RG-mapping

To substantiate our qualitative ideas we only have the cluster expansion at our disposal. Though this expansion is useless for a direct calculation of observables in the limit of large n, we can use it to construct the RG mapping, provided we make the additional assumption that the functions B(, dfj (Eq. (8.1)) allow" for a Taylor expansion in powers of, de ... 

Thus all seems perfect. We have constructed an RG mapping, wliich indeed shows a fixed point. However, the expression (8.32) for / is not satisfactory. It must be independent of A, otherwise dilatation by A2 does not lead to the same result as repeated dilatation by A. Now Eq. (8.32) is only approximate since in Eq. (8,31) we omitted terms O 0 2. This is justified only if 0 is small. We thus need a parameter which allows us to make. If arbitrarily small, irrespective of A. Only e — 4 — d can take this role. In all our results the dimension of the system occurs oidy in the form of explicit factors of d or It thus can be used formally as a continuous parameter. To make our expansion a consistent theory, we have to introduce the formal trick of expanding in powers of e — 4 — d. 3 vanishes for = 0, consistent with the observation (see Chap, fi) that the excluded volume is negligible above d = 4, not changing the Gaussian chain behavior qualitatively. For e > 0 Eq. (8.32) to first order in yields... 

Having constructed a global approximation to the RG mapping we are in a position to evaluate results like Eq. (8.26 i), which now takes the form... 

Any one-parameter set of transformations obeying this rule under successive applications can be said to be a representation of the dilatation group. Thus relation (8.5) identifies the RG mapping constructed in Sect. 8.1 as a representation of dilatations in the space of models parameterized by /, n, / . We now work out some properties of the dilatation group represented in some general space with coordinates Y = Yj = Yi. Y , Y3,.... This space for instance may be the space of all Hamiltonians, the coordinates Y> being the coupling constants. 

In this part,we first explain in general terms the construction of renormalized perturbation theory. We show how the RG results from the arbitrariness of r and establish the general scaling form (Chap. 11). We then turn to the specific technique of minimal subtraction and show how to calculate the scaling functions (Chap. 12). The RG mapping, used in the sequel, is presented and discussed in Chap. 13. We finally (Chap. 14) illustrate the theory with an evaluation of the tree approximation. 

The representation (11,39), (11.40) of the RG mapping, introducing two parameters z, Rq, is adequate outside the excluded volume limit. In the excluded volume limit u — u > i.e. / —+ 1, the two parameters combine into a single parameter. The limiting form of the RG mapping is best derived by... 

We finally note that this analysis resulted in an explicit identification of the renormalization factors Zv, Zn, Z of polymer theory with renormalization factors of field theory. We can thus take over higher order results for the renormalization factors and the resulting RG flow, established in field theory. This is a most useful result since a good representation of the RG mapping is crucial in the application of the theory, but higher order calculations are most complicated. 

This chapter is concerned with important but technical aspects of the actual calculations. A reader more interested in the results may go on directly to the next chapter, where the explicit form of the RG mapping is presented. 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

The construction of the quantitative RG mapping [SD89J starts from the perturbation expansion of W(u) evaluated to order uG or from the expansions for r (u) or 2 — l/i/(u) evaluated to order us. These expansions are only asymptotic, but they can be resumed, using the Borel method. Within... 

Our final expression for the RG mapping therefore reads RG Mapping (d = 3)... 

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