14 - Many-Body Theory

Abstract

The basic techniques of quantum field theory required to handle the many-body problem are introduced, starting with the fundamental concept of second quantization, construction of Fock space, and defining creation and annihilation operators. The Hamiltonian in Fock space in terms of products of annihilation and creation operators is developed. The field operators are defined, and the Hamiltonian as well as other important one and two-body operators are given in terms of field operators. Quantization of the radiation field that leads to the introduction of photons is presented, followed by a similar formalism for the quantization of crystal vibrations, leading to the concept of phonons. The formulation of quantum statistical mechanics within the second quantization formalism is worked out. Finally, an analysis of mean-field theory, an important method in the quest for non-perturbative approximate solutions of the many-body problem, is presented.