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Quantum Field Theory of Non-equilibrium States

Jørgen Rammer

Published Mar 1, 2011• 13 citations
OpenAlex

Abstract

Preface 1. Quantum fields 2. Operators on the multi-particle state space 3. Quantum dynamics and Green's functions 4. Non-equilibrium theory 5. Real-time formalism 6. Linear response theory 7. Quantum kinetic equations 8. Non-equilibrium superconductivity 9. Diagrammatics and generating functionals 10. Effective action 11. Disordered conductors 12. Classical statistical dynamics Appendices: A. Path integrals B. Retarded and advanced propagators C. Analytic properties of Green's functions Bibliography Index.

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Preface 1. Quantum fields 2. Operators on the multi-particle state space 3. Quantum dynamics and Green's functions 4. Non-equilibrium theory 5. Real-time formalism 6. Linear response theory 7. Quantum kinetic equations 8. Non-equilibrium superconductivity 9. Diagrammatics and generating functionals 10. Effective action 11. Disordered conductors 12. Classical statistical dynamics Appendices: A. Path integrals B. Retarded and advanced propagators C. Analytic properties of Green's functions Bibliography Index.
In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrödinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrödinger and Heisenberg pictures. The Schrödinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.
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 book presents an up-to-date formalism of non-equilibrium Green's functions covering different applications ranging from solid state physics, plasma physics, cold atoms in optical lattices up to relativistic transport and heavy ion collisions. Within the Green's function formalism, the basic sets of equations for these diverse systems are similar, and approximations developed in one field can be adapted to another field. The central object is the self-energy which includes all non-trivial aspects of the system dynamics. The focus is therefore on microscopic processes starting from elementary principles for classical gases and the complementary picture of a single quantum particle in a random potential. This provides an intuitive picture of the interaction of a particle with the medium formed by other particles, on which the Green's function is built on.
Part 1 Fundamentals and examples: systems of identical particles second quantization for fermions second quantization for bosons unitary transformations and field operators example - the Hamiltonian of translationally invariant systems in second quantization density operators the Hartree-Fock approximation restricted Hartree-Fock approximation and the symmetry dilemma Hartree-Fock for translationally invariant systems the homogeneous electron gas in the Hartree-Fock approximation long-range correlations in the electron-gas - plasmons phonons superconductivity. Part 2 Green's functions: pictures the single-particle Green's function and the hierarchy of equations of motion. Part 3 Perturbation theory: time-dependent perturbation theory with adiabatic turning-on of the interaction particle and hole operators and Wick's theorem Feynman diagrams diagrammatic calculation of the vacuum amplitude an example - the Gell-Mann-Brueckner correlation energy of a dense electron gas diagrammatic calculation of the single-particle Green's function - Dyson's equation diagrammatic analysis of the Green's function G(kappa, omega) self-consistent perturbation theory, higher-order Hartree-Fock theory the quasi-particle concept diagrammatic calculation of the two-particle Green's function and the polarization propagator effective interaction and dressed vertices plasmons in higher order perturbation theory at finite temperatures. Part 4 Fermi liquid theory: introduction equilibrium properties transport equation and collective modes microscopic derivation of the Landau fermi liquid theory applications to the Kondo problem.
Path Integrals in Classical Theory Brownian motion: Introduction to the concept of path integration Wiener path integrals and stochastic processes Path Integrals in Quantum Mechanics Feynman path integrals Path integrals in Hamiltonian formalism Quantizations, the operator ordering problem and path integrals Path integrals and quantizations in spaces with topologiecal constraints Path integrals in curved spaces, space time transformations and the Coulomb problem Path integrals over anticommuting variables for fermions and generalizations Appendices Bibliography Index