Continuous chain limit and the Edwards model

We must note also a second important restriction of the continuous chain model. As we will see. by construction it deals with infinitely long chains n — oo. infinitesimally close to the -point , 5C — 0. Thus naive two parameter theory is valid only very close to the -temperature. In later chapters we will see how further renormalization leads to a theory of excluded volume effects valid for all /%, 0. 

The organization of this chapter is as follows. In Sect. 7.1 wo carefully define the continuous chain limit and we introduce the appropriate modification of the Feynman rules. We. then establish the two parameter scheme by dimensional analysis. Section 7.2 is devoted to the question whether the continuous chain limit exists. The analysis is presented on the diagrammatic level. It exploits the field theoretic representation, which also is derived on the level of diagrams. All the analysis is based on the cluster expansion. Extension to the loop expansion is not difficult, but will not be considered, since it is not needed in the sequel. 

Finally, for completeness in Appendix A 7.1 we consider the formal relation of the continuous chain model to a field theoretic Hamiltonian, used to describe critical phenomena in ferrornagnets. It was this relation discovered by de Genries [dG72] and extended by Des Cloizeaux [Clo75, which initiated the application of the renormalization group to polymer solutions and led to the embedding into the larger realm of critical phenomena. 

In the sequel, for simplicity of notation, we take all chains to be of the same length n. 

Continuous Chain Model and Naive Two Parameter Theory 

The continuous chain limit is defined by taking but keeping 

Wc tlma take the segmorit size smaller and. smaller, at the same time increasing the number of segments so as to keep the unperturbed coil size constant. Thus 

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