ABSTRACT
This chapter aims to present the essence of the general theory in the particularly simple and convenient case of systems at finite temperature which may be treated in the grand canonical ensemble. It describes how thermal averages of time-ordered products of Heisenberg operators in imaginary time emerge naturally from path integrals and how to evaluate them in perturbation theory. The chapter considers at the outset what level of mathematical rigor one can actually expect and how in practice one may select and assess an approximation scheme. In view of the general lack of mathematical control on the convergence properties of the original series and the obvious ambiguity associated with regrouping divergent series, any such resummations must be understood ultimately on physical grounds. The ultimate objective of the quantum theory of many-particle systems is to understand experimentally observable properties of a diverse range of physical systems.
