deGennes Equations; 26. A Brief Excursus to the BCS-BEC Crossover; 27. Analytic Continuation from the Imaginary to the Real Time Axis; 28. Derivation of the Time-dependent Gross-Pitaevskii Equation for Composite Bosons in the BEC Limit of the BCS-BEC Crossover; 29. Derivation of the Time-dependent Ginzburg-Landau (TDGL) Equation for Cooper pairs in the BCS Limit of the BCS-BEC Crossover; 30. Real-frequency Green's Functions from the Kadanoff-Baym Equations in the Equilibrium Case; 31. Miscellany and Addenda to Part II; 32. An Overview on Applications: Yesterday, Today, and Tomorrow; 33. Driven Open Quantum Systems; 34. Extension to Superfluid Fermi systems; 35. Connection between the Schwinger-Keldysh Closed-contour Approach and the Lindblad Master Equation; 36. State-of-the-art Numerical Methods; 37. Miscellany and Addenda to Part III.

About the Author :
Giancarlo Calvenese Strinati is Emeritus Professor of Physics at the University of Camerino and his research is focused on condensed matter physics and ultra-cold atoms. He earned his Ph.D. at the University of Chicago in 1977 with support from the Fulbright Program, before spending a year as a Humboldt Fellow at the Max Planck Institute for Solid State Research. He joined the faculty of the Sapienza University of Rome – first as an Assistant Professor and then as an Associate Professor – and later worked at the Scuola Normale Superiore in Pisa. He has been a Fellow of the American Physical Society since 2010.