The system of coupled oscillators forms a very simple many-body system that can be treated exactly within elementary quantum mechanics. This system is used to introduce important many-body concepts, such as spectral function, propagator or Green function, sum rule, and elementary excitation. These concepts are briefly discussed for the real system of conduction electrons interacting with lattice vibrations.
The system of coupled oscillators forms a very simple many-body system that can be treated exactly within elementary quantum mechanics. This system is used to introduce important many-body concepts, such as spectral function, propagator or Green function, sum rule, and elementary excitation. These concepts are briefly discussed for the real system of conduction electrons interacting with lattice vibrations.
Green's functions are a cornerstone of quantum mechanics and many-body physics, providing a systematic framework to describe the propagation of particles and excitations in a system. They encode essential information about the evolution of a quantum particle in space and time, serving as the foundation for understanding properties such as energy levels, spectral functions and system responses to external perturbations. Their broad applicability makes them invaluable for studying systems ranging from simple, non-interacting particles, to highly correlated many-body systems.
This chapter presents field-theoretic methods and discusses three well-known "pictures" or representations used in discussing quantum mechanical problems. It explores how to build up the true one-electron and one-phonon Green's functions as a perturbation series involving the corresponding functions for the non-interacting system. In elementary discussions of quantum mechanics one usually works in the "Schrodinger picture" in which the dynamical variables are taken to be time independent so that the wave function contains the time dependence of the problem. The one-electron Green's function gives information about the spin and charge densities and the momentum distribution of the electrons as well as information about the excitation spectrum of the system. When the system consists only of fermions interacting via an instantaneous two-body potential, the one-particle Green's function also suffices to determine the ground-state energy of the system. The electrical conductivity, magnetic susceptibility, and many other non-equilibrium properties can be obtained from the two-particle Green's function.
A number of powerful methods exist in statistical mechanics to describe the interweaving of the variety of elementary modes of excitation taking place in a quantal many-body system. Among them, one can mention the Static Path approximation (cf., e.g., Feynmann and Hibbs ( 1965 )), the Monte-Carlo Shell Model Method (Koonin et al. ( 1997 )), and the thermal Green-function formalism (cf., e.g., Mahan ( 1981 )). The first two approaches have not yet been systematically used to describe nuclear motion at finite temperature. We shall briefly discuss them in the present chapter. The Matsubara techniques, which have been applied to the study of a large variety of phenomena, are discussed in Ch. 9 .
The method of many-body Green's functions is developed for arbitrary systems of electrons and nuclei starting from the full (beyond Born-Oppenheimer) Hamiltonian of Coulomb interactions and kinetic energies. The theory presented here resolves the problems arising from the translational and rotational invariance of this Hamiltonian that afflict the existing many-body Green's function theories. We derive a coupled set of exact equations for the electronic and nuclear Green's functions and provide a systematic way to approximately compute the properties of arbitrary many-body systems of electrons and nuclei beyond the Born-Oppenheimer approximation. The case of crystalline solids is discussed in detail.
Abstract The methods of coupled quantum field theory, which had great initial success in relativistic elementary particle physics and have subsequently played a major role in the extensive development of non-relativistic quantum many-particle theory and condensed matter physics, are at the core of this book. As an introduction to the subject, this presentation is intended to facilitate delivery of the material in an easily digestible form to students at a relatively early stage of their scientific development, specifically advanced undergraduates (rather than second or third year graduate students), who are mathematically strong physics majors. The mechanism to accomplish this is the early introduction of variational calculus with particle sources and the Schwinger Action Principle, accompanied by Green’s functions, and, in addition, a brief derivation of quantum mechanical ensemble theory introducing statistical thermodynamics. Important achievements of the theory in condensed matter and quantum statistical physics are reviewed in detail to help develop research capability. These include the derivation of coupled field Green’s function equations of motion for a model electron-hole-phonon system, extensive discussions of retarded, thermodynamic and non-equilibrium Green’s functions, and their associated spectral representations and approximation procedures. Phenomenology emerging in these discussions includes quantum plasma dynamic, nonlocal screening, plasmons, polaritons, linear electromagnetic response, excitons, polarons, phonons, magnetic Landau quantization, van der Waals interactions, chemisorption, etc. Considerable attention is also given to low-dimensional and nanostructured systems, including quantum wells, wires, dots and superlattices, as well as materials having exceptional conduction properties such as superconductors, superfluids and graphene.
This chapter introduces the basic theoretical tools for handling many-body quantum systems. Starting from second quantized operators, we discuss how it is possible to describe the composite wavefunction of multi-particle systems, and discuss representations in various bases. The algebra of Fock states is described for single and multi-mode systems, and how they relate to the eigenstates of the Schrodinger equation. Finally, we describe how interactions between particles can be introduced in a general way, and then describe the most common type of interaction in cold atom systems, the s-wave interaction