Fields associated with the standard continuous model

5 Fields associated with the standard continuous model 

We established the correspondence between polymers and fields in the most general case. In practice, we shall work as most physicists do, within the 

In all cases, in order to describe polymers, one has to take the limit n- 0. 

In the preceding sections, we associated IV fields with N polymers each field had n components, and we showed that a correspondence exists between partition functions and Green s functions in the limit n - 0. Of course, to calculate critical exponents only one-field is sufficient because, in this case, only isolated polymers can be considered. However, it is also possible to find a correspondence between a one-field model and a polymer ensemble. This correspondence played a decisive role in its time, because it provided the means by which renormalization theory could be applied to polymer solutions, and it led to the discovery of new scaling laws. The correspondence can be established by using a lattice model, but here we shall follow the historical approach. Thus, we shall deal with a continuous model, more useful for practical applications, without caring too much about the problems concerning short-distance divergences. 

Let us start from relation (11.4.1) defining the Laplace transforms of the partition functions, and let us consider the case where the aj are equal for all/ In 

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