Some fundamental concepts and mathematical tools of the quant um theory of many particle systems, which are indispensable to the study of solid state physics. are presented. The concepts of collective excitations and quasi-particles are stressed. Concerning the mathematical tools, Quantum Mechanics is presented in its general, representation independent formalism. The many-particle problem is approached in the occupation number representation, or second quantization. Finally, as an application the interaction between electrons and the vibrations of a crystal lattice, described in terms of elementary excitations of collective waves, i.e., phonons, is expressed and discussed within this framework.
Some fundamental concepts and mathematical tools of the quant um theory of many particle systems, which are indispensable to the study of solid state physics. are presented. The concepts of collective excitations and quasi-particles are stressed. Concerning the mathematical tools, Quantum Mechanics is presented in its general, representation independent formalism. The many-particle problem is approached in the occupation number representation, or second quantization. Finally, as an application the interaction between electrons and the vibrations of a crystal lattice, described in terms of elementary excitations of collective waves, i.e., phonons, is expressed and discussed within this framework.
This chapter introduces and applies the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. Second quantization formalism treats systems of bosons or fermions with a fixed or variable number of particles in a unified way. The chapter reviews the key aspects of this formalism. The formalism of second quantization treats systems of bosons with a fixed or variable number of particles in a unified manner. In many physical processes, the particle number does change. Different quantum operators are expressed most naturally in different representations, which make basis changes an essential issue in quantum physics. Quantum field theory is a universal tool that has applications ranging from atomic, molecular, and particle physics to condensed matter and statistical physics as well as modern quantum chemistry. The fundamental difference between classical and quantum mechanics relates to the concept of indistinguishability of identical particles.
In this article, we discuss some questions about the nature of elementary particles treated by quantum mechanics, in particular related to the concepts of identity and individuality of these particles. We started briefly exposing the philosophical and formal concepts of the identity and individuality, and then show how these notions are problematic when applied to elementary particles such as electrons, protons and neutrons. In particular, we emphasize that both philosophy, logic and set theory (and thus the mathematics) admit the usual assumption that things have a ‘type’ of identity and individuality (i.e., are individuals), in the sense that objects which have all the same properties are the same object (are equal). Nevertheless, we show that in the quantum universe is possible to find objects that share all their properties, but are not just one, constituting in something like “nonindividuals” (thesis defended by many physicists and philosophers of science). In sequence, we show how classical mathematics — which, as we said, assumes an individuality to your entities — handle this situation, in particular admitting assumptions external to the physical theories. To avoid this procedure, and look for a formalism more ‘natural’ and appropriate to work with these quantum characteristics, we suggest the use of a non-classical set theory called the quasi-set theory, based on a non-reflective logics, which admits objects devoid of identity and individuality ab initio. Finally, we show another application of this theory: the search to a quasi-set mathematical structure that describes the ‘behaviour’ of non-relativistic quantum theory, and discuss some advantages of this ‘quasi-structure’ to a classical one.
The quantum theory of many-particle systems is deeply connected to the mathematical formalism and techniques of quantum field theory, which are crucial in order describe the emergent phenomena that arise in condensed matter physics. The main goal of this thesis is to provide a unified description of quantum field theory methods in the context of many-particle physics in order to extend and develop these techniques to better understand the different quantum properties of matter. This both furthers the conceptual understanding of quantum many-body correlation and also allows for the development of novel theoretical techniques that combine the different formalisms employed in condensed matter theory, quantum chemistry and nuclear physics. To achieve this aim, the thesis is composed of two parts. Part I deals with many-body fermion-fermion interactions and is physically motivated by the non-relativistic electronic structure problem. Here, we introduce the Keldysh contour formulation of quantum field theory to provide a unified presentation of quantum systems both in and out-ofequilibrium. Then, we extend the conventional quantum theory of many-particle systems, based on hermitian quantum mechanics, to describe systems governed by non-hermitan Hamiltonians. This allows us to present a natural unification of two major pillars in the ab initio quantum theory of many-particle systems: coupled-cluster theory and the Green’s function formalism. This unification is further demonstrated through the emergence of the coupled-cluster amplitude equations from the Green’s function formalism. In particular, this analysis has direct bearing on the development of novel theoretical approaches to the electronic structure and nuclear many-body problems. Our findings are also relevant to the study of general non-hermitian as well as open quantum systems. Part II focuses on the quantum field theory of electron-nuclear interactions in infinite solids. Here, we use the Keldysh contour representation presented in Part I to understand excited state dynamics that include electron-electron and electron-lattice interactions in materials. We also derive the analytic continuation of the Green’s function for systems that have continuous eigenspectra. This analysis demonstrates the relationship between the Green’s function formalism and scattering theory, making the role of the continuum and causality conditions explicitly clear in the context of the pole condition of the Dyson equation. Using the analytic continuation of the propagator, we provide an in-depth analysis of electron-phonon and exciton-phonon scattering channels, with a particular focus on phonon-driven exciton dissociation scattering. Our theoretical analysis then firmly demonstrates the connection between the extension of the Bethe-Salpeter equation approach to include electron-phonon interactions and effective field theories based on composite exciton-phonon coupling. We subsequently combine the techniques presented in Part I, based on the Dyson supermatrix formulation, with the analysis presented in Part II to calculate, from first principles, the effects of including electron-phonon interactions on the excited states of heterogeneous semiconductors. Finally, we provide a summary of the fundamental contributions presented in this thesis and the potential further research directions to be pursued.
Art is the elimination of the unnecessary. Pablo Picasso Summary Condensed matter is constituted by a huge number of electrons and nuclei interacting with Coulomb potentials. The topic of this chapter is a way to deal with the full, coupled problem by separating it into a part that is tractable and the rest that is approximated. This naturally leads to the appearance of dynamical fields, and to the concept of “quasi-particles” that have the same quantum numbers as non-interacting electrons. The quasi-particles obey equations where the potential and bare interactions in the hamiltonian are replaced by dynamical self-energies and screened interactions that can describe many of the effects of correlation. In this chapter we discuss the intuitive concepts behind this approach in general terms to motivate the more rigorous formulations and approximations used in the Green's function methods of the following chapter and in Part III. The first equations of this book, Eqs. (1.1) and (1.2), express the fundamental theory of matter in terms of electrons and nuclei that interact with Coulomb potentials. For systems with only a few electrons, a solution of the problem can be obtained by exact diagonalization, using configuration interaction methods. For many-electron systems one has to resort to other methods. For example, QMC stochastic simulations (Chs. 22–25) are among the most accurate methods known to calculate certain expectation values, such as equilibrium thermodynamic properties, total energies, the density, and various static correlation functions like those in Ch. 5. Other properties such as excitation spectra are more difficult to access with QMC. Straightforward perturbation theory is, in general, not appropriate since interactions among the electrons are of the same order of magnitude as the independent-particle terms. Hence, we must develop other approaches. One strategy that gives access to equilibrium thermodynamic as well as dynamical excitation properties is to separate the interacting many-body problem into a part that is tractable by some means and the rest of the problem that is dealt with more approximately. The present chapter is devoted to this idea. It is a prelude to many-body Green's function methods; the aim is to provide a unified framework for the developments in Chs. 8–21, along with a qualitative description of the most relevant quantities.
Quantum theory comprises a number of physical concepts and methods of calculation applicable to bodies on an atomic scale and consequently to the explanation of the properties of macroscopic bodies in terms of the behaviour of their constituent particles. In advanced work the theory is highly mathematical, but at an elementary level the difficulties lie mostly in the appreciation of the relevant physical ideas. Wave-particle duality, indeterminacy, the nature of a quantized state of a system and the exclusion principle cause most conceptual problems.
The development of methods of quantum statistical mechanics is considered in light of their applications to quantum solid-state theory. We discuss fundamental problems of the physics of magnetic materials and the methods of the quantum theory of magnetism, including the method of two-time temperature Green’s functions, which is widely used in various physical problems of many-particle systems with interaction. Quantum cooperative effects and quasi-particle dynamics in the basic microscopic models of quantum theory of magnetism: the Heisenberg model, the Hubbard model, the Anderson Model, and the spinfermion model are considered in the framework of novel self-consistent-field approximation. We present a comparative analysis of these models; in particular, we compare their applicability for description of complex magnetic materials. The concepts of broken symmetry, quantum protectorate, and quasi-averages are analyzed in the context of quantum theory of magnetism and theory of superconductivity. The notion of broken symmetry is presented within the nonequilibrium statistical operator approach developed by D.N. Zubarev. In the framework of the latter approach we discuss the derivation of kinetic equations for a system in a thermal bath. Finally, the results of investigation of the dynamic behavior of a particle in an environment, taking into account dissipative effects, are presented.