Field theoretic renormalization group and universality

Let us now turn to the nontrivial fixed point. For d — 4 it merges into the Gaussian fixed point. Analyticity in the parameter d of the RG flow presupposed by the -expansion then guarantees that the scaling fields at u as well as their dimensions for d 4 smoothly develop from their equivalent at the Gaussian fixed point, d = 4, Indeed it has recently been shown [KWP93] that 

The serious problems encountered if we aim at a microscopically realistic calculation of the excluded volume forces have clearly been noted in the literature. See, for instance, [DLR96], 

The general renormalization group as a representation of dilatations in the space of all Hamiltonians has been introduced by Wilson (see [WK74] for a review). Much of the abstract structure has been worked out by Wegner, as reviewed in [Weg76]. See also [Ma76]. 

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 theory as presented here has been called the field theoretic approach , since it relies on powerful results derived in quantum field theory or in statistical field theory of critical phenomena. This aspect, however, will not be important to us. Yet a reader acquainted with critical phenomena will easily see close connections and hopefully may take some interest in the construction of crossover scaling functions as advocated here. 

---