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American Journal of Physics

Introduction to Many-Body Formalism

W. Jim Neidhardt

Published Sep 1, 1967• 0 citations
Publisher

Abstract

An elementary approach is presented to the many-body formalism needed to understand simple models of superfluid systems. In the framework of occupation number space-particle conserving operators that represent single-body and two-body observables are constructed in an intuitive manner. It is then shown that this formalism when applied to such observables as single-particle energy and two-body interaction potential yields results which are physically plausible. The basic formalism for the understanding of many-body systems is thereby presented without recourse to a complete development of quantum field theory.

DOI
10.1119/1.1974265

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An elementary approach is presented to the many-body formalism needed to understand simple models of superfluid systems. In the framework of occupation number space-particle conserving operators that represent single-body and two-body observables are constructed in an intuitive manner. It is then shown that this formalism when applied to such observables as single-particle energy and two-body interaction potential yields results which are physically plausible. The basic formalism for the understanding of many-body systems is thereby presented without recourse to a complete development of quantum field theory.
The book gives an introduction to the concepts and methods of many-body problems and quantum fields for graduate students and researchers. The formalism is developed in close conjunction with the description of a number of physical systems: cohesion and dielectric properties of the electron gas, superconductivity, superfluidity, nuclear matter and nucleon pairing, matter and radiation, interaction of fields by particle exchange and mass generation. Emphasis is put on analogies between the various systems rather than on advanced or specialized aspects, with the purpose of illustrating common ideas in different domains of physics. The exposition is self-contained and displays in a coherent way all details of derivations starting from a basic knowledge of quantum mechanics and classical electromagnetism.
A new fundamental theory of composite particles in nonrelativistic many-body systems is developed. The theory is constructed making use solely of the physical Fock space for the basic elementary particles which comprise the many-body system. In this manner exchange symmetry in the elementary particles is exact. Physical composite particle creation and annihilation operators are introduced and these operators satisfy exact Bose (Fermi) commutation (anticommutati$\stackrel{\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}{\mathrm{o}}$n) relations depending on whether the composites correspond to bosons or to fermions. Commuting physical occupation number operators for composite particles are then constructed in the usual manner from the composite creation and annihilation operators. These number operators are highly dressed in terms of the basic elementary particle operators. Finally, the foundations of statistical mechanics for bound composite particles are formulated in terms of the appropriate occupation number operators.
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.
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So far, we have dealt with one- and two-body systems in a variety of settings. These are quite simple, at least a priori, since the equations needed to solve one- and twoparticle scattering problems, as well as bound states, are usually easy enough to handle. The situation becomes much more intricate in the three-body problem, as we shall see in this chapter. Firstly, while two particles in free space can either scatter or bind together, three bodies have extra channels (for attractive interactions) corresponding to two bound particles scattering off a third. Moreover, the energies of these extra channels can be degenerate with those of three unbound particles, and the total momentum of a pair within the triplet is not conserved. Fortunately, we will see that getting our hands dirty with the three-body problem pays off, since there are a number of exotic phenomena associated with the quantum three-body problem, for both bound and scattering states. In this chapter, we introduce the formalism necessary to investigate the three-body problem and study in detail some of the most striking phenomenology known to date associated with, at least, the simplest interactions.
This selective review is written as an introduction to the mathematical theory of the Schrödinger equation for N particles. Characteristic for these systems are the cluster properties of the potential in configuration space, which are expressed in a simple geometric language. The methods developed over the last 40 years to deal with this primary aspect are described by giving full proofs of a number of basic and by now classical results. The central theme is the interplay between the spectral theory of N-body Hamiltonians and the space–time and phase-space analysis of bound states and scattering states.