Summations for an isolated chain Fixmans method

As we saw, a restricted partition function can be calculated either in real space or in reciprocal space. 

This renormalization is equivalent to an analytic continuation with respect to dimension A. 

If we work in real space and if we want to calculate a diagram (see Figs. 10.1 and 10.2), we start by fixing the correlation points and the interaction points both in space and on the polymer line. Subsequently, the interaction points are displaced in every possible way. 

It is possible to proceed more directly by observing that the interaction energy does not really depend on the relative positions of all the interaction points, but only on the relative positions of the points which belong to the same doublet or the same triplet of interaction points. This observation is the basis of Fixman s method.4 

However, in general, it is more convenient to work in reciprocal space by calculating the Fourier transforms of the restricted partition functions. In this case, the contribution of a diagram depends on a certain number of internal vectors. Applying Fixman s method is equivalent to summing the contributions over these wave vectors. The number Q of independent wave vectors is called the number of loops. In particular, for a diagram with p two-body interactions and p three-body interactions, we have 

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