Interactions and Feynman Diagrams

Abstract

In the previous chapter, we introduced the functional-integral formalism of quantum field theory for the treatment of many-body systems. In particular, we calculated the partition sum for the ideal quantum gas in three different ways. The last method involved the noninteracting Green's function, which was seen to form the bridge between the more familiar operator formalism and the newly obtained functional-integral formalism. In this chapter we extend the notion of the Green's function to interacting systems, starting with the Lehmann representation of the interacting Green's function. This representation is exact and shows that in general the poles of the interacting Green's function correspond to the single-particle excitations of the many-body system, which are also called the quasiparticle excitations. Realizing that the interacting Green's function gives us both the elementary excitations of the many-body system, as well as the expectation value of the one-particle observables, the question arises how to determine this important quantity in practice.

Since it is in general not possible to determine the interacting Green's function exactly, we need to develop approximate methods to take interaction effects into account. A systematic way to do so is by performing a perturbation theory in powers of the interaction, which is the many-body analogue of the perturbation theory known from quantum mechanics. The rather cumbersome expressions resulting from this expansion are then elegantly represented in terms of Feynman diagrams, which make it possible to see the general structure of the expansion. To lowest order in the interaction, the noninteracting Green's function is then found to be modified by the Hartree and Fock diagrams which consequently can be used to construct a self-consistent Hartree-Fock theory. The name of the resulting theory comes from the full analogy with the nonperturbative Hartree-Fock theory for many electrons in an atom. This theory is then not only studied diagrammatically, but also varia-tionally and finally with the Hubbard-Stratonovich transformation. The latter is the technique most frequently used in the following chapters when we are interested in phase transitions occurring in interacting quantum gases.