Comparison to the grand canonical formalism

For the free energy, disconnected vertices first contribute in two loop order (Fig. 5.20c). For correlation functions, however, disconnected vertices may contribute even to one loop order, and a calculation ignoring such contributions by keeping only naively connected diagrams would yield wrong results. 

Due to its more complicated structure the canonical expansion is not well adapted for a discussion of general features like the properties of the theory under renormalization. Thus in the sequel we exclusively work with the grand canonical approach. 

For a special choice of the chemical potential the grand canonical description as used here was suggested by Des Cloizeaux [Clo75], The generalization to arbitrary pp(n) is due to Schafer and Witten [SW80]. 

The importance of vertex iireducibility has been stressed by Dos Cloizeaux Clo80a]. Starting from a field theoretic formulation of the theory Knoll et al. KSW81] independently came to the same conclusion. Our discussion follows that work which in turn follows a solution of the so-called Goldstone mode problem in field theory of critical phenomena [SH78]. 

The auxiliary field method in the context of polymer physics has been introduced by Edwards [Edw66, Edw75] and has been widely used by Freed and coworkers. (See [Fre87]). 

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