Renormalization schemes

It is well known that it is difficult to solve numerically integral equations for models with Coulomb interaction [69,70]. One needs to develop a renormalization scheme for the long-range terms of ion-ion correlations. Here we must do that for ROZ equations. 

Thus different renormalization schemes are related by a so-called finite renormalization, which amounts to a redefinition of the renormalized parameters, or to a different parameterization of the field theoretic manifold, equivalently ... 

The results of the previous chapter are completely general. They are valid for any field theoretic renormalization scheme, i.e. independent of the specific choice of the renormalization factors, For quantitative calculations we of course have to specify the Z-factors, and as pointed out in Sect. 11,1, we have some freedom there. We will use the scheme of dimensional regularization and minimal subtraction . This scheme is most efficient for actual calcular tions, but its underlying basis is a little bit delicate, It needs some careful explanation. 

In this chapter we first show that the continuous chain model is renor-malizable by taking the naive continuous chain limit of the theorem of renor-malizability. We then argue that we can construct renormalization schemes for the continuous or the discrete chain models, equivalent in the sense that they yield the same renormalized theory (Sect. 12.1). In Sect. 12.2 we estab-... 

Concerning the first question we note that the result of any renormalization scheme based on the continuous chain model via a finite renormalization can be mapped on the renormalized theory derived from the discrete chain model, and vice versa. After renormalization the models are completely equivalent. 

Consider the e-expansion (12.27) of the renormalized end-to-end distribution. It contains the constant b and thus depends on our renormalization scheme. This dependence can be eliminated by replacing the chain length, which is a microscopic parameter, by the end-to-end distance... 

This calculation illustrates a general feature we may write down scaling laws for normalized quantities in terms of scaled momenta qRg (or qf e, equivalently), scaled concentrations cpRand ip replacing the coupling. Such relations involve only physically observable macroscopic quantities. They must have a uniquely defined -expansion, where ip = 0(e) acts as an expansion parameter. The result is necessarily independent of any conventions of the renormalization scheme. Not even the form of the RG flow equations matters. Furthermore, in establishing such results, we never have to invoke a condition like hr = 1. These are the great virtues of consistent e-expansion. 

For low order calculations of the scaling functions a variety of implementations of the RG have been used. The present formulation has grown out of the work [Sch84]. The basic philosophy is the same, but in this earlier work the renormalization scheme was based on field theoretic renormalization conditions1. This amounts to using a non-minimally subtracted theory, where the Z-factors are determined by imposing specific values to certain renormalized field theoretic vertex functions. The renormalized coupling, for instance, is defined as the value of (qi, qa, qg. qi) at some special momenta of order... 

In this framework, we present the repercussions on the physical properties of a renormalized indirect correlation function y (r) conjugated with an optimized division scheme. All the units are expressed in terms of the LJ parameters, that is, reduced temperature T = kBT/e and reduced density p = pa3. In order to examine the consequences of a renormalization scheme, the direct correlation function c(r) calculated from ZSEP conjugated with DHH splitting is compared in Fig. 7 to those obtained with the WCA separation. For high densities, the differences arise mainly in the core region for y(r) and c(r) [77]. These calculated quantities are in excellent agreement with simulation data. The reader has to note that similar results have been obtained with the ODS scheme (see Ref [80]). Since the acuracy of c(r) can be affected by the choice of a division scheme, the isothermal compressibility is affected too, as can be seen in Table III for the pkBTxT quantity. As compared to the values obtained with... 

In this section we will calculate the reducible contributions to the graph Fig. 1 a) and the total contributions to the graphs Figs. lb),lc). The general renormalization scheme for these graphs was presented in [30]. This scheme exploits the potential expansion for separating out the divergent terms and is suitable for the application of the PWE approach. Later these results were rederived in [31] by a different method. 

It is a statistical-mechanical theory of solutions to express the solvation free energy as a functional of distribution functions. Traditionally, the theory of solutions is formulated with a diagrammatic approach [13], in which an approximation is provided in a two-step procedure. In the first step, the free energy and/or distribution function is expanded with respect to the solute-solvent interaction potential function or its related function as an infinite, perturbation series. In the second step, a renormalization scheme is applied a set of functions are defined through partial summation of the series and are employed for substitution to make the infinite series more tractable. An approximation is typically introduced by neglecting diagrams of ill character. 

In this paper we adopt the PWE renormalization scheme. Accordingly, all bound terms in Eqs. (18)-(21) appear as double sums over partial waves. These double sums arise from the product of two matrix elements containing exp(i w r). Each individual partial wave contribution is finite and only the sum over partial waves diverges. In our calculation this divergency is removed by term-by-term subtraction of the corresponding counterterms. The PWE for the sum of all terms in Eq. (17) thus ensures a correct cancellation. 

This finite renormalization has two consequences. First, the nonuniver-sal parameters depend on both the microscopic system and the renormalized theory chosen. They thus have no direct microscopic meaning. Physical information is contained in the relative change upon changing the chemical microstructure or temperature, but not in the absolute values. Second, on a more technical level, numerical results of finite order calculations will differ for different renormalization schemes. This is a principle problem, unavoidable in low order calculations of scaling functions. Unambiguous results are foimd only for quantities not involving the nonuniversal constants, like exponents or critical ratios, or normalized scaling functions expressed in terms of RG-invariant variables. The function P pRa) (Eq. (11.52)) is an example. For such quantities the e-expansion is unique. This aspect will be discussed further in Sect. 12.4. 

There are more problems. The result (12.34), for instance, yields no information on chain-length or temperature dependence, which is hidden in or ip. To extract it, we have to write down RG flow equations for these variables. This results in the so-caUed direct renormalization scheme [Clo81], which however has not been pushed to the same high order as the flow equations derived for- n, Nfi in minimal subtraction. 

The mathematical problem associated with the Dirac Hamiltonian, i.e. the starting point of the relativistic theory of atoms, can be phrased in simple terms. The electron-positron field can have states of arbitrarily negative energy. As a general feature of the Dirac spectrum this instability occurs even in the case of extended nuclei and even in the absence of any nucleus (free Dirac spectrum), the energy is not bounded from below. This gives rise to the necessity of renormalization and well-established renormalization schemes have been around for many decades. Despite their successful applications in physics, we may ask instead whether there exist states that allow for positivity of the energy. 

An overview of relativistic density functional theory (RDFT) is presented with special emphasis on its field theoretical foundations and the construction of relativistic density functionals. A summary of quantum electrodynamics (QED) for bound states provides the background for the discussion of the relativistic generalization of the Hohenberg-Kohn theorem and the effective single-particle equations of RDFT. In particular, the renormalization procedure of bound state QED is reviewed in some detail. Knowledge of this renormalization scheme is pertinent for a careful derivation of the RDFT concept which necessarily has to reflect all the features of QED, such as transverse and vacuum corrections. This aspect not only shows up in the existence proof of RDFT, but also leads to an extended form of the single-particle equations which includes radiative corrections. The need for renormalization is also evident in the construction of explicit functionals. 

The proof given relies on a perturbation expansion with respect to both the electron-electron interaction and V. The necessity for these expansions originates from the recursive nature of the renormalization scheme which proceeds order by order in the fine-structure constant and from the fact that the treatment of inhomogeneous systems has to be derived from the renormalization procedure for the homogeneous QED vacuum. Only in this framework is it possible... 

In this first Appendix we consider the quantum field theoretical description of noninteracting spin-1/2 particles. In particular, we sununarize the quantization procedure, emphasizing the close relation between the minimum principle for the ground state energy and the renormalization scheme. At the same time this Appendix provides the background for the field theoretical treatment of the KS system, i.e. Eqs.(50)-(66). 

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