In this article a survey is given of the present status of the quantum theory of many-particle systems with special emphasis on the underlying principles. Very little is said about applications. A brief introduction to some models, which are of interest for an approximate or qualitative description of actual physical systems, is followed by a discussion of two important phenomena in many-particle systems: quasi-particles and collective excitations. Next, an outline is given of mathematical methods which have been used for a precise treatment of such phenomena. A rather thorough discussion of the Bogoliubov model for interacting bosons, and the BCS model for fermions, serves as an illustration. Some new features, typical of many-particle systems, are discussed on the basis of these models. Also another approach to the many-body problem is discussed which is more logical and mathematically more rigorous. This approach differs from standard procedures in that the limit Ω-> infinity, N -> infinity is not taken at the end of the calculations but in the beginning. This quantum theory of infinite systems leads, among other things, to a better understanding of the models mentioned above.
In this article a survey is given of the present status of the quantum theory of many-particle systems with special emphasis on the underlying principles. Very little is said about applications. A brief introduction to some models, which are of interest for an approximate or qualitative description of actual physical systems, is followed by a discussion of two important phenomena in many-particle systems: quasi-particles and collective excitations. Next, an outline is given of mathematical methods which have been used for a precise treatment of such phenomena. A rather thorough discussion of the Bogoliubov model for interacting bosons, and the BCS model for fermions, serves as an illustration. Some new features, typical of many-particle systems, are discussed on the basis of these models. Also another approach to the many-body problem is discussed which is more logical and mathematically more rigorous. This approach differs from standard procedures in that the limit Ω-> infinity, N -> infinity is not taken at the end of the calculations but in the beginning. This quantum theory of infinite systems leads, among other things, to a better understanding of the models mentioned above.
Identical Particles Second Quantization Independent-Particle Model for Fermions in Finite Systems Two-Particle States and Interactions Noninteracting Bosons and Fermions Propagators in One-Particle Quantum Mechanics Single-Particle Propagator in Many-Particle Systems Perturbation Expansion of the Single-Particle Propagator Dyson Equation and Self-consistent Green's Functions Mean-Field or Hartree-Fock Approximation Beyond the Mean-Field Approximation Interacting Boson Systems Excited States in Finite Systems Excited States in Infinite Systems Excited States in N 2 Systems and In-Medium Scattering Dynamical Treatment of the Self-energy in Infinite Systems Dynamical Treatment of the Self-energy in Finite Systems Bogoliubov Perturbation Expansion for the Bose Gas Boson Perturbation Theory Applied to Physical Systems In-Medium Interaction and Scattering of Dressed Particles Conserving Approximations and Excited States Pairing Phenomena Propagator Framework for Phenomenological Description of Many-Body Systems Finite Temperature Formalism and Applications.
Mathematical Methods of Many-Body Quantum Field Theory offers a comprehensive, mathematically rigorous treatment of many-body physics. It develops the mathematical tools for describing quantum many-body systems and applies them to the many-electron system. These tools include the formalism of second quantization, field theoretical perturbation theory, functional integral methods, bosonic and fermionic, and estimation and summation techniques for Feynman diagrams. Among the physical effects discussed in this context are BCS superconductivity, s-wave and higher l-wave, and the fractional quantum Hall effect. While the presentation is mathematically rigorous, the author does not focus solely on precise definitions and proofs, but also shows how to actually perform the computations.Presenting many recent advances and clarifying difficult concepts, this book provides the background, results, and detail needed to further explore the issue of when the standard approximation schemes in this field actually work and when they break down. At the same time, its clear explanations and methodical, step-by-step calculations shed welcome light on the established physics literature.
Summary A brief account is given of the reasons why the many-body problem has recently become the subject of a rapidly expanding literature. After a survey of the main ideas of recent theories, it is shown how these ideas have played a rôle in giving us a better understanding of such topics as superconductivity, plasmas, and liquid helium. In the theory of the many-body problem the main branches of theoretical physics applied are Hamiltonian mechanics or quantum mechanics (to describe the motion of the constituent particles of the many-body system under consideration) and statistical mechanics (to describe the collective action of the large number of particles in the system). As these subjects are relatively advanced we have discussed in an appendix the main concepts of Hamiltonian mechanics and some of those of quantum statistics, including the definition and some of the properties of fermions and bosons.
This book provides a comprehensive and pedagogical account of the various methods used in the quantum theory of finite systems, including molecular, atomic, nuclear, and particle phenomena. Covering both background material and advanced topics and including nearly 200 problems, Quantum Theory of Finite Systems has been designed to serve primarily as a text and will also prove useful as a reference in research.The first of the book's four parts introduces the basic mathematical apparatus: second quantization, canonical transformations, Wick theorems and the resulting diagram expansions, and oscillator models. The second part presents mean field approximations and the recently developed path integral methods for the quantization of collective modes. Part three develops perturbation theory in terms of both time-dependent Feynman diagrams and time-independent Goldstone diagrams. A fourth part discusses variational methods based on correlated wavefunctions, including spin correlations. The approximation schemes are formulated for fermions and bosons at eigher zero or non-zero temperature.Although the formalism developed applies to both finite and infinite systems, the book stresses those aspects of the theory that are specific to the description of finite systems. Thus special attention is given to mean field approximations, the ensuing broken symmetries, and the associated collective motions such as rotations. Conversely, some specific features of systems with infinite numbers of degrees of freedom (such as the thermodynamic limit, critical phenomena, and the elimination of ultraviolet divergencies) are deliberately omitted.Jean-Paul Blaizot and Georges Ripka are associated with the Centre d'Etudes Nucleaires de Saclay.
The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics. In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
Lattice models in theoretical physics A major part of theoretical physics involves the construction and systematic analysis of mathematical models as a description of the physical world. For microscopic phenomena, in particular, the reference to models becomes quite explicit. The phenomena are often complex, and any theory that attempts to include every detail soon becomes intractable. It is more fruitful to develop models which ignore irrelevant details but, hopefully, capture the essential physics of the phenomena of interest. Thus we have the Heisenberg model of magnetic order, the Bardeen–Cooper– Schrieffer (BCS) model of superconductivity, and so on. It is important to note, at the outset, that the models we shall be discussing describe strongly interacting, and therefore highly correlated, systems of particles. These are difficult and interesting problems. Where interactions are absent, or weak, elementary treatments are possible and the resulting phenomena are generally unspectacular. Why lattice models? In solid-state phenomena there is usually an underlying lattice structure, and the symmetry properties of this lattice play an important role in the analysis. Even the process of electrical conduction in metals or semiconductors can be equally well described in terms of localized quantum states or in terms of the more usual continuum picture. In quantum field theory, which is formulated in a space–time continuum, a lattice is often introduced for computational purposes. One can think of this in two ways: as an approximation to the continuum, which is then recovered as a limit at the end of the calculations, or as a necessary means of regulating the theory (i.e. controlling divergences) in calculating the Feynman path integral.