5 Structural Properties of Perturbation Theory 5.1 Generating Functionals

The structural properties of all diagrammatic expansions developed so far can be analyzed systematically with the help of functional equations. In Chapter 3 we have seen that the correlation functions obtained from the functional derivatives of Z[j] via relation (2.14), and the generating functional itself, contain many disconnected parts. Ultimately, however, we shall be interested only in the connected parts of Z[j]. Remember that a meaningful description of a very large thermodynamic system can only be given in terms of the free energy which is directly proportional to the total volume. In the limit of an infinite volume, also called thermodynamic limit, one has then a well-defined free energy density. The partition function, on the other hand, has no proper infinite-volume limit. We can observe this property directly in the diagrammatic expansion of Z[j]. Each component of a disconnected diagram is integrated over the entire space, thus contributing a volume factor. The expansion of Z[j] therefore diverges at an infinite volume. In thermodynamics, we form the free energy from the logarithm of the partition function, which carries only a single overall volume factor and contains only connected diagrams. Therefore we expect the logarithm of Z[j] to provide us with the desired generating functional W [j]: W [j] = log Z[j]. (5.1) In this chapter we shall see that the functional derivatives of W [j] produce, indeed, precisely the connected parts of the Feynman diagrams in each correlation function. Consider the connected correlation functions G