Renormalization techniques

With 0 3)h QED the major difference emerges from the effective photon bunching or interactions that can result in a photon loop, composed of an 4 photon and an photon. This loop will be associated with a quanta of field. Equation (149) illustrates how this fluctuation in the 4(l and 4(2) potentials are associated with this magnetic fluctuation. The other renormalization techniques in U(l) QED still apply, and are demonstrated below, and the renormalization of divergences associated with the Bii] magnetic fluctuation is also illustrated. 

We discuss some of the properties of maps and the techniques for analyzing them in Sections 10.1-10.5. The emphasis is on period-doubling and chaos in the logistic map. Section 10.6 introduces the amazing idea of universality, and summarizes experimental tests of the theory. Section 10.7 is an attempt to convey the basic ideas of Feigenbaum s renormalization technique. 

In 1970, K. Wilson discovered that to describe certain critical properties correctly it was necessary to resort to renormalization techniques borrowed from field theory. In 1972, P.G. de Gennes showed that a long polymer chain is a critical object and that the same techniques are applicable to its study. This new approach proved fruitful. Moreover, it explained the errors contained in the old theories. These errors arise partly from the fact that they take into account only one exponent, namely the size exponent v, overlooking the fact that other exponents may exist. Actually, renormalization theory shows that there are two fundamental exponents v and y. All the older theories implicitly state that y has... 

The same approach was used for polymers after 1972, owing to the analogy existing between polymer theory and the zero-component field theory. In the following years, the renormalization techniques became more adapted to the specific problems of polymer theory. Thus, renormalization techniques could be applied directly to polymers. In this way, difficulties related to the fact that polydispersion cannot be controlled with a one-field theory were solved, and... 

General description of renormalization techniques in real space... 

General principles common to all analytical renormalization techniques... 

In brief, the analytic renormalization techniques apply to continuous models depending on very few bare parameters, and they aim at expressing measurable quantities directly in terms of macroscopic and observable fundamental parameters. Thus, in spite of a certain mathematical complexity, this approach appears as essentially realistic. 

In the following, we shall not explain the technical points more precisely. In fact, in the preceding section we already described a renormalization technique in detail, and at the end of this chapter we shall show, also in detail, how we can obtain e-expansions by direct renormalization of polymers. Consequently, only results will be given here, and the reader will be referred to the relevant original articles. 

These series are obviously divergent, but by applying adequate renormalization techniques, it is possible to deduce from them precise evaluations of the swelling which is a well-defined function of z. We shall come back to this topic in Chapter 13, Section 1.4.1. 

However, the principles and the techniques of renormalization theory are not directly related to the existence of fields. They apply whenever one deals with a critical system, i.e. whenever one has to describe large-scale phenomena which depend only globally on the chemical microstructure. Thus, because an ensemble of long polymers in a solution constitutes a critical system, renormalization principles and renormalization techniques must be directly applicable to their study. Actually, this idea appeared quite naturally. It led to the decimation method which has been described previously and which lacks efficiency. However, the same idea can be applied in a much better way. This direct renormalization method (des Cloizeaux 1980)37,38 consists in adapting to polymers methods which had been successful in field theory.39 In other words, the aim is to bypass the Laplace de Gennes transformation (see Chapter 11). This method applies to semi-dilute solutions as well as to dilute solutions. 

Of course, approximations have to be made. Perturbation calculation provides a rather convenient framework, but other approximations are possible, and the renormalization techniques which we are going to describe have a very wide scope. 

In order to justify the conjecture made by Cardy and Hamber, Nienhuis used a cascade of models (including an unsolved six-vertex model)4 and equivalences that are more or less exact finally, he came down to a two-dimensional Coulomb gas. This gas is made of positively and negatively charged particles in interaction, the interaction potential being proportional to Inr where r is the distance between two charges. Then, Nienhuis could apply to this system approximate renormalization techniques which enabled him to predict the critical properties of the system. 

In this chapter, polymers are represented by means of the standard continuous model, described in Chapter 10. To calculate their properties in good solvent, we shall use the renormalization techniques described in Chapter 12. The principles and methods which serve to determine the properties of isolated polymers or of a small number of polymers (dilute solutions) also apply quite naturally to the study of polymer ensembles (solution with overlap) which we consider in the second part of the present chapter. In this way, we obtain all kinds of results which can be compared to experimental data. 

In fact, if we want to renormalize the diagrams to one-loop order, so as to calculate 3 (N x S), we must retain from the contributing diagrams only those which contain one-loop at most this is the step we shall take. By applying renormalization techniques to the result, we shall derive an expression of /7 valid for small values of g, i.e. actually for small values of e = 4 — d (nevertheless the value 3 for d can be considered close to 4). 

CONTINUOUS MODEL WITH ATTRACTION. APPLICATION OF RENORMALIZATION TECHNIQUES... 

Bergere, M.C. and Lam, Y.M.P. Unpublished course on renormalization techniques, given at Saclay (January 1975). 

We note that fab, k) diverges as fc 0. The numerical forward and back transforms are applied to c Q (,(r) and fj cab f )j respectively, and the divergent term is handled analytically. Since c cab r) and fj cab k) decay very rapidly (exponentially) as r —> oo and k oo, they can be accurately transformed numerically with a sufficiently small integration range. The special treatment of Coulomb potentials mentioned so far is simpler and faster than the so-called renormalization technique. 

There are basically four motivations for using renormalization techniques in the theory of fluids, namely, (1) a desire to remove divergences in individual graphs, (2) a desire to exploit extensive cancellation among terms in the series, (3) a desire to group terms in some particular way, such as to collect together all terms that are of the same order in some hopefully small parameter, and (4) a desire to generate tractable approximations for properties of the fluid. 

Hence, the renormalization technique applied in field theory is suitable in polymer theory as w cll. The chief idea is in the 9 dependence of the basic quantities at 2 -t oo. However, these hopes are realized only partially. 

These nanostructures are described using a single-band tight-binding Hamiltonian and their electronic conductance and density of states are calculated within the Green s function formalism based on real-space renormalization techniques (Rosales et al. 2008). 

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